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Acta Astronautica

Acta Astronautica 68 (2011) 1147–1155

0094-57

doi:10.1

n Corr

E-m

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

Dynamics analysis of space robot manipulator with joint clearance

Yang Zhao, Zheng Feng Bai n

Department of Astronautic Engineering, Harbin Institute of Technology, Harbin 150001, Heilongjiang, PR China

a r t i c l e i n f o

Article history:

Received 21 December 2009

Received in revised form

22 September 2010

Accepted 5 October 2010Available online 3 November 2010

Keywords:

Joint clearance

Space robotics

Contact

Robot dynamics

Dynamics simulation

65/$ - see front matter & 2010 Elsevier Ltd. A

016/j.actaastro.2010.10.004

esponding author. Tel./fax:+86 451 8640 207

ail address: baizhengfeng@126.com (Z.F. Bai).

a b s t r a c t

A computational methodology for analysis of space robot manipulator systems, con-

sidering the effects of the clearances in the joint, is presented. The contact dynamics

model in joint clearance is established using the nonlinear equivalent spring-damp model

and the friction effect is considered using the Coulomb friction model. The space robot

system dynamic equation of manipulator with clearance is established. Then the

dynamics simulation is presented and the dynamics characteristics of robot manipulator

with clearance are analyzed. This work provides a practical method to analyze the

dynamics characteristics of space robot manipulator with joint clearance and improves

the engineering application. The computational methodology can predict the effects of

clearance on space robot manipulator preferably, which is the basis of space robot

manipulator design, precision analysis and ground test.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In general dynamic analysis of robot space manipulatorsystems it is assumed that the kinematic joints are ideal orperfect, that is, clearance, local deformations and weareffects are neglected. However, the clearances in mechan-ism are unavoidable due to the assemblage, manufacturingerrors and wear [1–11]. Such clearance is necessary toallow the relative motion between the connected bodiesand to permit the components assemblage as well. Theclearance occurs in each active joint with the movement ofmechanism. The movements of the real mechanism aredeflected from the ideal mechanism and the motionaccuracy is decreased due to the joint clearances. Theexistence of clearance in joints also leads to the impactdynamic load, and affects the transfer of the system loadand the destruction and failure of the mechanism. Therequirement of predicting the system dynamics is veryurgent. There are two possible ways to solve the problem[12]: one is improving the manufacturing precision, but it isunrealistic for cost and the other is analyzing the relative

ll rights reserved.

9.

motion of the mechanism with clearances, confirming themain factor affecting the dynamic characteristics of themechanism, designing the rational mechanism to reducethe impact of the clearances as much as possible. Thesecond method has great value to mechanism design.Hence the study on mechanism dynamics with jointclearance is important and necessary.

Robot manipulators are widely used for help when indangerous, monotonous and tedious jobs. The dynamics ofmanipulator are studied in Refs. [13–17], in which theflexible link, dynamics modeling and control of manipu-lator are studied; however, the joints of robot manipulatorare considered perfect, and there are no considerations onthe clearances at the joints. These studies have been lessfocused on the effects of clearances on dynamics charac-teristics of the robot manipulator system. As a matter offact, the impacts within the machine joints with clearanceslead to the degradation of the overall system performance.

Kosuge et al. [18] analyzed the precision of planarmanipulator on silicon. They expressed the inaccuracy ofthe manipulator as the sum of the inaccuracies introducedby the clearance of different joints of the mechanism. Tinget al. [19] modeled the joints clearance by a small link withthe length equal to half of the joints clearance. Based on thisassumption, they gave the limits of the uncertain pose of

Fig. 1. Schematic representation of a revolute joint with clearance.

Oi

eij

JournalBear

Oj

X

Y

O

oi

r

oj

r

ex

ey

Fig. 2. Clearance model in revolution joint of mechanical system.

