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Simulation and Flight Control of a Tilt Duct UAV

Ozan Tekinalp1, Tugba Unlu2, and Ilkay Yavrucuk3

Middle East Technical University, Ankara, Turkey, 06531

Tilt duct VTOL UAV concept is presented. The equations of motion are given and, trim and simulation code is described. Trim flight conditions are given for hover, cruise and forward flight cases. A two loop SDRE control is proposed and explained. The blended inverse control allocation algorithm is used for allocating controllers during the transition flight phase, where there are redundant controls. Simulation results during transition phase are presented, and the success of the controller as well as the allocation algorithm is demonstrated.

Nomenclature

ijI = mass moments of inertias

nI = n dimensional unit matrix

, ,A A AM N L = aerodynamic moments p, q, r = body angular velocities u, v, w = translational velocities in the body fixed frame

, ,A A AX Y Z = aerodynamic forces

, ,A A AX Y Z = forces due to engine thrust

, ,A A AX Y Z = moments due to engine thrust

, ,φ θ ψ = Euler angles

η = main engine tilt angle

µ = exit guide vane angle of the aft propeller

I. Introduction

There has been an increased interest in the use of unmanned aerial vehicles for performing flights where the use

of manned flight vehicles is not appropriate or feasible for missions like delivery or supply, reconnaissance, target acquisition or designation, data acquisition. Present improvements are mainly on three types of UAV configurations; fixed-wing configuration, helicopter type configuration and tilt thrust type configurations. The latter two offers VTOL capabilities, removing the need for long runways, and permit operation in constrained areas. Among the tilt thrust configurations, the most feasible solutions are probably the tilt-rotor and tilt-duct concepts.

Tilt-rotor or tilt-duct type UAVs, provide translational flight, as well as vertical take-off and landing capabilities. Their ability to take-off and land vertically, combined with their ability to hover for extended periods of time over a point and operate in confined areas off steep slopes, make them ideally suited for real time tactical reconnaissance, target acquisition, surveillance, and ordnance delivery missions for front line tactical units. The rotors on these UAV’s are designed to provide the thrust necessary for both vertical and translational flights. Aircraft vertical motion of the UAV is provided by maintaining the vehicle fuselage substantially horizontal so that the thrust (downwash) of the rotors provides the necessary lift for the aircraft.

1 Professor, Aerospace Engineering Department, Member. 2 Graduate Student, Aerospace Engineering Department. 3 Faculty Member, Aerospace Engineering Department, Member.

AIAA Modeling and Simulation Technologies Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-6138

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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Ducted propellers have many advantages over tilt rotors1. They offer more thrust during vertical flight as well as forward flight. Reduced blade loading alleviate compressibility problems, cavitation and noise generation1. They are much more efficient in side winds, and ducted fans at the end of the wings also have the end plate effect. Finally, they are mechanically easy to construct since they do not require a swash plate mechanisms needed for helicopter or tilt rotor configurations.

A conceptual design of a tilt-duct UAV was presented previously1. Linear controllers were also designed2. It was shown that the system is over-actuated with redundant controls and it is unstable in transition mode2. The purpose of this study is to examine the flight control of the tilt-duct UAV further. The tilt-duct UAV is a highly nonlinear system, makes it a real challenge to come up with stable, robust, and simple controllers. In this manuscript we propose to use State Dependent Riccati Equation (SDRE) control, mainly because it is locally asymptotically stable and is expected to have similar robustness properties as the linear quadratic regulators3. One main issue with the SDRE control is the fact that although the system is controllable, it may become uncontrollable due to the choice of state factorization used4. To alleviate the controllability problem, a double loop control is used5. In the inner loop the translational and rotational velocities, in the outer loops, attitudes are controlled. Since during transition the system is over actuated, the ‘blended inverse’ control allocation algorithm, recently developed is used5. The advantage of blended inverse is that it can not only provide the necessary controls, but also allocate them to the actuators properly, avoiding saturation in the controls6.

In the next section, after a brief description of the tilt duct UAV concept, the equations of motion are given. It is followed by the description of the simulation program and trimmer. The flight control system is given next. First the SDRE controller and its implementation for the tilt duct UAV is described. Then the control allocation is explained. The trim results during different flight phases and simulation results during the transition period are presented and discussed. Finally conclusions are given.

Figure 1. Tilt duct UAV during vertical takeoff.

