KINEMATIC IDENTIFICATION OF PARALLEL MECHANISMS BYA DIVIDE AND CONQUER STRATEGY

Sebastian Durango a,§, David Restrepo a, Oscar Ruiz a, John Restrepo-Giraldo b, Sofiane Achiche b

a CAD CAM CAE research laboratory, EAFIT University, Medellın, Colombiab Management Engineering Dept., Technical University of Denmark, Lyngby, Denmark

sdurang1@eafit.edu.co, drestr21@eafit.edu.co, oruiz@eafit.edu.co, jdrg@man.dtu.dk, soac@man.dtu.dk§ Corresponding author

Keywords: Parallel mechanisms, kinematic identification, robot calibration.

Abstract: This paper presents a Divide and Conquer strategy to estimate the kinematic parameters of parallel symmetri-cal mechanisms. The Divide and Conquer kinematic identification is designed and performed independentlyfor each leg of the mechanism. The estimation of the kinematic parameters is performed using the inverse cali-bration method. The identification poses are selected optimizing the observability of the kinematic parametersfrom a Jacobian identification matrix. With respect to traditional identification methods the main advantagesof the proposed Divide and Conquer kinematic identification strategy are: (i) reduction of the kinematic iden-tification computational costs, (ii) improvement of the numerical efficiency of the kinematic identificationalgorithm and, (iii) improvement of the kinematic identification results. The contributions of the paper are:(i) The formalization of the inverse calibration method as the Divide and Conquer strategy for the kinematicidentification of parallel symmetrical mechanisms and, (ii) a new kinematic identification protocol based onthe Divide and Conquer strategy. As an application of the proposed kinematic identification protocol the iden-tification of a planar 5R symmetrical mechanism is simulated. The performance of the calibrated mechanismis evaluated by updating the kinematic model with the estimated parameters and developing simulations.

1 INTRODUCTION

In mechanisms and manipulators the accuracy of theend-effector critically depends on the knowledge ofthe kinematic model governing the control model(Zhuang et al., 1998). Therefore, to improve the accu-racy of a mechanism its kinematic parameters have tobe precisely estimated by means of a kinematic iden-tification procedure (Renaud et al., 2006).

Kinematic identification is an instance of the robotcalibration problem. The estimation of rigid-body in-ertial parameters and the estimation of sensor gainand offset are instances of calibration problems at thesame hierarchical level of the kinematic calibrationproblem (Hollerbach et al., 2008).

This paper is devoted to the kinematic identifi-cation of parallel symmetrical mechanisms. Parallelmechanisms are instances of closed-loop mechanismstypically formed by a moving platform connected toa fixed base by several legs. Each leg is a kinematicchain formed by a pattern of links, actuated and pas-

sive joints relating the moving platform with the fixedbase. If the pattern of joints and links is the same foreach leg and each leg is controlled by one actuator,then the parallel mechanism is denoted symmetrical(Tsai, 1999).

For parallel mechanisms the kinematic identifi-cation is usually performed minimizing an error be-tween the measured joint variables and their corre-sponding values calculated from the measured end-effector pose through an inverse kinematic model(Zhuang et al., 1998; Renaud et al., 2006). Thismethod is preferred for the identification of parallelmechanisms because:1. Inverse kinematics of parallel mechanisms is usu-ally derived analytically avoiding the numerical prob-lems associated with any forward kinematics solution(Zhuang et al., 1998; Renaud et al., 2006).2. The inverse calibration method is considered tobe the most numerically efficient among the identi-fication algorithms for parallel mechanisms (Renaudet al., 2006; Besnard and Khalil, 2001).

3. With respect to forward kinematic identificationno scaling is necessary to balance the contribution ofposition and orientation measurements (Zhuang et al.,1998).

