Research Article Dynamics and Control of a Flexible Solar...

Research ArticleDynamics and Control of a Flexible Solar Sail

Jiafu Liu1 Siyuan Rong2 Fan Shen2 and Naigang Cui2

1 Department of Aerospace Engineering Shenyang Aerospace University Shenyang 110136 China2Department of Aerospace Engineering Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Jiafu Liu liujiafuericking163com

Received 15 July 2014 Accepted 20 October 2014 Published 30 November 2014

Academic Editor Sebastian Anita

Copyright copy 2014 Jiafu Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Solar sail can merely make use of solar radiation pressure (SRP) force as the thrust for space missions The attitude dynamics isobtained for the highly flexible solar sail with control vanes sliding masses and a gimbaled control boomThe vibration equationsare derived considering the geometric nonlinearity of the sail structure subjected to the forces generated by the control vanessolar radiation pressure (SRP) and sliding masses Then the dynamic models for attitudevibration controller design and dynamicsimulation are obtained respectively The linear quadratic regulator (LQR) based and optimal proportional-integral (PI) basedcontrollers are designed for the coupled attitudevibration models with constant disturbance torques caused by the center-of-mass(cm)center-of-pressure (cp) offset respectively It can be concluded from the theoretical analysis and simulation results that theoptimal PI based controller performs better than the LQR based controller from the view of eliminating the steady-state errorsTheresponses with and without the geometrical nonlinearity are performed and the differences are observed and analyzed And somesuggestions are also presented

1 Introduction

Solar sail is a novel spacecraft with the unique propulsionstyle and has a great application potential in the near futureThe continuous thrust can be obtained using the huge andhighly flexible membrane to reflect solar photons thus themuch longer mission duration is possible by solar sail [1]It is reported that lots of basic scientific questions involvingthe impact cosmic rays have on the long-term conditions ofthe earth environment and on earth itself can be answeredby in situ exploration of the heliopause and the heliosphericinterface by solar sailing [2] This mission can hardly beaccomplished by spacecraft with chemical fuels such as thetwo Voyager spacecraft launched in the 1970s The spacemissions such as deorbiting [3] pole sitter [4] heliostormwarning [5] and many novel orbits [6ndash8] can be accom-plished by solar sail much more suitable than the traditionalspacecraft The IKAROS [9] and NanoSail-D [10] solar sailspacecraft have been launched into space and several basictheories concerning solar sail have been demonstrated andseveral space missions have been accomplished

One of the critical problems with respect to success orfailure for an orbiting solar sail is the attitude dynamics

and control problem The attitude control system (ACS) isproposed by Wie and Murphy consisting of a propellantlessprimary ACS and a microthruster-based secondary ACS [11]The sliding masses and roll stabilizer bars are used for yawand pitch and roll control respectively in the former ACSThe lightweight pulsed plasma thruster (PPT) is used forattitude recovery from off-nominal conditions in the latterone In addition the robust attitude controller is developed byemploying the attitude control actuators abovementioned [1213]Themodal data can be obtained for solar sail with attitudecontrol actuators as in [14] by using the finite elementmethodwith high fidelity The attitude controller considering thevibration of solar sail structure can be designed based on themodal coordinate state-space system and the effectivenessof the controller is verified The passive attitude stabilizationmethod (to spin solar sail) is proposed in the presence ofa cmcp offset in [15 16] The attitude control methodsare proposed and the simulations are carried out using thecontrol boom control vanes and sail shifting and tilting[17 18] in the presence of cmcp offset

The attitude dynamics are derived by employing thesliding masses and roll stabilizer bars (RSB) for attitude

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 868419 25 pageshttpdxdoiorg1011552014868419

2 Mathematical Problems in Engineering

control actuators The validity of the ACS is demonstratedby numerical simulations [19] A high performance solar sailattitude controller is presented by employing sliding massesinside the supporting beams and its ability of performingtime efficient reorientation maneuvers is demonstrated [20]The proposed controller combines a feedforward and a feed-back controller the former is a fast response controller whilethe latter can be used to respond to unpredicted disturbances[21] The solar sail ACS presented in [22] includes themovement of the small control mass in the solar sail planeand the rotation of ballast with two masses at the extremitiesto realize the pitchyaw and roll control respectively

The robust attitude controllers are designed using 119867infinQFT and input shaping methods respectively to allow forthe uncertainties of inertia natural frequencies dampingand modal constants of solar sail The performances ofthe controllers are analyzed using the linear and nonlineardynamics respectively [23]The reduced dynamic model of aflexible solar sail with foreshortening deformation couplingwith its attitude and vibration is derived in [24ndash26] TheBang-Bang control scheme combining input shapingmethodis used to eliminate the vibration for a time optimal attitudemaneuver The attitude control system is proposed usingsmall reactionwheels andmagnetic torquers for a solar sail onlow earth orbit and the validity is demonstrated by numericalsimulation [27]

The dynamic model of the multibody solar sail withcontrol boom and reaction wheel is derived in [28 29] Thecontrollability and stability are analyzed and the proposedattitude control scheme is demonstrated The effectiveness ofthe solar sail attitude control system employing four controlvanes is demonstrated by simulating deep space explorationmissions [30]The dynamics is derived and the controllabilityand stability are analyzed for solar sail with control boomreaction wheel and control vanes The effectiveness of theproposed ACS is demonstrated for the solar sail with cmcpoffset [31]

The trajectory tracking control is accomplished by usingsolar sail with four control vanes to control the solar angleof the sail The resultant sailcraft thrust vector amplitudecan be controlled by the additional degree of freedom ofthe control vanes [32] A robust nonlinear attitude controlalgorithm is developed for solar sail with four control vaneswith single degree-of-freedom rejecting disturbances by cmand cp offset The control allocation is studied by usingnonlinear programming [33]

The novel and practical ACS for solar sail is continuouslyproposed A bus-basedACS is presented in [34] including thehighly reflective panel actuator for roll control located at thefree end of the bus-based boom and the tether control massactuator for yawpitch control running along the bus-basedboomThis scalableACS can decrease the risk and complexityinvolved in the design of the sail deployment subsystem TheACS designed for IKAROS is muchmore novel and practicalThe ACS utilizes the reflectivity control device (RCD) torealize the yawpitch control for the spinning solar power sailIt is a fuel-free and oscillation-free ACS [35] But this ACSwill fail to generate the required torques when it is edge tothe sun (the sun angle approaches 90∘) And recently a novel

attitude control method is proposed in [36] The requiredtorque can be generated by adjusting the position of a wingtip along the boomThismethod presents an effective attitudeapproach for large sailcraft But the exact shape of the films isdifficult to determine

The dynamic modeling and control for the flexible sail-craft are studied in [37 38] In [37] the vibration equationsfor the axial and transverse deformations are establishedby considering the geometrical nonlinearity of the sailcraftBut merely the vibration analysis is insufficient for theorbiting sailcraft experiencing attitude motion The coupledattitudevibration analysis is required In [38] the coupledattitudevibration dynamics is established and the solvingprocess is also presented The controller is designed by usingthe Bang-Bang based PD theory The effectiveness of thecontroller is verified by numerical simulation But the largedeformation of the structure is not considered

The coupled attitude-orbit dynamics of a solar sail isstudied in [39]The equilibrium point of the dynamics can beobtained by designing the inertia of the sail and the stabilityof the equilibrium point is analyzed through a linearization

The attitude dynamics and control are studied thoroughlyin above references The ACS for solar sail by using actu-ators such as control boom and control vanes or by usingmeans such as spinning and translatingtilting sail panels arestudied Little research concerns attitude dynamic modelingwith highly flexible structure and vibration modeling withgeometrical nonlinearity and the attitudevibration controlBut this is a problem worth studying

This paper establishes the attitude dynamics for the largeflexible solar sail with control vanes control boom and slid-ing masses And the vibration equations are also presentedThe LQR and optimal PI based controllers are designed forthe attitudevibration dynamics By theoretical analysis andsimulation the controller with better performance is selectedand some discussions are presented And the differencesbetween the dynamics models with and without the geomet-rical nonlinearity are also inserted and analyzed

2 Attitude Dynamics

In this section the configuration and structure of the sailcraftare presented The related reference frames and coordinatetransformations are given The kinetic and gravitationalpotential energies and the generalized loads are obtainedTheLagrange equation method is adopted to derive the attitudedynamics with control vanes control boom and slidingmasses The attitude dynamics is an important part of thecoupled attitudevibration dynamics

21 The Configuration and Structure The configuration andstructure of the infinite-point connected square solar sailadopted in this paper are shown in Figure 1The ACS consistsof the control boom sliding masses and control vanes Thefour supporting beams are the most rigid component Thestiffness of the four triangular membranes can be neglectedby comparing with the supporting beams thus the kinetic

Mathematical Problems in Engineering 3

Controlvanes

Payload

Slidingmasses

Controlboom

Figure 1 The configuration and structure of solar sail

energy caused by vibrations and elastic potential energy ofthe membrane structure can be neglected

The paper mainly focuses on the dynamics for highlyflexible solar sail The following assumptions are made tosimplify the problems

(1) The detailed vibration of the membrane is neglectedin this paper as most of the kinetic energy causedby vibrations is by the vibrations of the supportingbeams

(2) The masses of control vanes and control boom areneglected We regard the control vanes and controlboom as rigid bodies

(3) The deformation of the structure is not affected by thethermal loads and the wrinkle effect is also neglected

(4) The torque by the cmcp offset is the disturbancetorque And other disturbance torques (such as grav-ity gradient torque) are neglected

22 The Reference Frames and Coordinate Transformations

The Inertial Reference Frame (E119868X119868Y119868Z119868120587

119868) It is an inertial

reference frame for solar sail attitude and orbital motionE119868X119868is in the vernal equinox direction E

119868Z119868is in the Earthrsquos

rotation axis perpendicular to equatorial plane E119868Y119868is in the

equatorial plane and finishes the ldquotriadrdquo of unit vectors Theunit vector is e

119868= (i

119868 j119868 k

119868)119879

The body reference frames of solar sail (O119874x119887y119887z119887120587

119887)

and control vanes (C119894x119894y119894z119894120587

119894 119894 = 1 2 3 4) O

119874is the geo-

metric center of the film O119874x119887is perpendicular to the film

pointing to the payload sideO119874y119887andO

119874z119887are in the plane

of the sail C119894is the geometric center of the 119894th control vane

When there exist no relative motions between solar sail andcontrol vanes120587

119894is coincident with120587

119887The related frames can

be found in Figure 2The attitude motion is described in Euler angles using

the yaw-roll-pitch notation In this notation a three-elementvector Θ = [120595 120593 120579]

119879 is used to describe the attitude of solarsail with respect to the inertial frame The transformation

EI

ZI

YI

XI

yb

zb

1

2

C2

C1

x2

z2

z1

y1

x1

y2 OO

xb

12

3

3

44

C4

C3

x3

y3z3

z4

y4x4

Figure 2 120587119868

120587119887

and 120587119894

Sun line

EI

OO

rO

ZI

XI

YI

ZI z998400

120595

120595

XI

EI(O)

Z998400998400998400(zb)

Z998400998400120593

120593

120579

120579

x998400x998400998400 x998400998400998400(xb)

y998400998400 y998400998400998400(yb)y998400

YI

The orbit

Figure 3 The orbit and the attitude motion

from 120587119868to 120587

119887can be realized by a 3-1-2 rotation as shown

in Figure 3 (right) with the following rotation matrix

C119887119868

= R2(120579)R

1(120593)R

3(120595)

= [

[

119888120579119888120595 minus 119904120579119904120593119904120595 119888120579119904120595 + 119904120579119904120593119888120595 minus119904120579119888120593

minus119888120593119904120595 119888120593119888120595 119904120593

119904120579119888120595 + 119888120579119904120593119904120595 119904120579119904120595 minus 119888120579119904120593119888120595 119888120579119888120593

]

]

(1)

where

R3(120595) = [

[

cos120595 sin120595 0

minus sin120595 cos120595 0

0 0 1

]

]

R1(120593) = [

[

1 0 0

0 cos120593 sin120593

0 minus sin120593 cos120593]

]

R2(120579) = [

[

cos 120579 0 minus sin 120579

0 1 0

sin 120579 0 cos 120579]

]

(2)

120595 120593 and 120579 are yaw roll and pitch angles respectively 119888120579 and119904120579 are short for cos 120579 and sin 120579

And the transformation from 120587119894to 120587

119887can be obtained as

119862119887119894= [

[

119888120579119894119888120595

119894minus 119904120579

119894119904120593

119894119904120595

119894119888120579

119894119904120595

119894+ 119904120579

119894119904120593

119894119888120595

119894minus119904120579

119894119888120593

119894

minus119888120593119894119904120595

119894119888120593

119894119888120595

119894119904120593

119894

119904120579119894119888120595

119894+ 119888120579

119894119904120593

119894119904120595

119894119904120579

119894119904120595

119894minus 119888120579

119894119904120593

119894119888120595

119894119888120579

119894119888120593

119894

]

]

(3)

The attitude control can be performed by varying 120595119894 120593

119894 and

120579119894


OO

xb mp

14

1205821

zbyb

dm32

3

1205822

EI

p

ch1

h2

31205883

O m3

r

r r

rr

R R

Figure 4 The positions and orientations of the components

23 The Positions and Velocities of the Components Theabsolute position of an arbitrary point on the supportingbeams sliding masses and payload are given in Figure 4where r

119888119894(119894 = 1 2 3 4) is the position vector of the cp of

the 119894th control vane The cp of each control vane is assumed

to locate at the free end of the supporting beams r119901is the

position vector of the payload O119874is the origin of 120587

119887 The

positions and velocities of an arbitrary point on each supportbeam can be obtained as

R119898119894

= R119874+ r

119894+ 120588

119894 k

119898119894= R

119874+∘

120588119894

+ 120596 times (r119894+ 120588

119894) (4)

whereR119898119894(v

119898119894) is the position (velocity) vector of an arbitrary

point on the deformed supporting beam and r119894is the position

vector of an arbitrary point on the undeformed beam relativetoO

119874R

119874(R

119874= v

119874) is the orbital position and velocity of the

sailcraft 120588119894is the transverse deformation of an arbitrary point

on the supporting beam (sdot) is the first time derivatives withrespect to 120587

119868and (∘) is the first time derivatives with respect

to 120587119887 120596 is the angular velocity of solar sail expressed as

[

[

120596119909

120596119910

120596119911

]

]119887

= [

[

cos 120579 minus sin 120579 cos120593120579 + sin120593

sin 120579 + cos 120579 cos120593]

]

(5)

If the orbital motion and its influence on attitude motionare neglected we have

R1198983

= r3+120588

3= [120588

3119909 120588

3119910 119911

3]119879

R1198981

= r1+ 120588

1= [120588

1119909 120588

1119910 119911

1]119879

R1198982

= r2+ 120588

2= [120588

2119909 119910

2 120588

2119911]119879

R1198984

= r4+ 120588

4= [120588

4119909 119910

4 120588

4119911]119879

(6)

R1198983

=∘

1205883

+ 120596 times (r3+ 120588

3) =

[[

[

1205883119909

1205883119910

0

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205883119909

1205883119910

1199113

]

]

=

[[[

[

1205883119909

+ 1199113

120579 + 1199113 sin120593 minus 120588

3119910 sin 120579 minus 120588

3119910 cos 120579 cos120593

1205883119910

+ 1205883119909 sin 120579 + 120588

3119909 cos 120579 cos120593 minus 119911

3 cos 120579 + 119911

3 sin 120579 cos120593

1205883119910 cos 120579 minus 120588

3119910 sin 120579 cos120593 minus 120588

3119909

120579 minus 1205883119909 sin120593

]]]

]

(7)

R1198981

=∘

1205881

+ 120596 times (r1+ 120588

1) =

[[

[

1205881119909

1205881119910

0

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205881119909

1205881119910

1199111

]

]

=

[[[

[

1205881119909

+ 1199111

120579 + 1199111 sin120593 minus 120588

1119910 sin 120579 minus 120588

1119910 cos 120579 cos120593

1205881119910

+ 1205881119909 sin 120579 + 120588

1119909 cos 120579 cos120593 minus 119911

1 cos 120579 + 119911

1 sin 120579 cos120593

1205881119910 cos 120579 minus 120588

1119910 sin 120579 cos120593 minus 120588

1119909

120579 minus 1205881119909 sin120593

]]]

]

(8)

R1198982

=∘

1205882

+ 120596 times (r2+ 120588

2) =

[[

[

1205882119909

0

1205882119911

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205882119909

1199102

1205882119911

]

]

=[[

[

1205882119909

+ 1205882119911

120579 + 1205882119911 sin120593 minus 119910

2 sin 120579 minus 119910

2 cos 120579 cos120593

1205882119909 sin 120579 + 120588

2119909 cos 120579 cos120593 minus 120588

2119911 cos 120579 + 120588

2119911 sin 120579 cos120593

1205882119911

+ 1199102 cos 120579 minus 119910

2 sin 120579 cos120593 minus 120588

2119909

120579 minus 1205882119909 sin120593

]]

]

(9)


R1198984

=∘

1205884

+ 120596 times (r4+ 120588

4) = [

[

1205884119909

0

1205884119911

]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205884119909

1199104

1205884119911

]

]

=[[

[

1205884119909

+ 1205884119911

120579 + 1205884119911 sin120593 minus 119910

4 sin 120579 minus 119910

4 cos 120579 cos120593

1205884119909 sin 120579 + 120588

4119909 cos 120579 cos120593 minus 120588

4119911 cos 120579 + 120588

4119911 sin 120579 cos120593

1205884119911

+ 1199104 cos 120579 minus 119910

4 sin 120579 cos120593 minus 120588

4119909

120579 minus 1205884119909 sin120593

]]

]

(10)

The components of the position and velocity vectors of anarbitrary point on the supporting beam can be computed byusing the above five equations

In Figure 4 1205821and 120582

2are the boom tilt and azimuth

angles relative to the sailcraft body axes respectively Thecomponents of r

119901can be expressed as follows by using these

two angles

r119901=10038161003816100381610038161003816r119901

10038161003816100381610038161003816cos 120582

1i119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816sin 120582

1cos 120582

2j119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816sin 120582

1sin 120582

2k119887

(11)

The position and velocity vectors of the payload can beobtained as

R119874119901

= R119874+ r

119901 k

119874119901= k

119874+ r

119901 (12)

where r119901=∘r119901+ 120596 times r

119901 And ∘r

119901can be expressed as

∘r119901= minus

10038161003816100381610038161003816r119901

100381610038161003816100381610038161sin 120582

1i119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816

times (1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2) j

119887

+10038161003816100381610038161003816r119901

10038161003816100381610038161003816(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2) k

119887

(13)

Thus r119901can be obtained as

r119901= [[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

times [119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1sin 120582

2( cos 120579 minus sin 120579 cos120593)]

times 119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

(14)

The position vectors of the two sliding masses can beobtained as

Rℎ119895

= R119874+ r

ℎ119895+ 120588

ℎ119895

kℎ2

= k119874+∘rℎ119895

+∘

120588ℎ119895

+ 120596 times (rℎ119895

+ 120588ℎ119895)

(119895 = 1 2)

(15)

If the orbital motion and its influence on attitude motioncan be neglected then the position and velocity vectors canbe obtained as follows

Rℎ2

= rℎ2

+ 120588ℎ2

= [120588ℎ2119909

120588ℎ2119910

119911ℎ2]119879

Rℎ1

= rℎ1

+ 120588ℎ1

= [120588ℎ1119909

119910ℎ1 120588

ℎ1119911]119879

Rℎ2

=[[

[

120588ℎ2119909

+ 119911ℎ2

( 120579 + sin120593) minus 120588ℎ2119910

( sin 120579 + cos 120579 cos120593)120588ℎ2119910

+ 120588ℎ2119909

( sin 120579 + cos 120579 cos120593) minus 119911ℎ2

( cos 120579 minus sin 120579 cos120593)ℎ2

+ 120588ℎ2119910

( cos 120579 minus sin 120579 cos120593) minus 120588ℎ2119909

( 120579 + sin120593)

]]

]

Rℎ1

=[[

[

120588ℎ1119909

+ 120588ℎ1119911

( 120579 + sin120593) minus 119910ℎ1

( sin 120579 + cos 120579 cos120593)119910ℎ1

+ 120588ℎ1119909


( cos 120579 minus sin 120579 cos120593)120588ℎ1119911

+ 119910ℎ1


( 120579 + sin120593)

]]

]

(16)


24 The Kinetic Energy The kinetic energy of the sail systemcan be obtained by neglecting the orbital motion and itsinfluence on attitude motion Consider

1198791198983

=1

2119898

119897int

119897

0

[∘

1205883

+ 120596 times (r3+ 120588

3)] sdot [

∘

1205883

+ 120596 times (r3+ 120588

3)] 119889119911

1198791198981

=1

2119898

119897int

0

minus119897

[∘

1205881

+ 120596 times (r1+ 120588

1)]

sdot [∘

1205881

+ 120596 times (r1+ 120588

1)] 119889119911

1198791198982

=1

2119898

119897int

119897

0

[∘

1205882

+ 120596 times (r2+ 120588

2)] sdot [

∘

1205882

+ 120596 times (r2+ 120588

2)] 119889119910

1198791198984

=1

2119898

119897int

0

minus119897

[∘

1205884

+ 120596 times (r4+ 120588

4)] sdot [

∘

1205884

+ 120596 times (r4+ 120588

4)] 119889119910

(17)

The above equations can be rewritten as

119879119898

=1

2120596 sdot J sdot 120596

+1

2119898

119897int

119897

0

2∘

1205883

sdot (120596 times r3) + (120596 times 120588

3) sdot (120596 times 120588

3)

+ 2 (120596 times r3) sdot (120596 times 120588

3) 119889119911

+1

2119898

119897int

0

minus119897

2∘

1205881


1) sdot (120596 times 120588

1)


1) 119889119911

+1

2119898

119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

+1

2119898

119897int

0

minus119897

2∘

1205884


4) sdot (120596 times 120588

4)


4) + (120596 times r

4)

sdot (120596 times r4) 119889119910

(18)

where J = sum4

119895=1

J119898119895

is the moment of inertia of sail systemJ119898119895

(119895 = 1ndash4) is the moment of inertia of the 119895th sail sys-tem and the expression of J can be expressed as J = diag[119869

119909

119869119910 119869

119911] inO

119874x119887y119887z119887

The kinetic energy of the payload is

119879119901=

1

2119898

119901

10038161003816100381610038161003816r119901

10038161003816100381610038161003816

2

=1

2119898

119901[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

2

(19)

The kinetic energy of the sliding masses can be obtainedas follows by neglecting the orbital motion and its influenceon attitude control

119879ℎ1

=1

2119898

ℎ1[ ( 120588

ℎ1119909+ 120579120588

ℎ1119911+ 120588

ℎ1119911sin120593

minus 119910ℎ1sin 120579 minus 119910

ℎ1cos 120579 cos120593)

2

]

+1

2119898

ℎ1[( 119910

ℎ1+ 120588

ℎ1119909sin 120579 + 120588

ℎ1119909cos 120579 cos120593

minus 120588ℎ1119911

cos 120579 + 120588ℎ1119911

sin 120579 cos120593)2]

+1

2119898

ℎ1[( 120588

ℎ1119911+ 119910

ℎ1cos 120579 minus 119910

ℎ1sin 120579 cos120593

minus 120579120588ℎ1119909

+ 120588ℎ1119909

sin120593)2

]

119879ℎ2

=1

2119898

ℎ2[( 120588

ℎ2119909+ 120579119911

ℎ2+ 119911

ℎ2sin120593 minus 120588

ℎ2119910sin 120579

minus 120588ℎ2119910

cos 120579 cos120593)2

]

+1

2119898

ℎ2[( 120588

ℎ2119910+ 120588

ℎ2119909sin 120579 + 120588

ℎ2119909cos 120579 cos120593

minus 119911ℎ2cos 120579 + 119911

ℎ2sin 120579 cos120593)

2

]

+1

2119898

ℎ2[(

ℎ2+ 120588

ℎ2119910cos 120579 minus 120588

ℎ2119910sin 120579 cos120593

minus 120579120588ℎ2119909

minus 120588ℎ2119909

sin120593)2

]

(20)

25 The Gravitational Potential Energy The gravitationalpotential energies of the sail system 119880

119892119891 payload 119880

119892119901 and

sliding masses 119880119892ℎ1

119880119892ℎ2

can be obtained as

119880119892119891

= minus

120583119898119891

1003816100381610038161003816R119874

1003816100381610038161003816

119880119892119901

= minus

120583119898119901

10038161003816100381610038161003816R119874+ r

119901

10038161003816100381610038161003816

119880119892ℎ1

= minus120583119898

ℎ1

1003816100381610038161003816R119874+ r

ℎ1

1003816100381610038161003816

119880119892ℎ2

= minus120583119898

ℎ2

1003816100381610038161003816R119874+ r

ℎ2

1003816100381610038161003816

(21)


26 The Generalized Forces The components of the sunlightunit vector S can be obtained as follows by referring toFigure 3 (left) in 120587

119868

S = j119868= [1 0 0]

119879

119868

(22)

Based on the abovementioned equations the componentsof S in 120587


S = C119887119868[1 0 0]

119879

119868

= [119878119909

119878119910

119878119911]119879

119887

(23)

The resultant SRP force can be simply modeled as followsfor a perfectly reflective control vane

F119894= 120578119875119860

119862119894(S sdot n

119862119894)2 n

119862119894= 119865

119862119894cos2120572

119862119894n119862119894 (24)

where 120578 is the overall sail-thrust coefficient (120578max = 2) 119875 =

4563 times 10minus6 Nm2 119860

119862119894is the area of the 119894th control vane

119865119862119894

= 120578119875119860119862119894is themaximumcontrol vane thrust andn

119862119894= i

119894

is the unit normal vector of the control vaneThe componentsof n

119862119894in 120587

119887can be written as

n119862119894

= C119887119894[1 0 0]

119879

119894

= [119878119909119894

119878119910119894

119878119911119894]119879

119887

(25)

The control torque by the 119894th control vane can beexpressed as

M119894= L

119894times F

119894 (26)

In fact L119894and F

119894are dependent on the deformations of

the supporting beams and control vanes In this paper it isassumed thatL

119894andF

119894are not influenced by the deformations

of the structure when computing the control torque by thecontrol vanes Thus L

119894and F


L1= [0 0 minus119897]

119879

L2= [0 119897 0]

119879

L3= [0 0 119897]

119879

L4= [0 minus119897 0]

119879

F119894= 120578119875119860

119862119894([119878

119909119878119910

119878119911]119887

sdot [119878119909119894

119878119910119894

119878119911119894]119879

119887

)

2

[119878119909119894

119878119910119894

119878119911119894]119879

119887

(27)

The force of each control vane can be written as

F119894= (119865

119862119894cos2120572

119862119894)C

119887119894[1 0 0]

119879

119894

= (119865119862119894cos2120572

119862119894) [119878

119909119894119878119910119894

119878119911119894]119879

119887

(28)

where 120572119862119894

is the sun angle of the 119894th control vane that canbe measured by the sun sensor And the components of thecontrol torques can be obtained as

M1198881

= (1198651198621cos2120572

1198621)[

[

0

0

minus119897

]

]

times (C1198871

[

[

1

0

0

]

]

)

= (1198651198621cos2120572

1198621)

times [

[

minus119897 cos1205931sin120595

1

minus119897 (cos 1205791cos120595

1minus sin 120579

1sin120593

1sin120595

1)

0

]

]

M1198882

= (1198651198622cos2120572

1198622)

times [

[

119897 (sin 1205792cos120595

2+ cos 120579

2sin120593

2sin120595

2)

0

minus119897 (cos 1205792cos120595

2minus sin 120579

2sin120593

2sin120595

2)

]

]

M1198883

= (1198651198623cos2120572

1198623)

times [

[

119897 cos1205933sin120595

3

119897 (cos 1205793cos120595

3minus sin 120579

3sin120593

3sin120595

3)

0

]

]

M1198884

= (1198651198624cos2120572

1198624)

times [

[

minus119897 (sin 1205794cos120595

4+ cos 120579

4sin120593

4sin120595

4)

0

119897 (cos 1205794cos120595

4minus sin 120579

4sin120593

4sin120595

4)

]

]

(29)

27 Attitude Dynamics of Solar Sail The attitude dynamicscan be obtained by using the Lagrange equation method as

119889

119889119905(

120597119871

120597 119902119895

) minus120597119871

120597119902119895

= 119876119895 (30)

where 119871 is the Lagrange function 119902119895(119895 = 1ndash3) is the vector

of the generalized coordinates and 119876119895is the vector of the

generalized force 119871 can be computed as

119871 = 119879 minus 119880 =

4

sum

119894=1

119879119898119894

+ 119879119901+

2

sum

119895=1

119879ℎ119895

minus (119880119892119891

+ 119880119892119901

+

2

sum

119895=1

119880119892ℎ119895

)