Figure 2. Tilt duct UAV in forward flight.

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II. The Tilt Duct UAV Concept Various views of the tilt duct UAV, in vertical takeoff and forward flight condition are shown in figures 1 and 2.

The aircraft has two main and one aft propeller. The main propellers are vertical during takeoff. Through differential thrust, they control the roll attitude. The aft propeller provides the pitch moment, as well as yaw moment needed during takeoff and transition phases of the flight. The pitch moment is generated by the direct downwash of the propeller, while yaw moment is created with the exit guide vanes acting on the aft propeller output. Thus, downward thrust force is vectored to the right or to the left. During transition main propellers are tilted gradually in the forward direction, causing a gradual increase in forward speed. As the speed increases, the control surfaces become more effective, and eventually replace the main controls. For these purpose conventional control surfaces, elevator, rudder and ailerons are used. During cruise aft propeller is turned of and only conventional control surfaces are used. The inlet and exit of the aft propeller is closed by vanes to provide a smooth fairing.

III. Equations of Motion Translational and rotational equations of the tilt duct UAV in the body fixed frame may be given as:

sin = A Tmu wq vr g X Xθ+ − + +& (1)

cos sin = A Tmv ur wp g Y Yθ φ+ − − +& (2)

cos cos = A Tmw vp uq g Z Zθ φ+ − − +& (3)

( ) =− + + − +& &xx xz xz zz yy A TI p I r I pq I I rq L L

(4)

2 2( ) ( ) =+ − + − +&yy xx zz xz A TI q I I pr I p r M M

(5)

( ) =− + − + +& &zz xz yy xx xz A TI r I p I I pq I rq N N

(6)

Aerodynamic forces and moments are calculated in a component buildup fashion. The necessary coefficients are estimated from semi-empirical formulas available in the literature7. Following stability derivative are estimated and used in the calculation of the aerodynamic forces and moments.

= ( ) ( ) ( )L L L elev L LWBT elev q

cC C C C q C

Vδ αα δ α+ + +

&

&

(7)

= ( ) ( )D L D elevWBT elev

C C Cδ

α δ+ (8)

0( ) = ( ) ( )D elev D elev D elevelevC C C

δ αδ δ δ α+

(9)

= ( ) ( ) ( )M M M elev M MWBT elev q

cC C C C q C

Vδ αα δ α+ + +

&

&

(10)

,= ( ) ( ( ) )

2Y Y Y rud Y YWBT rud p r

bC C C C p C r

Vβ δβ δ α+ + +

(11)

= ( ) ( ) ( ) ( ( ) ( ) )

2L L L ail L rud L LWBT ail rud p r

bC C C C C p C r

Vδ δα δ δ α α+ + + +

(12)

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,= ( , ) ( , ) ( ( ) ( ) )

2N N N ail N rud N NWBT ail rud p r

bC C C C C p C r

Vβ δ δβ α δ α δ α α+ + + +

(13)

Components of the thrust forces and moments are carried to the center of mass, and written in the body fixed coordinate frame:

= ( )cos( )µ+T MainL MainRX T T (14)

= ( )cos( )ηT AftY T

(15)

= ( )sin( ) ( )sin( )µ η+ +T MainL MainR AftZ T T T

(16)

= ( )sin( )µ+T ycg MainL MainRThmL L T T (17)

= ( )sin( ) ( )sin( )µ η+ −T xcg MainL MainR xcg AftThm ThaM L T T L T (18)

= ( )cos( ) ( )cos( )µ η+ +T xcg MainL MainR xcg AftTha ThmN L T T L T (19)

The upper and lower limits on the controls are listed in Table 1.

Table 1. Limits and units used for the controls

Control Limits Throttle, main left and right ( )0 ~ 100% max 640 N

Throttle aft aileron vertical ( )0 ~ 100% max 196.2 N

Throttle aft aileron lateral ( )100% max 196.2 N±

Elevator 20 ~ 20− +o o Aileron 15 ~ 10− +o o Rudder 25 ~ 25− +o o

IV. Simulation, Trim and Linearization Simulation model is constructed in the Matlab-Simulink environment. Each part such as dynamics and forces

acting on the aircraft are modeled in separate blocks and resultant forces and moments are calculated. Three different coordinate systems are used: body fixed coordinates for aircraft dynamics, Earth fixed coordinates for navigation, and stability axis coordinates to calculate aerodynamic forces and moments. Power plant models are also included. However, percent thrust is used in feedback control. Other models such as standard atmosphere model, gravity model, and wind-gust model are also included. The general structure of the simulation tool is shown in Figure 3.

