Dynamic formulation and performance evaluation of the...

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Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]]–]]]

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Robotics and Computer-Integrated Manufacturing

0736-58

doi:10.1

� Corr

E-m

(F. Gao)

PleasRobo

journal homepage: www.elsevier.com/locate/rcim

Dynamic formulation and performance evaluation of the redundantparallel manipulator

Yongjie Zhao a,b,�, Feng Gao b

a Department of Mechantronics Engineering, Shantou University, Shantou 515063, PR Chinab State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China

a r t i c l e i n f o

Article history:

Received 28 September 2007

Received in revised form

8 August 2008

Accepted 6 October 2008

Keywords:

Inverse dynamics

Redundant parallel manipulator

Performance index

Performance evaluation

Principle of virtual work

45/$ - see front matter & 2008 Elsevier Ltd. A

016/j.rcim.2008.10.001

esponding author.

ail addresses: [email protected] (Y. Zh

.

e cite this article as: Zhao Y, Gao F.t Comput Integr Manuf (2008), doi:

a b s t r a c t

The dynamic formulation and performance evaluation of the redundant parallel manipulator are

presented in this paper. By means of the principle of virtual work and the concept of link Jacobian

matrices, the inverse dynamic model of the redundant parallel manipulator is set up. It consists of six

linear consistent equations with eight unknown quantities. Then, the optimum solution of the actuating

torques is achieved by employing the Moore-Penrose inverse matrix. It is with minimum norm and least

quadratic sum among the possible actuating torque vectors. A series of new dynamic performance

indices with obvious physical meanings have been proposed in the paper. By decoupling the inverse

dynamics in the exhaustive way, a novel dynamic performance index combining the acceleration,

velocity and gravity terms of the dynamic equations has been presented to evaluate the dynamic

characteristic of the redundant parallel manipulator. With the index, it is possible to control the

performance in the different direction. The index has been applied to the dynamic characteristic

evaluation of the redundant parallel manipulator in the simulation. It is general and can be used for the

dynamic performance evaluation of other types of parallel manipulators.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

A parallel manipulator is a closed-loop kinematic chainmechanism whose end-effector is linked to the base by severalindependent kinematic chains [1–3]. Due to their potentialadvantages such as high accuracy, rigidity, speed and so on, manyinvestigations have been carried on the parallel manipulatorssince the concept was introduced. They have been successfullyapplied to the motion simulators, robotic end-effector and othercircumstances like fast pick-and-place operation.

Redundancy can usually improve the ability and performanceof the parallel manipulator [4–13]. There are many advantages forthe redundant parallel manipulator such as avoiding kinematicsingularities, increasing workspace, improving dexterity, enlar-ging load capability and so on. In general, there are two differenttypes of redundancy for the parallel manipulator: (a) kinematicredundancy and (b) actuation redundancy. A parallel manipulatoris said to be kinematically redundant manipulator when itsmobility of the mechanism is greater than the required degrees offreedom of the moving platform. On the other hand, a parallelmanipulator is called redundantly actuated manipulator when the

ll rights reserved.

ao), [email protected]

Dynamic formulation and10.1016/j.rcim.2008.10.001

number of actuators is greater than the mobility of the mechan-ism. Redundant actuation in parallel manipulator can be im-plemented by the following approaches. The first one is to actuatesome of the passive joints within the branches of the parallelmanipulator. The second one is to add some additional branchesbeyond the minimum necessary to actuate the parallel manip-ulator. The last one may be the hybrid of the above twoapproaches. Like anything has two sides, the redundantly actuatedparallel manipulator has also some disadvantages, though muchattention has been paid to it. It is noted that the actuatorredundancy does not affect the mobility of the parallel manip-ulator while maybe make it to be an over-constrained mechanism.The Jacobian matrix of this kind parallel manipulator which mapsan m-dimensional velocity vector X in the operation space into ann-dimensional joint velocity vector q is not a square matrix anymore. This will bring difficulties in the kinematics and dynamicsanalysis, especially for the dynamic performance evaluation.

Performance evaluation of the parallel manipulator is still anopen problem [14,15]. There are two basic criteria used for thegeneral parallel manipulator, (i) the conditioning indices and (ii)the manipulability measures. Salisbury and Craig [16] introducedthe condition number of the Jacobian matrix and Angeles [17,18]that used the inverse condition number as the kinematic dexterityindex for the serial manipulators. Gosselin [19,20] adopted it tothe parallel manipulators. The condition number should be assmall as possible in order to achieve the best performance and the

performance evaluation of the redundant parallel manipulator.

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Y. Zhao, F. Gao / Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]]–]]]2

manipulator is regarded as in the kinematic isotropy when thecondition number is one. A global conditioning index [21], whichcan be implemented with the numerical integration technique,was presented by Gosselin due to the following facts: (i) thecondition necessary to generate an isotropy cannot providesufficient design constraints to achieve a unique solution. (ii)The global kinematic performance cannot be guaranteed simplyby conducting local optimal design. However, as Lipkin and Duffy[22] and Doty [23,24] et al. pointed out, significant problems maybe caused when using the condition number of the Jacobianmatrix with non-homogeneous physical units in the optimaldesign and control. The condition number of the Jacobian matrixwill not be invariant under scaling of the dimensions of themanipulator when the position and orientation of the end-effectorare included in the kinematic equations. In order to avoid the unitinconsistency problem, some works [25–27] had been done basedon the principle that the plane is determined by three non-collinear points. Gosselin [25] derived a new Jacobian matrix thattransforms actuator velocities into the linear velocities of two orthree nonlinear points on the end-effector for the serial planar orspatial manipulators and then defined a new conditioning indexbased on the new Jacobian matrix. Kim [28] adopted the newconditioning index to the optimization of the Gough–Stewartparallel manipulator. The concept of kinematic manipulabilitywas introduced by Yoshikawa [29] as a measure of themanipulator capability for executing a specific task in a givenconfiguration. He defined the manipulability measure as ageneralized determinant of the Jacobian matrix and separatedthe total manipulability measure into translational and rotationalmanipulability measure later [30]. Hong and Kim [31] defined anew manipulability measure, which included the total volume ofmanipulability ellipsoid and the condition number of the Jacobianmatrix, and optimized the Eclipse parallel manipulator bymaximizing the manipulability measure.