(31)

where 119879119898119894 119879

119901 and 119879

ℎ119895are the kinetic energies of the

supporting beams payload and sliding masses respectively119880119892119891 119880

119892119901 and 119880

119892ℎ119895are the gravitational potential energies of

the sail system payload and sliding masses respectively


OOx

y

z

FCFCy

FCx

FCz

mh

q(x)

Figure 5 The vibration model

120590

120590e

120590120592

120590

Figure 6 The viscoelastic model

3 The Vibration of Solar Sail

The mechanical and material models of the sailcraft aregivenThe displacement filed stress and strain are presentedbased on the Euler beam assumptionThe Lagrange equationmethod is adopted to derive the vibration equations withlarge deformations based on the calculations of kineticenergies and works done by the external loads

31 The Mechanical Model The body frame O119874xyz is estab-

lished to describe the vibration (see Figure 5) F119862is the force

vector generated by the control vane used located at the tip ofthe supporting beam 119865

119862119909 119865

119862119910 and 119865

119862119911are the components

inO119874xyz 119902(119909) is the distributed load caused by SRP

32 The Material Model The carbon fibre enforced compos-ite material is used to construct the inflatable deploymentsupporting beam The model of the viscoelastic material canbe represented in Figure 6

The total stress is the summation of the elastic and viscousstresses It can be expressed as

120590 = 120590119890

+ 120590120592

= 119864120576 + 119888119887

120576 (32)

where 120590 120590119890 and 120590

120592 are the total elastic and viscous stressrespectively And 119864 is Youngrsquos modulus 120576 ( 120576) is the strain(strain rate) and 119888

119887is the coefficient of the internal damping

expressed as 119864120578119887 120578

119887is the proportionality constant of the

internal damping The dimensions of 119888119887and 120578

119887are N lowast sm2

and s respectively

OO x

y The support beam

Z

Figure 7 The schematic diagram of solar sail supporting beamdeformation

33 The Displacement Field Stress and Strain The deforma-tion of a spatial solar sail supporting beam is presented inFigure 7

The spatial displacement fields for supporting beam are as

119906 (119909 119910 119911 119905) = 1199060(119909 119905) minus 119910

120597V0

120597119909minus 119911

1205971199080

120597119909

V (119909 119910 119911 119905) = V0(119909 119905)

119908 (119909 119910 119911 119905) = 1199080(119909 119905)

(33)

119906(119909 119910 119911 119905) V(119909 119910 119911 119905) and 119908(119909 119910 119911 119905) are the displace-ments along 119909 119910 and 119911 directions respectively 119906

0(119909 119905)

V0(119909 119905) and 119908

0(119909 119905) are the axial and transverse displace-

ments of an arbitrary point on the neutral axisBy using Green strain definition von Karmanrsquos nonlinear

strain-displacement relationships based on assumptions oflarge deflections moderate rotations and small strains for a3D supporting beam are given as follows together with thestrain rate and normal stress

120576119909119909

=120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

120576119909119909

=120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909)

minus 1199111205972

0

1205971199092

120590119909119909

= 119864[120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

]

+ 119864120578119887(120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909) minus 119911

1205972

0

1205971199092

)

(34)


The single underlined terms in the preceding equationindicate the contribution by the geometrical nonlinearityThedynamics models and corresponding numerical simulationwithout considering the geometrical nonlinearity can beobtained and carried out by neglecting these terms

34 The Energies and Works Done by External Forces Thestrain energy can be derived as

119880119894=

1

2int

119897

0

119860119909119909

[120597119906

0

120597119909+

1

2(120597V

0

120597119909)

2

+1

2(120597119908

0

120597119909)

2

]

2

+ 119863119909119909

1205972V

0

1205971199092

+ 119863119909119909

1205972

1199080

1205971199092

119889119909

(35)

For an isotropic support beam the extensional stiffness119860

119909119909and bending stiffness119863

119909119909can be computed as

119860119909119909

= int119860

119864119889119860 119863119909119909

= int119860

1198641199112

119889119860 = int119860

1198641199102

119889119860 (36)

The dissipation function is as

Ψ119902=

1

2int

119897

0

[119860119909119909120578119887(120597

0

120597119909)

2

+ 119863119909119909120578119887(1205972V

0

1205971199092

)

2

+ 119863119909119909120578119887(1205972

0

1205971199092

)

2

]119889119909

(37)

The kinetic energy for the support beam is as

119879119890=

1

2int

119897

0

[119868119860(120597119906

0

120597119905)

2

+ 119868119860(120597119908

0

120597119905)

2

+ 119868119863(1205972

1199080

120597119909120597119905)

2

+ 119868119860(120597V

0

120597119905)

2

+ 119868119863(

1205972V

0

120597119909120597119905)

2

]119889119909

(38)

where 119868119860and 119868

119863are integration constant for the support

beam expressed as

119868119860= int

119860

120588119889119860 = 120588119860 119868119863= int

119860

1205881199112

119889119860 = int119860

1205881199102

119889119860 (39)

And the work done by the external force can be obtainedas119882 = 119865

119862119911sdot 119908 (119909

119888 119905) + 119865

119862119910sdot V (119909

119888 119905) + 119898

ℎ119886ℎ119910

(119909ℎ 119905) V (119909

ℎ 119905)

+ 119898ℎ119886ℎ119911

(119909ℎ 119905) 119908 (119909

ℎ 119905) + int

119897

0

119902 (119909) V (119909 119905) 119889119909

(40)where 119909

119888represents the position of the support beam

119908(119909119888 119905) V(119909

119888 119905) are the displacements along 119911 and 119910 direc-

tions respectively and the corresponding loads 119865119862119911 119865

119862119910are

known functions 119909ℎ 119897 and V

ℎare the position coordinate of

the sliding mass the length of the support beam and theconstant-velocity of the sliding mass The dynamic responseis studied only during the period [0 119897V

ℎ] 119898

ℎis the mass

and 119886ℎ119910(119909

ℎ 119905) and 119886

ℎ119911(119909

ℎ 119905) are the components of the

acceleration for a material point on the support beam theslidingmass just arrived at And the corresponding transversedeflections are 119908(119909

ℎ 119905) and V(119909

ℎ 119905) respectively

35 The Assumed Modes Method The axial displacement1199060(119909 119905) and the transverse deflections V

0(119909 119905) and119908

0(119909 119905) can

be represented as follows by considering the assumed modesmethod

1199060(119909 119905) =

infin

sum

119894=1

119880119894(119909)119880

119894(119905)

V0(119909 119905) =

infin

sum

119895=1

119881119895(119909) 119881

119895(119905)

1199080(119909 119905) =

infin

sum

119896=1

119882119896(119909)119882

119896(119905)

(41)

where 119880119894(119905) 119881

119895(119905) and 119882

119896(119905) are the time-dependent gen-

eralized coordinates and 119880119894(119909) 119881

119895(119909) and 119882

119896(119909) are the

assumedmodes For assumedmodemethod themode shapefunctions should satisfy the following requirements in thispaper

119880119894(119909)

1003816100381610038161003816119909=0= 0 119881

119895(119909)

10038161003816100381610038161003816119909=0= 0 119882

119896(119909)

1003816100381610038161003816119909=0= 0

1198811015840

119895

(119909)10038161003816100381610038161003816119909=0

= 0 1198821015840

119896

(119909)10038161003816100381610038161003816119909=0

= 0

(42)

The following first two assumed modes satisfying theassumed mode principle are used in this paper

1198801(119909) =

119909

119897 119880

2(119909) = (

119909

119897)

2

1198811(119909) = (

119909

119897)

2

1198812(119909) = (

119909

119897)

3

1198821(119909) = (

119909

119897)

2

1198822(119909) = (

119909

119897)

3

(43)

With the help of the assumedmodes the detailed expres-sions of 119880

119894 119879

119890119882

119911 and Ψ


119880119894=

9

2

119860119909119909

1198973(119881

2

2

+ 1198822

2

)2

+27

4

119860119909119909

1198973(119881

11198812+ 119882

1119882

2)

times (1198812

2

+ 1198822

2

)

+18

7

119860119909119909

1198973[(119881

2

1

+ 1198822

1

) (1198812

2

+ 1198822

2

) + (11988111198812+ 119882

1119882

2)2

]

+119860

119909119909

1198973[3

2119897119880

2(119881

2

2

+ 1198822

2

) + (1198812

1

+ 1198822

1

)

times (11988111198812+ 119882

1119882

2)]


+1

10

119860119909119909

1198973[9119880

1119897 (119881

2

2

+ 1198822

2

) + 241198802119897 (119881

11198812+ 119882

1119882

2)

+ 4 (1198812

1

+ 1198822

1

)2

]

+119860

119909119909

1198972[3

21198801(119881

11198812+ 119882

1119882

2) + 119880

2(119881

2

1

+ 1198822

1

)]

+6119863

119909119909119881

2

2

1198973+ 119860

119909119909(2119880

1119881

2

1

31198972+

21198801119882

2

1

31198972+

21198802

2

3119897)

+6119863

119909119909119882

2

2

1198973+

611986311990911990911988111198812

1198973+

11986011990911990911988011198802

119897+

6119863119909119909119882

1119882

2

1198973

+119860

119909119909119880

2

1

2119897+

2119863119909119909119881

2

1

1198973+

2119863119909119909119882

2

1

1198973

119879119890=

1

14(119897119868

119860

2

2

+ 119897119868119860

2

2

) +1

6(119897119868

119860

1

2+ 119897119868

11986012)

+1

10(119897119868

119860

2

2

+ 119897119868119860

2

1

+9119868

119863

2

2

119897+ 119897119868

119860

2

1

+9119868

119863

2

2

119897)

+1

4(119897119868

11986012+

6119868119863

1

2

119897+

611986811986312

119897)

+1

6(119897119868

119860

2

1

+4119868

119863

2

1

119897+

4119868119863

2

1

119897)

119882119911= 119865

119862119911sdot [119882

1(119905) + 119882

2(119905)] + 119865

119862119910sdot [119881

1(119905) + 119881

2(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119881

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119881

2(119905) +

6V3ℎ

1199052

1198973

sdot 2(119905) + (

Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198811(119905) + (

Vℎ119905

119897)

3

sdot 1198812(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119882

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119882

2(119905) +

6V3ℎ

1199052

1198973sdot

2(119905)

+ (Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198821(119905) + (

Vℎ119905

119897)

3

sdot 1198822(119905)]

+ 1198811(119905) sdot int

119897

0

119902 (119909) (119909

119897)

2

119889119909 + 1198812(119905)

sdot int

119897

0

119902 (119909) (119909

119897)

3

119889119909

Ψ119902=

1

6(36119863

119909119909120578119887

2

2

1198976+

4119860119909119909120578119887

2

2

1198974+

36119863119909119909120578119887

2

2

1198976) 119897

3

+1

4(4119860

11990911990912057811988712

1198973+

2411986311990911990912057811988712

1198975

+24119863

119909119909120578119887

1

2

1198975) 119897

2

+1

2

119860119909119909120578119887

2

1

119897+

2119863119909119909120578119887

2

1

1198973+

2119863119909119909120578119887

2

1

1198973

(44)

36 The Vibration Equation The Lagrange equation methodis used to derive the vibration equations

120597119871

120597119902119899

minus119889

119889119905(

120597119871

120597 119902119899

) + 119876119889= 0 119899 = 1 2 6 (45)

where 119871 is the Lagrange function 119902119899

= (1198801 119880

2 119881

1 119881

2

1198821119882

2)119879 is the vector of the generalized coordinates and

119902119899is the corresponding generalized velocity vector 119876

119889is the

generalized damping force The expressions for 119871 and 119876119889are

as

119871 = 119879119890minus (119880

119894minus 119882

119911)

119876119889= minus

120597Ψ119902

120597 119902119899

(119899 = 1 sim 6)

(46)

The axial vibration equations can be obtained as followsfor 119880

1 119880

2

1

31198971198681198601+

1

41198971198681198602+

120578119887119860

1199091199091

119897+

120578119887119860

1199091199092

119897+

1198601199091199091198801

119897

+119860

1199091199091198802

119897+

2119860119909119909119881

2

1

31198972+

311986011990911990911988111198812

21198972+

9119860119909119909119881

2

2

101198972

+2119860

119909119909119882

2

1

31198972+

3119860119909119909119882

1119882

2

21198972+

9119860119909119909119882

2

2

101198972= 0

(47a)

1

41198971198681198601+

1

51198971198681198602+

120578119887119860

1199091199091

119897+

4120578119887119860

1199091199092

3119897

+119860

1199091199091198801

119897+4119860

1199091199091198802

3119897+119860

119909119909119881

2

1

1198972+12119860

11990911990911988111198812

51198972+3119860

119909119909119881

2

2

21198972

+119860

119909119909119882

2

1

1198972+

12119860119909119909119882

1119882

2

51198972+

3119860119909119909119882

2

2

21198972= 0

(47b)

The following can be seen from the above equations

(1) Since the assumed modes rather than the naturalmodes are used the coupling between axial and trans-verse vibrations is rather severe for axial vibrations


(2) The nonlinearity is apparent in above equations onlyexisting in the terms of 119881

1 119881

2119882

1119882

2 We take (47a)

as an exampleThe terms including1198802and its first and

second derivative 1198811 119881

2119882

1119882

2 can be regarded as

the applied loads for 1198801

(3) The coupling between the axial and transverse vibra-tionswill disappear as long as the linear displacement-strain relationship is adopted

(4) There are no external loads for axial vibrationsaccording to above equations

The transverse vibration equations for 1198811 119881

2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)

The complexity high coupling and nonlinearity are obvi-ous in above equations

4 The Coupled Rigid-Flexible Dynamics

The dynamics for control and dynamic simulation shouldbe presented for solar sail based on the simplification ofequations derived in previous sections Some assumptions aremade before simplifying the equations as follows


(1) Only the attitude motion is affected by the slidingmass the vibration of solar sail structure is notaffected by the sliding mass

(2) Although the attitude control is accomplished byrotating control vanes and the gimbaled controlboom moving sliding masses the specific time his-tories of the actuators are not considered here

(3) The four supporting beams are regarded as an entirestructure thus the identical generalized coordinatesare adopted in the dynamic analysis

41 The Simplified Attitude Dynamics The kinetic energy ofsolar sail supporting beams can be obtained as

119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)

The subscripts 2 and 3 represent the second and thirdsupporting beams The displacements can be expressed as

V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)

The subscripts 119909 119910 and 119911 of the left terms in (50)represent the displacements along119909119910 and 119911directions119881

1(119905)

and 1198812(119905) are the generalized coordinates The displacements

1199080119910(119911 119905) V

0119909(119910 119905) and 119908

0119911(119910 119905) can be obtained using the

assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)

The vibration velocity can be obtained as

∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)

Without loss of generality 120572 = 120573 = 1205721= 120573

1= 1 and

120574 = 1205741= minus1 can be assumed in above expressions And the

kinetic energy of the slidingmasses can be obtained as followsby neglecting the vibration related terms

119879ℎ=

1

2119898

ℎ1(120596 times r

ℎ1) sdot (120596 times r

ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)

sdot (120596 times rℎ2)

(53)

The kinetic energy of the payload can be obtained as

119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)

The following simplified attitude dynamics can beobtained by neglecting the second and higher terms ofattitude angles and vibration modes in computing the kinetic


energies of the supporting beams payload and slidingmasses Consider

119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876

120595are the generalized nonpotential

attitude control torques generated by the control vanes Andthe attitude control torques generated by the gimbaled controlboom and sliding masses can be found on the left side ofthe above equation coupled with 120579 making the attitudecontroller design problem rather difficult

42 The Simplified Vibration Equations The couplingbetween the axial and transverse vibrations is neglectedand only the transverse vibration is considered The attitudedynamics for dynamic simulation can be obtained as followsbased on previous derivation by neglecting the action ofsolar-radiation pressure

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)

The following equations can be obtained by rearrangingthe above equations

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865

1198622(119905) are used to represent all the generalized

external forces they can be used for vibration control Thesecomplicated equations abovementioned are the basis forobtaining the simplified equations used for controller design

The following equations can be obtained by neglecting thetwo and higher terms

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)

43 The Rigid-Flexible Coupled Dynamics The simplifiedattitude and vibration dynamics can be written as

119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)

The related coefficients in the preceding equation can befound in (A1) in the appendix The dynamics (see (59)) isnot suitable for attitude controller design because the controlinputs exist in some coefficients of the angular accelerationsThis will make the controller design rather difficult Thusthe following dynamic model is obtained by putting theterms including the control inputs right side of the dynamicequations

119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)

The expressions for 119906 119906

120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579

and 119906are the functions of the positions of sliding

masses and payload and the angular accelerations The statevariables in (60) are defined as

1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)

Thedynamics (60) can bewritten as the state-spacemodelas following

1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906

120595are the generalized control inputs

for attitude control and 1199061198811

and 1199061198812

are the generalizedcontrol inputs for vibration control The controllability canbe satisfied by analyzing (63) The attitudevibration controlinputs and the coefficients will be presented later in thissection The form of a matrix differential equation can bewritten as follows for the above equation

x = Ax + Bu (64)

The expressions of the relatedmatrices and vectors can befound as

x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879

u (1199050) is known

(65)

where A xB and u are the system matrix state variablesvector control matrix and control inputs vector respectively


The complexity of the state-space model can be seen bypresenting the detailed expressions of control inputs forattitudevibration and related parameters in (A2) in theappendix

5 AttitudeVibration Control and Simulation

The controller for the attitude and vibration is developedusing the LQR and optimal PI theory The dynamic simula-tion is performed based on the coupled dynamics derived inthe previous sections

51 LQR and Optimal PI Based Controllers Design Thedynamics for solar sail can bewritten as follows togetherwithits quadratic cost function also defined as

x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879

gt 0 are constantThe control-lability of [AB] and the observability of [AD] can beensured when the equationDD119879

= Q holds for anymatrixDThe optimal control input ulowast(119905)within the scope of u(119905) isin R119903119905 isin [119905

0infin] can be computed as follows to make 119869minimum

ulowast (119905) = minusKxlowast (119905) K = Rminus1B119879P (67)

where P is the nonnegative definite symmetric solution of thefollowing Riccati algebraic equations

PA + A119879P minus PBRminus1B119879P +Q = 0 (68)

And the optimal performance index can be obtained asfollows for arbitrary initial states

119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)

The closed-loop system

x = (A minus BRminus1B119879P) x (70)

is asymptotically stableFor an orbiting solar sail the primary disturbance torque

is by the cmcp offset and the impact of the tiny dust inthe deep space The state regulator can be used to controlthe dynamic system affected by pulse disturbance torquewith good steady-state errors but will never achieve accuratesteady-state errors if the dynamics is affected by constantdisturbance torque Thus the optimal state regulator withan integrator to eliminate the constant disturbance torqueshould be presented This optimal proportional-integral reg-ulator not only eliminates the constant disturbance torque butalso possesses the property of optimal regulator

Table 1 The related parameters

Parameters ValuesCMCP offsetm [0 017678 017678]119879

The side length of square sailm 100Nominal solar-radiation pressureconstant at 1 AU from the sunNm2 4563 times 10minus6

Solar-radiation pressure forceN 912 times 10minus2

Roll inertiakglowastm2 50383Pitch and yaw inertiakglowastm2 251915Outer radiusm 0229Thickness of the supporting beamm 75 times 10minus6

Youngrsquos modulus of the supportingbeamGPa 124

Moment of inertia of the cross-sectionof the supporting beamm4 2829412 times 10minus7

Bending stiffness of the supportingbeamNlowastm2 3508471336

Proportional damping coefficient ofsupporting beams 001

Line density of the supportingbeamkgm 0106879

Cross-sectional area of the supportingbeamm2 1079 times 10minus5

Roll disturbance torqueNlowastm 0Pitch (yaw) disturbance torqueNlowastm 0016122 (minus0016122)

For the following linear system with the correspondingperformance index

x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879

gt 0 and S = S119879 gt 0 If [AB]is completely controllable and meanwhile [AD

11] is com-

pletely observable with the relationship D11D119879

11

= Q thesolution can be expressed as

ulowast = minusK1x minus K

2ulowast ulowast (119905

0) = u

1199050

(72)

If B119879B is full rank we can get


4x ulowast (119905

0) = u

1199050

(73)

K119894(119894 = 1 2 3 4) is the function of ABQR S


[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)

is asymptotically stableThe compromise between the state variables (the angle

angular velocity the vibration displacement and velocity)and control inputs (the torque by all the actuators) can


0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

Figure 8 The roll pitch and yaw errors

be achieved by using both the LQR based and optimal PIbased controllers for the coupled attitudevibration dynamicsystem The fact is that the attitude control ability is limitedto the control torque by the control actuators The change ofthe attitude angles and rates should not be frequent to voidexciting vibrations

Thus the weighting matrix Q and the control weightingmatrix R are selected as follows for LQR based controller

Q (1 1) = 9 times 10minus6



Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)

TheweightingmatrixQ and the control weightingmatrixR are selected as follows for the optimal PI based controller

Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4

Figure 9 The roll pitch and yaw rates

Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)

52 The Simulation Results The initial angle errors of rollpitch and yaw are adopted as 120593(119905

0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3

∘ the initial angular velocity errors of roll pitchand yaw are adopted as (119905

0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms

and 2(1199050) = 0ms are adopted in this paper for dynamic

simulationThe estimated cmcp offset is 025m for a 100 times 100m

square solar sail It is assumed that the components of cmcpoffset vector are [0 017678 017678]

119879 in the body frame

The relevant parameters for dynamic simulations are givenin Table 1

The dynamic simulation results are obtained as followsby using the LQR controller based on the abovementionedQ and R related parameters and initial conditions Thedynamics simulations with (denoted by ldquowith GFrdquo) andwithout (denoted by ldquowithout GFrdquo) considering the geo-metrical nonlinearity are presented And the correspondingdiscussions and analysis are also given according to thecalculated results

The roll pitch and yaw errors are presented based onthe dynamics models with and without considering thegeometrical nonlinearityThe control effect with zero steady-state error can be achieved for roll axis by referring tothe results But the steady-state errors exist for pitch andyaw axes from Figure 8 this is because there exist constantdisturbance torques for pitch and yaw axes caused by thecmcp offset Thus the LQR based controller cannot be usedfor attitude control with good steady-state performance Thesteady-state errors of pitch and yaw axes are about 0164

∘

and minus0164∘ respectively from the corresponding partial

enlarged drawing of the figures (see the corresponding left


minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Figure 10 The roll pitch and yaw torques

figures) A simple and practical controller should be designedto eliminate the steady-state errorsThese errorswill affect theprecision of the super long duration space mission deeply

The differences between the dynamics with and withoutthe geometrical nonlinearity effect can be observed Theamplitudes and phases change a bit by comparing the corre-sponding results

It can be observed by the partial enlarged figures(the right figures of Figure 9) that there are no steady-stateerrors for roll pitch and yaw rates The relative satisfieddynamic performance is achieved In fact a result with muchmore satisfied dynamic performance can be obtained but itwill make the desired control torques much larger duringthe initial period This will not only excite the vibration ofthe structure but also increase the burden of the controlactuators

And the differences between the dynamics models withand without the geometrical nonlinearity can also be seenmerely in the phase of the angular velocities

The desired attitude control torques are presented inFigure 10 The roll torque will tend to be zero with thedecrement of the state variable errors The pitch and yawtorques tend to be constant from the middle and bottom

figures The pitch and yaw steady-state errors can never beeliminated although the pitch and yaw torques (009Nlowastmand minus009Nlowastm) are applied continuously by theoreticalanalysis and simulations This is just a great disadvantage ofthe LQR based controller in this paper The initial controltorques can be afforded by vibrating the gimbaled controlboom andmoving the slidingmasses that can afford a relativelarge torque The control vanes can be used to afford thecontrol torques when solar sail is in steady-state process Theexpression of the attitude control inputs 119906

120593 119906

120579 and 119906

120595can be

found in previous sectionMoreover the differences between dynamics responses

with and without geometrical nonlinearity can also beobserved by comparing the corresponding results Therequired torques for the attitude control are a little largerfor the dynamics with geometrical nonlinearity And thedifference in the phase is also clear

It can be seen that the vibration of the supporting beamis depressed effectively by the simulation results in Figure 11It can be seen that the steady-state errors and the dynamicperformance are relatively satisfied It can also be seen that thedynamics responses between the model with and without the


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)

Figure 11 The displacement and velocity of the tip of the supporting beam

geometrical nonlinearity effect are different in the vibrationdisplacement and velocitymagnitudes and phases (Figure 15)

The simulation results can be obtained as follows by usingthe optimal PI based controller adopting the abovementionedparameters and initial conditions

The roll pitch and yaw errors are given by using theoptimal PI based controller These errors tend to disappearby observing the partially enlarged plotting of the figure (thecorresponding right figures) (shown in Figure 12) And therelative satisfied dynamic performance can also be obtainedFrom the view of eliminating the steady-state errors the opti-mal PI based controller performs better than the LQR basedcontroller by theoretical analysis and simulation results Theformer can achieve attitude control results without pitchand yaw steady-state errors From the view of dynamicperformance both the controllers perform well In additionit can be seen that the difference between the modelswith and without geometrical nonlinearity is merely inthe phase

The roll pitch and yaw rate errors are presented inFigure 13 The dynamic and steady-state performances aresatisfied by observing the simulation results And by observ-ing the difference between the dynamics responses with and

without the influence by the geometrical nonlinearity it canbe concluded that the merely difference is the phase

The desired control torques of the roll pitch and yaw areobtained in Figure 14 by using the optimal PI based controllerIt can be seen that the value of the desired roll control torquedecreases to zero as the values of the roll angle and ratedecrease to zero The desired pitch and yaw control torquestend to be the constant values 009Nlowastm and minus009Nlowastmrespectively The desired initial control torques are large byobserving Figure 14 It is suggested that the gimbaled controlboom combining the sliding masses is adopted to afford theinitial torque Then the control vanes can be used to controlthe solar sail after the initial phase The phase differencecan also be observed by inspecting the dynamics responsesduring the 4800sim5000 s period

The vibration can be controlled effectively based on theoptimal PI controller And the relative satisfied dynamicand steady-state performances can be achieved In additionthe tiny differences exist in the magnitudes of the requiredtorques And the phase differences also exist

It can be concluded as follows by analyzing and compar-ing the simulation results based on the LQR and optimal PItheories in detail


Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF


(1) From the view of eliminating the steady-state errorsof the pitch and yaw errors the optimal PI based con-troller performs better than the LQR based controllerby theoretical analysis and simulation results Thedynamic performances of the two controllers are bothsatisfied and the desired control torques are relativelysmall

(2) The weighting of the state variables is reflected by theweighting matrix according to the LQR and optimalPI theories The constant weighting matrices selectedin this paper can effectively reflect the initial statevariable errors but maybe reflect the final state vari-able errors deficiently The time-varying weightingmatrices should be selected

(3) For the actual orbiting solar sail the control actuators(control vanes gimbaled control boom and slidingmasses) cannot afford excessive large control torquesThus the control torques should be as small as possibleon the premise of acquiring satisfied dynamic perfor-mance and steady-state errors It will not only reducethe burden of the control actuators but also avoid

exciting structural vibration In engineering practicethe weighting matrices and the control weightingmatrices should be selected based on the attitudecontrol requirement (eg dynamic performance andsteady-state errors etc)

(4) The dynamics responses influenced by the geometri-cal nonlinearity can be observed In a word the tinydifferences between the magnitudes of the physicalvariables exist for the models with and withoutthe geometrical nonlinearity And the relative cleardifferences exist in the phases for the attitude anglesrates required torques and so forth

6 Conclusions

The coupled attitude and structural dynamics is establishedfor the highly flexible sailcraft by using the von-Karman von-Karmanrsquos nonlinear strain-displacement relationships basedon assumptions of large deflections moderate rotationsand


Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Figure 13 The roll pitch and yaw rate errors

small strains The Lagrange equation method is utilizedto derive the dynamics for controller design and dynamicsimulation The LQR and optimal PI based controllers aredesigned for the dynamics with the constant disturbancetorque caused by the cmcp offset By theoretical analysisand simulation it can be seen that both controllers cancompromise between the state variables and control inputsBecause the control actuators can hardly afford large controltorque the dynamic performance should be relaxed toreduce the requirement for the attitude control actuatorsBesides the optimal PI based controller can also eliminatethe constant disturbance torque caused by the cmcp offsetthat always exists while the LQR based controller fails toachieve the results with satisfied steady-state performancealthough the attitude control torque is always applied to thecoupled dynamic systemThe dynamics simulations with andwithout the geometrical nonlinearity effect are also carriedout And the differences between the models in magnitudesand phases are identified and analyzed It can be concludedthat the optimal PI based controller performs better than the