Y. Zhao, Z.F. Bai / Acta Astronautica 68 (2011) 1147–11551148

the end effector. Zhu and Ting [20] presented a probabilitydensity function (PDF) to examine the performance uncer-tainty caused by the joint clearance in robot. Chebbi et al.[21] dealt with the prediction of the pose error due to thejoints clearance for a 3-UPU parallel robot; they focused onthe development of an analytical predictive model for thepose error. Flores and Ambrosio [22] presented a computa-tional methodology for dynamic analysis of multibodymechanical systems with joint clearance. Parenti-Castelliand Venanzi [23] presented a new method based on theprinciple of virtual work for studying the kinematicinfluence of clearance affected pairs on the position andorientation of the links of spatial mechanisms. In conclu-sion, modeling and dynamics analysis of imperfect jointswith clearance are of great importance in the design andthe control of space robot manipulator system.

This paper focuses on the dynamics characteristics ofspace robot manipulator with joint clearance. The contactdynamics model in the clearance is established using thenonlinear spring-damp model and the friction effect isconsidered using the Coulomb friction model. Then thedynamics model of robot manipulator system with clear-ance is established and the dynamics simulation is pre-sented. Finally, the effects of clearance on the dynamicscharacteristics of manipulator are analyzed and discussed.The regularities of contact forces in clearance are studied indetail, and the effects of clearance size on the manipulatordynamics characteristic are analyzed.

2. Dynamics modeling of space robot manipulatorsystem with clearance

2.1. Definition of clearance

The existence of clearance in the joints of space robotmanipulator is inevitable. The joint clearance of robotmanipulator system is necessary to allow relative motionof the connected bodies, as well as to permit the assem-blage of the multibody system. Clearance also exists due tomanufacturing tolerances, imperfections, wear and mate-rial deformation. It is known that the performance of aspace robot manipulator is degraded by the presence of theclearance due to the impact force, which contributes tobearing failure due to the shock loading, to reduce the lifedue to the material fatigue, causing energy dissipation andto excite unwanted shake responses.

In general, a clearance joint can be included in a robotmanipulator much like a revolute joint. The classical approach,known as zero-clearance approach, considers that the con-necting points of two bodies linked by a revolute joint arecoincident. The introduction of the clearance in a jointseparates these two points. Fig. 1 depicts a revolute jointwith clearance. The difference in radius between the bearingand the journal defines the size of the radial clearance [3,9,24].Although, a revolute joint with clearance does not constraintany degree of freedom from the mechanical system like theideal joint, it imposes some kinematic restrictions, limiting thejournal to move within the bearing. Thus, when clearance ispresent in a revolute joint, the two kinematic constraints areremoved and two degrees of freedom are introduced instead.The dynamics of the joint is then controlled by forces working

on the journal and the bearing. Thus, whilst a perfect revolutejoint in a mechanical system imposes kinematic constraints, arevolute clearance joint leads to force constraints. Whencontact exists between the journal and bearing, a contactforce is applied perpendicular to the plane of collision.

Therefore, the motion of manipulator with clearancealways includes contact/impact process. The key point ofdynamics model for robot manipulator with clearance is toincorporate the clearance model and the dynamics model,which needs the accurate definition of the contact/impactprocess in the joint clearance. The clearance is defined asfollows:

c¼ RB�RJ ð1Þ

where RB and RJ are the radius of bearing and journal.

2.2. Mathematic model of joint with clearance

As shown in Fig. 2, Oi and Oj donate the center of bearand journal, respectively, ro

i and roj are the vectors donating

the positions of bear and journal in the global inertiacoordinate, respectively. Hence in Fig. 2, the clearance

Y. Zhao, Z.F. Bai / Acta Astronautica 68 (2011) 1147–1155 1149

vector [25,26] is

eij ¼ roi�ro

j ð2Þ

where eij represents the eccentric vector of shaft relative tobear. So the eccentricity of clearance vector can be stated as

eij ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

xþe2y

qð3Þ

The unit normal vector of bear and journal at the point oftheir contact is stated as:

n¼ eij=eij ð4Þ

that is n¼eijffiffiffiffiffiffiffiffiffiffiffi

e2x þe2

y

pFig. 3 describes the collision between bear and journal

and rQi and rQ

j are the positional vectors of the bear andjournal, respectively, which are described in global coor-dinate frame. The contact penetration depth caused bycollision between bear and journal can be described as