In the case of parallel symmetrical mechanismsthe inverse kinematic modeling can be formulated us-ing independent loop-closure equations. Each loop-closure equation relates the end-effector pose, the ge-ometry of a leg, and a fixed reference frame. In conse-quence, an independent kinematic constraint equationis formulated for each leg forming the mechanism.For the case of parallel symmetrical mechanisms theset of constraint equations is equal to the number oflegs and to the number of degrees of freedom of themechanisms. Each kinematic constraint equation canbe used for the independent identification of the pa-rameters of the leg correspondent to the equation.

The independent identification of the kinematicparameters of each leg in parallel mechanisms allowsto improve:1. The numerical efficiency of the identification algo-rithm (Zhuang et al., 1998).2. The kinematic calibration performance by the de-sign of independent experiments optimized for theidentification of each leg.

The independent identification of leg parametersin parallel mechanisms was sketched in (Zhuanget al., 1998) and developed for the specific case ofGough platforms in (Daney et al., 2002; Daney et al.,2005). However, the idea of the independence in thekinematic identification of each leg in a parallel mech-anism is not completely formalized.

This article presents a contribution to the im-provement of the pose accuracy in parallel symmet-rical mechanisms by a kinematic calibration protocolbased on inverse kinematic modeling and a divide andconquer strategy. The proposed divide and conquerstrategy takes advantage of the independent kinematicidentification of each leg in a parallel mechanism notonly from a numerical stand point but also from theselection of the optimal measurement set of poses thatimproves the kinematic identification of the parame-ters of the leg itself.

The layout for the rest of the document is as fol-lows: section 2 develops a literature review on the in-verse calibration of parallel mechanisms method, sec-tion 3 presents the divide and conquer identificationof parallel mechanisms strategy, section 4 develops akinematic identification of parallel mechanisms pro-tocol, section 5 presents the simulated kinematic iden-tification of a planar 5R symmetrical mechanism us-ing the identification protocol, finally, in section 6 theconclusions are developed.

2 LITERATURE REVIEW

The modeling of mechanical systems include the de-sign, analysis and control of mechanical devices. Anaccurate identification of the model parameters is re-quired in the case of control tasks (Hollerbach et al.,2008). Instances of models of mechanical systems in-cludes kinematic, dynamic, sensor, actuators and flex-ibility models. For parallel mechanisms updating thekinematic models with accurately estimated param-eters is essential to achieve precise motion at high-speed rates. This is the case when parallel mech-anisms are used in machining applications (Renaudet al., 2006).

The inverse calibration method is accepted as thenatural (Renaud et al., 2006; Zhuang et al., 1998) andmost numerically efficient (Besnard and Khalil, 2001)among the identification algorithms for parallel mech-anisms. The inverse calibration method is based oninverse kinematic modeling and a external metrolog-ical system. The calibration is developed minimiz-ing an error residual between the measured joint vari-ables and its estimated values from the end-effectorpose though the inverse kinematic model. The deriva-tion of the inverse kinematic model of parallel mech-anisms is usually straightforward obtained (Merlet,2006).

For our Divide and Conquer kinematic calibra-tion strategy we adopt the inverse calibration method.The method takes advantage of an intrinsic charac-teristic of parallel mechanisms: the straightforwardcalculation of the inverse kinematics. However, notall the intrinsic characteristics of parallel mechanismsare exploited. Specifically, (Zhuang et al., 1998; Ryuand Rauf, 2001) reported that for parallel mechanism,methods based on inverse kinematics allow to iden-tify error parameters of each leg of the mechanismindependently. The independent parameter identifica-tion of each leg is reported to improve the numericalefficiency of the kinematic identification algorithm,(Zhuang et al., 1998). However, it is not reported ageneral kinematic identification strategy based on theindependent identification of the legs and its advan-tages with respect to traditional identification meth-ods.

This article presents a contribution to the kine-matic calibration of parallel mechanisms developing akinematic identification protocol based on the inversecalibration method and on the independent identifica-tion of the parameters of each leg (Divide and Con-quer strategy).