LQRbased controller from the viewof eliminating the steady-state error caused by the disturbance torque by the cmcpoffset

Appendix

The related parameters in above equation are given as

119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)

The detailed expressions of control inputs for attitudevibration and related parameters in the state-space model ofthe dynamics (see (62) and (63)) are presented as follows

A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)

Figure 15 Vibration displacement and velocity of the tip

B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Project no 11302134 11272101)

References

[1] M Colin Solar Sailing Technology Dynamics and MissionApplications Springer-Praxis London UK 1st edition 1999

[2] V Lappas SWokesM Leipold A Lyngvi and P Falkner ldquoGui-dance and control for an Interstellar Heliopause Probe (IHP)solar sail mission to 200 AUrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and Exhibit pp2472ndash2492 San Francisco Calif USA August 2005

[3] L Charlotte C Camilla and C McInnes ldquoSolar radiation pres-sure augmented deorbiting fromhigh altitude sun-synchronousorbitsrdquo in Proceedings of the 4S Symposium 2012 Small SatellitesSystems and Services Portoroz Slovenia 2012

[4] M Ceriotti and C RMcInnes ldquoGeneration of optimal trajecto-ries for Earth hybrid pole sittersrdquo Journal of Guidance Controland Dynamics vol 34 no 3 pp 847ndash859 2011

[5] S Gong J Li andH Baoyin ldquoSolar sail transfer trajectory fromL1 point to sub-L1 pointrdquoAerospace Science and Technology vol15 no 7 pp 544ndash554 2011

[6] S Gong J Li and H Baoyin ldquoAnalysis of displaced solar sailorbits with passive controlrdquo Journal of Guidance Control andDynamics vol 31 no 3 pp 782ndash785 2008

[7] S Gong H Baoyin and J Li ldquoSolar sail three-body transfertrajectory designrdquo Journal of Guidance Control and Dynamicsvol 33 no 3 pp 873ndash886 2010


[8] S Gong J Li H Baoyin and J Simo ldquoA new solar sail orbitrdquoScience China Technological Sciences vol 55 no 3 pp 848ndash8552012

[9] Y Tsuda O Mori R Funase et al ldquoAchievement of IKAROS-Japanese deep space solar sail demonstration missionrdquo ActaAstronautica vol 82 no 2 pp 183ndash188 2013

[10] L Johnson MWhorton A Heaton R Pinson G Laue and CAdams ldquoNanoSail-D a solar sail demonstration missionrdquo ActaAstronautica vol 68 no 5-6 pp 571ndash575 2011

[11] B Wie and D Murphy ldquoSolar-sail attitude control design for asail flight validation missionrdquo Journal of Spacecraft and Rocketsvol 44 no 4 pp 809ndash821 2007

[12] B Wie D Murphy M Paluszek and S Thomas ldquoRobust atti-tude control systems design for solar sails part 1 propellantlessprimaryACSrdquo inProceedings of the AIAAGuidance Navigationand Control Conference and Exhibit Providence RI USAAugust 2004

[13] B Wie D Murphy M Paluszek and S Thomas ldquoRobust atti-tude control systems design for solar sails part 2 MicroPPT-based secondary ACSrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference and Exhibit pp 1325ndash1340Providence RI USA August 2004

[14] S Thomas M Paluszek B Wie and D Murphy ldquoAOCS per-formance and stability validation for large flexible solar sailspacecraftrdquo in Proceedings of the 41st AIAAASMESAEASEEJoint Propulsion Conference amp Exhibit Tucson Ariz USA 2005

[15] B Wie ldquoDynamic modeling and attitude control of solar sailspacecraft part Irdquo in Proceedings of the AIAAGuidance Naviga-tion and Control Conference and Exhibit Monterey Calif USAAugust 2002

[16] BWie ldquoSolar sail attitude control and dynamics part 1rdquo Journalof Guidance Control and Dynamics vol 27 no 4 pp 526ndash5352004

[17] B Wie ldquoDynamic modeling and attitude control of solar sailspacecraft part IIrdquo in Proceedings of the AIAA Guidance Nav-igation and Control Conference and Exhibit Monterey CalifUSA August 2002


[19] C Luo and J-H Zheng ldquoSolar-sail attitude control based onmoving masses and roll stabilizer barsrdquo Journal of HarbinInstitute of Technology vol 43 no 3 pp 95ndash101 2011

[20] D Romagnoli and T Oehlschlagel ldquoHigh performance twodegrees of freedom attitude control for solar sailsrdquo Advances inSpace Research vol 48 no 11 pp 1869ndash1879 2011

[21] C Scholz D Romagnoli B Dachwald and S Theil ldquoPerfor-mance analysis of an attitude control system for solar sails usingsliding massesrdquo Advances in Space Research vol 48 no 11 pp1822ndash1835 2011

[22] A Bolle and C Circi ldquoSolar sail attitude control through in-plane moving massesrdquo Journal of Aerospace Engineering vol222 no 1 pp 81ndash94 2008

[23] M RotunnoM Basso A B Pome andM Sallusti ldquoA compari-son of robust attitude control techniques for a solar sail space-craftrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference and Exhibit San Francisco Calif USA2005

[24] J Zhang K Zhai and T Wang ldquoControl of large angle maneu-vers for the flexible solar sailrdquo Science China Physics Mechanicsand Astronomy vol 54 no 4 pp 770ndash776 2011

[25] J Zhang andTWang ldquoCoupled attitude-orbit control of flexiblesolar sail for displaced solar orbitrdquo Journal of Spacecraft andRockets vol 50 no 3 pp 675ndash685 2013

[26] Q Li XMa and TWang ldquoReducedmodel for flexible solar saildynamicsrdquo Journal of Spacecraft and Rockets vol 48 no 3 pp446ndash453 2011

[27] M Polites J Kalmanson and D Mangus ldquoSolar sail attitudecontrol using small reaction wheels and magnetic torquersrdquoProceedings of the Institution of Mechanical Engineers G Journalof Aerospace Engineering vol 222 no 1 pp 53ndash62 2008

[28] H-T Cui J-H Luo J-H Feng and P-Y Cui ldquoAttitudecontrol of solar sail spacecraft with control boomrdquo Journal ofAstronautics vol 29 no 2 pp 560ndash566 2008

[29] E Mettler and S R Ploen ldquoSolar sail dynamics and controlusing a boom mounted bus articulated by a bi-state two-axisgimbal and reaction wheelsrdquo in Proceedings of the AIAAAASAstrodynamics Specialist Conference and Exhibit MontereyCalif USA August 2002

[30] J Luo X Li and J Feng ldquoPassive attitude control of solar sailspacecraftrdquo Flight Dynamics vol 26 no 5 pp 47ndash50 2008

[31] J Luo Study on attitude control of solar sail spaceraft [PhDthesis] School of Astronautics Harbin Institute of TechnologyHarbin China 2007

[32] D A Lawrence and S W Piggott ldquoIntegrated trajectory andattitude control for a four-vane solar sailrdquo in Proceedings ofthe AIAA Guidance Navigation and Control Conference andExhibit pp 2416ndash2431 San Francisco Calif USA August 2005

[33] E Mettler A B Acikmese and S R Ploen ldquoAttitude dynamicsand control of solar sails with articulated vanesrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit pp 2396ndash2415 San Francisco Calif USA August 2005

[34] S N Adeli V J Lappas and B Wie ldquoA scalable bus-basedattitude control system for Solar Sailsrdquo Advances in SpaceResearch vol 48 no 11 pp 1836ndash1847 2011

[35] R Funase Y Shirasawa Y Mimasu et al ldquoOn-orbit verificationof fuel-free attitude control system for spinning solar sailutilizing solar radiation pressurerdquo Advances in Space Researchvol 48 no 11 pp 1740ndash1746 2011

[36] B Fu and F O Eke ldquoAttitude control methodology for largesolar sailsrdquo in AIAA Guidance Navigation and Control (GNC)Conference 2013

[37] J Liu N Cui F Shen and S Rong ldquoDynamics of highly-flexiblesolar sail subjected to various forcesrdquoActaAstronautica vol 103pp 55ndash72 2014

[38] J F Liu S Y Rong and J G Li ldquoAttitude dynamics modelingand control of large flexible solar sail spacecraftrdquo in Proceedingsof the 3rd International Symposium on Systems and Control inAeronautics and Astronautics (ISSCAA rsquo10) pp 243ndash247 IEEEPublications Piscataway NJ USA 2010

[39] S Gong H Baoyin and J Li ldquoCoupled attitude-orbit dynamicsand control for displaced solar orbitsrdquo Acta Astronautica vol65 no 5-6 pp 730ndash737 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


control actuators The validity of the ACS is demonstratedby numerical simulations [19] A high performance solar sailattitude controller is presented by employing sliding massesinside the supporting beams and its ability of performingtime efficient reorientation maneuvers is demonstrated [20]The proposed controller combines a feedforward and a feed-back controller the former is a fast response controller whilethe latter can be used to respond to unpredicted disturbances[21] The solar sail ACS presented in [22] includes themovement of the small control mass in the solar sail planeand the rotation of ballast with two masses at the extremitiesto realize the pitchyaw and roll control respectively

The robust attitude controllers are designed using 119867infinQFT and input shaping methods respectively to allow forthe uncertainties of inertia natural frequencies dampingand modal constants of solar sail The performances ofthe controllers are analyzed using the linear and nonlineardynamics respectively [23]The reduced dynamic model of aflexible solar sail with foreshortening deformation couplingwith its attitude and vibration is derived in [24ndash26] TheBang-Bang control scheme combining input shapingmethodis used to eliminate the vibration for a time optimal attitudemaneuver The attitude control system is proposed usingsmall reactionwheels andmagnetic torquers for a solar sail onlow earth orbit and the validity is demonstrated by numericalsimulation [27]

The dynamic model of the multibody solar sail withcontrol boom and reaction wheel is derived in [28 29] Thecontrollability and stability are analyzed and the proposedattitude control scheme is demonstrated The effectiveness ofthe solar sail attitude control system employing four controlvanes is demonstrated by simulating deep space explorationmissions [30]The dynamics is derived and the controllabilityand stability are analyzed for solar sail with control boomreaction wheel and control vanes The effectiveness of theproposed ACS is demonstrated for the solar sail with cmcpoffset [31]

The trajectory tracking control is accomplished by usingsolar sail with four control vanes to control the solar angleof the sail The resultant sailcraft thrust vector amplitudecan be controlled by the additional degree of freedom ofthe control vanes [32] A robust nonlinear attitude controlalgorithm is developed for solar sail with four control vaneswith single degree-of-freedom rejecting disturbances by cmand cp offset The control allocation is studied by usingnonlinear programming [33]

The novel and practical ACS for solar sail is continuouslyproposed A bus-basedACS is presented in [34] including thehighly reflective panel actuator for roll control located at thefree end of the bus-based boom and the tether control massactuator for yawpitch control running along the bus-basedboomThis scalableACS can decrease the risk and complexityinvolved in the design of the sail deployment subsystem TheACS designed for IKAROS is muchmore novel and practicalThe ACS utilizes the reflectivity control device (RCD) torealize the yawpitch control for the spinning solar power sailIt is a fuel-free and oscillation-free ACS [35] But this ACSwill fail to generate the required torques when it is edge tothe sun (the sun angle approaches 90∘) And recently a novel

attitude control method is proposed in [36] The requiredtorque can be generated by adjusting the position of a wingtip along the boomThismethod presents an effective attitudeapproach for large sailcraft But the exact shape of the films isdifficult to determine

The dynamic modeling and control for the flexible sail-craft are studied in [37 38] In [37] the vibration equationsfor the axial and transverse deformations are establishedby considering the geometrical nonlinearity of the sailcraftBut merely the vibration analysis is insufficient for theorbiting sailcraft experiencing attitude motion The coupledattitudevibration analysis is required In [38] the coupledattitudevibration dynamics is established and the solvingprocess is also presented The controller is designed by usingthe Bang-Bang based PD theory The effectiveness of thecontroller is verified by numerical simulation But the largedeformation of the structure is not considered

The coupled attitude-orbit dynamics of a solar sail isstudied in [39]The equilibrium point of the dynamics can beobtained by designing the inertia of the sail and the stabilityof the equilibrium point is analyzed through a linearization

The attitude dynamics and control are studied thoroughlyin above references The ACS for solar sail by using actu-ators such as control boom and control vanes or by usingmeans such as spinning and translatingtilting sail panels arestudied Little research concerns attitude dynamic modelingwith highly flexible structure and vibration modeling withgeometrical nonlinearity and the attitudevibration controlBut this is a problem worth studying

This paper establishes the attitude dynamics for the largeflexible solar sail with control vanes control boom and slid-ing masses And the vibration equations are also presentedThe LQR and optimal PI based controllers are designed forthe attitudevibration dynamics By theoretical analysis andsimulation the controller with better performance is selectedand some discussions are presented And the differencesbetween the dynamics models with and without the geomet-rical nonlinearity are also inserted and analyzed

2 Attitude Dynamics

In this section the configuration and structure of the sailcraftare presented The related reference frames and coordinatetransformations are given The kinetic and gravitationalpotential energies and the generalized loads are obtainedTheLagrange equation method is adopted to derive the attitudedynamics with control vanes control boom and slidingmasses The attitude dynamics is an important part of thecoupled attitudevibration dynamics

21 The Configuration and Structure The configuration andstructure of the infinite-point connected square solar sailadopted in this paper are shown in Figure 1The ACS consistsof the control boom sliding masses and control vanes Thefour supporting beams are the most rigid component Thestiffness of the four triangular membranes can be neglectedby comparing with the supporting beams thus the kinetic


Controlvanes

Payload

Slidingmasses

Controlboom















119868= (i

119868 j119868 k

119868)119879


119887)


119894 119894 = 1 2 3 4) O

119874is the geo-











EI

ZI

YI

XI

yb

zb

1

2

C2

C1

x2

z2

z1

y1

x1

y2 OO

xb

12

3

3

44

C4

C3

x3

y3z3

z4

y4x4

Figure 2 120587119868

120587119887

and 120587119894

Sun line

EI

OO

rO

ZI

XI

YI

ZI z998400

120595

120595

XI

EI(O)

Z998400998400998400(zb)

Z998400998400120593

120593

120579

120579

x998400x998400998400 x998400998400998400(xb)

y998400998400 y998400998400998400(yb)y998400

YI

The orbit


from 120587119868to 120587



C119887119868

= R2(120579)R

1(120593)R

3(120595)

= [

[

119888120579119888120595 minus 119904120579119904120593119904120595 119888120579119904120595 + 119904120579119904120593119888120595 minus119904120579119888120593

minus119888120593119904120595 119888120593119888120595 119904120593

119904120579119888120595 + 119888120579119904120593119904120595 119904120579119904120595 minus 119888120579119904120593119888120595 119888120579119888120593

]

]

(1)

where

R3(120595) = [

[

cos120595 sin120595 0


0 0 1

]

]

R1(120593) = [

[

1 0 0

0 cos120593 sin120593

0 minus sin120593 cos120593]

]

R2(120579) = [

[

cos 120579 0 minus sin 120579

0 1 0

sin 120579 0 cos 120579]

]

(2)




119862119887119894= [

[

119888120579119894119888120595

119894minus 119904120579

119894119904120593

119894119904120595

119894119888120579

119894119904120595

119894+ 119904120579

119894119904120593

119894119888120595

119894minus119904120579

119894119888120593

119894

minus119888120593119894119904120595

119894119888120593

119894119888120595

119894119904120593

119894

119904120579119894119888120595

119894+ 119888120579

119894119904120593

119894119904120595

119894119904120579

119894119904120595

119894minus 119888120579

119894119904120593

119894119888120595

119894119888120579

119894119888120593

119894

]

]

(3)


119894 and

120579119894


OO

xb mp

14

1205821

zbyb

dm32

3

1205822

EI

p

ch1

h2

31205883

O m3

r

r r

rr

R R







119887 The


R119898119894

= R119874+ r

119894+ 120588

119894 k

119898119894= R

119874+∘

120588119894

+ 120596 times (r119894+ 120588

119894) (4)

whereR119898119894(v




119874R

119874(R

119874= v






[

[

120596119909

120596119910

120596119911

]

]119887

= [

[


sin 120579 + cos 120579 cos120593]

]

(5)


R1198983

= r3+120588

3= [120588

3119909 120588

3119910 119911

3]119879

R1198981

= r1+ 120588

1= [120588

1119909 120588

1119910 119911

1]119879

R1198982

= r2+ 120588

2= [120588

2119909 119910

2 120588

2119911]119879

R1198984

= r4+ 120588

4= [120588

4119909 119910

4 120588

4119911]119879

(6)

R1198983

=∘

1205883

+ 120596 times (r3+ 120588

3) =

[[

[

1205883119909

1205883119910

0

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205883119909

1205883119910

1199113

]

]

=

[[[

[

1205883119909

+ 1199113

120579 + 1199113 sin120593 minus 120588

3119910 sin 120579 minus 120588

3119910 cos 120579 cos120593

1205883119910

+ 1205883119909 sin 120579 + 120588

3119909 cos 120579 cos120593 minus 119911

3 cos 120579 + 119911

3 sin 120579 cos120593

1205883119910 cos 120579 minus 120588

3119910 sin 120579 cos120593 minus 120588

3119909

120579 minus 1205883119909 sin120593

]]]

]

(7)

R1198981

=∘

1205881

+ 120596 times (r1+ 120588

1) =

[[

[

1205881119909

1205881119910

0

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205881119909

1205881119910

1199111

]

]

=

[[[

[

1205881119909

+ 1199111

120579 + 1199111 sin120593 minus 120588

1119910 sin 120579 minus 120588

1119910 cos 120579 cos120593

1205881119910

+ 1205881119909 sin 120579 + 120588

1119909 cos 120579 cos120593 minus 119911

1 cos 120579 + 119911

1 sin 120579 cos120593

1205881119910 cos 120579 minus 120588

1119910 sin 120579 cos120593 minus 120588

1119909

120579 minus 1205881119909 sin120593

]]]

]

(8)

R1198982

=∘

1205882

+ 120596 times (r2+ 120588

2) =

[[

[

1205882119909

0

1205882119911

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205882119909

1199102

1205882119911

]

]

=[[

[

1205882119909

+ 1205882119911

120579 + 1205882119911 sin120593 minus 119910

2 sin 120579 minus 119910

2 cos 120579 cos120593

1205882119909 sin 120579 + 120588

2119909 cos 120579 cos120593 minus 120588

2119911 cos 120579 + 120588

2119911 sin 120579 cos120593

1205882119911

+ 1199102 cos 120579 minus 119910

2 sin 120579 cos120593 minus 120588

2119909

120579 minus 1205882119909 sin120593

]]

]

(9)


R1198984

=∘

1205884

+ 120596 times (r4+ 120588

4) = [

[

1205884119909

0

1205884119911

]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205884119909

1199104

1205884119911

]

]

=[[

[

1205884119909

+ 1205884119911

120579 + 1205884119911 sin120593 minus 119910

4 sin 120579 minus 119910

4 cos 120579 cos120593

1205884119909 sin 120579 + 120588

4119909 cos 120579 cos120593 minus 120588

4119911 cos 120579 + 120588

4119911 sin 120579 cos120593

1205884119911

+ 1199104 cos 120579 minus 119910

4 sin 120579 cos120593 minus 120588

4119909

120579 minus 1205884119909 sin120593

]]

]

(10)


In Figure 4 1205821and 120582




two angles

r119901=10038161003816100381610038161003816r119901

10038161003816100381610038161003816cos 120582

1i119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816sin 120582

1cos 120582

2j119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816sin 120582

1sin 120582

2k119887

(11)


R119874119901

= R119874+ r

119901 k

119874119901= k

119874+ r

119901 (12)

where r119901=∘r119901+ 120596 times r

119901 And ∘r


∘r119901= minus

10038161003816100381610038161003816r119901

100381610038161003816100381610038161sin 120582

1i119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816

times (1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2) j

119887

+10038161003816100381610038161003816r119901

10038161003816100381610038161003816(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2) k

119887

(13)


r119901= [[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

times [119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1sin 120582

2( cos 120579 minus sin 120579 cos120593)]

times 119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

(14)


Rℎ119895

= R119874+ r

ℎ119895+ 120588

ℎ119895

kℎ2

= k119874+∘rℎ119895

+∘

120588ℎ119895

+ 120596 times (rℎ119895

+ 120588ℎ119895)

(119895 = 1 2)

(15)


Rℎ2

= rℎ2

+ 120588ℎ2

= [120588ℎ2119909

120588ℎ2119910

119911ℎ2]119879

Rℎ1

= rℎ1

+ 120588ℎ1

= [120588ℎ1119909

119910ℎ1 120588

ℎ1119911]119879

Rℎ2

=[[

[

120588ℎ2119909

+ 119911ℎ2

( 120579 + sin120593) minus 120588ℎ2119910

( sin 120579 + cos 120579 cos120593)120588ℎ2119910

+ 120588ℎ2119909



+ 120588ℎ2119910


( 120579 + sin120593)

]]

]

Rℎ1

=[[

[

120588ℎ1119909

+ 120588ℎ1119911

( 120579 + sin120593) minus 119910ℎ1

( sin 120579 + cos 120579 cos120593)119910ℎ1

+ 120588ℎ1119909



+ 119910ℎ1


( 120579 + sin120593)

]]

]

(16)



1198791198983

=1

2119898

119897int

119897

0

[∘

1205883

+ 120596 times (r3+ 120588

3)] sdot [

∘

1205883

+ 120596 times (r3+ 120588

3)] 119889119911

1198791198981

=1

2119898

119897int

0

minus119897

[∘

1205881

+ 120596 times (r1+ 120588

1)]

sdot [∘

1205881

+ 120596 times (r1+ 120588

1)] 119889119911

1198791198982

=1

2119898

119897int

119897

0

[∘

1205882

+ 120596 times (r2+ 120588

2)] sdot [

∘

1205882

+ 120596 times (r2+ 120588

2)] 119889119910

1198791198984

=1

2119898

119897int

0

minus119897

[∘

1205884

+ 120596 times (r4+ 120588

4)] sdot [

∘

1205884

+ 120596 times (r4+ 120588

4)] 119889119910

(17)


119879119898

=1

2120596 sdot J sdot 120596

+1

2119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+1

2119898

119897int

0

minus119897

2∘

1205881


1) sdot (120596 times 120588

1)


1) 119889119911

+1

2119898

119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

+1

2119898

119897int

0

minus119897

2∘

1205884


4) sdot (120596 times 120588

4)


4) + (120596 times r

4)

sdot (120596 times r4) 119889119910

(18)

where J = sum4

119895=1

J119898119895



119909

119869119910 119869

119911] inO

119874x119887y119887z119887


119879119901=

1

2119898

119901

10038161003816100381610038161003816r119901

10038161003816100381610038161003816

2

=1

2119898

119901[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

2

(19)


119879ℎ1

=1

2119898

ℎ1[ ( 120588

ℎ1119909+ 120579120588

ℎ1119911+ 120588

ℎ1119911sin120593

minus 119910ℎ1sin 120579 minus 119910

ℎ1cos 120579 cos120593)

2

]

+1

2119898

ℎ1[( 119910

ℎ1+ 120588

ℎ1119909sin 120579 + 120588

ℎ1119909cos 120579 cos120593

minus 120588ℎ1119911

cos 120579 + 120588ℎ1119911

sin 120579 cos120593)2]

+1

2119898

ℎ1[( 120588

ℎ1119911+ 119910

ℎ1cos 120579 minus 119910

ℎ1sin 120579 cos120593

minus 120579120588ℎ1119909

+ 120588ℎ1119909

sin120593)2

]

119879ℎ2

=1

2119898

ℎ2[( 120588

ℎ2119909+ 120579119911

ℎ2+ 119911

ℎ2sin120593 minus 120588

ℎ2119910sin 120579

minus 120588ℎ2119910

cos 120579 cos120593)2

]

+1

2119898

ℎ2[( 120588

ℎ2119910+ 120588

ℎ2119909sin 120579 + 120588

ℎ2119909cos 120579 cos120593

minus 119911ℎ2cos 120579 + 119911

ℎ2sin 120579 cos120593)

2

]

+1

2119898

ℎ2[(

ℎ2+ 120588

ℎ2119910cos 120579 minus 120588

ℎ2119910sin 120579 cos120593

minus 120579120588ℎ2119909

minus 120588ℎ2119909

sin120593)2

]

(20)


119892119891 payload 119880

119892119901 and


119880119892ℎ2

can be obtained as

119880119892119891

= minus

120583119898119891

1003816100381610038161003816R119874

1003816100381610038161003816

119880119892119901

= minus

120583119898119901

10038161003816100381610038161003816R119874+ r

119901

10038161003816100381610038161003816

119880119892ℎ1

= minus120583119898

ℎ1

1003816100381610038161003816R119874+ r

ℎ1

1003816100381610038161003816

119880119892ℎ2

= minus120583119898

ℎ2

1003816100381610038161003816R119874+ r

ℎ2

1003816100381610038161003816

(21)



119868

S = j119868= [1 0 0]

119879

119868

(22)



S = C119887119868[1 0 0]

119879

119868

= [119878119909

119878119910

119878119911]119879

119887

(23)


F119894= 120578119875119860

119862119894(S sdot n

119862119894)2 n

119862119894= 119865

119862119894cos2120572

119862119894n119862119894 (24)




119865119862119894


119862119894= i

119894


119862119894in 120587


n119862119894

= C119887119894[1 0 0]

119879

119894

= [119878119909119894

119878119910119894

119878119911119894]119879

119887

(25)


M119894= L

119894times F

119894 (26)




119894andF



119894and F


L1= [0 0 minus119897]

119879

L2= [0 119897 0]

119879

L3= [0 0 119897]

119879

L4= [0 minus119897 0]

119879

F119894= 120578119875119860

119862119894([119878

119909119878119910

119878119911]119887

sdot [119878119909119894

119878119910119894

119878119911119894]119879

119887

)

2

[119878119909119894

119878119910119894

119878119911119894]119879

119887

(27)


F119894= (119865

119862119894cos2120572

119862119894)C

119887119894[1 0 0]

119879

119894

= (119865119862119894cos2120572

119862119894) [119878

119909119894119878119910119894

119878119911119894]119879

119887

(28)

where 120572119862119894


M1198881

= (1198651198621cos2120572

1198621)[

[

0

0

minus119897

]

]

times (C1198871

[

[

1

0

0

]

]

)

= (1198651198621cos2120572

1198621)

times [

[

minus119897 cos1205931sin120595

1

minus119897 (cos 1205791cos120595

1minus sin 120579

1sin120593

1sin120595

1)

0

]

]

M1198882

= (1198651198622cos2120572

1198622)

times [

[

119897 (sin 1205792cos120595

2+ cos 120579

2sin120593

2sin120595

2)

0

minus119897 (cos 1205792cos120595

2minus sin 120579

2sin120593

2sin120595

2)

]

]

M1198883

= (1198651198623cos2120572

1198623)

times [

[

119897 cos1205933sin120595

3

119897 (cos 1205793cos120595

3minus sin 120579

3sin120593

3sin120595

3)

0

]

]

M1198884

= (1198651198624cos2120572

1198624)

times [

[

minus119897 (sin 1205794cos120595

4+ cos 120579

4sin120593

4sin120595

4)

0

119897 (cos 1205794cos120595

4minus sin 120579

4sin120593

4sin120595

4)

]

]

(29)


119889

119889119905(

120597119871

120597 119902119895

) minus120597119871

120597119902119895

= 119876119895 (30)




119871 = 119879 minus 119880 =

4

sum

119894=1

119879119898119894

+ 119879119901+

2

sum

119895=1

119879ℎ119895

minus (119880119892119891

+ 119880119892119901

+

2

sum

119895=1

119880119892ℎ119895

)

(31)

where 119879119898119894 119879

119901 and 119879



119892119901 and 119880




OOx

y

z

FCFCy

FCx

FCz

mh

q(x)


120590

120590e

120590120592

120590







119862119909 119865

119862119910 and 119865





120590 = 120590119890

+ 120590120592

= 119864120576 + 119888119887

120576 (32)

where 120590 120590119890 and 120590



expressed as 119864120578119887 120578




and s respectively

OO x

y The support beam

Z




119906 (119909 119910 119911 119905) = 1199060(119909 119905) minus 119910

120597V0

120597119909minus 119911

1205971199080

120597119909

V (119909 119910 119911 119905) = V0(119909 119905)

119908 (119909 119910 119911 119905) = 1199080(119909 119905)

(33)


0(119909 119905)

V0(119909 119905) and 119908




120576119909119909

=120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

120576119909119909

=120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909)

minus 1199111205972

0

1205971199092

120590119909119909

= 119864[120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

]

+ 119864120578119887(120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909) minus 119911

1205972

0

1205971199092

)

(34)