Trim code uses the Matlab function ‘linmod’ to linearized the nonlinear Simulink model. Using the linearized model, next candidate trim states and inputs are found. The linearized equations are obtained at the new candidate point and iteration repeated. The iteration is stopped until the derivatives are sufficiently close to zero. The trim conditions considered assume horizontal flight. The following is sought for in all cases:

= = = = = = 0p q r u v w& & & & & & (20)

Additional conditions are imposed depending on the trim flight condition requested (i.e., steady wings level flight, symmetric pull up, coordinated turn etc.).

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V. Flight Control System Design

A. SDRE Control Given a system, that is full state-observable, autonomous, nonlinear in state, and affine in input represented as3:

0( ) ( ) , (0)= + =&x f x B x u x x (21)

The state x may be factored as: ( )=f x A(x)x . Then the system equations becomes,

0( ) ( ) , (0)= + =&x A x x B x u x x (22)

This is called extended linearization3. Note that this factorization is not unique. The control of this equation may be sought by freezing the state instantaneously and posing the following infinite horizon quadratic performance index:

{ }0

1( ) ( )

2

∞

= +∫T TJ dtx Q x x u R x u (23)

where, ( )= tx x , ( )= tu u , ( ) ≥Q x 0 , and ( ) >R x 0 . The above system may be controlled in a similar fashion as the LQR control. Provided that the factorized plant is fully controllable, the feedback control may be given as, 1( ) ( ) ( ) ( )−= − = − Tu K x x R x B x P x x , where ( )P x is the solution of the following Algebraic State Dependent Riccati Equation:

1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )−+ − + =T TP x A x A x P x P x B x R x B x P x Q x 0 (24)

This approach is expected to have the usual robustness and asymptotic stability properties of the classical LQR. The controller is a nonlinear controller which does not require the linearization of the system matrices. One important issue is to make sure that the system matrices factorized as shown is fully controllable. A physically controllable system may become uncontrollable from time to time due to the choice of factorization carried out5. To avoid uncontrollable circumstances, the equations of motion are treated as inner and outer loop states and inputs. In the inner loop, translational and rotational velocities are considered. The state and input vector of the inner loop are:

[ ]1 =T

u v w p q rx (25)

1 =T

x y z x y zF F F M M M u (26)

Then the inner loop state dependent system matrix,1( )A x , and input matrix, 1( )B x are:

11 2 1 2 4 3 4 3

7 8 7 8

5 2 5 2 6 6 4 4

/ 2 / 2 / 2 / 2 / 2

/ 2 / 2 / 2 / 2 0

/ 2 / 2 / 2 / 2 0( ) =

0 / 2 ( ) / 2 / 2

0

0 / 2 ( ) / 2 / 2

u q

u v p

u w q

u u v v

u w q

u u v v

X r w r X w v

Y q Y p Y w u

Z q p Z v Z ux

I L I N I L I N I q I r I p I q

M M I p I r M I p I r

I N I L I N I L I q I p I r I q

−

− + − + − − + + + +

+ − + + − −

A (27)

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11 2

2 5

1/ 0 0 0 0 0

0 1/ 0 0 0 0

0 0 1/ 0 0 0( ) =

0 0 0 0

0 0 0 0 1/ 0

0 0 0 0

m

m

mx

I I

Iyy

I I

B (28)

where,

( )( ) ( )

2 21 2 3

24 5 6

7 8

= , = / , = / , = ( ) /

= ( ) / , = / , = ( ) /

( ) / , /

num xx zz xz zz num xz num yy zz zz xz num

xx yy zz xz num xx num xx yy xx xz num

I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I

I Izz Ixx Iyy I Ixz Iyy

− − −

− + − +

= − =

(29)

The outer loop only uses the Euler angle kinematics relations. In addition integral states are also added to avoid steady state error. Hence,

[ ]2 = , , , , ,T

I I Iφ θ ψ φ θ ψx , [ ]2 = , ,T

p q ru (30)

Figure 5. Block diagram of the flight control system

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0 0 0 0 0 0 cos sin sin sin cos