With respect to traditional identification methods,our Divide and Conquer strategy has the following ad-vantages:

1. The identification poses can be optimized to theidentification of reduced sets of parameters (the setscorresponding to each leg).2. The independent identification of the parametersof each leg improves the numerical efficiency of theidentification algorithms.3. By 1. and 2. the identified set of parameters iscloser to the real (unknown) set of parameters thansets identified by other traditional calibration meth-ods.

The divide and Conquer strategy for the inde-pendent kinematic identification of the parameters ofeach leg in a parallel symmetrical mechanism is pre-sented in section 3.

3 DIVIDE AND CONQUERIDENTIFICATION STRATEGY

Parallel symmetrical mechanisms satisfy (Tsai,1999):1. The number of legs is equal to the number of de-grees of freedom of the end-effector.2. All the legs have an identical structure. This is,each leg has the same number of active and passivejoints and the joints are arranged in an identical pat-tern.

In a practical way, the definition of parallel sym-metrical mechanism covers most of the industrial par-allel structures. For parallel symmetrical mechanismsthe kinematic identification by inverse kinematics anda divide and conquer strategy is stated for each leg κ

independently, κ = 1,2, . . . ,nlimbs:Given

1. A set of nominal kinematic parameters of the κthleg (ϕκ). nκ parameters are assumed to be identified:

ϕκ =[ϕκ,1 . . .ϕκ,nκ

]T. (1)

2. An inverse kinematic function gκ relating the κthactive joint variable (qκ) with the end-effector pose(r). For the jth pose of the mechanism the inversefunction of the κth leg is defined to be:

g jκ : ϕκ× r j→ q j

κ,

j = 1,2, . . . ,N.(2)

3. A set of N end-effector measured configurations(Rκ) for the identification of the κth leg:

Rκ =[r1

κ · · · rNκ

]T. (3)

4. A set of measured input variables (Qκ) correspond-ing to the set of end-effector measurements (Rκ):

Qκ =[q1

κ · · · qNκ

]T. (4)

GoalTo find the set of unknown (real) kinematic pa-rameters (ϕκ) that minimizes an error between themeasured joint variables (Qκ) and their correspond-ing values (Qκ) estimated from the measured end-effector poses by the inverse kinematic model gκ. Theproblem can be formally stated as the following non-linear minimization problem:

ϕκ :N

∑j=1

∥∥∥q jκ− qκ

(r j

κ,ϕκ

)∥∥∥2is minimum,

subject to : Rκ ⊂WR,

WR is the usable end− effector workspace.

(5)

The optimization problem is constrained by theuseful workspace (a workspace without singularities)of the mechanism.

A kinematic identification of parallel symmetricalmechanisms protocol based on the Divide and Con-quer identification strategy is developed in section 4.

4 KINEMATIC IDENTIFICATIONPROTOCOL

Based on the Divide and Conquer strategy forthe kinematic identification of parallel symmetricalmechanisms (section 3) the following kinematic iden-tification protocol (Fig. 1) is proposed.1. Given the nominal parameters of the κth leg (ϕκ,Eq. 1) and the correspondent inverse kinematic func-tion (gκ, Eq. 2) to calculate the κth Jacobian identifi-cation matrix of a representative set of postures of theusable workspace:

Cκ(WR,ϕκ) =∂gκ(WR,ϕκ)

∂ϕTκ

. (6)

2. Given the Jacobian identification matrix calculatedin the first step to select an optimal set of postures(Rκ(Cκ)) for the kinematic identification of the κthleg. The set of postures is selected searching the im-provement of the observability of the set of parame-ters ϕκ. To select the poses we adopt the active cali-bration algorithm developed by (Sun and Hollerbach,2008) that reduces the complexity of computing anobservability index reducing computational time forfinding optimal poses. The optimized identificationset of postures is then defined in the following man-ner:

Rκ : O1(Rκ) is maximal,

O1(Rκ) =nκ√s1 s2 · · ·snκ

nκ

,

Rκ ⊂WR,

(7)

Calculation of the Jacobian

identification matrix

Pose selection by Active Robot Calibration Algorithm (Sun and Hollerbach, 2008)

Estimation of kinematic parameters process

is the end-effector usable workspace

End-effector usable workspace, WR

Identified parameters,

Nominal parameters,Inverse kinematics equationfor the κth leg:

x

Jacobian identification matrix, CΚ

Updated kinematic model

Update kinematic model

Set of optimal identification postures, RΚ

Set of end-effector external measurements, Set of active-joint measurements,

is minimum,

,.

q q rj j

Figure 1: Kinematic identification of parallel symmetricalmechanisms protocol.

were O1 is an observability index of the identifica-tion matrix (Cκ(Rκ,ϕκ)) of the κth leg, nκ is the num-ber of parameters to be identified in the κth leg, ands1, s2 . . . ,snκ

are the singular values of the identifica-tion matrix Cκ. As a rule of thumb, in order to sup-press the influence of measurement noise, the numberof identification poses should be two or three timeslarger than the number of parameters to be estimated(Jang et al., 2001).3. Given the optimized set of identification posturesobtained in the second step and the correspondent setsof active joint (Qκ) and end-effector (Rκ) measure-ments to solve the optimization problem defined onEq. 5 for the identification of the kinematic parame-ters (ϕκ) of the κth leg.4. Given the identified set of parameters of the κthleg obtained in the third step to update the kinematicmodel of the parallel mechanism.

The protocol is repeated until all the legs in themechanism are identified.

With respect to traditional identification algo-rithms for the kinematic identification of parallelmechanism (Renaud et al., 2006; Zhuang et al., 1998)the proposed kinematic identification protocol has the

following advantages:1. Reduction of the kinematic identification compu-tational costs. If a linear least-squares estimation ofthe kinematic parameters is used to solve the identi-fication problem (Eq. 5), then the correction to beapplied to the kinematic parameters (∆ϕ) can be esti-mated iteratively as (Hollerbach and Wampler, 1996):

∆ϕ =(CTC

)−1CT

∆Q. (8)

The computational cost of the matrix inversion(CTC)−1 is reduced proportionally to the square ofthe number of legs of the mechanism, Table 1.2. Improvement of the numerical efficiency of thekinematic identification algorithm by the independentidentification of the parameters of each leg.3. Improvement of the kinematic identification by thedesign of independent experiments optimized for theidentification of each leg.

Table 1: Computational and measurement costs of kine-matic identification.

Traditional kinematic Divide and conqueridentification identification

Regressor CTC(N nlimbs × N nlimbs) CTκ Cκ(N× N)

Computationalcost (Matrix ∝ N3 nlimbs

3 ∝ N3 nlimbsinversion)

The kinematic identification of parallel mecha-nisms protocol is used in the simulated identificationof a planar 5R symmetrical mechanism in section 5.

5 RESULTS

The results on kinematic identification of parallelmechanisms by a Divide and Conquer strategy arepresented using a case study: the simulated kinematicidentification of the planar 5R symmetrical parallelmechanism.

The planar 5R symmetrical mechanism (Fig. 2)has two degrees-of-freedom (DOF) that allows it toposition the end-effector point (P) in the plane thatcontains the mechanism. The mechanism is formedby two driving links (l1 and l2) and a conducted dyad(L1 and L2), Fig. 2. The planar 5R symmetricalmechanism is an instance of the parallel symmetricalmechanisms defined in section 3. A complete charac-terization of the assembly configurations (Cervantes-Sanchez et al., 2000), kinematic design (Cervantes-Sanchez et al., 2001; Liu et al., 2006a; Liu et al.,2006b; Liu et al., 2006c), workspace (Cervantes-Sanchez et al., 2001; Cervantes-Sanchez et al., 2000;Liu et al., 2006a), singularities (Cervantes-Sanchez

et al., 2001; Cervantes-Sanchez et al., 2000; Liu et al.,2006a) and performance atlases (Liu et al., 2006b) arereported. However, no research is reported on kine-matic identification.