119880119894=

1

2int

119897

0

119860119909119909

[120597119906

0

120597119909+

1

2(120597V

0

120597119909)

2

+1

2(120597119908

0

120597119909)

2

]

2

+ 119863119909119909

1205972V

0

1205971199092

+ 119863119909119909

1205972

1199080

1205971199092

119889119909

(35)




119860119909119909

= int119860

119864119889119860 119863119909119909

= int119860

1198641199112

119889119860 = int119860

1198641199102

119889119860 (36)


Ψ119902=

1

2int

119897

0

[119860119909119909120578119887(120597

0

120597119909)

2

+ 119863119909119909120578119887(1205972V

0

1205971199092

)

2

+ 119863119909119909120578119887(1205972

0

1205971199092

)

2

]119889119909

(37)


119879119890=

1

2int

119897

0

[119868119860(120597119906

0

120597119905)

2

+ 119868119860(120597119908

0

120597119905)

2

+ 119868119863(1205972

1199080

120597119909120597119905)

2

+ 119868119860(120597V

0

120597119905)

2

+ 119868119863(

1205972V

0

120597119909120597119905)

2

]119889119909

(38)

where 119868119860and 119868


beam expressed as

119868119860= int

119860

120588119889119860 = 120588119860 119868119863= int

119860

1205881199112

119889119860 = int119860

1205881199102

119889119860 (39)


119862119911sdot 119908 (119909

119888 119905) + 119865

119862119910sdot V (119909

119888 119905) + 119898

ℎ119886ℎ119910

(119909ℎ 119905) V (119909

ℎ 119905)

+ 119898ℎ119886ℎ119911

(119909ℎ 119905) 119908 (119909

ℎ 119905) + int

119897

0

119902 (119909) V (119909 119905) 119889119909

(40)where 119909


119908(119909119888 119905) V(119909



119862119910are




ℎ] 119898

ℎis the mass

and 119886ℎ119910(119909

ℎ 119905) and 119886

ℎ119911(119909



ℎ 119905) and V(119909



0(119909 119905) and119908

0(119909 119905) can


1199060(119909 119905) =

infin

sum

119894=1

119880119894(119909)119880

119894(119905)

V0(119909 119905) =

infin

sum

119895=1

119881119895(119909) 119881

119895(119905)

1199080(119909 119905) =

infin

sum

119896=1

119882119896(119909)119882

119896(119905)

(41)

where 119880119894(119905) 119881

119895(119905) and 119882



119895(119909) and 119882

119896(119909) are the


119880119894(119909)

1003816100381610038161003816119909=0= 0 119881

119895(119909)

10038161003816100381610038161003816119909=0= 0 119882

119896(119909)

1003816100381610038161003816119909=0= 0

1198811015840

119895

(119909)10038161003816100381610038161003816119909=0

= 0 1198821015840

119896

(119909)10038161003816100381610038161003816119909=0

= 0

(42)


1198801(119909) =

119909

119897 119880

2(119909) = (

119909

119897)

2

1198811(119909) = (

119909

119897)

2

1198812(119909) = (

119909

119897)

3

1198821(119909) = (

119909

119897)

2

1198822(119909) = (

119909

119897)

3

(43)


119894 119879

119890119882

119911 and Ψ


119880119894=

9

2

119860119909119909

1198973(119881

2

2

+ 1198822

2

)2

+27

4

119860119909119909

1198973(119881

11198812+ 119882

1119882

2)

times (1198812

2

+ 1198822

2

)

+18

7

119860119909119909

1198973[(119881

2

1

+ 1198822

1

) (1198812

2

+ 1198822

2

) + (11988111198812+ 119882

1119882

2)2

]

+119860

119909119909

1198973[3

2119897119880

2(119881

2

2

+ 1198822

2

) + (1198812

1

+ 1198822

1

)

times (11988111198812+ 119882

1119882

2)]


+1

10

119860119909119909

1198973[9119880

1119897 (119881

2

2

+ 1198822

2

) + 241198802119897 (119881

11198812+ 119882

1119882

2)

+ 4 (1198812

1

+ 1198822

1

)2

]

+119860

119909119909

1198972[3

21198801(119881

11198812+ 119882

1119882

2) + 119880

2(119881

2

1

+ 1198822

1

)]

+6119863

119909119909119881

2

2

1198973+ 119860

119909119909(2119880

1119881

2

1

31198972+

21198801119882

2

1

31198972+

21198802

2

3119897)

+6119863

119909119909119882

2

2

1198973+

611986311990911990911988111198812

1198973+

11986011990911990911988011198802

119897+

6119863119909119909119882

1119882

2

1198973

+119860

119909119909119880

2

1

2119897+

2119863119909119909119881

2

1

1198973+

2119863119909119909119882

2

1

1198973

119879119890=

1

14(119897119868

119860

2

2

+ 119897119868119860

2

2

) +1

6(119897119868

119860

1

2+ 119897119868

11986012)

+1

10(119897119868

119860

2

2

+ 119897119868119860

2

1

+9119868

119863

2

2

119897+ 119897119868

119860

2

1

+9119868

119863

2

2

119897)

+1

4(119897119868

11986012+

6119868119863

1

2

119897+

611986811986312

119897)

+1

6(119897119868

119860

2

1

+4119868

119863

2

1

119897+

4119868119863

2

1

119897)

119882119911= 119865

119862119911sdot [119882

1(119905) + 119882

2(119905)] + 119865

119862119910sdot [119881

1(119905) + 119881

2(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119881

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119881

2(119905) +

6V3ℎ

1199052

1198973

sdot 2(119905) + (

Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198811(119905) + (

Vℎ119905

119897)

3

sdot 1198812(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119882

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119882

2(119905) +

6V3ℎ

1199052

1198973sdot

2(119905)

+ (Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198821(119905) + (

Vℎ119905

119897)

3

sdot 1198822(119905)]

+ 1198811(119905) sdot int

119897

0

119902 (119909) (119909

119897)

2

119889119909 + 1198812(119905)

sdot int

119897

0

119902 (119909) (119909

119897)

3

119889119909

Ψ119902=

1

6(36119863

119909119909120578119887

2

2

1198976+

4119860119909119909120578119887

2

2

1198974+

36119863119909119909120578119887

2

2

1198976) 119897

3

+1

4(4119860

11990911990912057811988712

1198973+

2411986311990911990912057811988712

1198975

+24119863

119909119909120578119887

1

2

1198975) 119897

2

+1

2

119860119909119909120578119887

2

1

119897+

2119863119909119909120578119887

2

1

1198973+

2119863119909119909120578119887

2

1

1198973

(44)


120597119871

120597119902119899

minus119889

119889119905(

120597119871

120597 119902119899

) + 119876119889= 0 119899 = 1 2 6 (45)


= (1198801 119880

2 119881

1 119881

2

1198821119882



119889is the


as

119871 = 119879119890minus (119880

119894minus 119882

119911)

119876119889= minus

120597Ψ119902

120597 119902119899

(119899 = 1 sim 6)

(46)


1 119880

2

1

31198971198681198601+

1

41198971198681198602+

120578119887119860

1199091199091

119897+

120578119887119860

1199091199092

119897+

1198601199091199091198801

119897

+119860

1199091199091198802

119897+

2119860119909119909119881

2

1

31198972+

311986011990911990911988111198812

21198972+

9119860119909119909119881

2

2

101198972

+2119860

119909119909119882

2

1

31198972+

3119860119909119909119882

1119882

2

21198972+

9119860119909119909119882

2

2

101198972= 0

(47a)

1

41198971198681198601+

1

51198971198681198602+

120578119887119860

1199091199091

119897+

4120578119887119860

1199091199092

3119897

+119860

1199091199091198801

119897+4119860

1199091199091198802

3119897+119860

119909119909119881

2

1

1198972+12119860

11990911990911988111198812

51198972+3119860

119909119909119881

2

2

21198972

+119860

119909119909119882

2

1

1198972+

12119860119909119909119882

1119882

2

51198972+

3119860119909119909119882

2

2

21198972= 0

(47b)





1 119881

2119882

1119882

2 We take (47a)



2119882

1119882






2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)









119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Controlvanes

Payload

Slidingmasses

Controlboom















119868= (i

119868 j119868 k

119868)119879


119887)


119894 119894 = 1 2 3 4) O

119874is the geo-











EI

ZI

YI

XI

yb

zb

1

2

C2

C1

x2

z2

z1

y1

x1

y2 OO

xb

12

3

3

44

C4

C3

x3

y3z3

z4

y4x4

Figure 2 120587119868

120587119887

and 120587119894

Sun line

EI

OO

rO

ZI

XI

YI

ZI z998400

120595

120595

XI

EI(O)

Z998400998400998400(zb)

Z998400998400120593

120593

120579

120579

x998400x998400998400 x998400998400998400(xb)

y998400998400 y998400998400998400(yb)y998400

YI

The orbit


from 120587119868to 120587



C119887119868

= R2(120579)R

1(120593)R

3(120595)

= [

[

119888120579119888120595 minus 119904120579119904120593119904120595 119888120579119904120595 + 119904120579119904120593119888120595 minus119904120579119888120593

minus119888120593119904120595 119888120593119888120595 119904120593

119904120579119888120595 + 119888120579119904120593119904120595 119904120579119904120595 minus 119888120579119904120593119888120595 119888120579119888120593

]

]

(1)

where

R3(120595) = [

[

cos120595 sin120595 0


0 0 1

]

]

R1(120593) = [

[

1 0 0

0 cos120593 sin120593

0 minus sin120593 cos120593]

]

R2(120579) = [

[

cos 120579 0 minus sin 120579

0 1 0

sin 120579 0 cos 120579]

]

(2)




119862119887119894= [

[

119888120579119894119888120595

119894minus 119904120579

119894119904120593

119894119904120595

119894119888120579

119894119904120595

119894+ 119904120579

119894119904120593

119894119888120595

119894minus119904120579

119894119888120593

119894

minus119888120593119894119904120595

119894119888120593

119894119888120595

119894119904120593

119894

119904120579119894119888120595

119894+ 119888120579

119894119904120593

119894119904120595

119894119904120579

119894119904120595

119894minus 119888120579

119894119904120593

119894119888120595

119894119888120579

119894119888120593

119894

]

]

(3)


119894 and

120579119894


OO

xb mp

14

1205821

zbyb

dm32

3

1205822

EI

p

ch1

h2

31205883

O m3

r

r r

rr

R R







119887 The


R119898119894

= R119874+ r

119894+ 120588

119894 k

119898119894= R

119874+∘

120588119894

+ 120596 times (r119894+ 120588

119894) (4)

whereR119898119894(v




119874R

119874(R

119874= v






[

[

120596119909

120596119910

120596119911

]

]119887

= [

[


sin 120579 + cos 120579 cos120593]

]

(5)


R1198983

= r3+120588

3= [120588

3119909 120588

3119910 119911

3]119879

R1198981

= r1+ 120588

1= [120588

1119909 120588

1119910 119911

1]119879

R1198982

= r2+ 120588

2= [120588

2119909 119910

2 120588

2119911]119879

R1198984

= r4+ 120588

4= [120588

4119909 119910

4 120588

4119911]119879

(6)

R1198983

=∘

1205883

+ 120596 times (r3+ 120588

3) =

[[

[

1205883119909

1205883119910

0

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205883119909

1205883119910

1199113

]

]

=

[[[

[

1205883119909

+ 1199113

120579 + 1199113 sin120593 minus 120588

3119910 sin 120579 minus 120588

3119910 cos 120579 cos120593

1205883119910

+ 1205883119909 sin 120579 + 120588

3119909 cos 120579 cos120593 minus 119911

3 cos 120579 + 119911

3 sin 120579 cos120593

1205883119910 cos 120579 minus 120588

3119910 sin 120579 cos120593 minus 120588

3119909

120579 minus 1205883119909 sin120593

]]]

]

(7)

R1198981

=∘

1205881

+ 120596 times (r1+ 120588

1) =

[[

[

1205881119909

1205881119910

0

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205881119909

1205881119910

1199111

]

]

=

[[[

[

1205881119909

+ 1199111

120579 + 1199111 sin120593 minus 120588

1119910 sin 120579 minus 120588

1119910 cos 120579 cos120593

1205881119910

+ 1205881119909 sin 120579 + 120588

1119909 cos 120579 cos120593 minus 119911

1 cos 120579 + 119911

1 sin 120579 cos120593

1205881119910 cos 120579 minus 120588

1119910 sin 120579 cos120593 minus 120588

1119909

120579 minus 1205881119909 sin120593

]]]

]

(8)

R1198982

=∘

1205882

+ 120596 times (r2+ 120588

2) =

[[

[

1205882119909

0

1205882119911

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205882119909

1199102

1205882119911

]

]

=[[

[

1205882119909

+ 1205882119911

120579 + 1205882119911 sin120593 minus 119910

2 sin 120579 minus 119910

2 cos 120579 cos120593

1205882119909 sin 120579 + 120588

2119909 cos 120579 cos120593 minus 120588

2119911 cos 120579 + 120588

2119911 sin 120579 cos120593

1205882119911

+ 1199102 cos 120579 minus 119910

2 sin 120579 cos120593 minus 120588

2119909

120579 minus 1205882119909 sin120593

]]

]

(9)


R1198984

=∘

1205884

+ 120596 times (r4+ 120588

4) = [

[

1205884119909

0

1205884119911

]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205884119909

1199104

1205884119911

]

]

=[[

[

1205884119909

+ 1205884119911

120579 + 1205884119911 sin120593 minus 119910

4 sin 120579 minus 119910

4 cos 120579 cos120593

1205884119909 sin 120579 + 120588

4119909 cos 120579 cos120593 minus 120588

4119911 cos 120579 + 120588

4119911 sin 120579 cos120593

1205884119911

+ 1199104 cos 120579 minus 119910

4 sin 120579 cos120593 minus 120588

4119909

120579 minus 1205884119909 sin120593

]]

]

(10)


In Figure 4 1205821and 120582




two angles

r119901=10038161003816100381610038161003816r119901

10038161003816100381610038161003816cos 120582

1i119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816sin 120582

1cos 120582

2j119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816sin 120582

1sin 120582

2k119887

(11)


R119874119901

= R119874+ r

119901 k

119874119901= k

119874+ r

119901 (12)

where r119901=∘r119901+ 120596 times r

119901 And ∘r


∘r119901= minus

10038161003816100381610038161003816r119901

100381610038161003816100381610038161sin 120582

1i119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816

times (1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2) j

119887

+10038161003816100381610038161003816r119901

10038161003816100381610038161003816(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2) k

119887

(13)


r119901= [[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

times [119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1sin 120582

2( cos 120579 minus sin 120579 cos120593)]

times 119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

(14)


Rℎ119895

= R119874+ r

ℎ119895+ 120588

ℎ119895

kℎ2

= k119874+∘rℎ119895

+∘

120588ℎ119895

+ 120596 times (rℎ119895

+ 120588ℎ119895)

(119895 = 1 2)

(15)


Rℎ2

= rℎ2

+ 120588ℎ2

= [120588ℎ2119909

120588ℎ2119910

119911ℎ2]119879

Rℎ1

= rℎ1

+ 120588ℎ1

= [120588ℎ1119909

119910ℎ1 120588

ℎ1119911]119879

Rℎ2

=[[

[

120588ℎ2119909

+ 119911ℎ2

( 120579 + sin120593) minus 120588ℎ2119910

( sin 120579 + cos 120579 cos120593)120588ℎ2119910

+ 120588ℎ2119909



+ 120588ℎ2119910


( 120579 + sin120593)

]]

]

Rℎ1

=[[

[

120588ℎ1119909

+ 120588ℎ1119911

( 120579 + sin120593) minus 119910ℎ1

( sin 120579 + cos 120579 cos120593)119910ℎ1

+ 120588ℎ1119909



+ 119910ℎ1


( 120579 + sin120593)

]]

]

(16)



1198791198983

=1

2119898

119897int

119897

0

[∘

1205883

+ 120596 times (r3+ 120588

3)] sdot [

∘

1205883

+ 120596 times (r3+ 120588

3)] 119889119911

1198791198981

=1

2119898

119897int

0

minus119897

[∘

1205881

+ 120596 times (r1+ 120588

1)]

sdot [∘

1205881

+ 120596 times (r1+ 120588

1)] 119889119911

1198791198982

=1

2119898

119897int

119897

0

[∘

1205882

+ 120596 times (r2+ 120588

2)] sdot [

∘

1205882

+ 120596 times (r2+ 120588

2)] 119889119910

1198791198984

=1

2119898

119897int

0

minus119897

[∘

1205884

+ 120596 times (r4+ 120588

4)] sdot [

∘

1205884

+ 120596 times (r4+ 120588

4)] 119889119910

(17)


119879119898

=1

2120596 sdot J sdot 120596

+1

2119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+1

2119898

119897int

0

minus119897

2∘

1205881


1) sdot (120596 times 120588

1)


1) 119889119911

+1

2119898

119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

+1

2119898

119897int

0

minus119897

2∘

1205884


4) sdot (120596 times 120588

4)


4) + (120596 times r

4)

sdot (120596 times r4) 119889119910

(18)

where J = sum4

119895=1

J119898119895



119909

119869119910 119869

119911] inO

119874x119887y119887z119887


119879119901=

1

2119898

119901

10038161003816100381610038161003816r119901

10038161003816100381610038161003816

2

=1

2119898

119901[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

2

(19)


119879ℎ1

=1

2119898

ℎ1[ ( 120588

ℎ1119909+ 120579120588

ℎ1119911+ 120588

ℎ1119911sin120593

minus 119910ℎ1sin 120579 minus 119910

ℎ1cos 120579 cos120593)

2

]

+1

2119898

ℎ1[( 119910

ℎ1+ 120588

ℎ1119909sin 120579 + 120588

ℎ1119909cos 120579 cos120593

minus 120588ℎ1119911

cos 120579 + 120588ℎ1119911

sin 120579 cos120593)2]

+1

2119898

ℎ1[( 120588

ℎ1119911+ 119910

ℎ1cos 120579 minus 119910

ℎ1sin 120579 cos120593

minus 120579120588ℎ1119909

+ 120588ℎ1119909

sin120593)2

]

119879ℎ2

=1

2119898

ℎ2[( 120588

ℎ2119909+ 120579119911

ℎ2+ 119911

ℎ2sin120593 minus 120588

ℎ2119910sin 120579

minus 120588ℎ2119910

cos 120579 cos120593)2

]

+1

2119898

ℎ2[( 120588

ℎ2119910+ 120588

ℎ2119909sin 120579 + 120588

ℎ2119909cos 120579 cos120593

minus 119911ℎ2cos 120579 + 119911

ℎ2sin 120579 cos120593)

2

]

+1

2119898

ℎ2[(

ℎ2+ 120588

ℎ2119910cos 120579 minus 120588

ℎ2119910sin 120579 cos120593

minus 120579120588ℎ2119909

minus 120588ℎ2119909

sin120593)2

]

(20)


119892119891 payload 119880

119892119901 and


119880119892ℎ2

can be obtained as

119880119892119891

= minus

120583119898119891

1003816100381610038161003816R119874

1003816100381610038161003816

119880119892119901

= minus

120583119898119901

10038161003816100381610038161003816R119874+ r

119901

10038161003816100381610038161003816

119880119892ℎ1

= minus120583119898

ℎ1

1003816100381610038161003816R119874+ r

ℎ1

1003816100381610038161003816

119880119892ℎ2

= minus120583119898

ℎ2

1003816100381610038161003816R119874+ r

ℎ2

1003816100381610038161003816

(21)



119868

S = j119868= [1 0 0]

119879

119868

(22)



S = C119887119868[1 0 0]

119879

119868

= [119878119909

119878119910

119878119911]119879

119887

(23)


F119894= 120578119875119860

119862119894(S sdot n

119862119894)2 n

119862119894= 119865

119862119894cos2120572

119862119894n119862119894 (24)




119865119862119894


119862119894= i

119894


119862119894in 120587


n119862119894

= C119887119894[1 0 0]

119879

119894

= [119878119909119894

119878119910119894

119878119911119894]119879

119887

(25)


M119894= L

119894times F

119894 (26)




119894andF



119894and F


L1= [0 0 minus119897]

119879

L2= [0 119897 0]

119879

L3= [0 0 119897]

119879

L4= [0 minus119897 0]

119879

F119894= 120578119875119860

119862119894([119878

119909119878119910

119878119911]119887

sdot [119878119909119894

119878119910119894

119878119911119894]119879

119887

)

2

[119878119909119894

119878119910119894

119878119911119894]119879

119887

(27)


F119894= (119865

119862119894cos2120572

119862119894)C

119887119894[1 0 0]

119879

119894

= (119865119862119894cos2120572

119862119894) [119878

119909119894119878119910119894

119878119911119894]119879

119887

(28)

where 120572119862119894


M1198881

= (1198651198621cos2120572

1198621)[

[

0

0

minus119897

]

]

times (C1198871

[

[

1

0

0

]

]

)

= (1198651198621cos2120572

1198621)

times [

[

minus119897 cos1205931sin120595

1

minus119897 (cos 1205791cos120595

1minus sin 120579

1sin120593

1sin120595

1)

0

]

]

M1198882

= (1198651198622cos2120572

1198622)

times [

[

119897 (sin 1205792cos120595

2+ cos 120579

2sin120593

2sin120595

2)

0

minus119897 (cos 1205792cos120595

2minus sin 120579

2sin120593

2sin120595

2)

]

]

M1198883

= (1198651198623cos2120572

1198623)

times [

[

119897 cos1205933sin120595

3

119897 (cos 1205793cos120595

3minus sin 120579

3sin120593

3sin120595

3)

0

]

]

M1198884

= (1198651198624cos2120572

1198624)

times [

[

minus119897 (sin 1205794cos120595

4+ cos 120579

4sin120593

4sin120595

4)

0

119897 (cos 1205794cos120595

4minus sin 120579

4sin120593

4sin120595

4)

]

]

(29)


119889

119889119905(

120597119871

120597 119902119895

) minus120597119871

120597119902119895

= 119876119895 (30)




119871 = 119879 minus 119880 =

4

sum

119894=1

119879119898119894

+ 119879119901+

2

sum

119895=1

119879ℎ119895

minus (119880119892119891

+ 119880119892119901

+

2

sum

119895=1

119880119892ℎ119895

)

(31)

where 119879119898119894 119879

119901 and 119879



119892119901 and 119880




OOx

y

z

FCFCy

FCx

FCz

mh

q(x)


120590

120590e

120590120592

120590







119862119909 119865

119862119910 and 119865





120590 = 120590119890

+ 120590120592

= 119864120576 + 119888119887

120576 (32)

where 120590 120590119890 and 120590



expressed as 119864120578119887 120578




and s respectively

OO x

y The support beam

Z




119906 (119909 119910 119911 119905) = 1199060(119909 119905) minus 119910

120597V0

120597119909minus 119911

1205971199080

120597119909

V (119909 119910 119911 119905) = V0(119909 119905)

119908 (119909 119910 119911 119905) = 1199080(119909 119905)

(33)


0(119909 119905)

V0(119909 119905) and 119908




120576119909119909

=120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

120576119909119909

=120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909)

minus 1199111205972

0

1205971199092

120590119909119909

= 119864[120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

]

+ 119864120578119887(120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909) minus 119911

1205972

0

1205971199092

)

(34)




119880119894=

1

2int

119897

0

119860119909119909

[120597119906

0

120597119909+

1

2(120597V

0

120597119909)

2

+1

2(120597119908

0

120597119909)

2

]

2

+ 119863119909119909

1205972V

0

1205971199092

+ 119863119909119909

1205972

1199080

1205971199092

119889119909

(35)




119860119909119909

= int119860

119864119889119860 119863119909119909

= int119860

1198641199112

119889119860 = int119860

1198641199102

119889119860 (36)


Ψ119902=

1

2int

119897

0

[119860119909119909120578119887(120597

0

120597119909)

2

+ 119863119909119909120578119887(1205972V

0

1205971199092

)

2

+ 119863119909119909120578119887(1205972

0

1205971199092

)

2

]119889119909

(37)


119879119890=

1

2int

119897

0

[119868119860(120597119906

0

120597119905)

2

+ 119868119860(120597119908

0

120597119905)

2

+ 119868119863(1205972

1199080

120597119909120597119905)

2

+ 119868119860(120597V

0

120597119905)

2

+ 119868119863(

1205972V

0

120597119909120597119905)

2

]119889119909

(38)

where 119868119860and 119868


beam expressed as

119868119860= int

119860

120588119889119860 = 120588119860 119868119863= int

119860

1205881199112

119889119860 = int119860

1205881199102

119889119860 (39)


119862119911sdot 119908 (119909

119888 119905) + 119865

119862119910sdot V (119909

119888 119905) + 119898

ℎ119886ℎ119910

(119909ℎ 119905) V (119909

ℎ 119905)

+ 119898ℎ119886ℎ119911

(119909ℎ 119905) 119908 (119909

ℎ 119905) + int

119897

0

119902 (119909) V (119909 119905) 119889119909

(40)where 119909


119908(119909119888 119905) V(119909



119862119910are




ℎ] 119898

ℎis the mass

and 119886ℎ119910(119909

ℎ 119905) and 119886

ℎ119911(119909



ℎ 119905) and V(119909



0(119909 119905) and119908

0(119909 119905) can


1199060(119909 119905) =

infin

sum

119894=1

119880119894(119909)119880

119894(119905)

V0(119909 119905) =

infin

sum

119895=1

119881119895(119909) 119881

119895(119905)

1199080(119909 119905) =

infin

sum

119896=1

119882119896(119909)119882

119896(119905)

(41)

where 119880119894(119905) 119881

119895(119905) and 119882



119895(119909) and 119882

119896(119909) are the


119880119894(119909)

1003816100381610038161003816119909=0= 0 119881

119895(119909)

10038161003816100381610038161003816119909=0= 0 119882

119896(119909)

1003816100381610038161003816119909=0= 0

1198811015840

119895

(119909)10038161003816100381610038161003816119909=0

= 0 1198821015840

119896

(119909)10038161003816100381610038161003816119909=0

= 0

(42)


1198801(119909) =

119909

119897 119880

2(119909) = (

119909

119897)

2

1198811(119909) = (

119909

119897)

2

1198812(119909) = (

119909

119897)

3

1198821(119909) = (

119909

119897)

2

1198822(119909) = (

119909

119897)

3

(43)


119894 119879

119890119882

119911 and Ψ


119880119894=

9

2

119860119909119909

1198973(119881

2

2

+ 1198822

2

)2

+27

4

119860119909119909

1198973(119881

11198812+ 119882

1119882

2)

times (1198812

2

+ 1198822

2

)

+18

7

119860119909119909

1198973[(119881

2

1

+ 1198822

1

) (1198812

2

+ 1198822

2

) + (11988111198812+ 119882

1119882

2)2

]

+119860

119909119909

1198973[3

2119897119880

2(119881

2

2

+ 1198822

2

) + (1198812

1

+ 1198822

1

)

times (11988111198812+ 119882

1119882

2)]


+1

10

119860119909119909

1198973[9119880

1119897 (119881

2

2

+ 1198822

2

) + 241198802119897 (119881

11198812+ 119882

1119882

2)

+ 4 (1198812

1

+ 1198822

1

)2

]

+119860

119909119909

1198972[3

21198801(119881

11198812+ 119882

1119882

2) + 119880

2(119881

2

1

+ 1198822

1

)]

+6119863

119909119909119881

2

2

1198973+ 119860

119909119909(2119880

1119881

2

1

31198972+

21198801119882

2

1

31198972+

21198802

2

3119897)

+6119863

119909119909119882

2

2

1198973+

611986311990911990911988111198812

1198973+

11986011990911990911988011198802

119897+

6119863119909119909119882

1119882

2

1198973

+119860

119909119909119880

2

1

2119897+

2119863119909119909119881

2

1

1198973+

2119863119909119909119882

2

1

1198973

119879119890=

1

14(119897119868

119860

2

2

+ 119897119868119860

2

2

) +1

6(119897119868

119860

1

2+ 119897119868

11986012)

+1

10(119897119868

119860

2

2

+ 119897119868119860

2

1

+9119868

119863

2

2

119897+ 119897119868

119860

2

1

+9119868

119863

2

2

119897)

+1

4(119897119868

11986012+

6119868119863

1

2

119897+

611986811986312

119897)

+1

6(119897119868

119860

2

1

+4119868

119863

2

1

119897+

4119868119863

2

1

119897)

119882119911= 119865

119862119911sdot [119882

1(119905) + 119882

2(119905)] + 119865

119862119910sdot [119881

1(119905) + 119881

2(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119881

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119881

2(119905) +

6V3ℎ

1199052

1198973

sdot 2(119905) + (

Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198811(119905) + (

Vℎ119905

119897)

3

sdot 1198812(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119882

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119882

2(119905) +

6V3ℎ

1199052

1198973sdot

2(119905)