0 0 0 0 0 0 0 cos cos cos sin

0 0 0 0 0 0 0 sin cos11 0 0 0 0 0 0 0 0cos

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

II

II

II

p

φ θ θ φ θ φφθ θ φ θ φθψ φ φψφ θφθθψψ

− = +

&

&

&

&

&

&

q

r

(31)

2 2 2 2 2 2( )= +x A x B x u& (32)

The inner loop feedback gain is computed by solving the SDRE equation. The inputs to the inner loop are the differences with the reference and current states. This difference is multiplied by SDRE gain calculated online. The outer loop, on the other hand, uses eigenvalue assignment. The block diagram of the controller is shown in Figure 3. Those two loops are observable from this figure. In addition, inner loop Riccati equation solver block is shown.

B. Control Allocation In Eq. (26) controls are defined in terms of forces and moments. They are related to the actual inputs such as

aileron, rudder, elevator, left main engine thrust, right main engine thrust, aft engine thrust, exit guide vane angle, duct angle etc. In total there are eight physical controls as opposed to three forces and three moments of 1u , making the system over actuated when all the controls are turned on. The relation between the forces and moments and actual physical controls may be written as:

1 =u Fδ

(33)

Since the matrix F is a rectangular matrix, one way to invert it is to use Moore-Penrose pseudo inverse (MP-inverse), which finds the minimum norm solution of the vector.

1

1T T

MP

− =

δF FF u

(34)

However, with MP-inverse the allocation is not controllable. The only control allocation routine, that gives the desired output, 1u , while controlling the physical inputs, δ , is the Blended inverse (B-inverse)6.

1

1T T

BI n desiredq q−

= + + δ

I F Fδ

F u (35)

Here, q is the blending coefficient, usually taken as a scalar. In this manuscript the desired controls are taken as trim values at the given flight condition. To help controls stay in the neighborhood of the trim values, and prevent them from saturating, the blending coefficient may be increased exponentially as the controls wander away from their desire values. For example,

_( exp( ))i i errorq δ=q diag (36)

( )_ _ _/i error i i desired i desiredδ δ δ δ= −

(37)

where q is a diagonal matrix dynamically changed according to the error between the desired and actual controls.

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VI. Results and Discussion

A. Trim Code Results For this aircraft three basic trim modes are present: hover, transition, and forward flight. The hover trim

conditions for the tilt duct UAV are: 67.68%Tmainδ = , 71.98%Taftδ = , 90µ = o , 90η = o .

During transition, the forward speed of the tilt duct UAV is increased towards the cruise speed. As the speed is increased, the wings start generating more lift force; while the need for vertical thrust is decreased, the need for horizontal thrust is increased. Thus, the ducted fans shall be tilted forward, as the aircraft gains speed. To obtain this condition, the tilt angle is changed linearly with the speed, and trim conditions shown in Figure 6 are obtained. The transition is completed at three different velocities: Best range speed, cruise speed, and maximum speed. In each case, the duct tilt angle is linearly driven to zero. Corresponding main and aft thrust values are plotted in a separate figure. In this figure, it is also observable that aft engine is gradually and set to zero closed between 10 to 20 m/s. At the same time, the elevator is brought into play and becomes more and more effective as the forward velocity increases. The angle of attack is also plotted as a function of airspeed in this figures.

Another approach in programming the transition phase would be to change the duct angle and speed such that at every transition speed, pitch angle is zero. This trim condition is also sought for. In Figure 7 this trim condition is given presented. It may be observed from this figure that the duct angle now changes in a nonlinear fashion with the airspeed. It is large up to a certain velocity and goes to zero in a very steep fashion. In the mean while, the main and aft throttles also drops. The elevator brought into play around 10 m/s. This causes the aft throttle to drop sharply.

B. Flight Control Results The inner loop of the flight control system uses SDRE control while in the outer loop eigenvalues are assigned.