d¼ eij�c ð5Þ

where c represents the clearance, it is a constant.Therefore, d can be used to decide whether the bear and

journal have collided. The kinematic contact conditionbetween the bear and journal can be given by

d¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

xþe2y

q�cZ0 ð6Þ

It is important to estimate the loss of energy in thecourse of contact between bear and journal; hence it isnecessary to calculate the relative velocity between theirsurfaces. When we project d to the facet of contact, we canget the normal velocity and tangential velocity of thepotential points of contact between the bear and journal:

vn ¼ ðdÞT _nvt ¼ ðdÞT _t ð7Þ

X

Oi

eij

Bear

Journal

Oj

Y

Qi

rQj

r

QiQj

O

nt

�

Fig. 3. Revolution joint clearance with contact.

where the unit tangential vector t can be obtained byreversing the unit normal vector n for 901.

2.3. Model for contact force in clearance

To evaluate efficiently the contact forces between thebearing and journal in a revolute joint with clearance,special attention must be given to the numerical descrip-tion of the contact model [3,27,28]. Information on theimpact velocity, material properties of the colliding bodiesand geometry characteristics of the surfaces in contactmust be included into the force contact model. Thesecharacteristics are observed with a continuous contactforce, in which the deformation and contact force areconsidered as continuous functions.

The simplest contact force model is spring-damp model,which is modeled by a parallel spring-damp element. Thespring represents the elasticity of the contacting bodies,and the damper describes the loss of kinetic energy duringthe impact. The contact force depends on the stiff and dampproperties of the spring. The expression of the contactmodel considering the clearance using the spring-dampmodel is generally as follows [9,29–31]:

Fn ¼ FkþFd ¼ kndmþcðdÞ _d ð8Þ

where Fn is the normal contact force, Fd the damp force, kn

the generalized contact stiffness, d the penetration depth,c(d) the damp factor related with d and m the exponent.

The best-known contact force law between two spheres ofisotropic material is due to the pioneering work of Hertz. Thenormal contact/impact force is defined by the Hertz contacttheory. The nonlinear normal contact/impact force Fk, knownas the Hertz force-displacement law, is evaluated as a functionof indentation between two colliding bodies as follows:

Fk ¼ knd1:5

ð9Þ

where d is the penetration depth. The exponent m is generallyset to 1.5.

In order to overcome the limitations of the linear dampmodel and to satisfy the boundary conditions, the dampforce Fd is defined, using the nonlinear damp model, asfollows:

Fd ¼ cðdÞ _d ð10Þ

where c(d) is the damp factor related with d. In this paper,c(d) is defined as follows:

cðdÞ ¼ 0:75ð1�e2Þknd3=2=v0 ð11Þ

where e is the coefficient of restitution and v0 the initialimpact velocity.

The tangential friction force, known as the Coulomb, isevaluated as follows:

Ft ¼�mdFnsgnðvÞ

vðq, _q,tÞa0 ð12Þ

where sgn(v) is a sign function, vðq, _q,tÞa0 the relative slipvelocity between the contact points of the bodies and md

the coefficient of sliding friction.

Table 2Mass and inertia properties of spacecraft.

Mass (kg) 1738.6

Jx (kg/m2) 2903.4

Jy (kg/m2) 1902.03

Jz (kg/m2) 3315.39

Jxy (kg/m2) 2.9

Jyz (kg/m2) 82.7

Jzx (kg/m2) 82.7

Table 1Geometric and mass properties of the manipulator system.

Arm 1 Arm_2

Length (m) 0.8 1.0

Width (m) 0.02 0.02

Thickness (m) 0.02 0.02

Density (kg/m3) 2699.0 2699.0

Modulus of elasticity, E (N/m2) 6.895�1010 6.895�1010

Poisson ratio, g 0.3 0.3

Table 3Parameters used in the dynamics simulation of the space robot

manipulator.