In order to investigate the dynamic characteristic of theparallel manipulator, the dynamic model must be set up firstly.Many works [32–77] can be found on the dynamics of the parallelmanipulators. Several approaches have been applied to thedynamics analysis of parallel manipulators. They mainly can beclassified into four categories: the Newton-Euler method [32–41],Lagrangian method [42–48], Kane’s method [49,50] and virtualwork principle method [51–62]. In fact, the inverse dynamics ofparallel manipulators almost involve all the mechanics principles.Along with these mechanics principles, many mathematicmethods such as screw theory [63], Lie algebra [64], naturalorthogonal complement [65–67], motor algebra [68,69], grouptheory [70], symbolic programming [70,71], geometric approach[72], parallel computational algorithms [73] and system identifi-cation [74] have also been adopted to the dynamics of parallelmanipulators. As a matter of fact, the results of actuate forces/torques computed by different methods are equivalent [75].Theoretically, it can be concluded that there is no trouble inmodeling the dynamics of parallel manipulators. Much attentionshould be focused on the model accuracy, computation efficiencyand practical application. It should be also pointed that thenumber of the unknown quantities is larger than that of the linearconsistent equations for the inverse dynamics of the redundantparallel manipulator. So the solution is infinite on this condition.The most common strategy to solve this kind of problem is byminimizing the Euclidian norm of the actuating torque vectors.

The dynamic models are strongly nonlinear equations, whichmake it difficult to predict the dynamic performance of themanipulator. However, the dynamic performance of the parallelmanipulator should be taken into account for the design, moreadvanced control and trajectory planning. Several dynamic perfor-mance indices have been presented in the literatures [78–87]. Ma

Please cite this article as: Zhao Y, Gao F. Dynamic formulation anRobot Comput Integr Manuf (2008), doi:10.1016/j.rcim.2008.10.001

and Angeles [78,79] introduced the concept of dynamic condition-ing index, which was defined as the least-square difference betweenthe generalized inertia matrix and an isotropic matrix. Asadaintroduced the generalized inertia ellipsoid as a tool to measure thecapability of changing end-effector’s velocity in different directionsfor the given kinetic energy [80,81]. Yoshikawa [82] extended theconcept of kinematic manipulability ellipsoid to dynamic manipul-ability ellipsoid for measuring the case of changing the end-effector’s configuration by a set of joint torques with fixedmagnitude (Euclidian norm). Huang [51,83] adopted the maximumsingular value of the generalized inertia matrix or its row vectormatrix to evaluate the dynamic characteristic of the parallelmanipulator. Since the velocity term and the gravity term of thedynamic model are neglected in the indices, they are not very suitedfor the dynamic performance evaluation of the high speedmanipulator or the heavy manipulator used for the application ofthe large load requirement. In addition, the generalized inertiamatrix is an approximate expression [83]. The key at the basis of thedefinition of the above performance measures is the generalizedinertia matrix which describes the mapping between the jointforces/torques and the accelerations.

According to the matrix theory, the condition number is theratio of the maximum singular value to the minimum singularvalue of the matrix and the manipulability can be represented bythe continued multiplication of all the singular values. Thoughthey have been used widely in robotics, they are not veryappropriate for the parallel manipulators [88]. Some problemsarise when applying them to the performance evaluation: (i) forthe out-parallel manipulator, there are three kinds of singularconfigurations which make the manipulability ellipsoid thatcannot measure the performance quantitatively. Furthermore,when take the manipulability ellipsoid is taken as the perfor-mance index of the parallel manipulator, the volume and theshape of the ellipsoid should be taken into account which makesthe performance prediction become complex. As for the conditionnumber, it cannot be a measurement of distance from asingularity while used widely in the in-parallel manipulators.Huang [89] found that the 2-dof out-parallel manipulator mayalso be nearly at the singular configuration when the conditionnumber of the Jacobian matrix is one. In addition, the denomi-nator of the Jacobian matrix will be lost in the computation of thecondition number when the end-effector is on the center line ofthe workspace. (ii) The distributions of the singular values cannotbe achieved from the condition number, which represents the ill-condition of the matrix and can be thought as a factor amplifyingthe errors of the manipulator or the measure of the matrixreversibility. (iii) The condition indices and the manipulabilitymeasures do not have very obvious physical meanings. (iv) Theconditioning indices and the manipulability measures cannot beapplied to the redundant parallel manipulator directly since theJacobian matrix and the generalized inertia matrix are not squarematrix any more. In fact, there is little work on the dynamiccharacteristic evaluation of the redundant parallel manipulator.