The gains of the cost function are selected constant throughout the flight. These matrices are:

[ ]

[ ]6

diagonal 12.3 12.3 12.3 0.001 0.001 0.001

10 diagonal 6 4 25 4.5 5 0.1−

=

=

Q

R (38)

The eigenvalues assigned to the outer loop are listed below: [ ]2 12 0.1 0.5 9 0.3= − − − − − −

λ (39)

With these parameters, the transition phase simulation is carried out. The states of the simulation that are input and realized are given in Figure 9. Only the longitudinal velocity is increase with time. The reference inputs to all other translational and rotational velocities as well as Euler angles are set to zero. It may be observed from Figure 9, that the longitudinal velocity follows the reference very closely, while other states are all zero. The necessary controls during this transition flight are given in Figure 10. Beside the nominal transition flight, a disturbance is also added to the simulation. Two separate horizontal winds of reaching 10 m/s is added. The profiles of these disturbance inputs are given in Figure 8. Two separate simulations are also carried out with these disturbances. The controls generated by the flight control system under the effect of these disturbances are also given in Figure 10. It may be observed from this figure that although the controls are slightly disturbed during the disturbance inputs, they return back to their nominal values as soon as the disturbance is alleviated.

However, it should be remembered that these forces and moments are the result of the actual physical controls allocated according to the requirements of the control system. We tested two allocation methods: MP-inverse and B-inverse. The MP-inverse always saturated one of the controls, and was not useful. To use B-inverse, the desired controls are selected as the trim values shown in Figure 7. Thus, the pitch attitude during this transition phase trim was zero. The results of the allocation are observable in Figure 11. Careful examination shows that the nominal plant controls follows the trim values closely. When the longitudinal gust disturbance is added, the physical controls are moved from their trim values to counteract the gust. However, as soon as the gust input ended the controls returned to their nominal trim values. This shows that the B-inverse algorithm is reversible. The reversibility of the B-inverse control allocation algorithm was also demonstrated previously in the context of control moment gyros3.

VII. Conclusion Modeling, flight simulation and control of a tilt-duct UAV is presented. The trim flight conditions while the

UAV goes from hover to forward flight are calculated and presented. A two loop control algorithm is proposed. The inner loop uses SDRE control where in the outer loop, eigenvalue assignment is done. It is shown that the control system successfully controlled the tilt-duct UAV during transition, while resisting nose wind input. Hence

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SDRE control is very effective in controlling this nonlinear tilt-duct UAV. It also may be concluded that the B-inverse control allocation algorithm realizes the flight stability without saturating the controls and it is reversible.

Acknowledgments This work is supported by The Scientific and Technological Research Council of Turkey, Project No: 107M346.

References 1Armutcuoglu, O, Kavsaoglu, M.S. and O. Tekinalp “ilt Duct Vertical Takeoff and Landing Uninhabited Aerial Vehicle

Concept Design Study,” AIAA Journal of Aircraft, Vol. 41, No. 2, 2004, pp. 215-223. 2Okan, A., Tekinalp, O., and Kavsaoglu, M., “Flight Control of a Tilt-Duct UAV,” 1st International Conference on

Unmanned Aerospace Vehicles, AIAA-2002-3466, May 2002. 3Tayfun Cimen “State-Dependent Ricatti Equation (SDRE) Control : A Survey,” The International Federation of Automatic

Control, 2008, pp. 3761 - 3775. 4Hammet,K.D., Hall, C.D., and Ridgley, D.B., “Controllability issues in Nonlinear State-Dependent Ricatti Equation

Control,” Journal of Guidance Control and Dynamics, Vol. 21, No.5, 1998, pp. 767-773. 5Coultier, J.R., Stansbery, D.T., “Nonlinear, Hybrid Bank-to-Turn / Skid-to-Turn Missile Autopilot Design,” AIAA Guidance

Navigation and Control Conference, AIAA-01-5929, 2001. 6Tekinalp, O., Yavuzoglu, E., “A New Steering Law for Redundant Control Moment Gyroscope Clusters,” Aerospace

Science and Technology, V. 9, 2005, pp. 626-634. 7Lan, C-T.E., Roskam, J., Airplane Aerodynamics and Performance, U. of Kansas, Lawrence, Kansas, 1988.