Drive torque, T1 (Nm) 0.2 sin(2pt)

Drive torque, T2 (Nm) 0.05 sin(2pt)

Contact stiffness, K (N/m) 6.75�106

Damp factor, C (Ns/m) 175

Friction coefficient, md 0.1

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2.4. Equations of motion for robot manipulator with clearance

The dynamics model of robot manipulator is establishedconsidering the clearance model. The clearance of jointleads to two different motion phases of bodies connectedwith the clearance joint: one is the bodies move free in theclearance and the other is the bodies contact and interact.So the robot manipulator system with clearances betweenbodies is a variable topology system.

The variable topology system of manipulator system issolved using the dynamic segmentation modeling method.Therefore, the dynamic equation is obtained using theLagrange multiplier method.

In the free motion phase, the dynamic equation is

M €qþC _qþKqþUTql¼ f

Fðq,tÞ ¼ 0 ð13Þ

where q is the generalized coordinate column matrix, M, C

and K are the generalized mass matrix, generalized dampmatrix and generalized stiffness matrix, respectively, Fq isthe Jacobin matrix of constraint equation, f the generalizedforce matrix and l the Lagrange multiplier column matrix.

In contact phase, the bodies contact and interact. Thusthe contact forces exist in the clearances. The dynamicequation is

M €qþC _qþKqþUTql¼ f þFc

Fðq,tÞ ¼ 0 ð14Þ

where Fc is the contact force relative to q, which containsboth normal contact force Fn and tangential friction force Ft.

3. Dynamics simulation of robot manipulator withclearance

3.1. Properties of space robot manipulator system

The space robot manipulator system in this work has aspacecraft body and two flexible link arms. Fig. 4 shows theconfiguration of the space robot manipulator system, whichconsists of two flexible manipulator arms. In this work, thearms are considered as flexible bodies and the spacecraft isconsidered as rigid body. The joint between arm 1 andmanipulator root is considered as ideal joint, i.e. there is noclearance in this joint, and the joint between arms 1 and 2 isconsidered as real joint, i.e. there is a clearance in this joint.The mass and geometric properties of the space robotmanipulator system are shown as follows (Tables 1 and 2).

Arm_1Spacecraft

ideal joint_1

Fig. 4. Sketch map of space robot manipulator system

Table 3 presents the parameters used in the dynamicssimulation of the space robot manipulator. In this work,the simulation of space robot manipulator system with jointclearance is considered as plane mechanical system and thedynamic simulation used ADAMS, a mechanical systemsimulation software package.

3.2. Simulation results

3.2.1. Dynamics characteristic analysis of space robot

manipulator with clearance

In order to study the effects of clearance on thedynamics characteristics of space robot manipulator, thiswork chooses clearance c=0.2 mm and makes dynamicsimulation for the manipulator with clearances and the

Arm_2

clearance joint_2

clearance

with clearance ( the joint clearance zoom-in).

Y. Zhao, Z.F. Bai / Acta Astronautica 68 (2011) 1147–1155 1151

ideal manipulator without clearance, the simulationresults are as follows.

Fig. 5 shows that the angle displacement curves of robotmanipulator are almost completely a coincidence betweenthe real manipulator with clearance and the ideal manip-ulator without clearance. It indicates that the clearance hasless effect on angle displacement of manipulator. Fig. 6shows that the effects of clearance on the angular velocityof robot manipulator are obvious. The angular velocity ofmanipulator is shaky around the ideal angular velocity ofmanipulator, which will lead to effects on the movementaccuracy. Fig. 7 shows that the effects of clearance on theangular acceleration are extraordinarily obvious. The angu-lar acceleration of manipulator with clearance is obviouslyshake and the amplitude is increased from the manipulatorwithout clearance, which indicate that the existence ofclearance will lead to oscillation of the manipulator move-ment and influence the movement stability of the manip-ulator. Fig. 8 shows that the existence of clearance leads to

Fig. 5. Angular displacement of space robot manipulator (clearan

Fig. 6. Angular velocity of space robot manipulator (clearance

impulse type clearance contact force and the contact forceis high-frequency shake, and the contact force is increased.