The aim of this paper is to present the work on the dynamicformulation and performance evaluation of the redundant parallelmanipulator. The dynamical equations of the redundant parallelmanipulator have been formulated by mean of the principle ofvirtual work and the concept of link Jacobian matrices [2,56]. Byemploying the Moore-Penrose inverse matrix, the optimumsolution of the actuating torques is achieved. It is with minimumnorm and least quadratic sum among the possible actuatingtorque vectors. Then, a series of new dynamic performance indiceswith obvious physical meanings have been proposed in the paper.By decoupling the inverse dynamics in the exhaustive way, a noveldynamic performance index combining the acceleration, velocityand gravity terms of the dynamic equations is presented to

d performance evaluation of the redundant parallel manipulator.

dx.doi.org/10.1016/j.rcim.2008.10.001

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Y. Zhao, F. Gao / Robotics and Computer-Integrated Manufacturing ] (]]]]) ]]]–]]] 3

evaluate the dynamic characteristic of the redundant parallelmanipulator. It has been applied to the dynamic perfor-mance evaluation of the redundant parallel manipulator in thesimulation.

xy

z

O

uv

w

O'

bi

ai

r

qiei

liwiw

di

Bi

Ai

Ci

Fig. 2. Vector diagram of a PSS kinematic chain.

Ci

Ai

yi′yi

zi′z

x

zi

y

xi′xi

iφ

iϕ

Fig. 3. The local coordinate system of the ith strut.

2. Kinematics

The 8PSS redundant parallel manipulator is shown in Figs. 1and 2. It consists of a moving platform and eight sliders. In eachkinematic chain, the platform and the slider are connected viaspherical ball bearing joints by a strut of fixed length. Each slideris driven by DC motor via linear ball screw. The lead screw of B1,B2, B3 and B4 are vertical to the ground. The lead screw of B5, B6

and the lead screw of B7, B8 are parallel with the ground. They areorthogonal for each other.

For the purpose of analysis, the following coordinate systemsare defined: the coordinate system O�xyz is attached to the fixedbase and another moving coordinate frame O0�uvw is located atthe center of mass of the moving platform. The pose of the movingplatform can be described by a position vector, r, and a rotationmatrix, oRo0 . Let the rotation matrix be defined by the roll, pitchand yaw angles, namely, a rotation of fx about the fixed x axis,followed by a rotation of fy about the fixed y-axis, and a rotationof fz about the fixed z-axis. Thus, the rotation matrix is

oRo0 ¼ Rotðz;fzÞRotðy;fyÞRotðx;fxÞ

¼

cfzcfy cfzsfysfx � sfzcfx cfzsfycfx þ sfzsfx

sfzcfy sfzsfysfx þ cfzcfx sfzsfycfx � cfzsfx

�sfy cfysfx cfycfx

264

375 (1)

where sf denotes the sine of angle f, while cf denotes the cosineof angle f. The angular velocity of the moving platform is given byRef. [56]

x ¼ _fx_fy

_fz

h iT(2)

The orientation of each kinematic strut with respect to the fixedbase can be described by two Euler angles. As shown in Fig. 3, thelocal coordinate system of the ith strut can be thought of as arotation of fi about the z-axis resulting in a Ci�x0iy

0iz0i system

B

B

B

C

C

C

A

A

A

xyz

O

v

w

O'

2a

A

A

C

C

B

B

2b

2c

h

h

2

3

4

2

3

4

2

3

4

7

8

7

8

7

8

2

1-a

Fig. 1. Schematic of an 8PSS redu

Please cite this article as: Zhao Y, Gao F. Dynamic formulation andRobot Comput Integr Manuf (2008), doi:10.1016/j.rcim.2008.10.001

followed by another rotation of ji about the rotated y0 i-axis. Sothe rotation matrix of the ith strut can be written as

oRi ¼ Rotðz;fiÞRotðy0i;jiÞ ¼

cficji �sfi cfisji

sficji cfi sfisji

�sji 0 cji

264

375 (3)

B

C

Au

2a

A

AC

C B

B

2b

2c

h

h

1

1

1

5

65

6 5

62

1-a

h

ndant parallel manipulator.


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The unit vector along the lead screw in the coordinate systemO�xyz is

wi¼oRi

iwi¼oRi

0

0

1

264

375 ¼

cfisji

sfisji

cji

264

375 (4)

So the Euler angles fi and ji can be computed as the following

cji ¼ wiz

sji ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2

ix þw2iy

q; ð0pjiopÞ

sfi ¼ wiy=sji

cfi ¼ wix=sji

if ji ¼ 0; then fi ¼ 0 (5)

2.1. Position analysis

As shown in Fig. 2, the closed-loop position equationassociated with the ith kinematic chain can be written as

r þ ai ¼ liwi þ bi þ di þ qiei; i ¼ 1;2; . . . ;8 (6)

where r, qi, ei, wi, ai bi and di denote the vector OO0, the jointvariable, the unit vector along the lead screw, the unit vector alongstrut CiAi, the vector O0Ai, the vector OBi and the vector from thelead screw to the center point of the joint Ci, respectively.

So the inverse position solution can be achieved

qi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr þ ai � liwi � bi � diÞ

Tðr þ ai � liwi � bi � diÞ

q,

i ¼ 1;2; . . . ;8 (7)

2.2. Velocity analysis

Taking the derivative of Eq. (6) with respects to time yields

_qiei þxi � liwi ¼ v þx� ai (8)

where xi and v denote the angular velocity of the strut CiAi, thelinear velocity of the moving platform.