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0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90Main Duct vs Speed

Airspeed (m/s)

Mai

n du

ct µ

(deg

)

Trans. up to Best Range SpeedTrans. up to Cuise SpeedTrans. up to Max Speed

0 10 20 30 40 50 60 70 80 90-10

0

10

20

30

40

50

60

70

80

Airspeed (m/s)

Mai

n-A

ft T

hrus

t (%

Thr

ottle

)

Main-Aft Thrust vs Speed

main thr up to Best Range Speedaft thr up to Best Range Speedmain thr up to Cruise Speedaft thr up to Cruise Speedmain thr up to Max Speedaft thr up to Max Speed

0 10 20 30 40 50 60 70 80 90-5

0

5

10

15

20Alpha - Theta vs Speed

Airspeed (m/s)

Alp

ha -

The

ta (d

eg)

Best Range SpeedCruise SpeedMax Speed

0 10 20 30 40 50 60 70 80 90

-5

-4

-3

-2

-1

0

1Elevator vs Speed

Airspeed (m/s)

Ele

vato

r (de

g)

Best Range Speed

Cruise Speed

Max Speed

Figure 6. Trim flight conditions during transition flight. Tilt angle is linearly changed with speed.

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0 10 20 30 40 500

20

40

60

80

100Main duct vs Speed

Airspeed (m/s)

Mai

n du

ct µ

(de

g)

0 10 20 30 40 50-20

0

20

40

60

80

Airspeed (m/s)M

ain-

Aft

Thr

ust

(%T

hrot

tle)

Main-Aft Thrust vs Speed

main throttle (%)

aft throttle (%)

0 10 20 30 40 500

5

10

15

20Elevator vs Speed

Airspeed (m/s)

Ele

vato

r (d

eg)

0 10 20 30 40 50-1

-0.5

0

0.5

1Alpha - Theta vs Speed

Airspeed (m/s)

Alp

ha -

The

ta (

deg)

Figure 7. Trim flight conditions during transition flight. Pitch attitude is fixed.

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0 5 10 15 20 25 30 35 40 45

-10

-5

0

5

10

time[sec]

Hor

izon

tal W

ind

[m/s

]

gust horizontal -10 m/s

gust horizontal 10 m/s

Figure 8. Characteristic of the gust applied. Pitch attitude is fixed.

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0 5 10 15 20 25 30 35 40 45-10

0

10

20

time[sec]θ

[deg

]

θinput[deg]

θoutput[deg]

0 5 10 15 20 25 30 35 40 450

20

40

60

time[sec]

u [m

/s]

uinput[m/s]

uoutput[m/s]

0 5 10 15 20 25 30 35 40 45-10

0

10

time[sec]

w [m

/s]

winput[m/s]

woutput[m/s]

0 20 40 600

200

400

600

800

1000

time[sec]

X D

ista

nce

[m]

0 20 40 60-1

-0.5

0

0.5

1

time[sec]

Y D

ista

nce

[m]

0 20 40 60280

285

290

295

300

305

310

315

320

time[sec]

Alti

tude

[m

]

0 20 40 60-100

-80

-60

-40

-20

0

20

time[sec]

γ [d

eg]

0 20 40 60-1

-0.5

0

0.5

1

time[sec]

φ [d

eg]

0 20 40 60-1

-0.5

0

0.5

1

time[sec]

ψ [d

eg]

Figure 9. States during feedback controlled transition flight phase.

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..

0 10 20 30 40 50-500

0

500

1000

time[sec]

Fx [N

]

0 10 20 30 40 50-1

0

1

time[sec]

Fy [N

]

0 10 20 30 40 50-3000

-2000

-1000

0

time[sec]

Fz [N

]

0 10 20 30 40 50-1

0

1

time[sec]M

x [Nm

]

0 10 20 30 40 50-400

-200

0

200

time[sec]

My [N

m]

0 10 20 30 40 50-1

0

1

time[sec]

Mz [N

m]

no gust

hor gust 10 m/shor gust -10 m/s

Figure 10. Control forces and moments during transition flight. Simulation results with nominal and horizontal gust.

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……

0 10 20 30 40 500

50

100

time[sec]

δ ThM

L

0 10 20 30 40 500

50

100

time[sec]

δ ThM

R

0 10 20 30 40 500

50

100

time[sec]

δ ThA

lon

0 10 20 30 40 500

50

100

time[sec]

δ µ [deg

]

0 10 20 30 40 50-1

0

1

time[sec]

δ ThA

lat

0 10 20 30 40 50-20

0

20

time[sec]

δ ele

v [deg

]

0 10 20 30 40 50-1

0

1

time[sec]

δ ail [d

eg]

0 10 20 30 40 50-1

0

1

time[sec]

δ rud [d

eg]

no gust

gust 10m/sgust -10 m/s

Figure 11. Allocated physical controls during transition phase of the flight. Nominal and horizontal gust acted upon.