+ (Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198821(119905) + (

Vℎ119905

119897)

3

sdot 1198822(119905)]

+ 1198811(119905) sdot int

119897

0

119902 (119909) (119909

119897)

2

119889119909 + 1198812(119905)

sdot int

119897

0

119902 (119909) (119909

119897)

3

119889119909

Ψ119902=

1

6(36119863

119909119909120578119887

2

2

1198976+

4119860119909119909120578119887

2

2

1198974+

36119863119909119909120578119887

2

2

1198976) 119897

3

+1

4(4119860

11990911990912057811988712

1198973+

2411986311990911990912057811988712

1198975

+24119863

119909119909120578119887

1

2

1198975) 119897

2

+1

2

119860119909119909120578119887

2

1

119897+

2119863119909119909120578119887

2

1

1198973+

2119863119909119909120578119887

2

1

1198973

(44)


120597119871

120597119902119899

minus119889

119889119905(

120597119871

120597 119902119899

) + 119876119889= 0 119899 = 1 2 6 (45)


= (1198801 119880

2 119881

1 119881

2

1198821119882



119889is the


as

119871 = 119879119890minus (119880

119894minus 119882

119911)

119876119889= minus

120597Ψ119902

120597 119902119899

(119899 = 1 sim 6)

(46)


1 119880

2

1

31198971198681198601+

1

41198971198681198602+

120578119887119860

1199091199091

119897+

120578119887119860

1199091199092

119897+

1198601199091199091198801

119897

+119860

1199091199091198802

119897+

2119860119909119909119881

2

1

31198972+

311986011990911990911988111198812

21198972+

9119860119909119909119881

2

2

101198972

+2119860

119909119909119882

2

1

31198972+

3119860119909119909119882

1119882

2

21198972+

9119860119909119909119882

2

2

101198972= 0

(47a)

1

41198971198681198601+

1

51198971198681198602+

120578119887119860

1199091199091

119897+

4120578119887119860

1199091199092

3119897

+119860

1199091199091198801

119897+4119860

1199091199091198802

3119897+119860

119909119909119881

2

1

1198972+12119860

11990911990911988111198812

51198972+3119860

119909119909119881

2

2

21198972

+119860

119909119909119882

2

1

1198972+

12119860119909119909119882

1119882

2

51198972+

3119860119909119909119882

2

2

21198972= 0

(47b)





1 119881

2119882

1119882

2 We take (47a)



2119882

1119882






2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)









119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










OO

xb mp

14

1205821

zbyb

dm32

3

1205822

EI

p

ch1

h2

31205883

O m3

r

r r

rr

R R







119887 The


R119898119894

= R119874+ r

119894+ 120588

119894 k

119898119894= R

119874+∘

120588119894

+ 120596 times (r119894+ 120588

119894) (4)

whereR119898119894(v




119874R

119874(R

119874= v






[

[

120596119909

120596119910

120596119911

]

]119887

= [

[


sin 120579 + cos 120579 cos120593]

]

(5)


R1198983

= r3+120588

3= [120588

3119909 120588

3119910 119911

3]119879

R1198981

= r1+ 120588

1= [120588

1119909 120588

1119910 119911

1]119879

R1198982

= r2+ 120588

2= [120588

2119909 119910

2 120588

2119911]119879

R1198984

= r4+ 120588

4= [120588

4119909 119910

4 120588

4119911]119879

(6)

R1198983

=∘

1205883

+ 120596 times (r3+ 120588

3) =

[[

[

1205883119909

1205883119910

0

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205883119909

1205883119910

1199113

]

]

=

[[[

[

1205883119909

+ 1199113

120579 + 1199113 sin120593 minus 120588

3119910 sin 120579 minus 120588

3119910 cos 120579 cos120593

1205883119910

+ 1205883119909 sin 120579 + 120588

3119909 cos 120579 cos120593 minus 119911

3 cos 120579 + 119911

3 sin 120579 cos120593

1205883119910 cos 120579 minus 120588

3119910 sin 120579 cos120593 minus 120588

3119909

120579 minus 1205883119909 sin120593

]]]

]

(7)

R1198981

=∘

1205881

+ 120596 times (r1+ 120588

1) =

[[

[

1205881119909

1205881119910

0

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205881119909

1205881119910

1199111

]

]

=

[[[

[

1205881119909

+ 1199111

120579 + 1199111 sin120593 minus 120588

1119910 sin 120579 minus 120588

1119910 cos 120579 cos120593

1205881119910

+ 1205881119909 sin 120579 + 120588

1119909 cos 120579 cos120593 minus 119911

1 cos 120579 + 119911

1 sin 120579 cos120593

1205881119910 cos 120579 minus 120588

1119910 sin 120579 cos120593 minus 120588

1119909

120579 minus 1205881119909 sin120593

]]]

]

(8)

R1198982

=∘

1205882

+ 120596 times (r2+ 120588

2) =

[[

[

1205882119909

0

1205882119911

]]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205882119909

1199102

1205882119911

]

]

=[[

[

1205882119909

+ 1205882119911

120579 + 1205882119911 sin120593 minus 119910

2 sin 120579 minus 119910

2 cos 120579 cos120593

1205882119909 sin 120579 + 120588

2119909 cos 120579 cos120593 minus 120588

2119911 cos 120579 + 120588

2119911 sin 120579 cos120593

1205882119911

+ 1199102 cos 120579 minus 119910

2 sin 120579 cos120593 minus 120588

2119909

120579 minus 1205882119909 sin120593

]]

]

(9)


R1198984

=∘

1205884

+ 120596 times (r4+ 120588

4) = [

[

1205884119909

0

1205884119911

]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205884119909

1199104

1205884119911

]

]

=[[

[

1205884119909

+ 1205884119911

120579 + 1205884119911 sin120593 minus 119910

4 sin 120579 minus 119910

4 cos 120579 cos120593

1205884119909 sin 120579 + 120588

4119909 cos 120579 cos120593 minus 120588

4119911 cos 120579 + 120588

4119911 sin 120579 cos120593

1205884119911

+ 1199104 cos 120579 minus 119910

4 sin 120579 cos120593 minus 120588

4119909

120579 minus 1205884119909 sin120593

]]

]

(10)


In Figure 4 1205821and 120582




two angles

r119901=10038161003816100381610038161003816r119901

10038161003816100381610038161003816cos 120582

1i119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816sin 120582

1cos 120582

2j119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816sin 120582

1sin 120582

2k119887

(11)


R119874119901

= R119874+ r

119901 k

119874119901= k

119874+ r

119901 (12)

where r119901=∘r119901+ 120596 times r

119901 And ∘r


∘r119901= minus

10038161003816100381610038161003816r119901

100381610038161003816100381610038161sin 120582

1i119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816

times (1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2) j

119887

+10038161003816100381610038161003816r119901

10038161003816100381610038161003816(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2) k

119887

(13)


r119901= [[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

times [119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1sin 120582

2( cos 120579 minus sin 120579 cos120593)]

times 119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

(14)


Rℎ119895

= R119874+ r

ℎ119895+ 120588

ℎ119895

kℎ2

= k119874+∘rℎ119895

+∘

120588ℎ119895

+ 120596 times (rℎ119895

+ 120588ℎ119895)

(119895 = 1 2)

(15)


Rℎ2

= rℎ2

+ 120588ℎ2

= [120588ℎ2119909

120588ℎ2119910

119911ℎ2]119879

Rℎ1

= rℎ1

+ 120588ℎ1

= [120588ℎ1119909

119910ℎ1 120588

ℎ1119911]119879

Rℎ2

=[[

[

120588ℎ2119909

+ 119911ℎ2

( 120579 + sin120593) minus 120588ℎ2119910

( sin 120579 + cos 120579 cos120593)120588ℎ2119910

+ 120588ℎ2119909



+ 120588ℎ2119910


( 120579 + sin120593)

]]

]

Rℎ1

=[[

[

120588ℎ1119909

+ 120588ℎ1119911

( 120579 + sin120593) minus 119910ℎ1

( sin 120579 + cos 120579 cos120593)119910ℎ1

+ 120588ℎ1119909



+ 119910ℎ1


( 120579 + sin120593)

]]

]

(16)



1198791198983

=1

2119898

119897int

119897

0

[∘

1205883

+ 120596 times (r3+ 120588

3)] sdot [

∘

1205883

+ 120596 times (r3+ 120588

3)] 119889119911

1198791198981

=1

2119898

119897int

0

minus119897

[∘

1205881

+ 120596 times (r1+ 120588

1)]

sdot [∘

1205881

+ 120596 times (r1+ 120588

1)] 119889119911

1198791198982

=1

2119898

119897int

119897

0

[∘

1205882

+ 120596 times (r2+ 120588

2)] sdot [

∘

1205882

+ 120596 times (r2+ 120588

2)] 119889119910

1198791198984

=1

2119898

119897int

0

minus119897

[∘

1205884

+ 120596 times (r4+ 120588

4)] sdot [

∘

1205884

+ 120596 times (r4+ 120588

4)] 119889119910

(17)


119879119898

=1

2120596 sdot J sdot 120596

+1

2119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+1

2119898

119897int

0

minus119897

2∘

1205881


1) sdot (120596 times 120588

1)


1) 119889119911

+1

2119898

119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

+1

2119898

119897int

0

minus119897

2∘

1205884


4) sdot (120596 times 120588

4)


4) + (120596 times r

4)

sdot (120596 times r4) 119889119910

(18)

where J = sum4

119895=1

J119898119895



119909

119869119910 119869

119911] inO

119874x119887y119887z119887


119879119901=

1

2119898

119901

10038161003816100381610038161003816r119901

10038161003816100381610038161003816

2

=1

2119898

119901[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

2

(19)


119879ℎ1

=1

2119898

ℎ1[ ( 120588

ℎ1119909+ 120579120588

ℎ1119911+ 120588

ℎ1119911sin120593

minus 119910ℎ1sin 120579 minus 119910

ℎ1cos 120579 cos120593)

2

]

+1

2119898

ℎ1[( 119910

ℎ1+ 120588

ℎ1119909sin 120579 + 120588

ℎ1119909cos 120579 cos120593

minus 120588ℎ1119911

cos 120579 + 120588ℎ1119911

sin 120579 cos120593)2]

+1

2119898

ℎ1[( 120588

ℎ1119911+ 119910

ℎ1cos 120579 minus 119910

ℎ1sin 120579 cos120593

minus 120579120588ℎ1119909

+ 120588ℎ1119909

sin120593)2

]

119879ℎ2

=1

2119898

ℎ2[( 120588

ℎ2119909+ 120579119911

ℎ2+ 119911

ℎ2sin120593 minus 120588

ℎ2119910sin 120579

minus 120588ℎ2119910

cos 120579 cos120593)2

]

+1

2119898

ℎ2[( 120588

ℎ2119910+ 120588

ℎ2119909sin 120579 + 120588

ℎ2119909cos 120579 cos120593

minus 119911ℎ2cos 120579 + 119911

ℎ2sin 120579 cos120593)

2

]

+1

2119898

ℎ2[(

ℎ2+ 120588

ℎ2119910cos 120579 minus 120588

ℎ2119910sin 120579 cos120593

minus 120579120588ℎ2119909

minus 120588ℎ2119909

sin120593)2

]

(20)


119892119891 payload 119880

119892119901 and


119880119892ℎ2

can be obtained as

119880119892119891

= minus

120583119898119891

1003816100381610038161003816R119874

1003816100381610038161003816

119880119892119901

= minus

120583119898119901

10038161003816100381610038161003816R119874+ r

119901

10038161003816100381610038161003816

119880119892ℎ1

= minus120583119898

ℎ1

1003816100381610038161003816R119874+ r

ℎ1

1003816100381610038161003816

119880119892ℎ2

= minus120583119898

ℎ2

1003816100381610038161003816R119874+ r

ℎ2

1003816100381610038161003816

(21)



119868

S = j119868= [1 0 0]

119879

119868

(22)



S = C119887119868[1 0 0]

119879

119868

= [119878119909

119878119910

119878119911]119879

119887

(23)


F119894= 120578119875119860

119862119894(S sdot n

119862119894)2 n

119862119894= 119865

119862119894cos2120572

119862119894n119862119894 (24)




119865119862119894


119862119894= i

119894


119862119894in 120587


n119862119894

= C119887119894[1 0 0]

119879

119894

= [119878119909119894

119878119910119894

119878119911119894]119879

119887

(25)


M119894= L

119894times F

119894 (26)




119894andF



119894and F


L1= [0 0 minus119897]

119879

L2= [0 119897 0]

119879

L3= [0 0 119897]

119879

L4= [0 minus119897 0]

119879

F119894= 120578119875119860

119862119894([119878

119909119878119910

119878119911]119887

sdot [119878119909119894

119878119910119894

119878119911119894]119879

119887

)

2

[119878119909119894

119878119910119894

119878119911119894]119879

119887

(27)


F119894= (119865

119862119894cos2120572

119862119894)C

119887119894[1 0 0]

119879

119894

= (119865119862119894cos2120572

119862119894) [119878

119909119894119878119910119894

119878119911119894]119879

119887

(28)

where 120572119862119894


M1198881

= (1198651198621cos2120572

1198621)[

[

0

0

minus119897

]

]

times (C1198871

[

[

1

0

0

]

]

)

= (1198651198621cos2120572

1198621)

times [

[

minus119897 cos1205931sin120595

1

minus119897 (cos 1205791cos120595

1minus sin 120579

1sin120593

1sin120595

1)

0

]

]

M1198882

= (1198651198622cos2120572

1198622)

times [

[

119897 (sin 1205792cos120595

2+ cos 120579

2sin120593

2sin120595

2)

0

minus119897 (cos 1205792cos120595

2minus sin 120579

2sin120593

2sin120595

2)

]

]

M1198883

= (1198651198623cos2120572

1198623)

times [

[

119897 cos1205933sin120595

3

119897 (cos 1205793cos120595

3minus sin 120579

3sin120593

3sin120595

3)

0

]

]

M1198884

= (1198651198624cos2120572

1198624)

times [

[

minus119897 (sin 1205794cos120595

4+ cos 120579

4sin120593

4sin120595

4)

0

119897 (cos 1205794cos120595

4minus sin 120579

4sin120593

4sin120595

4)

]

]

(29)


119889

119889119905(

120597119871

120597 119902119895

) minus120597119871

120597119902119895

= 119876119895 (30)




119871 = 119879 minus 119880 =

4

sum

119894=1

119879119898119894

+ 119879119901+

2

sum

119895=1

119879ℎ119895

minus (119880119892119891

+ 119880119892119901

+

2

sum

119895=1

119880119892ℎ119895

)

(31)

where 119879119898119894 119879

119901 and 119879



119892119901 and 119880




OOx

y

z

FCFCy

FCx

FCz

mh

q(x)


120590

120590e

120590120592

120590







119862119909 119865

119862119910 and 119865





120590 = 120590119890

+ 120590120592

= 119864120576 + 119888119887

120576 (32)

where 120590 120590119890 and 120590



expressed as 119864120578119887 120578




and s respectively

OO x

y The support beam

Z




119906 (119909 119910 119911 119905) = 1199060(119909 119905) minus 119910

120597V0

120597119909minus 119911

1205971199080

120597119909

V (119909 119910 119911 119905) = V0(119909 119905)

119908 (119909 119910 119911 119905) = 1199080(119909 119905)

(33)


0(119909 119905)

V0(119909 119905) and 119908




120576119909119909

=120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

120576119909119909

=120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909)

minus 1199111205972

0

1205971199092

120590119909119909

= 119864[120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

]

+ 119864120578119887(120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909) minus 119911

1205972

0

1205971199092

)

(34)




119880119894=

1

2int

119897

0

119860119909119909

[120597119906

0

120597119909+

1

2(120597V

0

120597119909)

2

+1

2(120597119908

0

120597119909)

2

]

2

+ 119863119909119909

1205972V

0

1205971199092

+ 119863119909119909

1205972

1199080

1205971199092

119889119909

(35)




119860119909119909

= int119860

119864119889119860 119863119909119909

= int119860

1198641199112

119889119860 = int119860

1198641199102

119889119860 (36)


Ψ119902=

1

2int

119897

0

[119860119909119909120578119887(120597

0

120597119909)

2

+ 119863119909119909120578119887(1205972V

0

1205971199092

)

2

+ 119863119909119909120578119887(1205972

0

1205971199092

)

2

]119889119909

(37)


119879119890=

1

2int

119897

0

[119868119860(120597119906

0

120597119905)

2

+ 119868119860(120597119908

0

120597119905)

2

+ 119868119863(1205972

1199080

120597119909120597119905)

2

+ 119868119860(120597V

0

120597119905)

2

+ 119868119863(

1205972V

0

120597119909120597119905)

2

]119889119909

(38)

where 119868119860and 119868


beam expressed as

119868119860= int

119860

120588119889119860 = 120588119860 119868119863= int

119860

1205881199112

119889119860 = int119860

1205881199102

119889119860 (39)


119862119911sdot 119908 (119909

119888 119905) + 119865

119862119910sdot V (119909

119888 119905) + 119898

ℎ119886ℎ119910

(119909ℎ 119905) V (119909

ℎ 119905)

+ 119898ℎ119886ℎ119911

(119909ℎ 119905) 119908 (119909

ℎ 119905) + int

119897

0

119902 (119909) V (119909 119905) 119889119909

(40)where 119909


119908(119909119888 119905) V(119909



119862119910are




ℎ] 119898

ℎis the mass

and 119886ℎ119910(119909

ℎ 119905) and 119886

ℎ119911(119909



ℎ 119905) and V(119909



0(119909 119905) and119908

0(119909 119905) can


1199060(119909 119905) =

infin

sum

119894=1

119880119894(119909)119880

119894(119905)

V0(119909 119905) =

infin

sum

119895=1

119881119895(119909) 119881

119895(119905)

1199080(119909 119905) =

infin

sum

119896=1

119882119896(119909)119882

119896(119905)

(41)

where 119880119894(119905) 119881

119895(119905) and 119882



119895(119909) and 119882

119896(119909) are the


119880119894(119909)

1003816100381610038161003816119909=0= 0 119881

119895(119909)

10038161003816100381610038161003816119909=0= 0 119882

119896(119909)

1003816100381610038161003816119909=0= 0

1198811015840

119895

(119909)10038161003816100381610038161003816119909=0

= 0 1198821015840

119896

(119909)10038161003816100381610038161003816119909=0

= 0

(42)


1198801(119909) =

119909

119897 119880

2(119909) = (

119909

119897)

2

1198811(119909) = (

119909

119897)

2

1198812(119909) = (

119909

119897)

3

1198821(119909) = (

119909

119897)

2

1198822(119909) = (

119909

119897)

3

(43)


119894 119879

119890119882

119911 and Ψ


119880119894=

9

2

119860119909119909

1198973(119881

2

2

+ 1198822

2

)2

+27

4

119860119909119909

1198973(119881

11198812+ 119882

1119882

2)

times (1198812

2

+ 1198822

2

)

+18

7

119860119909119909

1198973[(119881

2

1

+ 1198822

1

) (1198812

2

+ 1198822

2

) + (11988111198812+ 119882

1119882

2)2

]

+119860

119909119909

1198973[3

2119897119880

2(119881

2

2

+ 1198822

2

) + (1198812

1

+ 1198822

1

)

times (11988111198812+ 119882

1119882

2)]


+1

10

119860119909119909

1198973[9119880

1119897 (119881

2

2

+ 1198822

2

) + 241198802119897 (119881

11198812+ 119882

1119882

2)

+ 4 (1198812

1

+ 1198822

1

)2

]

+119860

119909119909

1198972[3

21198801(119881

11198812+ 119882

1119882

2) + 119880

2(119881

2

1

+ 1198822

1

)]

+6119863

119909119909119881

2

2

1198973+ 119860

119909119909(2119880

1119881

2

1

31198972+

21198801119882

2

1

31198972+

21198802

2

3119897)

+6119863

119909119909119882

2

2

1198973+

611986311990911990911988111198812

1198973+

11986011990911990911988011198802

119897+

6119863119909119909119882

1119882

2

1198973

+119860

119909119909119880

2

1

2119897+

2119863119909119909119881

2

1

1198973+

2119863119909119909119882

2

1

1198973

119879119890=

1

14(119897119868

119860

2

2

+ 119897119868119860

2

2

) +1

6(119897119868

119860

1

2+ 119897119868

11986012)

+1

10(119897119868

119860

2

2

+ 119897119868119860

2

1

+9119868

119863

2

2

119897+ 119897119868

119860

2

1

+9119868

119863

2

2

119897)

+1

4(119897119868

11986012+

6119868119863

1

2

119897+

611986811986312

119897)

+1

6(119897119868

119860

2

1

+4119868

119863

2

1

119897+

4119868119863

2

1

119897)

119882119911= 119865

119862119911sdot [119882

1(119905) + 119882

2(119905)] + 119865

119862119910sdot [119881

1(119905) + 119881

2(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119881

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119881

2(119905) +

6V3ℎ

1199052

1198973

sdot 2(119905) + (

Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198811(119905) + (

Vℎ119905

119897)

3

sdot 1198812(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119882

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119882

2(119905) +

6V3ℎ

1199052

1198973sdot

2(119905)

+ (Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198821(119905) + (

Vℎ119905

119897)

3

sdot 1198822(119905)]

+ 1198811(119905) sdot int

119897

0

119902 (119909) (119909

119897)

2

119889119909 + 1198812(119905)

sdot int

119897

0

119902 (119909) (119909

119897)

3

119889119909

Ψ119902=

1

6(36119863

119909119909120578119887

2

2

1198976+

4119860119909119909120578119887

2

2

1198974+

36119863119909119909120578119887

2

2

1198976) 119897

3

+1

4(4119860

11990911990912057811988712

1198973+

2411986311990911990912057811988712

1198975

+24119863

119909119909120578119887

1

2

1198975) 119897

2

+1

2

119860119909119909120578119887

2

1

119897+

2119863119909119909120578119887

2

1

1198973+

2119863119909119909120578119887

2

1

1198973

(44)


120597119871

120597119902119899

minus119889

119889119905(

120597119871

120597 119902119899

) + 119876119889= 0 119899 = 1 2 6 (45)


= (1198801 119880

2 119881

1 119881

2

1198821119882



119889is the


as

119871 = 119879119890minus (119880

119894minus 119882

119911)

119876119889= minus

120597Ψ119902

120597 119902119899

(119899 = 1 sim 6)

(46)


1 119880

2

1

31198971198681198601+

1

41198971198681198602+

120578119887119860

1199091199091

119897+

120578119887119860

1199091199092

119897+

1198601199091199091198801

119897

+119860

1199091199091198802

119897+

2119860119909119909119881

2

1

31198972+

311986011990911990911988111198812

21198972+

9119860119909119909119881

2

2

101198972

+2119860

119909119909119882

2

1

31198972+

3119860119909119909119882

1119882

2

21198972+

9119860119909119909119882

2

2

101198972= 0

(47a)

1

41198971198681198601+

1

51198971198681198602+

120578119887119860

1199091199091

119897+

4120578119887119860

1199091199092

3119897

+119860

1199091199091198801

119897+4119860

1199091199091198802

3119897+119860

119909119909119881

2

1

1198972+12119860

11990911990911988111198812

51198972+3119860

119909119909119881

2

2

21198972

+119860

119909119909119882

2

1

1198972+

12119860119909119909119882

1119882

2

51198972+

3119860119909119909119882

2

2

21198972= 0

(47b)





1 119881

2119882

1119882

2 We take (47a)



2119882

1119882






2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)









119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R1198984

=∘

1205884

+ 120596 times (r4+ 120588

4) = [

[

1205884119909

0

1205884119911

]

]

+[[

[


sin 120579 + cos 120579 cos120593

]]

]

times [

[

1205884119909

1199104

1205884119911

]

]

=[[

[

1205884119909

+ 1205884119911

120579 + 1205884119911 sin120593 minus 119910

4 sin 120579 minus 119910

4 cos 120579 cos120593

1205884119909 sin 120579 + 120588

4119909 cos 120579 cos120593 minus 120588

4119911 cos 120579 + 120588

4119911 sin 120579 cos120593

1205884119911

+ 1199104 cos 120579 minus 119910

4 sin 120579 cos120593 minus 120588

4119909

120579 minus 1205884119909 sin120593

]]

]

(10)


In Figure 4 1205821and 120582




two angles

r119901=10038161003816100381610038161003816r119901

10038161003816100381610038161003816cos 120582

1i119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816sin 120582

1cos 120582

2j119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816sin 120582

1sin 120582

2k119887

(11)


R119874119901

= R119874+ r

119901 k

119874119901= k

119874+ r

119901 (12)

where r119901=∘r119901+ 120596 times r

119901 And ∘r


∘r119901= minus

10038161003816100381610038161003816r119901

100381610038161003816100381610038161sin 120582

1i119887+10038161003816100381610038161003816r119901

10038161003816100381610038161003816

times (1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2) j

119887

+10038161003816100381610038161003816r119901

10038161003816100381610038161003816(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2) k

119887

(13)


r119901= [[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

times [119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1sin 120582

2( cos 120579 minus sin 120579 cos120593)]

times 119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

(14)


Rℎ119895

= R119874+ r

ℎ119895+ 120588

ℎ119895

kℎ2

= k119874+∘rℎ119895

+∘

120588ℎ119895

+ 120596 times (rℎ119895

+ 120588ℎ119895)

(119895 = 1 2)

(15)


Rℎ2

= rℎ2

+ 120588ℎ2

= [120588ℎ2119909

120588ℎ2119910

119911ℎ2]119879

Rℎ1

= rℎ1

+ 120588ℎ1

= [120588ℎ1119909

119910ℎ1 120588

ℎ1119911]119879

Rℎ2

=[[

[

120588ℎ2119909

+ 119911ℎ2

( 120579 + sin120593) minus 120588ℎ2119910

( sin 120579 + cos 120579 cos120593)120588ℎ2119910

+ 120588ℎ2119909



+ 120588ℎ2119910


( 120579 + sin120593)

]]

]

Rℎ1

=[[

[

120588ℎ1119909

+ 120588ℎ1119911

( 120579 + sin120593) minus 119910ℎ1

( sin 120579 + cos 120579 cos120593)119910ℎ1

+ 120588ℎ1119909



+ 119910ℎ1


( 120579 + sin120593)

]]

]

(16)



1198791198983

=1

2119898

119897int

119897

0

[∘

1205883

+ 120596 times (r3+ 120588

3)] sdot [

∘

1205883

+ 120596 times (r3+ 120588

3)] 119889119911

1198791198981

=1

2119898

119897int

0

minus119897

[∘

1205881

+ 120596 times (r1+ 120588

1)]

sdot [∘

1205881

+ 120596 times (r1+ 120588

1)] 119889119911

1198791198982

=1

2119898

119897int

119897

0

[∘

1205882

+ 120596 times (r2+ 120588

2)] sdot [

∘

1205882

+ 120596 times (r2+ 120588

2)] 119889119910

1198791198984

=1

2119898

119897int

0

minus119897

[∘

1205884

+ 120596 times (r4+ 120588

4)] sdot [

∘

1205884

+ 120596 times (r4+ 120588

4)] 119889119910

(17)


119879119898

=1

2120596 sdot J sdot 120596

+1

2119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+1

2119898

119897int

0

minus119897

2∘

1205881


1) sdot (120596 times 120588

1)


1) 119889119911

+1

2119898

119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

+1

2119898

119897int

0

minus119897

2∘

1205884


4) sdot (120596 times 120588

4)


4) + (120596 times r

4)

sdot (120596 times r4) 119889119910

(18)

where J = sum4

119895=1

J119898119895



119909

119869119910 119869

119911] inO

119874x119887y119887z119887


119879119901=

1

2119898

119901

10038161003816100381610038161003816r119901

10038161003816100381610038161003816

2

=1

2119898

119901[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

2

(19)


119879ℎ1

=1

2119898

ℎ1[ ( 120588

ℎ1119909+ 120579120588

ℎ1119911+ 120588

ℎ1119911sin120593

minus 119910ℎ1sin 120579 minus 119910

ℎ1cos 120579 cos120593)

2

]

+1

2119898

ℎ1[( 119910

ℎ1+ 120588

ℎ1119909sin 120579 + 120588

ℎ1119909cos 120579 cos120593

minus 120588ℎ1119911

cos 120579 + 120588ℎ1119911

sin 120579 cos120593)2]

+1

2119898

ℎ1[( 120588

ℎ1119911+ 119910

ℎ1cos 120579 minus 119910

ℎ1sin 120579 cos120593

minus 120579120588ℎ1119909

+ 120588ℎ1119909

sin120593)2

]

119879ℎ2

=1

2119898

ℎ2[( 120588

ℎ2119909+ 120579119911

ℎ2+ 119911

ℎ2sin120593 minus 120588

ℎ2119910sin 120579

minus 120588ℎ2119910

cos 120579 cos120593)2

]