The reason is that the value of radial clearance in thejoint is very small, which is 0.2 mm; the force of clearancecollisions is only a local effect; each collision has very shorttime, small impulse; so it leads to slight effects on theangular displacement of manipulator, but more andseverer effects on the angular velocity and angular accel-eration of manipulator. The results verify that the angularvelocity and angular acceleration are sensitive to theclearance and the effects of clearance on angular velocitiesand angular acceleration of the robot manipulator systemare severer than the effects of clearance on angulardisplacement of the robot manipulator system.

It indicates that the existence of clearance will lead tothe impact dynamic load and affect the movement accu-racy and stability of space robot manipulator, which willlead to the destruction and failure of the space robotmanipulator.

ce is 0.2 mm in nonideal joint): (a) Arm_1 and (b) Arm_2.

is 0.2 mm in nonideal joint): (a) Arm_1 and (b) Arm_2.

Fig. 7. Angular acceleration of space robot manipulator (clearance is 0.2 mm in nonideal joint): contact force (a) without joint clearance (Arm_1) and

(b) with joint clearance (Arm_2) (clearance is 0.2 mm in nonideal joint).

Fig. 8. Contact force in clearance joint 2: (a) contact force without joint clearance and (b) contact force with joint clearance (clearance is 0.2 mm in

nonideal joint).

Y. Zhao, Z.F. Bai / Acta Astronautica 68 (2011) 1147–11551152

Y. Zhao, Z.F. Bai / Acta Astronautica 68 (2011) 1147–1155 1153

3.2.2. Effects of clearance size on dynamics characteristic of

space robot manipulator

In order to study the effects of clearance size on the spacerobot manipulator dynamics characteristics, this workincreases the size of clearance to c=0.4 mm and makesdynamic simulation. The simulation results are shown asfollows.

Fig. 9 shows that the clearance size has slight effects onthe angular displacement of space robot manipulator.Figs. 10 and 11 show the higher size of clearance, themore obvious shake and higher amplitude of angularvelocity and angular acceleration of robot manipulator,but lower shake frequency, which is due to the bearing andthe journal that have longer time to contact with each otherwhen the clearance is higher. Thus the shake periodic islonger, and the angular velocity and angular acceleration ofthe manipulator have lower shake frequency. Fig. 12 showsthat the existence of clearance leads to impulse typeclearance contact force and the contact force is high-

Fig. 9. Angular displacement of space robot m

Fig. 10. Angular velocity of space robot ma

frequency shake. It indicates the higher size of clearance,the higher contact force and the lower contact frequency,which are due to the clearance that leads to translationaldisplacement and velocity of manipulator arms, when thesize of clearance is big, the translational time of manip-ulator arms is long, and hence there is long translationalvelocity. Therefore, the higher the size of clearance, thelower the contact frequency, but the contact force is higher.The results indicate that the dynamics characteristics ofspace robot manipulator are obviously different with thedifferent sizes of clearance, and higher size of clearance hassevere effects on the dynamics characteristics of the spacerobot manipulator.

4. Conclusions

In this paper, the effects of joint clearance on space robotmanipulator system are studied. A computational metho-dology for the analysis of space robot manipulator systems,

anipulator: (a) Arm_1 and (b) Arm_2.

nipulator: (a) Arm_1 and (b) Arm_2.

Fig. 11. Angular acceleration of space robot manipulator: contact force (a) without joint clearance (Arm_1) and (b) with joint clearance (Arm_2).

Fig. 12. Contact force in clearance joint 2 (X direction): (a) contact force without joint clearance and (b) contact force with joint clearance..