Taking the dot product of both sides of Eq. (8) with wi yields

_qi ¼wT

i

wTi ei

ðai �wiÞT

wTi ei

" #v

x

� �(9)

Rewriting Eq. (9) in the matrix form yields

_q ¼ J�1q Jx

_X ¼ J _X (10)

where

_q ¼ _q1 _q2 _q3 _q4 _q5 _q6 _q7 _q8

h iT(11)

_X ¼v

x

� �(12)

Jq ¼ diagðwT1e1 wT

2e2 wT3e3 wT

4e4 wT5e5 wT

6e6 wT7e7 wT

8e8 Þ

(13)

Jx ¼w1 w2 w3 w4 w5 w6 w7 w8

a1 �w1 a2 �w2 a3 �w3 a4 �w4 a5 �w5 a6 �w6 a7 �w7 a8 �w8

" #T

(14)

J ¼ J�1q Jx ¼ ½ J

T1 JT

2 JT3 JT

4 JT5 JT

6 JT7 JT

8 �T (15)

where J is the Jacobian matrix which maps the velocity vector Xinto the joint velocity vector q.


2.3. Link velocity analysis

The linear velocity of the point Ai in the Ci�x0 iy0iz0i coordinate

system is

ivAi ¼ _qiiei þ

ixi � liiwi ¼

iv þ ix� ia (16)

Since ixiTiwi ¼ 0, cross multiplying both sides of Eq. (16) with

iwi, so the angular velocity of the driven parallelogram arm isobtained as

ixi ¼1

liðiwi �

ivAi �iwi � _qi

ieiÞ ¼1

liðSðiwiÞ

ivAi � SðiwiÞ _qiieiÞ (17)

where

SðiwiÞ ¼

0 �iwiziwiy

iwiz 0 �iwix

�iwiyiwix 0

2664

3775 (18)

Substituting Eqs. (9) and (16) into Eq. (17) then yields

ixi ¼1

liSðiwiÞ

iRo �SðiwiÞSðiaiÞ

iRo

h i(

� ðiwi �ieiÞ

wTi

wTi ei

ðai �wiÞT

wTi ei

" #)v

x

" #

¼ J iov

x

" #(19)

where

SðiaiÞ ¼

0 �iaiziaiy

iaiz 0 �iaix

�iaiyiaix 0

2664

3775 (20)

iRo¼oR�1

i ¼oRT

i (21)

The velocity of the center of the ith strut in the Ci�x0iy0iz0i

coordinate system is

ivi ¼ivAi �

ixi �li2

iwi (22)

Substituting Eqs. (9), (16) and (19) into Eq. (22) then yields

ivi ¼iRo �SðiaiÞ

iRo

h iþ

li2

SðiwiÞJio

� �v

x

� �¼ J iv

v

x

� �(23)

Rewrite the linear and angular velocity of the ith strut in thematrix form:

ivi

ixi

" #¼

Jiv

J io

" #v

x

� �¼ Jivo

v

x

� �(24)

where Jivo is the link Jacobian matrix which maps thevelocity of the moving platform in the task space intothe velocity of the ith strut in the Ci�x0 iy

0iz0i coordinate

system.


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2.4. Acceleration analysis

Taking the derivative of Eq. (8) with respects to time gives

_v ¼ €qiei � _x� ai �x� ðx� aiÞ þ _xi � liwi þxi � ðxi � liwiÞ

(25)

Taking the dot product of both sides of Eq. (25) with wi andsimplifying yields

€qi ¼1

wTi ei

ðwTi_v þ ðai �wiÞ

T _xþwTi ðx� ðx� aiÞÞ

�wTi ðxi � ðxi � liwiÞÞÞ

¼ JTi

_v

_x

" #þ

1

wTi ei

ððwTi xÞða

Ti xÞ � ðw

Ti aiÞðx

TxÞ þ lijxi �wij2Þ

(26)

Rewriting Eq. (26) in the matrix form yields

€q ¼ J €X þ V (27)

where

V ¼ V1 V2 V3 V4 V5 V6 V7 V8� �T

(28)

Vi ¼1

wTi ei

ððwTi xÞða

Ti xÞ � ðw

Ti aiÞðx

TxÞ þ lijxi �wij2Þ (29)

2.5. Link acceleration analysis

Taking the derivative of Eq. (8) with respects to time in theCi�x0iy

0iz0i coordinate system yields

i _v ¼ €qiiei �

i _x� iai �ix� ðix� iaiÞ

þ i _xi � liiwi þ

ixi � ðixi � li

iwiÞ (30)

Cross multiplying both sides of Eq. (30) with iwi then yields

i _x ¼1

liðiwi �

i _v � ðiwi �ieiÞ €qi �

iwi � ðiai �

i _xÞ

þ iwi � ðix� ðix� iaiÞÞ

� iwi � ðixi � ð

ixi �iwiÞÞÞ (31)

Substituting Eq. (26) into Eq. (31) and simplifying yields

i _x ¼ Jio_v

_x

" #þ

1

liðD1 þD2Þ (32)

where

D1 ¼ �ðiwi �

ieiÞ

wTi ei

ððwTi xÞða

Ti xÞ � ðw

Ti aiÞðx

TxÞ þ lijxi �wij2Þ (33)

D2 ¼ ðixT

iiaiÞð

iwi �ixiÞ � ð

ixTixÞðiwi �iaiÞ (34)

Taking the derivative of Eq. (22) with respects to time yields

i _vi ¼i _vAi �

li2

i _xi �iwi �

ixi �ixi �

li2

iwi

¼ i _v � SðiaiÞ

i _xþ Sði _xÞSðixÞiai

þli2

SðiwiÞi _xi �

li2

SðixiÞSðixiÞ

iwi (35)

Substituting Eq. (32) into Eq. (35) and simplifying yields

i _vi ¼ J iv

_v

_x

" #þ SðixiÞSð

ixiÞiai

þ1

2SðiwiÞðD1 þD2Þ �

li2

SðixiÞSðixiÞ

iwi (36)


3. Inverse dynamic formulation

3.1. Applied and inertia wrenches

According to the D’Alembert’s principle, the force acting on thecenter of mass of each link consists of two parts: the inertial forceand the gravity force. Similarly, the moment acting on each rigidbody is the inertial moment. Assuming there is no frictional forcefor the whole robot system. The resultant of applied and inertiaforces exerted at the center of the mass of the moving platform is

Q P ¼f P

nP

" #¼

f e þmpg �mp _v

ne �oIp _x�x� ðoIpxÞ

" #(37)

where fe and ne are the external force and moment exerted at thecenter of mass of the moving platform, oIp ¼

oRo0o0Ip

o0Ro is theinertia matrix of the moving platform taken about the center ofmass expressed in the O�xyz coordinate system and mp is itsmass.