+1

2119898

ℎ2[( 120588

ℎ2119910+ 120588

ℎ2119909sin 120579 + 120588

ℎ2119909cos 120579 cos120593

minus 119911ℎ2cos 120579 + 119911

ℎ2sin 120579 cos120593)

2

]

+1

2119898

ℎ2[(

ℎ2+ 120588

ℎ2119910cos 120579 minus 120588

ℎ2119910sin 120579 cos120593

minus 120579120588ℎ2119909

minus 120588ℎ2119909

sin120593)2

]

(20)


119892119891 payload 119880

119892119901 and


119880119892ℎ2

can be obtained as

119880119892119891

= minus

120583119898119891

1003816100381610038161003816R119874

1003816100381610038161003816

119880119892119901

= minus

120583119898119901

10038161003816100381610038161003816R119874+ r

119901

10038161003816100381610038161003816

119880119892ℎ1

= minus120583119898

ℎ1

1003816100381610038161003816R119874+ r

ℎ1

1003816100381610038161003816

119880119892ℎ2

= minus120583119898

ℎ2

1003816100381610038161003816R119874+ r

ℎ2

1003816100381610038161003816

(21)



119868

S = j119868= [1 0 0]

119879

119868

(22)



S = C119887119868[1 0 0]

119879

119868

= [119878119909

119878119910

119878119911]119879

119887

(23)


F119894= 120578119875119860

119862119894(S sdot n

119862119894)2 n

119862119894= 119865

119862119894cos2120572

119862119894n119862119894 (24)




119865119862119894


119862119894= i

119894


119862119894in 120587


n119862119894

= C119887119894[1 0 0]

119879

119894

= [119878119909119894

119878119910119894

119878119911119894]119879

119887

(25)


M119894= L

119894times F

119894 (26)




119894andF



119894and F


L1= [0 0 minus119897]

119879

L2= [0 119897 0]

119879

L3= [0 0 119897]

119879

L4= [0 minus119897 0]

119879

F119894= 120578119875119860

119862119894([119878

119909119878119910

119878119911]119887

sdot [119878119909119894

119878119910119894

119878119911119894]119879

119887

)

2

[119878119909119894

119878119910119894

119878119911119894]119879

119887

(27)


F119894= (119865

119862119894cos2120572

119862119894)C

119887119894[1 0 0]

119879

119894

= (119865119862119894cos2120572

119862119894) [119878

119909119894119878119910119894

119878119911119894]119879

119887

(28)

where 120572119862119894


M1198881

= (1198651198621cos2120572

1198621)[

[

0

0

minus119897

]

]

times (C1198871

[

[

1

0

0

]

]

)

= (1198651198621cos2120572

1198621)

times [

[

minus119897 cos1205931sin120595

1

minus119897 (cos 1205791cos120595

1minus sin 120579

1sin120593

1sin120595

1)

0

]

]

M1198882

= (1198651198622cos2120572

1198622)

times [

[

119897 (sin 1205792cos120595

2+ cos 120579

2sin120593

2sin120595

2)

0

minus119897 (cos 1205792cos120595

2minus sin 120579

2sin120593

2sin120595

2)

]

]

M1198883

= (1198651198623cos2120572

1198623)

times [

[

119897 cos1205933sin120595

3

119897 (cos 1205793cos120595

3minus sin 120579

3sin120593

3sin120595

3)

0

]

]

M1198884

= (1198651198624cos2120572

1198624)

times [

[

minus119897 (sin 1205794cos120595

4+ cos 120579

4sin120593

4sin120595

4)

0

119897 (cos 1205794cos120595

4minus sin 120579

4sin120593

4sin120595

4)

]

]

(29)


119889

119889119905(

120597119871

120597 119902119895

) minus120597119871

120597119902119895

= 119876119895 (30)




119871 = 119879 minus 119880 =

4

sum

119894=1

119879119898119894

+ 119879119901+

2

sum

119895=1

119879ℎ119895

minus (119880119892119891

+ 119880119892119901

+

2

sum

119895=1

119880119892ℎ119895

)

(31)

where 119879119898119894 119879

119901 and 119879



119892119901 and 119880




OOx

y

z

FCFCy

FCx

FCz

mh

q(x)


120590

120590e

120590120592

120590







119862119909 119865

119862119910 and 119865





120590 = 120590119890

+ 120590120592

= 119864120576 + 119888119887

120576 (32)

where 120590 120590119890 and 120590



expressed as 119864120578119887 120578




and s respectively

OO x

y The support beam

Z




119906 (119909 119910 119911 119905) = 1199060(119909 119905) minus 119910

120597V0

120597119909minus 119911

1205971199080

120597119909

V (119909 119910 119911 119905) = V0(119909 119905)

119908 (119909 119910 119911 119905) = 1199080(119909 119905)

(33)


0(119909 119905)

V0(119909 119905) and 119908




120576119909119909

=120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

120576119909119909

=120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909)

minus 1199111205972

0

1205971199092

120590119909119909

= 119864[120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

]

+ 119864120578119887(120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909) minus 119911

1205972

0

1205971199092

)

(34)




119880119894=

1

2int

119897

0

119860119909119909

[120597119906

0

120597119909+

1

2(120597V

0

120597119909)

2

+1

2(120597119908

0

120597119909)

2

]

2

+ 119863119909119909

1205972V

0

1205971199092

+ 119863119909119909

1205972

1199080

1205971199092

119889119909

(35)




119860119909119909

= int119860

119864119889119860 119863119909119909

= int119860

1198641199112

119889119860 = int119860

1198641199102

119889119860 (36)


Ψ119902=

1

2int

119897

0

[119860119909119909120578119887(120597

0

120597119909)

2

+ 119863119909119909120578119887(1205972V

0

1205971199092

)

2

+ 119863119909119909120578119887(1205972

0

1205971199092

)

2

]119889119909

(37)


119879119890=

1

2int

119897

0

[119868119860(120597119906

0

120597119905)

2

+ 119868119860(120597119908

0

120597119905)

2

+ 119868119863(1205972

1199080

120597119909120597119905)

2

+ 119868119860(120597V

0

120597119905)

2

+ 119868119863(

1205972V

0

120597119909120597119905)

2

]119889119909

(38)

where 119868119860and 119868


beam expressed as

119868119860= int

119860

120588119889119860 = 120588119860 119868119863= int

119860

1205881199112

119889119860 = int119860

1205881199102

119889119860 (39)


119862119911sdot 119908 (119909

119888 119905) + 119865

119862119910sdot V (119909

119888 119905) + 119898

ℎ119886ℎ119910

(119909ℎ 119905) V (119909

ℎ 119905)

+ 119898ℎ119886ℎ119911

(119909ℎ 119905) 119908 (119909

ℎ 119905) + int

119897

0

119902 (119909) V (119909 119905) 119889119909

(40)where 119909


119908(119909119888 119905) V(119909



119862119910are




ℎ] 119898

ℎis the mass

and 119886ℎ119910(119909

ℎ 119905) and 119886

ℎ119911(119909



ℎ 119905) and V(119909



0(119909 119905) and119908

0(119909 119905) can


1199060(119909 119905) =

infin

sum

119894=1

119880119894(119909)119880

119894(119905)

V0(119909 119905) =

infin

sum

119895=1

119881119895(119909) 119881

119895(119905)

1199080(119909 119905) =

infin

sum

119896=1

119882119896(119909)119882

119896(119905)

(41)

where 119880119894(119905) 119881

119895(119905) and 119882



119895(119909) and 119882

119896(119909) are the


119880119894(119909)

1003816100381610038161003816119909=0= 0 119881

119895(119909)

10038161003816100381610038161003816119909=0= 0 119882

119896(119909)

1003816100381610038161003816119909=0= 0

1198811015840

119895

(119909)10038161003816100381610038161003816119909=0

= 0 1198821015840

119896

(119909)10038161003816100381610038161003816119909=0

= 0

(42)


1198801(119909) =

119909

119897 119880

2(119909) = (

119909

119897)

2

1198811(119909) = (

119909

119897)

2

1198812(119909) = (

119909

119897)

3

1198821(119909) = (

119909

119897)

2

1198822(119909) = (

119909

119897)

3

(43)


119894 119879

119890119882

119911 and Ψ


119880119894=

9

2

119860119909119909

1198973(119881

2

2

+ 1198822

2

)2

+27

4

119860119909119909

1198973(119881

11198812+ 119882

1119882

2)

times (1198812

2

+ 1198822

2

)

+18

7

119860119909119909

1198973[(119881

2

1

+ 1198822

1

) (1198812

2

+ 1198822

2

) + (11988111198812+ 119882

1119882

2)2

]

+119860

119909119909

1198973[3

2119897119880

2(119881

2

2

+ 1198822

2

) + (1198812

1

+ 1198822

1

)

times (11988111198812+ 119882

1119882

2)]


+1

10

119860119909119909

1198973[9119880

1119897 (119881

2

2

+ 1198822

2

) + 241198802119897 (119881

11198812+ 119882

1119882

2)

+ 4 (1198812

1

+ 1198822

1

)2

]

+119860

119909119909

1198972[3

21198801(119881

11198812+ 119882

1119882

2) + 119880

2(119881

2

1

+ 1198822

1

)]

+6119863

119909119909119881

2

2

1198973+ 119860

119909119909(2119880

1119881

2

1

31198972+

21198801119882

2

1

31198972+

21198802

2

3119897)

+6119863

119909119909119882

2

2

1198973+

611986311990911990911988111198812

1198973+

11986011990911990911988011198802

119897+

6119863119909119909119882

1119882

2

1198973

+119860

119909119909119880

2

1

2119897+

2119863119909119909119881

2

1

1198973+

2119863119909119909119882

2

1

1198973

119879119890=

1

14(119897119868

119860

2

2

+ 119897119868119860

2

2

) +1

6(119897119868

119860

1

2+ 119897119868

11986012)

+1

10(119897119868

119860

2

2

+ 119897119868119860

2

1

+9119868

119863

2

2

119897+ 119897119868

119860

2

1

+9119868

119863

2

2

119897)

+1

4(119897119868

11986012+

6119868119863

1

2

119897+

611986811986312

119897)

+1

6(119897119868

119860

2

1

+4119868

119863

2

1

119897+

4119868119863

2

1

119897)

119882119911= 119865

119862119911sdot [119882

1(119905) + 119882

2(119905)] + 119865

119862119910sdot [119881

1(119905) + 119881

2(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119881

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119881

2(119905) +

6V3ℎ

1199052

1198973

sdot 2(119905) + (

Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198811(119905) + (

Vℎ119905

119897)

3

sdot 1198812(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119882

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119882

2(119905) +

6V3ℎ

1199052

1198973sdot

2(119905)

+ (Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198821(119905) + (

Vℎ119905

119897)

3

sdot 1198822(119905)]

+ 1198811(119905) sdot int

119897

0

119902 (119909) (119909

119897)

2

119889119909 + 1198812(119905)

sdot int

119897

0

119902 (119909) (119909

119897)

3

119889119909

Ψ119902=

1

6(36119863

119909119909120578119887

2

2

1198976+

4119860119909119909120578119887

2

2

1198974+

36119863119909119909120578119887

2

2

1198976) 119897

3

+1

4(4119860

11990911990912057811988712

1198973+

2411986311990911990912057811988712

1198975

+24119863

119909119909120578119887

1

2

1198975) 119897

2

+1

2

119860119909119909120578119887

2

1

119897+

2119863119909119909120578119887

2

1

1198973+

2119863119909119909120578119887

2

1

1198973

(44)


120597119871

120597119902119899

minus119889

119889119905(

120597119871

120597 119902119899

) + 119876119889= 0 119899 = 1 2 6 (45)


= (1198801 119880

2 119881

1 119881

2

1198821119882



119889is the


as

119871 = 119879119890minus (119880

119894minus 119882

119911)

119876119889= minus

120597Ψ119902

120597 119902119899

(119899 = 1 sim 6)

(46)


1 119880

2

1

31198971198681198601+

1

41198971198681198602+

120578119887119860

1199091199091

119897+

120578119887119860

1199091199092

119897+

1198601199091199091198801

119897

+119860

1199091199091198802

119897+

2119860119909119909119881

2

1

31198972+

311986011990911990911988111198812

21198972+

9119860119909119909119881

2

2

101198972

+2119860

119909119909119882

2

1

31198972+

3119860119909119909119882

1119882

2

21198972+

9119860119909119909119882

2

2

101198972= 0

(47a)

1

41198971198681198601+

1

51198971198681198602+

120578119887119860

1199091199091

119897+

4120578119887119860

1199091199092

3119897

+119860

1199091199091198801

119897+4119860

1199091199091198802

3119897+119860

119909119909119881

2

1

1198972+12119860

11990911990911988111198812

51198972+3119860

119909119909119881

2

2

21198972

+119860

119909119909119882

2

1

1198972+

12119860119909119909119882

1119882

2

51198972+

3119860119909119909119882

2

2

21198972= 0

(47b)





1 119881

2119882

1119882

2 We take (47a)



2119882

1119882






2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)









119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198791198983

=1

2119898

119897int

119897

0

[∘

1205883

+ 120596 times (r3+ 120588

3)] sdot [

∘

1205883

+ 120596 times (r3+ 120588

3)] 119889119911

1198791198981

=1

2119898

119897int

0

minus119897

[∘

1205881

+ 120596 times (r1+ 120588

1)]

sdot [∘

1205881

+ 120596 times (r1+ 120588

1)] 119889119911

1198791198982

=1

2119898

119897int

119897

0

[∘

1205882

+ 120596 times (r2+ 120588

2)] sdot [

∘

1205882

+ 120596 times (r2+ 120588

2)] 119889119910

1198791198984

=1

2119898

119897int

0

minus119897

[∘

1205884

+ 120596 times (r4+ 120588

4)] sdot [

∘

1205884

+ 120596 times (r4+ 120588

4)] 119889119910

(17)


119879119898

=1

2120596 sdot J sdot 120596

+1

2119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+1

2119898

119897int

0

minus119897

2∘

1205881


1) sdot (120596 times 120588

1)


1) 119889119911

+1

2119898

119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

+1

2119898

119897int

0

minus119897

2∘

1205884


4) sdot (120596 times 120588

4)


4) + (120596 times r

4)

sdot (120596 times r4) 119889119910

(18)

where J = sum4

119895=1

J119898119895



119909

119869119910 119869

119911] inO

119874x119887y119887z119887


119879119901=

1

2119898

119901

10038161003816100381610038161003816r119901

10038161003816100381610038161003816

2

=1

2119898

119901[minus119897

1199011sin 120582

1+ 119897

119901sin 120582

1sin 120582

2( 120579 + sin120593)

minus 119897119901sin 120582

1cos 120582

2( sin 120579 + cos 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1cos 120582

2minus

2sin 120582

1sin 120582

2)

+ 119897119901cos 120582

1( sin 120579 + cos 120579 cos120593)

minus 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)]

2

+1

2119898

119901[119897119901(

1cos 120582

1sin 120582

2+

2sin 120582

1cos 120582

2)

+ 119897119901sin 120582

1cos 120582

2( cos 120579 minus sin 120579 cos120593)

minus 119897119901cos 120582

1( 120579 + sin120593)]

2

(19)


119879ℎ1

=1

2119898

ℎ1[ ( 120588

ℎ1119909+ 120579120588

ℎ1119911+ 120588

ℎ1119911sin120593

minus 119910ℎ1sin 120579 minus 119910

ℎ1cos 120579 cos120593)

2

]

+1

2119898

ℎ1[( 119910

ℎ1+ 120588

ℎ1119909sin 120579 + 120588

ℎ1119909cos 120579 cos120593

minus 120588ℎ1119911

cos 120579 + 120588ℎ1119911

sin 120579 cos120593)2]

+1

2119898

ℎ1[( 120588

ℎ1119911+ 119910

ℎ1cos 120579 minus 119910

ℎ1sin 120579 cos120593

minus 120579120588ℎ1119909

+ 120588ℎ1119909

sin120593)2

]

119879ℎ2

=1

2119898

ℎ2[( 120588

ℎ2119909+ 120579119911

ℎ2+ 119911

ℎ2sin120593 minus 120588

ℎ2119910sin 120579

minus 120588ℎ2119910

cos 120579 cos120593)2

]

+1

2119898

ℎ2[( 120588

ℎ2119910+ 120588

ℎ2119909sin 120579 + 120588

ℎ2119909cos 120579 cos120593

minus 119911ℎ2cos 120579 + 119911

ℎ2sin 120579 cos120593)

2

]

+1

2119898

ℎ2[(

ℎ2+ 120588

ℎ2119910cos 120579 minus 120588

ℎ2119910sin 120579 cos120593

minus 120579120588ℎ2119909

minus 120588ℎ2119909

sin120593)2

]

(20)


119892119891 payload 119880

119892119901 and


119880119892ℎ2

can be obtained as

119880119892119891

= minus

120583119898119891

1003816100381610038161003816R119874

1003816100381610038161003816

119880119892119901

= minus

120583119898119901

10038161003816100381610038161003816R119874+ r

119901

10038161003816100381610038161003816

119880119892ℎ1

= minus120583119898

ℎ1

1003816100381610038161003816R119874+ r

ℎ1

1003816100381610038161003816

119880119892ℎ2

= minus120583119898

ℎ2

1003816100381610038161003816R119874+ r

ℎ2

1003816100381610038161003816

(21)



119868

S = j119868= [1 0 0]

119879

119868

(22)



S = C119887119868[1 0 0]

119879

119868

= [119878119909

119878119910

119878119911]119879

119887

(23)


F119894= 120578119875119860

119862119894(S sdot n

119862119894)2 n

119862119894= 119865

119862119894cos2120572

119862119894n119862119894 (24)




119865119862119894


119862119894= i

119894


119862119894in 120587


n119862119894

= C119887119894[1 0 0]

119879

119894

= [119878119909119894

119878119910119894

119878119911119894]119879

119887

(25)


M119894= L

119894times F

119894 (26)




119894andF



119894and F


L1= [0 0 minus119897]

119879

L2= [0 119897 0]

119879

L3= [0 0 119897]

119879

L4= [0 minus119897 0]

119879

F119894= 120578119875119860

119862119894([119878

119909119878119910

119878119911]119887

sdot [119878119909119894

119878119910119894

119878119911119894]119879

119887

)

2

[119878119909119894

119878119910119894

119878119911119894]119879

119887

(27)


F119894= (119865

119862119894cos2120572

119862119894)C

119887119894[1 0 0]

119879

119894

= (119865119862119894cos2120572

119862119894) [119878

119909119894119878119910119894

119878119911119894]119879

119887

(28)

where 120572119862119894


M1198881

= (1198651198621cos2120572

1198621)[

[

0

0

minus119897

]

]

times (C1198871

[

[

1

0

0

]

]

)

= (1198651198621cos2120572

1198621)

times [

[

minus119897 cos1205931sin120595

1

minus119897 (cos 1205791cos120595

1minus sin 120579

1sin120593

1sin120595

1)

0

]

]

M1198882

= (1198651198622cos2120572

1198622)

times [

[

119897 (sin 1205792cos120595

2+ cos 120579

2sin120593

2sin120595

2)

0

minus119897 (cos 1205792cos120595

2minus sin 120579

2sin120593

2sin120595

2)

]

]

M1198883

= (1198651198623cos2120572

1198623)

times [

[

119897 cos1205933sin120595

3

119897 (cos 1205793cos120595

3minus sin 120579

3sin120593

3sin120595

3)

0

]

]

M1198884

= (1198651198624cos2120572

1198624)

times [

[

minus119897 (sin 1205794cos120595

4+ cos 120579

4sin120593

4sin120595

4)

0

119897 (cos 1205794cos120595

4minus sin 120579

4sin120593

4sin120595

4)

]

]

(29)


119889

119889119905(

120597119871

120597 119902119895

) minus120597119871

120597119902119895

= 119876119895 (30)




119871 = 119879 minus 119880 =

4

sum

119894=1

119879119898119894

+ 119879119901+

2

sum

119895=1

119879ℎ119895

minus (119880119892119891

+ 119880119892119901

+

2

sum

119895=1

119880119892ℎ119895

)

(31)

where 119879119898119894 119879

119901 and 119879



119892119901 and 119880




OOx

y

z

FCFCy

FCx

FCz

mh

q(x)


120590

120590e

120590120592

120590







119862119909 119865

119862119910 and 119865





120590 = 120590119890

+ 120590120592

= 119864120576 + 119888119887

120576 (32)

where 120590 120590119890 and 120590



expressed as 119864120578119887 120578




and s respectively

OO x

y The support beam

Z




119906 (119909 119910 119911 119905) = 1199060(119909 119905) minus 119910

120597V0

120597119909minus 119911

1205971199080

120597119909

V (119909 119910 119911 119905) = V0(119909 119905)

119908 (119909 119910 119911 119905) = 1199080(119909 119905)

(33)


0(119909 119905)

V0(119909 119905) and 119908




120576119909119909

=120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

120576119909119909

=120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909)

minus 1199111205972

0

1205971199092

120590119909119909

= 119864[120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

]

+ 119864120578119887(120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909) minus 119911

1205972

0

1205971199092

)

(34)




119880119894=

1

2int

119897

0

119860119909119909

[120597119906

0

120597119909+

1

2(120597V

0

120597119909)

2

+1

2(120597119908

0

120597119909)

2

]

2

+ 119863119909119909

1205972V

0

1205971199092

+ 119863119909119909

1205972

1199080

1205971199092

119889119909

(35)




119860119909119909

= int119860

119864119889119860 119863119909119909

= int119860

1198641199112

119889119860 = int119860

1198641199102

119889119860 (36)


Ψ119902=

1

2int

119897

0

[119860119909119909120578119887(120597

0

120597119909)

2

+ 119863119909119909120578119887(1205972V

0

1205971199092

)

2

+ 119863119909119909120578119887(1205972

0

1205971199092

)

2

]119889119909

(37)


119879119890=

1

2int

119897

0

[119868119860(120597119906

0

120597119905)

2

+ 119868119860(120597119908

0

120597119905)

2

+ 119868119863(1205972

1199080

120597119909120597119905)

2

+ 119868119860(120597V

0

120597119905)

2

+ 119868119863(

1205972V

0

120597119909120597119905)

2

]119889119909

(38)

where 119868119860and 119868


beam expressed as

119868119860= int

119860

120588119889119860 = 120588119860 119868119863= int

119860

1205881199112

119889119860 = int119860

1205881199102

119889119860 (39)


119862119911sdot 119908 (119909

119888 119905) + 119865

119862119910sdot V (119909

119888 119905) + 119898

ℎ119886ℎ119910

(119909ℎ 119905) V (119909

ℎ 119905)

+ 119898ℎ119886ℎ119911

(119909ℎ 119905) 119908 (119909

ℎ 119905) + int

119897

0

119902 (119909) V (119909 119905) 119889119909

(40)where 119909


119908(119909119888 119905) V(119909



119862119910are




ℎ] 119898

ℎis the mass

and 119886ℎ119910(119909

ℎ 119905) and 119886

ℎ119911(119909



ℎ 119905) and V(119909



0(119909 119905) and119908

0(119909 119905) can


1199060(119909 119905) =

infin

sum

119894=1

119880119894(119909)119880

119894(119905)

V0(119909 119905) =

infin

sum

119895=1

119881119895(119909) 119881

119895(119905)

1199080(119909 119905) =

infin

sum

119896=1

119882119896(119909)119882

119896(119905)

(41)

where 119880119894(119905) 119881

119895(119905) and 119882



119895(119909) and 119882

119896(119909) are the


119880119894(119909)

1003816100381610038161003816119909=0= 0 119881

119895(119909)

10038161003816100381610038161003816119909=0= 0 119882

119896(119909)

1003816100381610038161003816119909=0= 0

1198811015840

119895

(119909)10038161003816100381610038161003816119909=0

= 0 1198821015840

119896

(119909)10038161003816100381610038161003816119909=0

= 0

(42)


1198801(119909) =

119909

119897 119880

2(119909) = (

119909

119897)

2

1198811(119909) = (

119909

119897)

2

1198812(119909) = (

119909

119897)

3

1198821(119909) = (

119909

119897)

2

1198822(119909) = (

119909

119897)

3

(43)


119894 119879

119890119882

119911 and Ψ


119880119894=

9

2

119860119909119909

1198973(119881

2

2

+ 1198822

2

)2

+27

4

119860119909119909

1198973(119881

11198812+ 119882

1119882

2)

times (1198812

2

+ 1198822

2

)

+18

7

119860119909119909

1198973[(119881

2

1

+ 1198822

1

) (1198812

2

+ 1198822

2

) + (11988111198812+ 119882

1119882

2)2

]

+119860

119909119909

1198973[3

2119897119880

2(119881

2

2

+ 1198822

2

) + (1198812

1

+ 1198822

1

)

times (11988111198812+ 119882

1119882

2)]


+1

10

119860119909119909

1198973[9119880

1119897 (119881

2

2

+ 1198822

2

) + 241198802119897 (119881

11198812+ 119882

1119882

2)

+ 4 (1198812

1

+ 1198822

1

)2

]

+119860

119909119909

1198972[3

21198801(119881

11198812+ 119882

1119882

2) + 119880

2(119881

2

1

+ 1198822

1

)]

+6119863

119909119909119881

2

2

1198973+ 119860

119909119909(2119880

1119881

2

1

31198972+

21198801119882

2

1

31198972+

21198802

2

3119897)

+6119863

119909119909119882

2

2

1198973+

611986311990911990911988111198812

1198973+

11986011990911990911988011198802

119897+

6119863119909119909119882

1119882

2

1198973

+119860

119909119909119880

2

1

2119897+

2119863119909119909119881

2

1

1198973+

2119863119909119909119882

2

1

1198973

119879119890=

1

14(119897119868

119860

2

2

+ 119897119868119860

2

2

) +1

6(119897119868

119860

1

2+ 119897119868

11986012)

+1

10(119897119868

119860

2

2

+ 119897119868119860

2

1

+9119868

119863

2

2

119897+ 119897119868

119860

2

1

+9119868

119863

2

2

119897)

+1

4(119897119868

11986012+

6119868119863

1

2

119897+

611986811986312

119897)

+1

6(119897119868

119860

2

1

+4119868

119863

2

1

119897+

4119868119863

2

1

119897)

119882119911= 119865

119862119911sdot [119882

1(119905) + 119882

2(119905)] + 119865

119862119910sdot [119881

1(119905) + 119881

2(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119881

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119881

2(119905) +

6V3ℎ

1199052

1198973

sdot 2(119905) + (

Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198811(119905) + (

Vℎ119905

119897)

3

sdot 1198812(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119882

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119882

2(119905) +

6V3ℎ

1199052

1198973sdot

2(119905)

+ (Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198821(119905) + (

Vℎ119905

119897)

3

sdot 1198822(119905)]

+ 1198811(119905) sdot int

119897

0

119902 (119909) (119909

119897)

2

119889119909 + 1198812(119905)

sdot int

119897

0

119902 (119909) (119909

119897)

3

119889119909

Ψ119902=

1

6(36119863

119909119909120578119887

2

2

1198976+

4119860119909119909120578119887

2

2

1198974+

36119863119909119909120578119887

2

2

1198976) 119897

3

+1

4(4119860

11990911990912057811988712

1198973+

2411986311990911990912057811988712

1198975

+24119863

119909119909120578119887

1

2

1198975) 119897

2

+1

2

119860119909119909120578119887

2

1

119897+

2119863119909119909120578119887

2

1

1198973+

2119863119909119909120578119887

2

1

1198973

(44)


120597119871

120597119902119899

minus119889

119889119905(

120597119871

120597 119902119899

) + 119876119889= 0 119899 = 1 2 6 (45)


= (1198801 119880

2 119881

1 119881

2

1198821119882



119889is the


as

119871 = 119879119890minus (119880

119894minus 119882

119911)

119876119889= minus

120597Ψ119902

120597 119902119899

(119899 = 1 sim 6)

(46)


1 119880

2

1

31198971198681198601+

1

41198971198681198602+

120578119887119860

1199091199091

119897+

120578119887119860

1199091199092

119897+

1198601199091199091198801

119897

+119860

1199091199091198802

119897+

2119860119909119909119881

2

1

31198972+

311986011990911990911988111198812

21198972+

9119860119909119909119881

2

2

101198972

+2119860

119909119909119882

2

1

31198972+

3119860119909119909119882

1119882

2

21198972+

9119860119909119909119882

2

2

101198972= 0

(47a)

1

41198971198681198601+

1

51198971198681198602+

120578119887119860

1199091199091

119897+

4120578119887119860

1199091199092

3119897

+119860

1199091199091198801

119897+4119860

1199091199091198802

3119897+119860

119909119909119881

2

1

1198972+12119860

11990911990911988111198812

51198972+3119860

119909119909119881

2

2

21198972

+119860

119909119909119882

2

1

1198972+

12119860119909119909119882

1119882

2

51198972+

3119860119909119909119882

2

2

21198972= 0

(47b)