Y. Zhao, Z.F. Bai / Acta Astronautica 68 (2011) 1147–11551154

considering the effects of the clearance in the joint, ispresented. The contact dynamics model in the joint clear-ance is established using the nonlinear equivalent spring-damp model and the friction effect is considered using theCoulomb friction model. Then the dynamics simulation ispresented and the results show that the clearance hasobvious effects on the dynamics characteristics. (1) Theclearance has slight effects on angle displacement of robotmanipulator. (2) The effects of clearance on the angularvelocity of robot manipulator are obvious. The angularvelocity of robot manipulator is shaky around the idealangular velocity of robot manipulator. (3) The effects ofclearance on the angular acceleration are extraordinarilyobvious. The angular acceleration of manipulator withclearance is obviously shaky and the amplitude isincreased. It can be found that the angular velocity andangular acceleration are sensitive to the clearance, and itindicates that the existence of clearance will lead to theimpact dynamic load, and influences the movement accu-racy and stability of the space robot manipulator, which

will lead to the destruction and failure of the space robotmanipulator.

The existence of clearance leads to the joint contactforce increase, and the clearance induces impulse typeclearance contact force and the contact force is high-frequency shake. Higher size of clearance induces high-frequency shakes and larger shake amplitudes of theangular velocity and angular acceleration of manipulator,but lower shake frequency; higher size of clearance leads tohigher contact force and lower contact frequency.

The dynamics model of the space robot manipulatorwith joint clearance describes the dynamics characteristicsof the space robot manipulator system more realistically.The dynamics analysis of space robot manipulator withjoint clearance is the basis of the motion accuracy analysisof space robot manipulator system and it is useful for thecontrol system design of space robot manipulator andground text. This work provides a practical method toanalyze the dynamics characteristics of space robot manip-ulator with joint clearance and improves the engineering

Y. Zhao, Z.F. Bai / Acta Astronautica 68 (2011) 1147–1155 1155

application, which put forward up space robot system tothe practical engineering.

Acknowledgments

This research is supported by the National NaturalScience Foundation of China (Grant no. 50975056) andthe Program for Changjiang Scholars and InnovativeResearch Team in University (PCSIRT).

References

[1] B.B.A. Olivier, R. Jesus, Modeling of Joints with clearance in flexiblemulti-body system, International Journal of Solids and Structures 39(2002) 41–63.

[2] G.S. Nurre, J.P. Sharkey, B.J.D. Nelson, Preserving mission, on-orbitmodifications to Hubble space telescope pointing control system,Journal of Guidance Control and Dynamics 18 (2) (1995) 222–229.

[3] P. Flores, J. Ambrosio, H.C.P. Claro, H.M. Lankarani, C.S. Koshy, A studyon dynamics of mechanical systems including joints with clearanceand lubrication, Mechanism and Machine Theory 41 (2006)247–261.

[4] J.C.G. Oeden, Analysis of clearance in multibody system, Multi-bodySystem Dynamics 13 (2005) 401–420.

[5] T.S. Wang, X.R. Knog, B.L. Wang, X.R. Ma, Analysis of deployment ofspacecraft appendages with hinge gap, Journal of Harbin Institute ofTechnology 33 (3) (2001) 283–286.

[6] S. Erkaya, I. Uzmay, A neural–genetic (NN–GA) approach for optimis-ing mechanisms having joints with clearance, Multibody SystemDynamics 20 (1) (2008) 69–83.

[7] S. Erkaya, I. Uzmay, Investigation on effect of joint clearance ondynamics of four-bar mechanism, Nonlinear Dynamics 2009 58 (1-2)(2009) 179–198.

[8] P. Flores, J. Ambrosio, J.P. Claro, Dynamic analysis for planar multi-body mechanical systems with lubricated Joints, Multibody SystemDynamics 12 (2004) 47–74.

[9] P. Flores, J. Ambrosio, J.C.P. Claro, H.M. Lankarani, Dynamic behaviourof planar rigid multibody systems including revolute joints withclearance, Proceedings of the Institution of Mechanical Engineers,Part-K. Journal of Multibody Dynamics 221 (2) (2007) 161–174.

[10] P. Flores, Modeling and simulation of wear in revolute clearancejoints in multibody systems, Mechanism and Machine Theory 44 (6)(2009) 1211–1222.

[11] I. Khemili, L. Romdhane, Dynamic analysis of a flexible slider-crankmechanism with clearance, European Journal of Mechanics A/Solids.27 (2008) 882–898.