The resultant of applied and inertia forces exerted at the centerof mass of the ith strut can be expressed in the Ci�x0iy

0iz0i

coordinate system as

iQ i ¼

if iini

" #¼

miiRog �mi

i _vi

�iIii _xi �

ixi � ðiIi

ixiÞ

" #(38)

where iIi is the inertia matrix of the ith cylindrical strut abouttheir respective centers of mass expressed in the Ci�x0iy

0iz0i

coordinate system and mi is its mass.There is pure translational motion for the slider, so the

resultant of applied and inertia forces exerted at the center ofmass of the slider can be expressed in the O�xyz coordinatesystem as

f qi ¼ ðmqig �mqi €qiÞTei (39)

where mqi and qi are the mass and the acceleration of the slider.There is pure rotation motion for the lead screw, coupler and

motor rotor, so the resultant of applied and inertia forces exertedat the screw–coupler–rotor is

Ni ¼ ti � ðILi þ ICi þ IMiÞ€yi (40)

where ILi, ICi and IMi are the rotary inertia of the lead screw, couplerand motor rotor, respectively, ti is the input torque actuated by themotor, €yi is the angular acceleration of the screw–coupler–rotor.The relationship between the lead screw motion and the slidermotion is €yi ¼ ð2p=piÞ €qi. Where pi ¼ 0.05 m�1 is the lead of thelinear ball screw. Other than the speed reduction caused by thepitch of the lead screws there is no speed reducer for theredundant parallel manipulator, otherwise the reduction ratioshould be included in Eq. (40).

3.2. Equations of motion

According to the D’Alembert’s principle, the principle of thevirtual work can be extended from the static to the dynamic case.It can be stated as: the virtual work of the external forces appliedto the system must be zero. For this redundant parallelmanipulator it can be expressed in the formula:

dxTpQ p þ

X8

i¼1

dixTi

iQ i þ dqTf q þ dhTN ¼ 0 (41)

where

dq ¼ dq1 dq2 dq3 dq4 dq5 dq6 dq7 dq8

h iT(42)


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dh ¼ dy1 dy2 dy3 dy4 dy5 dy6 dy7 dy8� �T¼ diag

2pp1

2pp2

2pp3

2pp4

2pp5

2pp6

2pp7

2pp8

!dq

¼ Adq (43)

f q ¼f q1 f q2 f q3 f q4 f q5 f q6 f q7 f q8

h iT(44)

N ¼ N1 N2 N3 N4 N5 N6 N7 N8� �T

(45)

In Eq. (41), the resultant of the applied and inertia forces iQi andits corresponding virtual displacement dixi are expressed in theCi�x0iy

0iz0i coordinate system. From Eq. (24), the relationship

between the above virtual displacement dixi and the virtualdisplacement dxp is determined by

dixTi ¼ dxT

pJTivo (46)

The relationship between the virtual displacement dq and dxp is

dqT ¼ dxTpJT (47)

Substituting Eqs. (43), (46) and (47) into Eq. (41) yields

dxTpQ p þ

X8

i¼1

dxTpJT

ivoiQ i þ dxT

pJTf q þ dxTpJTATN ¼ 0 (48)

Since Eq. (48) is always valid for any dxp, it must follow that

Q p þX8

i¼1

JTivo

iQ i þ JTf q þ JTATN ¼ 0 (49)

Eq. (49) can be written as

JTATs ¼ JTATILCM€h� Q p �

X8

i¼1

JTivo

iQ i � JTf (50)

where

ILCM ¼ diagð ILCM1 ILCM2 ILCM3 ILCM4 ILCM5 ILCM6 ILCM7 ILCM8 Þ

(51)

ILCMi ¼ ILi þ ICi þ IMi (52)

There are eight unknown quantities in the six linear consistentequations. So the solution is infinite. The most common strategyto solve this kind of problem is by minimizing the Euclidian normof the actuating torque vector:

s ¼ �A�TðJTÞþ Q p þ

X8

i¼1

JTivo

iQ i þ JTf q

!þ ILCM

€h

¼ �A�TðJTÞþ Q p þ

X8

i¼1

JTivo

iQ i þ JTf q

!þ ILCMA €q (53)

The physical significance of the above equation is that amongthe possible actuating torque vectors, the optimum solution iswith minimum norm and least quadratic sum. J+ is the Moore-Penrose inverse of the Jacobian matrix. As for any Jacobian matrix,there exists a unitary matrices UAUm�m and VAUn�n such that

D ¼ UHJV ¼

s1

s2

. ..

sr

0

266666664

377777775

(54)

is diagonal. UH is the conjugate transpose of U.So

Jþ ¼ VDþUH (55)


Dþ ¼

s�11

s�12

. ..

s�1r

0

266666664

377777775

(56)

when JARmm�n, J+

¼ JT(JJT)�1¼ Jx

T(JxJxT)�1Jq.