1 119881

2119882

1119882

2 We take (47a)



2119882

1119882






2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)









119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119868

S = j119868= [1 0 0]

119879

119868

(22)



S = C119887119868[1 0 0]

119879

119868

= [119878119909

119878119910

119878119911]119879

119887

(23)


F119894= 120578119875119860

119862119894(S sdot n

119862119894)2 n

119862119894= 119865

119862119894cos2120572

119862119894n119862119894 (24)




119865119862119894


119862119894= i

119894


119862119894in 120587


n119862119894

= C119887119894[1 0 0]

119879

119894

= [119878119909119894

119878119910119894

119878119911119894]119879

119887

(25)


M119894= L

119894times F

119894 (26)




119894andF



119894and F


L1= [0 0 minus119897]

119879

L2= [0 119897 0]

119879

L3= [0 0 119897]

119879

L4= [0 minus119897 0]

119879

F119894= 120578119875119860

119862119894([119878

119909119878119910

119878119911]119887

sdot [119878119909119894

119878119910119894

119878119911119894]119879

119887

)

2

[119878119909119894

119878119910119894

119878119911119894]119879

119887

(27)


F119894= (119865

119862119894cos2120572

119862119894)C

119887119894[1 0 0]

119879

119894

= (119865119862119894cos2120572

119862119894) [119878

119909119894119878119910119894

119878119911119894]119879

119887

(28)

where 120572119862119894


M1198881

= (1198651198621cos2120572

1198621)[

[

0

0

minus119897

]

]

times (C1198871

[

[

1

0

0

]

]

)

= (1198651198621cos2120572

1198621)

times [

[

minus119897 cos1205931sin120595

1

minus119897 (cos 1205791cos120595

1minus sin 120579

1sin120593

1sin120595

1)

0

]

]

M1198882

= (1198651198622cos2120572

1198622)

times [

[

119897 (sin 1205792cos120595

2+ cos 120579

2sin120593

2sin120595

2)

0

minus119897 (cos 1205792cos120595

2minus sin 120579

2sin120593

2sin120595

2)

]

]

M1198883

= (1198651198623cos2120572

1198623)

times [

[

119897 cos1205933sin120595

3

119897 (cos 1205793cos120595

3minus sin 120579

3sin120593

3sin120595

3)

0

]

]

M1198884

= (1198651198624cos2120572

1198624)

times [

[

minus119897 (sin 1205794cos120595

4+ cos 120579

4sin120593

4sin120595

4)

0

119897 (cos 1205794cos120595

4minus sin 120579

4sin120593

4sin120595

4)

]

]

(29)


119889

119889119905(

120597119871

120597 119902119895

) minus120597119871

120597119902119895

= 119876119895 (30)




119871 = 119879 minus 119880 =

4

sum

119894=1

119879119898119894

+ 119879119901+

2

sum

119895=1

119879ℎ119895

minus (119880119892119891

+ 119880119892119901

+

2

sum

119895=1

119880119892ℎ119895

)

(31)

where 119879119898119894 119879

119901 and 119879



119892119901 and 119880




OOx

y

z

FCFCy

FCx

FCz

mh

q(x)


120590

120590e

120590120592

120590







119862119909 119865

119862119910 and 119865





120590 = 120590119890

+ 120590120592

= 119864120576 + 119888119887

120576 (32)

where 120590 120590119890 and 120590



expressed as 119864120578119887 120578




and s respectively

OO x

y The support beam

Z




119906 (119909 119910 119911 119905) = 1199060(119909 119905) minus 119910

120597V0

120597119909minus 119911

1205971199080

120597119909

V (119909 119910 119911 119905) = V0(119909 119905)

119908 (119909 119910 119911 119905) = 1199080(119909 119905)

(33)


0(119909 119905)

V0(119909 119905) and 119908




120576119909119909

=120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

120576119909119909

=120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909)

minus 1199111205972

0

1205971199092

120590119909119909

= 119864[120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

]

+ 119864120578119887(120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909) minus 119911

1205972

0

1205971199092

)

(34)




119880119894=

1

2int

119897

0

119860119909119909

[120597119906

0

120597119909+

1

2(120597V

0

120597119909)

2

+1

2(120597119908

0

120597119909)

2

]

2

+ 119863119909119909

1205972V

0

1205971199092

+ 119863119909119909

1205972

1199080

1205971199092

119889119909

(35)




119860119909119909

= int119860

119864119889119860 119863119909119909

= int119860

1198641199112

119889119860 = int119860

1198641199102

119889119860 (36)


Ψ119902=

1

2int

119897

0

[119860119909119909120578119887(120597

0

120597119909)

2

+ 119863119909119909120578119887(1205972V

0

1205971199092

)

2

+ 119863119909119909120578119887(1205972

0

1205971199092

)

2

]119889119909

(37)


119879119890=

1

2int

119897

0

[119868119860(120597119906

0

120597119905)

2

+ 119868119860(120597119908

0

120597119905)

2

+ 119868119863(1205972

1199080

120597119909120597119905)

2

+ 119868119860(120597V

0

120597119905)

2

+ 119868119863(

1205972V

0

120597119909120597119905)

2

]119889119909

(38)

where 119868119860and 119868


beam expressed as

119868119860= int

119860

120588119889119860 = 120588119860 119868119863= int

119860

1205881199112

119889119860 = int119860

1205881199102

119889119860 (39)


119862119911sdot 119908 (119909

119888 119905) + 119865

119862119910sdot V (119909

119888 119905) + 119898

ℎ119886ℎ119910

(119909ℎ 119905) V (119909

ℎ 119905)

+ 119898ℎ119886ℎ119911

(119909ℎ 119905) 119908 (119909

ℎ 119905) + int

119897

0

119902 (119909) V (119909 119905) 119889119909

(40)where 119909


119908(119909119888 119905) V(119909



119862119910are




ℎ] 119898

ℎis the mass

and 119886ℎ119910(119909

ℎ 119905) and 119886

ℎ119911(119909



ℎ 119905) and V(119909



0(119909 119905) and119908

0(119909 119905) can


1199060(119909 119905) =

infin

sum

119894=1

119880119894(119909)119880

119894(119905)

V0(119909 119905) =

infin

sum

119895=1

119881119895(119909) 119881

119895(119905)

1199080(119909 119905) =

infin

sum

119896=1

119882119896(119909)119882

119896(119905)

(41)

where 119880119894(119905) 119881

119895(119905) and 119882



119895(119909) and 119882

119896(119909) are the


119880119894(119909)

1003816100381610038161003816119909=0= 0 119881

119895(119909)

10038161003816100381610038161003816119909=0= 0 119882

119896(119909)

1003816100381610038161003816119909=0= 0

1198811015840

119895

(119909)10038161003816100381610038161003816119909=0

= 0 1198821015840

119896

(119909)10038161003816100381610038161003816119909=0

= 0

(42)


1198801(119909) =

119909

119897 119880

2(119909) = (

119909

119897)

2

1198811(119909) = (

119909

119897)

2

1198812(119909) = (

119909

119897)

3

1198821(119909) = (

119909

119897)

2

1198822(119909) = (

119909

119897)

3

(43)


119894 119879

119890119882

119911 and Ψ


119880119894=

9

2

119860119909119909

1198973(119881

2

2

+ 1198822

2

)2

+27

4

119860119909119909

1198973(119881

11198812+ 119882

1119882

2)

times (1198812

2

+ 1198822

2

)

+18

7

119860119909119909

1198973[(119881

2

1

+ 1198822

1

) (1198812

2

+ 1198822

2

) + (11988111198812+ 119882

1119882

2)2

]

+119860

119909119909

1198973[3

2119897119880

2(119881

2

2

+ 1198822

2

) + (1198812

1

+ 1198822

1

)

times (11988111198812+ 119882

1119882

2)]


+1

10

119860119909119909

1198973[9119880

1119897 (119881

2

2

+ 1198822

2

) + 241198802119897 (119881

11198812+ 119882

1119882

2)

+ 4 (1198812

1

+ 1198822

1

)2

]

+119860

119909119909

1198972[3

21198801(119881

11198812+ 119882

1119882

2) + 119880

2(119881

2

1

+ 1198822

1

)]

+6119863

119909119909119881

2

2

1198973+ 119860

119909119909(2119880

1119881

2

1

31198972+

21198801119882

2

1

31198972+

21198802

2

3119897)

+6119863

119909119909119882

2

2

1198973+

611986311990911990911988111198812

1198973+

11986011990911990911988011198802

119897+

6119863119909119909119882

1119882

2

1198973

+119860

119909119909119880

2

1

2119897+

2119863119909119909119881

2

1

1198973+

2119863119909119909119882

2

1

1198973

119879119890=

1

14(119897119868

119860

2

2

+ 119897119868119860

2

2

) +1

6(119897119868

119860

1

2+ 119897119868

11986012)

+1

10(119897119868

119860

2

2

+ 119897119868119860

2

1

+9119868

119863

2

2

119897+ 119897119868

119860

2

1

+9119868

119863

2

2

119897)

+1

4(119897119868

11986012+

6119868119863

1

2

119897+

611986811986312

119897)

+1

6(119897119868

119860

2

1

+4119868

119863

2

1

119897+

4119868119863

2

1

119897)

119882119911= 119865

119862119911sdot [119882

1(119905) + 119882

2(119905)] + 119865

119862119910sdot [119881

1(119905) + 119881

2(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119881

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119881

2(119905) +

6V3ℎ

1199052

1198973

sdot 2(119905) + (

Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198811(119905) + (

Vℎ119905

119897)

3

sdot 1198812(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119882

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119882

2(119905) +

6V3ℎ

1199052

1198973sdot

2(119905)

+ (Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198821(119905) + (

Vℎ119905

119897)

3

sdot 1198822(119905)]

+ 1198811(119905) sdot int

119897

0

119902 (119909) (119909

119897)

2

119889119909 + 1198812(119905)

sdot int

119897

0

119902 (119909) (119909

119897)

3

119889119909

Ψ119902=

1

6(36119863

119909119909120578119887

2

2

1198976+

4119860119909119909120578119887

2

2

1198974+

36119863119909119909120578119887

2

2

1198976) 119897

3

+1

4(4119860

11990911990912057811988712

1198973+

2411986311990911990912057811988712

1198975

+24119863

119909119909120578119887

1

2

1198975) 119897

2

+1

2

119860119909119909120578119887

2

1

119897+

2119863119909119909120578119887

2

1

1198973+

2119863119909119909120578119887

2

1

1198973

(44)


120597119871

120597119902119899

minus119889

119889119905(

120597119871

120597 119902119899

) + 119876119889= 0 119899 = 1 2 6 (45)


= (1198801 119880

2 119881

1 119881

2

1198821119882



119889is the


as

119871 = 119879119890minus (119880

119894minus 119882

119911)

119876119889= minus

120597Ψ119902

120597 119902119899

(119899 = 1 sim 6)

(46)


1 119880

2

1

31198971198681198601+

1

41198971198681198602+

120578119887119860

1199091199091

119897+

120578119887119860

1199091199092

119897+

1198601199091199091198801

119897

+119860

1199091199091198802

119897+

2119860119909119909119881

2

1

31198972+

311986011990911990911988111198812

21198972+

9119860119909119909119881

2

2

101198972

+2119860

119909119909119882

2

1

31198972+

3119860119909119909119882

1119882

2

21198972+

9119860119909119909119882

2

2

101198972= 0

(47a)

1

41198971198681198601+

1

51198971198681198602+

120578119887119860

1199091199091

119897+

4120578119887119860

1199091199092

3119897

+119860

1199091199091198801

119897+4119860

1199091199091198802

3119897+119860

119909119909119881

2

1

1198972+12119860

11990911990911988111198812

51198972+3119860

119909119909119881

2

2

21198972

+119860

119909119909119882

2

1

1198972+

12119860119909119909119882

1119882

2

51198972+

3119860119909119909119882

2

2

21198972= 0

(47b)





1 119881

2119882

1119882

2 We take (47a)



2119882

1119882






2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)









119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










OOx

y

z

FCFCy

FCx

FCz

mh

q(x)


120590

120590e

120590120592

120590







119862119909 119865

119862119910 and 119865





120590 = 120590119890

+ 120590120592

= 119864120576 + 119888119887

120576 (32)

where 120590 120590119890 and 120590



expressed as 119864120578119887 120578




and s respectively

OO x

y The support beam

Z




119906 (119909 119910 119911 119905) = 1199060(119909 119905) minus 119910

120597V0

120597119909minus 119911

1205971199080

120597119909

V (119909 119910 119911 119905) = V0(119909 119905)

119908 (119909 119910 119911 119905) = 1199080(119909 119905)

(33)


0(119909 119905)

V0(119909 119905) and 119908




120576119909119909

=120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

120576119909119909

=120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909)

minus 1199111205972

0

1205971199092

120590119909119909

= 119864[120597119906

0

120597119909+1

2(120597V

0

120597119909)

2

minus 1199101205972V

0

1205971199092

+1

2(120597119908

0

120597119909)

2

minus 1199111205972

1199080

1205971199092

]

+ 119864120578119887(120597

0

120597119909+(

120597V0

120597119909)(

120597V0

120597119909) minus 119910

1205972V

0

1205971199092

+(120597119908

0

120597119909)(

1205970

120597119909) minus 119911

1205972

0

1205971199092

)

(34)




119880119894=

1

2int

119897

0

119860119909119909

[120597119906

0

120597119909+

1

2(120597V

0

120597119909)

2

+1

2(120597119908

0

120597119909)

2

]

2

+ 119863119909119909

1205972V

0

1205971199092

+ 119863119909119909

1205972

1199080

1205971199092

119889119909

(35)




119860119909119909

= int119860

119864119889119860 119863119909119909

= int119860

1198641199112

119889119860 = int119860

1198641199102

119889119860 (36)


Ψ119902=

1

2int

119897

0

[119860119909119909120578119887(120597

0

120597119909)

2

+ 119863119909119909120578119887(1205972V

0

1205971199092

)

2

+ 119863119909119909120578119887(1205972

0

1205971199092

)

2

]119889119909

(37)


119879119890=

1

2int

119897

0

[119868119860(120597119906

0

120597119905)

2

+ 119868119860(120597119908

0

120597119905)

2

+ 119868119863(1205972

1199080

120597119909120597119905)

2

+ 119868119860(120597V

0

120597119905)

2

+ 119868119863(

1205972V

0

120597119909120597119905)

2

]119889119909

(38)

where 119868119860and 119868


beam expressed as

119868119860= int

119860

120588119889119860 = 120588119860 119868119863= int

119860

1205881199112

119889119860 = int119860

1205881199102

119889119860 (39)


119862119911sdot 119908 (119909

119888 119905) + 119865

119862119910sdot V (119909

119888 119905) + 119898

ℎ119886ℎ119910

(119909ℎ 119905) V (119909

ℎ 119905)

+ 119898ℎ119886ℎ119911

(119909ℎ 119905) 119908 (119909

ℎ 119905) + int

119897

0

119902 (119909) V (119909 119905) 119889119909

(40)where 119909


119908(119909119888 119905) V(119909



119862119910are




ℎ] 119898

ℎis the mass

and 119886ℎ119910(119909

ℎ 119905) and 119886

ℎ119911(119909



ℎ 119905) and V(119909



0(119909 119905) and119908

0(119909 119905) can


1199060(119909 119905) =

infin

sum

119894=1

119880119894(119909)119880

119894(119905)

V0(119909 119905) =

infin

sum

119895=1

119881119895(119909) 119881

119895(119905)

1199080(119909 119905) =

infin

sum

119896=1

119882119896(119909)119882

119896(119905)

(41)

where 119880119894(119905) 119881

119895(119905) and 119882



119895(119909) and 119882

119896(119909) are the


119880119894(119909)

1003816100381610038161003816119909=0= 0 119881

119895(119909)

10038161003816100381610038161003816119909=0= 0 119882

119896(119909)

1003816100381610038161003816119909=0= 0

1198811015840

119895

(119909)10038161003816100381610038161003816119909=0

= 0 1198821015840

119896

(119909)10038161003816100381610038161003816119909=0

= 0

(42)


1198801(119909) =

119909

119897 119880

2(119909) = (

119909

119897)

2

1198811(119909) = (

119909

119897)

2

1198812(119909) = (

119909

119897)

3

1198821(119909) = (

119909

119897)

2

1198822(119909) = (

119909

119897)

3

(43)


119894 119879

119890119882

119911 and Ψ


119880119894=

9

2

119860119909119909

1198973(119881

2

2

+ 1198822

2

)2

+27

4

119860119909119909

1198973(119881

11198812+ 119882

1119882

2)

times (1198812

2

+ 1198822

2

)

+18

7

119860119909119909

1198973[(119881

2

1

+ 1198822

1

) (1198812

2

+ 1198822

2

) + (11988111198812+ 119882

1119882

2)2

]

+119860

119909119909

1198973[3

2119897119880

2(119881

2

2

+ 1198822

2

) + (1198812

1

+ 1198822

1

)

times (11988111198812+ 119882

1119882

2)]


+1

10

119860119909119909

1198973[9119880

1119897 (119881

2

2

+ 1198822

2

) + 241198802119897 (119881

11198812+ 119882

1119882

2)

+ 4 (1198812

1

+ 1198822

1

)2

]

+119860

119909119909

1198972[3

21198801(119881

11198812+ 119882

1119882

2) + 119880

2(119881

2

1

+ 1198822

1

)]

+6119863

119909119909119881

2

2

1198973+ 119860

119909119909(2119880

1119881

2

1

31198972+

21198801119882

2

1

31198972+

21198802

2

3119897)

+6119863

119909119909119882

2

2

1198973+

611986311990911990911988111198812

1198973+

11986011990911990911988011198802

119897+

6119863119909119909119882

1119882

2

1198973

+119860

119909119909119880

2

1

2119897+

2119863119909119909119881

2

1

1198973+

2119863119909119909119882

2

1

1198973

119879119890=

1

14(119897119868

119860

2

2

+ 119897119868119860

2

2

) +1

6(119897119868

119860

1

2+ 119897119868

11986012)

+1

10(119897119868

119860

2

2

+ 119897119868119860

2

1

+9119868

119863

2

2

119897+ 119897119868

119860

2

1

+9119868

119863

2

2

119897)

+1

4(119897119868

11986012+

6119868119863

1

2

119897+

611986811986312

119897)

+1

6(119897119868

119860

2

1

+4119868

119863

2

1

119897+

4119868119863

2

1

119897)

119882119911= 119865

119862119911sdot [119882

1(119905) + 119882

2(119905)] + 119865

119862119910sdot [119881

1(119905) + 119881

2(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119881

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119881

2(119905) +

6V3ℎ

1199052

1198973

sdot 2(119905) + (

Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198811(119905) + (

Vℎ119905

119897)

3

sdot 1198812(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119882

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119882

2(119905) +

6V3ℎ

1199052

1198973sdot

2(119905)

+ (Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198821(119905) + (

Vℎ119905

119897)

3

sdot 1198822(119905)]

+ 1198811(119905) sdot int

119897

0

119902 (119909) (119909

119897)

2

119889119909 + 1198812(119905)

sdot int

119897

0

119902 (119909) (119909

119897)

3

119889119909

Ψ119902=

1

6(36119863

119909119909120578119887

2

2

1198976+

4119860119909119909120578119887

2

2

1198974+

36119863119909119909120578119887

2

2

1198976) 119897

3

+1

4(4119860

11990911990912057811988712

1198973+

2411986311990911990912057811988712

1198975

+24119863

119909119909120578119887

1

2

1198975) 119897

2

+1

2

119860119909119909120578119887

2

1

119897+

2119863119909119909120578119887

2

1

1198973+

2119863119909119909120578119887

2

1

1198973

(44)


120597119871

120597119902119899

minus119889

119889119905(

120597119871

120597 119902119899

) + 119876119889= 0 119899 = 1 2 6 (45)


= (1198801 119880

2 119881

1 119881

2

1198821119882



119889is the


as

119871 = 119879119890minus (119880

119894minus 119882

119911)

119876119889= minus

120597Ψ119902

120597 119902119899

(119899 = 1 sim 6)

(46)


1 119880

2

1

31198971198681198601+

1

41198971198681198602+

120578119887119860

1199091199091

119897+

120578119887119860

1199091199092

119897+

1198601199091199091198801

119897

+119860

1199091199091198802

119897+

2119860119909119909119881

2

1

31198972+

311986011990911990911988111198812

21198972+

9119860119909119909119881

2

2

101198972

+2119860

119909119909119882

2

1

31198972+

3119860119909119909119882

1119882

2

21198972+

9119860119909119909119882

2

2

101198972= 0

(47a)

1

41198971198681198601+

1

51198971198681198602+

120578119887119860

1199091199091

119897+

4120578119887119860

1199091199092

3119897

+119860

1199091199091198801

119897+4119860

1199091199091198802

3119897+119860

119909119909119881

2

1

1198972+12119860

11990911990911988111198812

51198972+3119860

119909119909119881

2

2

21198972

+119860

119909119909119882

2

1

1198972+

12119860119909119909119882

1119882

2

51198972+

3119860119909119909119882

2

2

21198972= 0

(47b)





1 119881

2119882

1119882

2 We take (47a)



2119882

1119882






2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)









119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119880119894=

1

2int

119897

0

119860119909119909

[120597119906

0

120597119909+

1

2(120597V

0

120597119909)

2

+1

2(120597119908

0

120597119909)

2

]

2

+ 119863119909119909

1205972V

0

1205971199092

+ 119863119909119909

1205972

1199080

1205971199092

119889119909

(35)




119860119909119909

= int119860

119864119889119860 119863119909119909

= int119860

1198641199112

119889119860 = int119860

1198641199102

119889119860 (36)


Ψ119902=

1

2int

119897

0

[119860119909119909120578119887(120597

0

120597119909)

2

+ 119863119909119909120578119887(1205972V

0

1205971199092

)

2

+ 119863119909119909120578119887(1205972

0

1205971199092

)

2

]119889119909

(37)


119879119890=

1

2int

119897

0

[119868119860(120597119906

0

120597119905)

2

+ 119868119860(120597119908

0

120597119905)

2

+ 119868119863(1205972

1199080

120597119909120597119905)

2

+ 119868119860(120597V

0

120597119905)

2

+ 119868119863(

1205972V

0

120597119909120597119905)

2

]119889119909

(38)

where 119868119860and 119868


beam expressed as

119868119860= int

119860

120588119889119860 = 120588119860 119868119863= int

119860

1205881199112

119889119860 = int119860

1205881199102

119889119860 (39)


119862119911sdot 119908 (119909

119888 119905) + 119865

119862119910sdot V (119909

119888 119905) + 119898

ℎ119886ℎ119910

(119909ℎ 119905) V (119909

ℎ 119905)

+ 119898ℎ119886ℎ119911

(119909ℎ 119905) 119908 (119909

ℎ 119905) + int

119897

0

119902 (119909) V (119909 119905) 119889119909

(40)where 119909


119908(119909119888 119905) V(119909



119862119910are




ℎ] 119898

ℎis the mass

and 119886ℎ119910(119909

ℎ 119905) and 119886

ℎ119911(119909



ℎ 119905) and V(119909



0(119909 119905) and119908

0(119909 119905) can


1199060(119909 119905) =

infin

sum

119894=1

119880119894(119909)119880

119894(119905)

V0(119909 119905) =

infin

sum

119895=1

119881119895(119909) 119881

119895(119905)

1199080(119909 119905) =

infin

sum

119896=1

119882119896(119909)119882

119896(119905)

(41)

where 119880119894(119905) 119881

119895(119905) and 119882



119895(119909) and 119882

119896(119909) are the


119880119894(119909)

1003816100381610038161003816119909=0= 0 119881

119895(119909)

10038161003816100381610038161003816119909=0= 0 119882

119896(119909)

1003816100381610038161003816119909=0= 0

1198811015840

119895

(119909)10038161003816100381610038161003816119909=0

= 0 1198821015840

119896

(119909)10038161003816100381610038161003816119909=0

= 0

(42)


1198801(119909) =

119909

119897 119880

2(119909) = (

119909

119897)

2

1198811(119909) = (

119909

119897)

2

1198812(119909) = (

119909

119897)

3

1198821(119909) = (

119909

119897)

2

1198822(119909) = (

119909

119897)

3

(43)


119894 119879

119890119882

119911 and Ψ


119880119894=

9

2

119860119909119909

1198973(119881

2

2

+ 1198822

2

)2

+27

4

119860119909119909

1198973(119881

11198812+ 119882

1119882

2)

times (1198812

2

+ 1198822

2

)

+18

7

119860119909119909

1198973[(119881

2

1

+ 1198822

1

) (1198812

2

+ 1198822

2

) + (11988111198812+ 119882

1119882

2)2

]

+119860

119909119909

1198973[3

2119897119880

2(119881

2

2

+ 1198822

2

) + (1198812

1

+ 1198822

1

)

times (11988111198812+ 119882

1119882

2)]


+1

10

119860119909119909

1198973[9119880

1119897 (119881

2

2

+ 1198822

2

) + 241198802119897 (119881

11198812+ 119882

1119882

2)

+ 4 (1198812

1

+ 1198822

1

)2

]

+119860

119909119909

1198972[3

21198801(119881

11198812+ 119882

1119882

2) + 119880

2(119881

2

1

+ 1198822

1

)]

+6119863

119909119909119881

2

2

1198973+ 119860

119909119909(2119880

1119881

2

1

31198972+

21198801119882

2

1

31198972+

21198802

2

3119897)

+6119863

119909119909119882

2

2

1198973+

611986311990911990911988111198812

1198973+

11986011990911990911988011198802

119897+

6119863119909119909119882

1119882

2

1198973

+119860

119909119909119880

2

1

2119897+

2119863119909119909119881

2

1

1198973+

2119863119909119909119882

2

1

1198973

119879119890=

1

14(119897119868

119860

2

2

+ 119897119868119860

2

2

) +1

6(119897119868

119860

1

2+ 119897119868

11986012)

+1

10(119897119868

119860

2

2

+ 119897119868119860

2

1

+9119868

119863

2

2

119897+ 119897119868

119860

2

1

+9119868

119863

2

2

119897)

+1

4(119897119868

11986012+

6119868119863

1

2

119897+

611986811986312

119897)

+1

6(119897119868

119860

2

1

+4119868

119863

2

1

119897+

4119868119863

2

1

119897)

119882119911= 119865

119862119911sdot [119882

1(119905) + 119882

2(119905)] + 119865

119862119910sdot [119881

1(119905) + 119881

2(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119881

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119881

2(119905) +

6V3ℎ

1199052

1198973

sdot 2(119905) + (

Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198811(119905) + (

Vℎ119905

119897)

3

sdot 1198812(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119882

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119882

2(119905) +

6V3ℎ

1199052

1198973sdot

2(119905)

+ (Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198821(119905) + (

Vℎ119905

119897)

3

sdot 1198822(119905)]

+ 1198811(119905) sdot int

119897

0

119902 (119909) (119909

119897)

2

119889119909 + 1198812(119905)

sdot int

119897

0

119902 (119909) (119909

119897)

3

119889119909

Ψ119902=

1

6(36119863

119909119909120578119887

2

2

1198976+

4119860119909119909120578119887

2

2

1198974+

36119863119909119909120578119887

2

2

1198976) 119897

3

+1

4(4119860

11990911990912057811988712

1198973+

2411986311990911990912057811988712

1198975

+24119863

119909119909120578119887

1

2

1198975) 119897

2

+1

2

119860119909119909120578119887

2

1

119897+

2119863119909119909120578119887

2

1

1198973+

2119863119909119909120578119887

2

1

1198973

(44)


120597119871

120597119902119899

minus119889

119889119905(

120597119871

120597 119902119899

) + 119876119889= 0 119899 = 1 2 6 (45)


= (1198801 119880

2 119881

1 119881

2

1198821119882



119889is the


as

119871 = 119879119890minus (119880

119894minus 119882

119911)

119876119889= minus

120597Ψ119902

120597 119902119899

(119899 = 1 sim 6)

(46)


1 119880

2

1

31198971198681198601+

1

41198971198681198602+

120578119887119860

1199091199091

119897+

120578119887119860

1199091199092

119897+

1198601199091199091198801

119897

+119860

1199091199091198802

119897+

2119860119909119909119881

2

1

31198972+

311986011990911990911988111198812

21198972+

9119860119909119909119881

2

2

101198972

+2119860

119909119909119882

2

1

31198972+

3119860119909119909119882

1119882

2

21198972+

9119860119909119909119882

2

2

101198972= 0

(47a)