[12] G.Q. Wang, H.Z. Liu, Dynamic response of 4 R four-bar mechanismwith a clearance joint, Mechanical Transmission 25 (3) (2001) 18–20.

[13] X.R. Xu, W.J. Chung, Y.H. Chou, Modeling of kineto-elastodynamicrobots with flexible links, in: Proceedings of the IEEE InternationalConference on Robots and Automation, 1999, pp. 753–758.

[14] B. Subudhi, A.S. Morris, Dynamic modeling, simulation and control ofa manipulator with flexible links and joints, Robots and AutonomousSystems 41 (2002) 257–270.

[15] M.C. Reedman, Approaches for dynamic modeling of flexible manip-ulator systems, Control Theory Application 4 (6) (2003) 401–411.

[16] Y. Aoustin, A. Formalsky, On the feed forward torques and referencetrajectory for flexible two-link arm, Multibody System Dynamics 3(1999) 241–265.

[17] B.O. Xu, K. Fujimoto, Y. Hayakawa, Control of two-link flexiblemanipulators via generalized canonical transformation, in: Proceed-ings of the IEEE Conference on Robots, Automation and Mechatro-nics, 2004, pp. 107–112.

[18] K. Kosuge, T. Fukuda, M. Mehregany, Kinematic analysis of precisionplanar manipulator on silicon, IEEE Transaction on Transducers. 91(1991) 618–621.

[19] K.L. Ting, J. Zhu, D. Watkins, The effect of joints clearance on positionand orientation deviation of linkages and manipulators, Mechanismand Machine Theory 35 (2000) 391–401.

[20] J.M. Zhu, K.L. Ting, Uncertainty analysis of planar and spatial robotswith joint clearances, Mechanism and Machine Theory 35 (2000)1239–1256.

[21] A.H. Chebbi, Z. Affi, L. Romdhane, Prediction of the pose errorsproduced by joints clearance for a 3-UPU parallel robot, Mechanismand Machine Theory 44 (2009) 1768–1783.

[22] P. Flores, J. Ambrosio, Revolute joints with clearance in multibodysystems, Computers and Structures 82 (2004) 1359–1369.

[23] V. Parenti-Castelli, S. Venanzi, Clearance influence analysis onmechanisms, Mechanism and Machine Theory 40 (2005)1316–1329.

[24] P. Flores, J. Ambrosio, J.C.P. Claro, et al., Influence of the contact-impact force model on the dynamic response of multibody systems,Journal of Multibody Dynamics 220 (1) (2006) 21–34.

[25] S. Bing, J. Ye, Dynamic analysis of the reheat-stop-valve mechanismwith revolute clearance joint in consideration of thermal effect,Mechanism and Machine Theory 43 (12) (2008) 1625–1638.

[26] T. Qiang, Y. Zhang, L. Chen, P. Flores, Dynamics of spatial flexiblemultibody systems with clearance and lubricated spherical joints,Computers and Structures 87 (13-14) (2009) 913–929.

[27] S. Erkaya, I. Uzmay, Optimization of transmission angle for slider-crank mechanism with joint clearances, Structural and Multidisci-plinary Optimization 37 (2009) 493–508.

[28] S. Mukras, N.H. Kim, N.A. Mauntler, T. Schmitz, W.G. Sawyer, Analysisof planar multibody systems with revolute joint wear, Wear 268 (5-6)(2010) 643–652.

[29] H.M. Lankarani, P.E.A. Nikravesh, Contact force model with hyster-esis damping for impact analysis of multibody systems, Journal ofMechanical Design 112 (1990) 369–376.

[30] P. Flores, J. Ambrosio, J.C.P. Claro, H.M. Lankarani, Influence of thecontact-impact force model on the dynamic response of multibodysystems, Proceedings of the Institution of Mechanical Engineers,Part-K. Journal of Multibody Dynamics 220 (2006) 21–34.

[31] P. Flores, R. Leine, R. Glocker, Modeling and analysis of rigid multi-body systems with translational clearance joints based on thenonsmooth dynamics approach, Multibody System Dynamics 23(2) (2010) 165–190.