Substituting Eqs. (37)–(40) into Eq. (53) yields

s ¼ � A�T JT� �þ f e

ne

" #� A�T JT

� �þ mpg

0

" #þX8

i¼1

JTivo

miiRog

0

" #(

þ JT ðmq1gÞTe1 ðmq2gÞTe2 ðmq3gÞTe3 ðmq4gÞTe4

h

ðmq5gÞTe5 ðmq6gÞTe6 ðmq7gÞTe7 ðmq8gÞTe8

iT)

þ A�T JT� �þ mp _v

oIp _x

" #þX8

i¼1

JTivo

mii _vi

iIii _xi

" #(

þJT mq1 €q1 mq2 €q2 mq3 €q3 mq4 €q4 mq5 €q5 mq6 €q6 mq7 €q7 mq8 €q8

h iT)

þ ILCMA €qþ A�T JT� �þ 0

x� ðoIpxÞ

" #þX8

i¼1

JTivo

0

ixi � ðiIi

ixiÞ

" #( )

(57)

where 0 ¼ ½0 0 0 �T.From Eq. (57), it is shown that the input torques are determined

by the structure parameters, the position parameters, the kine-matics parameters (including the velocity and the acceleration), thegravity term and the term caused by the external force and momentexerted at the moving platform. Substituting Eqs. (24), (31) and (36)into Eq. (57) and simplifying, the inverse dynamic model of theredundant parallel manipulator is achieved

s ¼ DðqÞ €qþ hðq; _qÞ þ GðqÞ � A�TJ�Tf e

ne

" #(58)

where D(q)q is the acceleration term, h(q,q) is the velocity term andG(q) is the gravity term of the inverse dynamic equations.

4. Performance evaluation

4.1. Kinematic performance evaluation

The relationship between the slider velocity and the inputangular velocity of the motor rotor is

_h ¼ A _q (59)

where _h and q are the input rotor angular velocity and the velocityof the slider, respectively.

So

_h ¼ A _q ¼ AJ _X ¼ Jy _X ¼ ½ Jyv Jyo �v

x

" #

¼ ½ Jy1 Jy2 Jy3 Jy4 Jy5 Jy6 �v

x

" #(60)

where Jy is the matrix which maps the velocity of the movingplatform into the angular velocity of the motor rotor, Jyj is thecolumn vector component of the matrix Jy, Jyv and Jyo are thepartitioned matrix component of the matrix Jy.

When the moving platform translates along or rotates aboutthe x-, y- and z-axis in the unit velocity, the input angular velocityof the motor rotor is

_hj ¼ Jyvj ¼ Jyj; ðj ¼ 1;2; . . . ;6Þ (61)


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ARTICLE IN PRESS


where

v1 ¼ ½1 0 0 0 0 0 �T (62a)

v2 ¼ ½0 1 0 0 0 0 �T (62b)

v3 ¼ ½0 0 1 0 0 0 �T (62c)

v4 ¼ ½0 0 0 1 0 0 �T (62d)

v5 ¼ ½0 0 0 0 1 0 �T (62e)

v6 ¼ ½0 0 0 0 0 1 �T (62f)

are used to represent the unit (angular) velocity of the movingplatform when it translates along or rotates about the x-, y- and z-

axis. They will also be used later to denote the unit (angular)acceleration of the moving platform when it translates along orrotates about the x-, y- and z-axis.

In general, when the moving platform moves in the arbitraryunit velocity

v ¼X6

j¼1

gjvj;X6

j¼1

g2j ¼ 1; kvk2

2 ¼ 1

0@

1A (63)

The input angular velocity vector of the motor rotor is

_hw ¼ Jyv ¼X6

j¼1

gjJyj (64)

The maximum input angular velocity is

_ywmax ¼_hw

1pX6

j¼1

gj

�� Jyj

1¼X6

j¼1

gj

�� _hj

1

pmax _h1

1

_h2

1

_h3

1

_h4

1

_h5

1

_h6

1

¼ max Jy1

1

Jy2

1

Jy3

1

Jy4

1

Jy5

1

Jy6

1

� �¼ max

i¼1;2;...;8;j¼1;2;...;6Jyij

� �¼ Jy 0

1¼ _ymax (65)

So _hmax ¼ Jy 0

1is taken as the kinematic performance index. It

is the maximum input angular velocity when the moving platformtranslates along or rotates about the x-, y- and z-axis in the unitvelocity.

Similarly, when the moving platform has a pure translationalmotion, the following index is adopted to evaluate the transla-tional capability of the redundant parallel manipulator.

_hv max ¼ kJyvk01 (66)

If the moving platform has a pure rotational motion

_homax ¼ kJyok01 (67)

is taken as the index to evaluate the rotational capability of theredundant parallel manipulator.

4.2. Dynamic performance evaluation

From Eqs. (57) and (58), the dynamic model of the redundantparallel manipulator can be written as

s ¼MðXÞ €X þ VðX; _XÞ þ gðXÞ � A�TJ�Tf e

ne

" #

¼ sa þ sv þ sg þ se (68)


where

MðXÞ ¼ A�TðJTÞþ

mpE3

oIp

" #þX8

i¼1

JTivmiJ iv þ

X8

i¼1

JTio

iIiJ io

!

þ A�Tdiagðmq1 mq2 mq3 mq4 mq5 mq6 mq7 mq8 ÞJ

þ AILCMJ

¼ ½M1 M2 M3 M4 M5 M6 �

¼ Mv Mo� �

(69)

is the generalized inertia matrix of the redundant parallelmanipulator which maps the acceleration of the moving platforminto the actuating torques. E3 denotes the unit matrix of orderthree. V(X,X) and g(X) are the velocity term and the gravity termof the inverse dynamic equations, respectively.