1

41198971198681198601+

1

51198971198681198602+

120578119887119860

1199091199091

119897+

4120578119887119860

1199091199092

3119897

+119860

1199091199091198801

119897+4119860

1199091199091198802

3119897+119860

119909119909119881

2

1

1198972+12119860

11990911990911988111198812

51198972+3119860

119909119909119881

2

2

21198972

+119860

119909119909119882

2

1

1198972+

12119860119909119909119882

1119882

2

51198972+

3119860119909119909119882

2

2

21198972= 0

(47b)





1 119881

2119882

1119882

2 We take (47a)



2119882

1119882






2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)









119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+1

10

119860119909119909

1198973[9119880

1119897 (119881

2

2

+ 1198822

2

) + 241198802119897 (119881

11198812+ 119882

1119882

2)

+ 4 (1198812

1

+ 1198822

1

)2

]

+119860

119909119909

1198972[3

21198801(119881

11198812+ 119882

1119882

2) + 119880

2(119881

2

1

+ 1198822

1

)]

+6119863

119909119909119881

2

2

1198973+ 119860

119909119909(2119880

1119881

2

1

31198972+

21198801119882

2

1

31198972+

21198802

2

3119897)

+6119863

119909119909119882

2

2

1198973+

611986311990911990911988111198812

1198973+

11986011990911990911988011198802

119897+

6119863119909119909119882

1119882

2

1198973

+119860

119909119909119880

2

1

2119897+

2119863119909119909119881

2

1

1198973+

2119863119909119909119882

2

1

1198973

119879119890=

1

14(119897119868

119860

2

2

+ 119897119868119860

2

2

) +1

6(119897119868

119860

1

2+ 119897119868

11986012)

+1

10(119897119868

119860

2

2

+ 119897119868119860

2

1

+9119868

119863

2

2

119897+ 119897119868

119860

2

1

+9119868

119863

2

2

119897)

+1

4(119897119868

11986012+

6119868119863

1

2

119897+

611986811986312

119897)

+1

6(119897119868

119860

2

1

+4119868

119863

2

1

119897+

4119868119863

2

1

119897)

119882119911= 119865

119862119911sdot [119882

1(119905) + 119882

2(119905)] + 119865

119862119910sdot [119881

1(119905) + 119881

2(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119881

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119881

2(119905) +

6V3ℎ

1199052

1198973

sdot 2(119905) + (

Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198811(119905) + (

Vℎ119905

119897)

3

sdot 1198812(119905)]

+ 119898ℎ[2V2

ℎ

1198972sdot 119882

1(119905) +

4V2ℎ

119905

1198972sdot

1(119905) + (

Vℎ119905

119897)

2

sdot 1(119905)

+6V3

ℎ

119905

1198973sdot 119882

2(119905) +

6V3ℎ

1199052

1198973sdot

2(119905)

+ (Vℎ119905

119897)

3

sdot 2(119905)]

times [(Vℎ119905

119897)

2

sdot 1198821(119905) + (

Vℎ119905

119897)

3

sdot 1198822(119905)]

+ 1198811(119905) sdot int

119897

0

119902 (119909) (119909

119897)

2

119889119909 + 1198812(119905)

sdot int

119897

0

119902 (119909) (119909

119897)

3

119889119909

Ψ119902=

1

6(36119863

119909119909120578119887

2

2

1198976+

4119860119909119909120578119887

2

2

1198974+

36119863119909119909120578119887

2

2

1198976) 119897

3

+1

4(4119860

11990911990912057811988712

1198973+

2411986311990911990912057811988712

1198975

+24119863

119909119909120578119887

1

2

1198975) 119897

2

+1

2

119860119909119909120578119887

2

1

119897+

2119863119909119909120578119887

2

1

1198973+

2119863119909119909120578119887

2

1

1198973

(44)


120597119871

120597119902119899

minus119889

119889119905(

120597119871

120597 119902119899

) + 119876119889= 0 119899 = 1 2 6 (45)


= (1198801 119880

2 119881

1 119881

2

1198821119882



119889is the


as

119871 = 119879119890minus (119880

119894minus 119882

119911)

119876119889= minus

120597Ψ119902

120597 119902119899

(119899 = 1 sim 6)

(46)


1 119880

2

1

31198971198681198601+

1

41198971198681198602+

120578119887119860

1199091199091

119897+

120578119887119860

1199091199092

119897+

1198601199091199091198801

119897

+119860

1199091199091198802

119897+

2119860119909119909119881

2

1

31198972+

311986011990911990911988111198812

21198972+

9119860119909119909119881

2

2

101198972

+2119860

119909119909119882

2

1

31198972+

3119860119909119909119882

1119882

2

21198972+

9119860119909119909119882

2

2

101198972= 0

(47a)

1

41198971198681198601+

1

51198971198681198602+

120578119887119860

1199091199091

119897+

4120578119887119860

1199091199092

3119897

+119860

1199091199091198801

119897+4119860

1199091199091198802

3119897+119860

119909119909119881

2

1

1198972+12119860

11990911990911988111198812

51198972+3119860

119909119909119881

2

2

21198972

+119860

119909119909119882

2

1

1198972+

12119860119909119909119882

1119882

2

51198972+

3119860119909119909119882

2

2

21198972= 0

(47b)





1 119881

2119882

1119882

2 We take (47a)



2119882

1119882






2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)









119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 119881

2119882

1119882

2 We take (47a)



2119882

1119882






2and119882

1 119882

2

can be obtained as

4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602minus

119898ℎV5ℎ

1199055

2

1198975+

31198681198632

2119897

+4119898

ℎV4ℎ

1199053

1

1198974+

4120578119887119863

1199091199091

1198973minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

1199091199092

1198973

+8119898

ℎV4ℎ

1199052

1198811

1198974+

41198631199091199091198811

1198973+

8119898ℎV5ℎ

1199053

1198812

1198975+

61198631199091199091198812

1198973

+27119860

1199091199091198812119882

2

2

81198973+

541198601199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973

+6119860

119909119909119881

2

1

1198812

1198973+

21198601199091199091198812119882

2

1

1198973+

1211986011990911990911988021198812

51198972

+8119860

1199091199091198811119882

2

1

51198973+

311986011990911990911988011198812

21198972+

211986011990911990911988021198811

1198972

+8119860

119909119909119881

3

1

51198973+

27119860119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973

+4119860

1199091199091198811119882

1119882

2

1198973+

411986011990911990911988011198811

31198972minus 119865

119888119910(119905) minus 05119865

025119865= 0

1

61198971198681198601+

3

2

1198681198631

119897minus

119898ℎV5ℎ

1199055

1

1198975+

1

71198971198681198602+

9

5

1198681198632

119897

+6120578

119887119863

1199091199091

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

12120578119887119863

1199091199092

1198973+

6119898ℎV6ℎ

1199055

2

1198976

+6119863

1199091199091198811

1198973+

16119898ℎV5ℎ

1199053

1198811

1198975+

121198631199091199091198812

1198973+

18119898ℎV6ℎ

1199054

1198812

1198976

+9119860

1199091199091198812119882

2

2

21198973+

811198601199091199091198811119881

2

2

81198973+

311986011990911990911988011198811

21198972

+27119860

1199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973+

181198601199091199091198812119882

2

1

71198973

+3119860

11990911990911988021198812

1198972+

21198601199091199091198811119882

2

1

1198973+

911986011990911990911988011198812

51198972+

1211986011990911990911988021198811

51198972

+9119860

119909119909119881

3

2

21198973+

2119860119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973

+36119860

1199091199091198811119882

1119882

2

71198973minus 04119865

025119865minus 119865

119888119910(119905) = 0

4

3

119868119863

1

119897+

1

5119897119868119860

1+

3

2

119868119863

2

119897+

1

6119897119868119860

2minus

119898ℎV5ℎ

1199055

2

1198975

+4120578

119887119863

119909119909

1

1198973+

4119898ℎV4ℎ

1199053

1

1198974minus

2119898ℎV5ℎ

1199054

2

1198975+

6120578119887119863

119909119909

2

1198973

+4119863

119909119909119882

1

1198973+

8119898ℎV4ℎ

1199052

1198821

1198974+

8119898ℎV5ℎ

1199053

1198822

1198975+

6119863119909119909119882

2

1198973

+36119860

119909119909119882

211988111198812

71198973+

4119860119909119909119882

111988111198812

1198973+

27119860119909119909119882

2119881

2

2

81198973

+18119860

119909119909119882

1119881

2

2

71198973+

54119860119909119909119882

1119882

2

2

71198973+

6119860119909119909119882

2

1

1198822

1198973

+2119860

119909119909119882

2119881

2

1

1198973+

121198601199091199091198802119882

2

51198972+

8119860119909119909119882

1119881

2

1

51198973

+3119860

1199091199091198801119882

2

21198972+

21198601199091199091198802119882

1

1198972+

27119860119909119909119882

3

2

81198973

+8119860

119909119909119882

3

1

51198973+

41198601199091199091198801119882

1

31198972minus 119865

119888119911(119905) = 0

1

7119897119868119860

2+

9

5

119868119863

2

119897+

1

6119897119868119860

1minus

119898ℎV5ℎ

1199055

1

1198975+

3

2

119868119863

1

119897

+6120578

119887119863

119909119909

1

1198973+

2119898ℎV5ℎ

1199054

1

1198975+

6119898ℎV6ℎ

1199055

2

1198976

+12120578

119887119863

119909119909

2

1198973+

6119863119909119909119882

1

1198973+

16119898ℎV5ℎ

1199053

1198821

1198975

+12119863

119909119909119882

2

1198973+

18119898ℎV6ℎ

1199054

1198822

1198976+

31198601199091199091198801119882

1

21198972

+27119860

119909119909119882

211988111198812

41198973+

36119860119909119909119882

111988111198812

71198973+

9119860119909119909119882

2119881

2

2

21198973

+27119860

119909119909119882

1119881

2

2

81198973+

81119860119909119909119882

1119882

2

2

81198973+

18119860119909119909119882

2119881

2

1

71198973

+54119860

119909119909119882

2119882

2

1

71198973+

31198601199091199091198802119882

2

1198972+

2119860119909119909119882

1119881

2

1

1198973

+9119860

1199091199091198801119882

2

51198972+

121198601199091199091198802119882

1

51198972+

2119860119909119909119882

3

1

1198973

+9119860

119909119909119882

3

2

21198973minus 119865

119888119911(119905) = 0

(48)









119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119879119898

=1

2120596 sdot J sdot 120596 + 119898

119897int

119897

0

2∘

1205883


3) sdot (120596 times 120588

3)


3) 119889119911

+ 119898119897int

119897

0

2∘

1205882


2) sdot (120596 times 120588

2)


2) 119889119910

(49)


V0119909

(119911 119905) = (119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

1199080119910

(119911 119905) = 120572V0119909

(119911 119905) = (119911

119897)

2

sdot 1205721198811(119905) + (

119911

119897)

3

sdot 1205721198812(119905)

for the 3rd beam

V0119909

(119910 119905) = 120573V0119909

(119911 119905) = (119910

119897)

2

sdot 1205731198811(119905) + (

119910

119897)

3

sdot 1205731198812(119905)

1199080119911

(119910 119905) = 120574V0119909

(119910 119905) = (119910

119897)

2

sdot 1205741198811(119905) + (

119910

119897)

3

sdot 1205741198812(119905)

for the 2nd beam

(50)


1(119905)


1199080119910(119911 119905) V

0119909(119910 119905) and 119908


assumed modes as

1205883= V

0119909(119911 119905) i

119887+ 119908

0119910(119911 119905) j

119887

=

[[[[[

[

(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)

120572 [(119911

119897)

2

sdot 1198811(119905) + (

119911

119897)

3

sdot 1198812(119905)]

0

]]]]]

]

1205882= V

0119909(119910 119905) i

119887+ 119908

0119911(119910 119905) k

119887

=

[[[[[

[

120573[(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

0

120574 [(119910

119897)

2

sdot 1198811(119905) + (

119910

119897)

3

sdot 1198812(119905)]

]]]]]

]

(51)


∘

1205883

= 1205883119909i119887+ 120588

3119910j119887=

[[[[[

[

(119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)

1205721[(

119911

119897)

2

sdot 1(119905) + (

119911

119897)

3

sdot 2(119905)]

0

]]]]]

]

∘

1205882

= 1205882119909i119887+ 120588

2119911k119887=

[[[[[

[

1205731[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

0

1205741[(

119910

119897)

2

sdot 1(119905) + (

119910

119897)

3

sdot 2(119905)]

]]]]]

]

(52)


1= 1 and



119879ℎ=

1

2119898

ℎ1(120596 times r


ℎ1) +

1

2119898

ℎ2(120596 times r

ℎ2)


(53)


119879119901=

1

2119898

119901(120596 times r

119901) sdot (120596 times r

119901) (54)




119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

minus 119898119901119903119901119909119903119901119911 + 119898

1199011199032

119901119911

+ 1198981199011199032

119901119910

minus 119898119901119903119901119909119903119901119910

120579 = 119876120593

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1+ 119898

ℎ21199112

ℎ2

120579

+ 1198981199011199032

119901119911

120579 minus 119898119901119903119901119910119903119901119911 minus 119898

119901119903119901119909119903119901119910

+ 1198981199011199032

119901119909

120579 = 119876120579

119869119911 minus

2

51198972

1198981198972minus

1

21198972

1198981198971+ 119898

ℎ11199102

ℎ1

minus 119898119901119903119901119910119903119901119911

120579 + 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

minus 119898119901119903119901119909119903119901119911 = 119876

120595

(55)

where 119876120593 119876

120579 and 119876




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

271198601199091199091198812119882

2

2

81198973

+54119860

1199091199091198811119881

2

2

71198973+

181198601199091199091198811119882

2

2

71198973+

6119860119909119909119881

2

1

1198812

1198973

+2119860

1199091199091198812119882

2

1

1198973+

81198601199091199091198811119882

2

1

51198973+

8119860119909119909119881

3

1

51198973

+27119860

119909119909119881

3

2

81198973+

361198601199091199091198812119882

1119882

2

71198973+

41198601199091199091198811119882

1119882

2

1198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973+

91198601199091199091198812119882

2

2

21198973

+81119860

1199091199091198811119881

2

2

81198973+

271198601199091199091198811119882

2

2

81198973+

541198601199091199091198812119881

2

1

71198973

+18119860

1199091199091198812119882

2

1

71198973+

21198601199091199091198811119882

2

1

1198973+

9119860119909119909119881

3

2

21198973

+2119860

119909119909119881

3

1

1198973+

271198601199091199091198812119882

1119882

2

41198973+

361198601199091199091198811119882

1119882

2

71198973

minus 1198651198622

(119905) = 0

(56)


4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973+

16119860119909119909119881

3

1

51198973

+12119860

119909119909119881

2

1

1198812

1198973+

1081198601199091199091198811119881

2

2

71198973+

54119860119909119909119881

3

2

81198973

minus 1198651198621

(119905) = 0

1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973

+4119860

119909119909119881

3

1

1198973+

108119860119909119909119881

2

1

1198812

71198973+

811198601199091199091198811119881

2

2

41198973

+9119860

119909119909119881

3

2

1198973minus 119865

1198622(119905) = 0

(57)

1198651198621(119905) and 119865




4

3

1198681198631

119897+

1

51198971198681198601+

1

61198971198681198602+

31198681198632

2119897+

4120578119887119863

1199091199091

1198973

+6120578

119887119863

1199091199092

1198973+

41198631199091199091198811

1198973+

61198631199091199091198812

1198973= 119865

1198621

(119905)


1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

61198971198681198601+

3

2

1198681198631

119897+

1

71198971198681198602+

9

5

1198681198632

119897+

6120578119887119863

1199091199091

1198973

+12120578

119887119863

1199091199092

1198973+

61198631199091199091198811

1198973+

121198631199091199091198812

1198973= 119865

1198622

(119905)

(58)


119886120593 + 119886

120579

120579 + 119886120595 + 119886

1198811

1+ 119886

1198812

2= 119876

120593

119887120579

120579 + 119887120595 + 119887

120593 + 119887

1198811

1+ 119887

1198812

2= 119876

120579

119888120595 + 119888

120579

120579 + 119888120593 + 119888

1198811

1+ 119888

1198812

2= 119876

120595

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1198811

+ 1198891198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1198811

+ 1198901198812

1198812= 119876

1198812

(59)


119869119909 minus 119898

1198971198972

1minus

4

5119898

1198971198972

2= 119876

120593+ 119906

119869119910

120579 +2

5119898

1198971198972

2+

1

2119898

1198971198972

1= 119876

120579+ 119906

120579

119869119911 minus

1

21198972

1198981198971minus

2

51198972

1198981198972= 119876

120595+ 119906

119889

1198811

1+ 119889

1198812

2+ 119889

1198811

1+ 119889

1198812

2

+ 1198891198811

1198811+ 119889

1198812

1198812= 119876

1198811

119890

1198811

1+ 119890

1198812

2+ 119890

1198811

1+ 119890

1198812

2

+ 1198901198811

1198811+ 119890

1198812

1198812= 119876

1198812

(60)


120579

and 119906are as

119906= minus119898

ℎ11199102

ℎ1

minus 119898ℎ21199112

ℎ2

+ 119898119901119903119901119909119903119901119911 minus 119898

1199011199032

119901119911

minus 1198981199011199032

119901119910

+ 119898119901119903119901119909119903119901119910

120579

119906

120579

= minus119898ℎ21199112

ℎ2

120579 minus 1198981199011199032

119901119911

120579 + 119898119901119903119901119910119903119901119911 + 119898

119901119903119901119909119903119901119910

minus 1198981199011199032

119901119909

120579

119906= minus119898

ℎ11199102

ℎ1

+ 119898119901119903119901119910119903119901119911

120579 minus 1198981199011199032

119901119910

minus 1198981199011199032

119901119909

+ 119898119901119903119901119909119903119901119911

(61)

119906 119906

120579



1199091= 120593 119909

2= 119909

3= 120579 119909

4= 120579

1199095= 120595 119909

6= 119909

7= 119881

1

1199098=

1 119909

9= 119881

2 119909

10=

2

(62)


1= 119909

2

2= 119886

11199097+ 119886

21199098+ 119886

31199099+ 119886

411990910

+ 119906120593

3= 119909

4

4= 119886

51199097+ 119886

61199098+ 119886

71199099+ 119886

811990910

+ 119906120579

5= 119909

6

6= 119886

91199097+ 119886

101199098+ 119886

111199099+ 119886

1211990910

+ 119906120595

7= 119909

8

8= 119886

131199097+ 119886

141199098+ 119886

151199099+ 119886

1611990910

+ 1199061198811

9= 119909

10

10

= 119886171199097+ 119886

181199098+ 119886

191199099+ 119886

2011990910

+ 1199061198812

(63)

where 119906120579 119906

120593 and 119906



and 1199061198812


x = Ax + Bu (64)


x = [1

2

3

4

5

6

7

8

9

10]119879 x (119905

0) = x

1199050

u = [119906120593 119906

120579 119906

120595 119906

1198811

1199061198812

]119879


(65)







x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














x = Ax + Bu x (1199050) = x

1199050

(known)

119869 =1

2int

infin

1199050

[x119879 (119905)Qx (119905) + u119879 (119905)Ru (119905)] 119889119905

(66)

whereQ = Q119879

⩾ 0 andR = R119879








119869lowast

[x (1199050) 119905

0] =

1

2x1198790

Px0 (69)


















x = Ax + Bu x (1199050) = x

1199050

u (1199050) = u

1199050

119869 =1

2int

infin

1199050

(x119879Qx + u119879Ru + u119879Su) 119889119905(71)

where Q = Q119879

⩾ 0 R = R119879


11] is com-


11




0) = u

1199050

(72)



4x ulowast (119905

0) = u

1199050

(73)



[xulowast] = [

A BminusK

1minusK

2

] [xulowast] [

[

x (1199050)

u (1199050)]

]

= [x1199050

u1199050

] (74)




0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000minus5

0

5

minus5

0

5

minus5

0

5

minus4

minus2

0

2

4

times10minus4

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

00162

00164

00166

4800 4900 5000

4800 4900 5000

4800 4900 5000

minus00165

minus0016

Time (s) Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)







Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7) = Q (8 8)

= Q (9 9) = Q (10 10) = 1 times 10minus6

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = 900 R (3 3) = 100

R (4 4) = R (5 5) = 10000

R119894119895

= 0 (119894 = 119895 119894 119895 = 1 sim 5)

(75)


Q (1 1) = Q (2 2) = 4 times 10minus8

Q (3 3) = Q (4 4) = Q (5 5) = Q (6 6) = Q (7 7)

= Q (8 8) = 16 times 10minus8


minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus02

0

02

Roll

rate

(∘s)

Roll

rate

(∘s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Pitc

h ra

te(∘s)

Pitc

h ra

te(∘s)

0 1000 2000 3000 4000 5000

minus05

0

05

minus05

0

05

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Time (s)

Yaw

rate

(∘

s)

Yaw

rate

(∘

s)

4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

Time (s)4800 4850 4900 4950 5000

minus2

0

2

minus2

0

2

minus2

0

2

times10minus4

times10minus4

times10minus4


Q (9 9) = Q (10 10) = 1 times 10minus8

Q (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 10)

R (1 1) = R (2 2) = R (3 3) = 1 times 10minus8

R (4 4) = R (5 5) = 4 times 10minus8

R (119894 119895) = 0 (119894 = 119895 119894 119895 = 1 sim 5)

(76)


0) = 3

∘ 120579(1199050) = 3

∘ and120595(119905

0) = 3


0) = 0

∘

s 120579(1199050) = 0

∘

s and(119905

0) = 0

∘

s 1198811(1199050) = 01m 119881

2(1199050) = 01m

1(1199050) = 0ms








∘




minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus20

0

20

Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

minus50

0

50

minus50

0

50

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

0 1000 2000 3000 4000 5000

0 2000 4000 6000

0 2000 4000 6000

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)

minus002

0

002

008

01

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus009

minus008

Time (s)Time (s)

Time (s)Time (s)

Time (s)Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF








120593 119906

120579 and 119906

120595can be





0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

Time (s) Time (s)

Time (s) Time (s)

minus5

0

5

times10minus4

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

The d

ispla

cem

ent o

f the

tip

(m)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloc

ity o

f the

tip

(ms

)











Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Roll

erro

r120593(∘

)

Roll

erro

r120593(∘

)

minus5

0

5

minus5

0

5

Pitc

h er

ror120579

(∘)

Pitc

h er

ror120579

(∘)

0 1000 2000 3000 4000 5000minus10

0

10

Time (s) Time (s)

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

Yaw

erro

r120595(∘

)

Yaw

erro

r120595(∘

)

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus1

0

1

times10minus3

minus1

0

1

times10minus3

minus1

0

1

times10minus3

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF







6 Conclusions



Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Roll

rate

erro

r(∘ s

)

Roll

rate

erro

r(∘ s

)

minus01

0

01

minus01

0

01

Pitc

h ra

te er

ror (

∘ s)

Pitc

h ra

te er

ror (

∘ s)

0 1000 2000 3000 4000 5000minus02

0

02

Yaw

rate

erro

r (∘ s

)

Yaw

rate

erro

r (∘ s

)

4800 4850 4900 4950 5000

minus5

0

5times10minus4

minus5

0

5

times10minus4

minus5

0

5times10minus4

Time (s)

4800 4850 4900 4950 5000

Time (s)

4800 4850 4900 4950 5000

Time (s)

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

0 1000 2000 3000 4000 5000

Time (s)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF




Appendix


119886120593= 119869

119909+ 119898

ℎ11199102

ℎ1

+ 119898ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119910

119886120579= minus 119898

119901119903119901119909119903119901119910 119886

120595= minus119898

119901119903119901119909119903119901119911

1198861198811

= minus 1198981198971198972

1198861198812

= minus4

5119898

1198971198972

119887120579= 119869

119910+ 119898

ℎ21199112

ℎ2

+ 1198981199011199032

119901119911

+ 1198981199011199032

119901119909

119887120595= minus 119898

119901119903119901119910119903119901119911 119887

120593= minus119898

119901119903119901119909119903119901119910


Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Roll

torq

ue (N

lowastm

)

Roll

torq

ue (N

lowastm

)

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

0 2000 4000 6000

0 2000 4000 6000

0 2000 4000 6000

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

minus01

minus008

008

009

01

minus002

0

002

minus5

0

5

minus2

0

2

Pitc

h to

rque

(Nlowast

m)

Pitc

h to

rque

(Nlowast

m)

minus2

0

2

Yaw

torq

ue (N

lowastm

)

Yaw

torq

ue (N

lowastm

)


1198871198811

=1

2119898

1198971198972

1198871198812

=2

5119898

1198971198972

119888120595= 119869

119911+ 119898

ℎ11199102

ℎ1

+ 1198981199011199032

119901119910

+ 1198981199011199032

119901119909

119888120579= minus 119898

119901119903119901119910119903119901119911 119888

120593= minus119898

119901119903119901119909119903119901119911

1198881198811

= minus1

21198972

119898119897 119888

1198812

= minus2

51198972

119898119897

119889

1198811

= (4

3

119868119863

119897+

1

5119897119868119860) 119889

1198812

= (1

6119897119868119860+

3119868119863

2119897)

119889

1198811

=4120578

119887119863

119909119909

1198973 119889

1198812

=6120578

119887119863

119909119909

1198973

1198891198811

=4119863

119909119909

1198973 119889

1198812

=6119863

119909119909

1198973

1198761198811

= 1198651198621

(119905)

119890

1198811

= (3

2

119868119863

119897+

1

6119897119868119860)

119890

1198812

= (1

7119897119868119860+

9

5

119868119863

119897) 119890

1198811

=6120578

119887119863

119909119909

1198973

119890

1198812

=12120578

119887119863

119909119909

1198973 119890

1198811

=6119863

119909119909

1198973

1198901198812

=12119863

119909119909

1198973 119876

1198812

= 1198651198622

(119905)

(A1)


A =

[[[[[[[[[[[[[[

[

0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1198861

1198862

1198863

1198864

0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1198865

1198866

1198867

1198868

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1198869

11988610

11988611

11988612

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 11988613

11988614

11988615

11988616

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 11988617

11988618

11988619

11988620

]]]]]]]]]]]]]]

]


0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

minus01

0

01

minus02

0

02

4800 4850 4900 4950 5000

4800 4850 4900 4950 5000

With GFWithout GF

With GFWithout GF

With GFWithout GF

With GFWithout GF

Time (s)

Time (s)Time (s)

Time (s)

minus1

0

1

minus1

0

1

times10minus3

times10minus3Th

e disp

lace

men

t of t

he ti

p (m

)

The d

ispla

cem

ent o

f the

tip

(m)

The v

eloci

ty o

f the

tip

(ms

)

The v

eloci

ty o

f the

tip

(ms

)


B =

[[[[[[[[[[[[[[

[

0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 1 0

0 0 0 0 0

0 0 0 0 1

]]]]]]]]]]]]]]

]

1198861= minus

1198981198971198972

119869119909

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198862= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198863= minus

1198981198971198972

119869119909

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198864= minus

1198981198971198972

119869119909

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus4

5119869119909

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120593=

119876120593

119869119909

+

119906

119869119909

+119898

1198971198972

119869119909

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+4

5119869119909

1198981198971198972

+

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198865=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198866=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198867=

1

2119869119910

1198981198971198972

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119910

1198981198971198972

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198868=

1

2119869119910

1198981198971198972

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)


+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+2

5119869119910

1198981198971198972

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120579=

119876120579

119869119910

+119906

120579

119869119910

minus1

2119869119910

1198981198971198972

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119910

1198981198971198972

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1198869= minus

1

2119869119911

1198972

119898119897

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988610

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988611

= minus1

2119869119911

1198972

119898119897

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988612

= minus1

2119869119911

1198972

119898119897

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

minus2

5119869119911

1198972

119898119897

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

119906120595=

119876120595

119869119911

+

119906

119869119911

+1

2119869119911

1198972

119898119897

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

+2

5119869119911

1198972

119898119897

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988613

= minus

(119890

1198812

1198891198811

minus 1198901198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988614

= minus

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988615

= minus

(119890

1198812

1198891198812

minus 1198901198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988616

= minus

(119890

1198812

119889

1198812

minus 119890

1198812

119889

1198812

)

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

1199061198811

=

119890

1198812

1198761198811

minus 119889

1198812

1198761198812

(119890

1198812

119889

1198811

minus 119890

1198811

119889

1198812

)

11988617

= minus

(119890

1198811

1198891198811

minus 1198901198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988618

= minus

(119890

1198811

119889

1198811

minus 119890

1198811

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988619

= minus

(119890

1198811

1198891198812

minus 1198901198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

11988620

= minus

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

1199061198812

=

119890

1198811

1198761198811

minus 119889

1198811

1198761198812

(119890

1198811

119889

1198812

minus 119890

1198812

119889

1198811

)

(A2)



Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Dynamics and Control of a Flexible Solar...

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Transcript of Research Article Dynamics and Control of a Flexible Solar...