4.2.1. Torques caused by the acceleration term

When the moving platform accelerates from the stationary inthe unit acceleration, translates along or rotates about the x-, y-

and z-axis, the input torque caused by the acceleration term is

saj ¼MðXÞvj; j ¼ 1;2; . . . ;6 (70)

So the maximum input torque caused by the acceleration termis

ta max ¼ maxðksajk1Þ ¼ maxðkMjk1Þ ¼ maxðkMijkÞ ¼ kMk01;

i ¼ 1;2; . . . ;8; j ¼ 1;2; . . . ;6

(71)

It is taken as the dynamic performance index to evaluate theacceleration capability of the redundant parallel manipulator. Whenthe moving platform has a pure linear acceleration motion, themaximum input torque caused by the acceleration term is taken asthe index to evaluate the translational acceleration capability

tav max ¼ kMvk01 (72)

If the moving platform has a pure angular acceleration motion,the following maximum input torque caused by the accelerationterm is adopted to evaluate the rotational acceleration capability ofthe parallel manipulator

tao max ¼ kMok01 (73)

4.2.2. Torques caused by the velocity term

From Eqs. (57) and (68), the torque caused by the velocity termis

svj ¼ VðX; _XÞj _X¼vj j¼1;2;...;6 (74)

So the maximum input torque caused by the velocity is

tv max ¼ maxðksvjk1Þ; j ¼ 1;2; . . . ;6 (75)

When the moving platform has a pure translational or a purerotational motion, the maximum input torques caused by thevelocity term are the following, respectively

svv max ¼maxðksvjk1Þ; j ¼ 1;2;3 (76)

tvo max ¼ maxðksvjk1Þ; j ¼ 4;5;6 (77)

4.2.3. Torques caused by the gravity term

The torque caused by the gravity term is

tg ¼ gðXÞ (78)

So the maximum input torques caused by the gravity term is

tg max ¼ ksgk1 (79)


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Table 3The length of the strut CiAi (m)

1 2 3 4 5 6 7 8

li 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

Table 4The mass parameters of the manipulator (kg)

1 2 3 4 5 6 7 8

mi 20 20 20 20 20 20 20 20

mqi 50 50 50 50 50 50 50 50


4.2.4. Dynamic performance index

When the moving platform has a maximum (angular) accel-eration Zi ¼ 1, 2, y, 6 m/s2(rad/s2) and a maximum (angular)velocity lj ¼ 1, 2, y, 6 m/s(rad/s) in the operation space, theprobable maximum input torque is

tavgmax ¼ maxðkjZijjsaij þ l2j jsvjj þ jsg jk1Þ,

i ¼ 1;2; . . . ;6; j ¼ 1;2; . . . ;6 (80)

It is taken as the dynamic performance index which includesthe acceleration, velocity and gravity component of the dynamicequations to evaluate the dynamic characteristic of the redundantparallel manipulator. The physical meaning of the dynamicperformance index can be interpreted as the probable maximuminput torque of one of the actuated joint required for producing amaximum (angular) acceleration Zi i ¼ 1, 2, y, 6 m/s2(rad/s2) anda maximum (angular) velocity lj j ¼ 1, 2, y, 6 m/s(rad/s) of themoving platform in the operation space. It should be as small aspossible to achieve the best dynamic performance for theredundant parallel manipulator.

For Eq. (80), the Zi and lj can be written as

Zi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik _vk2

2 þ k _xk22

q; i ¼ 1;2; . . . ;6 (81)

lj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikvk2

2 þ kxk22

q; j ¼ 1;2; . . . ;6 (82)

where _V and _x are the maximum (angular) acceleration, v and xare the maximum (angular) velocity of the moving platformrequired for the design task.

In order to evaluate the dynamic performance of the redundantparallel manipulator in the different direction, the index ismodified as

savg ¼ kjZijjsaij þ l2j jsvjj þ jsg jk1 (83)

It is the maximum input torque of one of the actuated jointrequired for producing a (angular) acceleration Zi m/s2(rad/s2) anda maximum (angular) velocity lj m/s(rad/s) of the moving plat-form. In order to control the dynamic performance of theredundant parallel manipulator in a different direction, the tavg

can be used for the object function or the constraint function inthe dynamic optimum design.

The contributions of the dynamic performance index are thefollowing: (i) it combines the acceleration, velocity and gravityterms of the dynamic equations to evaluate the dynamiccharacteristic of the redundant parallel manipulator; (ii) itevaluates the dynamic characteristic of the redundant parallelmanipulator by combining a series of decoupled indices, so itmakes it possible to control the performance of the redundant

Table 1The parameters of the base platform (m)

1 2 3 4

xBi 0.400000 0.400000 �0.400000 �0.400000

yBi �0.400000 0.400000 0.400000 �0.400000

zBi 0.000000 0.000000 0.000000 0.000000

Table 2The parameters of the moving platform which are measured in the coordinate frame O

1 2 3 4

xAi 0.400000 0.400000 �0.400000 �0.400000

yAi �0.400000 0.400000 0.400000 �0.400000

zAi �0.166000 �0.166000 �0.166000 �0.166000


parallel manipulator in the different direction by dynamicoptimum design.

5. Simulation

In this section, simulations for the performance evaluation ofthe 8PSS redundant parallel manipulator are presented. Theparameters of the manipulator are given in Tables 1–4.

The mass of the moving platform is mp ¼ 200 kg. The inertiaparameters used in the simulation are given as

o0 Ip ¼

17:333 0 0

0 17:333 0

0 0 33:333

264

375kg m2;

iIi ¼

1:279 0 0

0 1:279 0

0 0 0:005

264

375kg m2;

ILi ¼ 10.5�10�4 kg m2, ICi+IMi ¼ 248�10�4 kg m2. Other para-meters used in the simulation are given as di ¼ 0.244 m andz0 ¼ 1.744 m.

The distribution of the kinematic performance index is shownin Fig. 4 when the pose of the moving platform arefx ¼ fy ¼ fz ¼ 0 and z ¼ z0.

It is shown from Fig. 4(b) that the translational capability ofthe redundant parallel manipulator is equivalent in the investi-gated small workspace. It is consistent with the practical case.From Fig. 4(c), the difference between the maximum andminimum of the performance distribution in the workspace isvery small. In a sense of engineering, the rotational capability ofthe redundant parallel manipulator is approximately equivalent inthe workspace.

5 6 7 8

�0.400000 �0.400000 �2.000000 �2.000000

�2.000000 �2.000000 �0.400000 0.400000

1.500000 1.500000 1.500000 1.500000

0�uvw (m)

5 6 7 8

0.400000 �0.400000 �0.681000 �0.681000

�0.681000 �0.681000 �0.400000 0.400000

�0.037500 �0.037500 �0.037500 �0.037500


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Fig. 4. The distribution of the kinematic performance index (a) _ymax, (b) _yv max and (c) _yo max.

Fig. 5. The distribution of the dynamic performance index (a) tamax, (b) tavmax and (c) taomax.


The distribution of the dynamic performance index to evaluatethe acceleration capability of the redundant parallel manipulatoris shown in Fig. 5, when the pose of the moving platform arefx ¼ fy ¼ fz ¼ 0 and z ¼ z0.

In a sense of engineering, the acceleration capability of theredundant parallel manipulator is approximately equivalent in theworkspace from Fig. 5.

The distributions of the dynamic performance indices relativeto the maximum torques caused by the velocity term and thegravity term are shown in Figs. 6 and 7.


From Figs. 5 and 6, the tvmax is little smaller than the tamax,which means that the torque caused by the velocity term issmaller than that caused by the acceleration term as for thisredundant parallel manipulator. From Fig. 7, the torque causedby the gravity term is also not too small to be neglected inthe dynamic performance index, especially for the heavyparallel manipulator used for the application of the large loadrequirement.

From Eq. (80), the distributions of the dynamic performanceindices are shown in Figs. 8 and 9 with two cases: (i) the desired


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Fig. 6. The distribution of the dynamic performance index (a) tvmax, (b) tvvmax and (c) tvomax.

Fig. 7. The distribution of the dynamic performance index tgmax.

Fig. 8. The distribution of the dynamic performance index tavgmax.

Fig. 9. The distribution of the dynamic performance index tavgmax.


maximum acceleration and velocity of the moving platform areZ1 ¼1, Z2 ¼ 1, Z3 ¼ 1 (m/s2) Z4 ¼ 1, Z5 ¼ 1, Z6 ¼ 1 (rad/s2) andl1 ¼1, l2 ¼ 1, l3 ¼ 1 (m/s) l4 ¼ 1, l5 ¼ 1, l6 ¼ 1 (rad/s); (ii) thedesired maximum acceleration and velocity are Z1 ¼10, Z2 ¼ 10,Z3 ¼ 10 (m/s2) Z4 ¼ 10, Z5 ¼ 10, Z6 ¼ 10 (rad/s2) and l1 ¼10,l2 ¼ 10, l3 ¼ 10 (m/s) l4 ¼ 10, l5 ¼ 10, l6 ¼ 10 (rad/s). The resultsalso show that the index can be applied to the performance


evaluation of the redundant parallel manipulator in the differentdirection since it is combined by a series of decoupled indices.

6. Conclusions

The dynamic formulation and performance evaluation of the8PSS redundant parallel manipulator have been investigated inthe paper. The dynamical equations of the redundant parallelmanipulator have been formulated by the mean of the principle ofvirtual work and the concept of link Jacobian matrices. Byemploying the Moore-Penrose inverse matrix, the optimumsolution of the actuating torques is achieved. It is with minimumnorm and least quadratic sum among the possible actuatingtorque vectors. Then, a series of new dynamic performance indiceswith obvious physical meanings have been proposed in the paper.By decoupling the inverse dynamics in the exhaustive way, a noveldynamic performance index combining the acceleration, velocityand gravity terms of the dynamic equations has been presented toevaluate the dynamic characteristic of the redundant parallelmanipulator. It has been applied to the dynamic performanceevaluation of the redundant parallel manipulator in the simula-tions. The contributions of the dynamic performance index are the


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ARTICLE IN PRESS


following: (i) it combines the acceleration, velocity and gravityterms of the dynamic equations to evaluate the dynamiccharacteristic of the redundant parallel manipulator; (ii) itevaluates the dynamic characteristic of the redundant parallelmanipulator by combining a series of decoupled indices, so itmakes it possible to control the performance index of theredundant parallel manipulator in the different direction bydynamic optimum design. The dynamic performance index isgeneral and can be used for the dynamic performance evaluationof other types of the parallel manipulators.

Future investigation will be paid on the dynamic optimumdesign of the redundant parallel manipulator by adopting thepresented indices. However, these related issues will be addressedby separate papers.

Acknowledgments

This research is sponsored by the China Postdoctoral ScienceFoundation (Grant no. 20070410719). We would also like to thankthe anonymous reviewers for their very useful comments.

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