A1

P(x,y)

X

Yl1 l2A2

L1 L2

P(x,y)

l2

A2

L2

P(x,y)

A1

l1

L1

Leg1Leg2

Figure 2: Planar 5R symmetrical mechanism

The kinematic identification of the planar 5Rsymmetrical mechanism is simulated using the kine-matic identification of parallel symmetrical mecha-nisms protocol (section 4) under the following con-ditions:1. A linear model is assumed for the active joint Aκ:

θκ = kψκ + γκ, (9)

where the kκ represent the joint gain, γκ is the jointoffset, ψκ is the measured active joint angle and θκ isthe active joint angle, κ = 1,2.2. In parallel mechanisms the principal source of er-ror in positioning is due to limited knowledge of thejoint centers, leg lengths and active joint parameters(Daney et al., 2002). In consequence, the parametersto be estimated are the attachment points (Aκ), the leglengths (lκ, Lκ), and the joint gain and offset (kκ, γκ),κ = 1,2:

ϕκ = [lκ Lκ Aκx Aκy kκ γκ]T . (10)

3. The external parameters associated with the mea-suring device will not be identified. For the exter-nal measuring system this implies that its position isknown and coincident with the reference frame X−Yand the measurement target is coincident with theend-effector point.4. The nominal kinematic parameters of the mecha-nism are disturbed adding a random error with normaldistribution and a standard deviation σ. The nominaland disturbed parameters are shown in Table 2.5. The constrain equation of the inverse kinematicsis defined independently for the κth leg (Liu et al.,2006a), κ = 1,2:

Lκ2 = (x− lκ cosθκ−Aκx)

2+

(y− lκ sinθκ−Aκy)2. (11)

6. The end-effector and joint workspace are limitedby the maximal inscribed workspace (MIW), Fig. 2.

Table 2: Planar 5R symmetrical mechanism. Nominal andreal (disturbed) parameters.

Nominal Real Nominal Realvalue value value value

A1x [m] -0.5000 -0.4988 A2x [m] 0.5000 0.4961A1y [m] 0.0000 0.0028 A2y [m] 0.0000 0.0066k1 1.0000 1.004 k2 -1.0000 -0.9984γ1 [rad] 0.0000 0.0048 γ2 [rad] 3.1416 3.1418l1 [m] 0.7500 0.7507 l2 [m] 0.7500 0.7559L1 [m] 1.1000 1.0995 L2 [m] 1.1000 1.0959

The MIW corresponds to the maximum singularity-free-end-effector workspace limited by a circle (Liuet al., 2006c).7. Each leg is identified using a set of 18 posturesof the mechanism to measure the end-effector posi-tion and the corresponding active joint variable. Thedesigned sets of identification postures in the end-effector workspace are presented in Fig. 4b.8. The set of end-effector measurements (Rκ) andits corresponding active joint measurements (Qκ) aresimulated using forward kinematics and adding ran-dom disturbances with normal distribution and stan-dard deviation σ = 1 ·10−4.9. A linearization of the inverse kinematics is used foriteratively solving the non-linear optimization prob-lem (Eq. 5), then, for the jth identification pose theidentification problem of the κth leg is in the follow-ing form:

∆q jκ =

∂g jκ

∂ϕκ

∆ϕκ =C jκ∆ϕ,

∆q jκ = q j

κ− q jκ,

∆ϕκ = ϕκ−ϕκ.

(12)

Using N = 18 measurements to identify the set of pa-rameters ϕκ the identification problem is stated in thefollowing manner:

∆Qκ =Cκ∆ϕκ,

Cκ =[C1

κ · · ·CNκ

]T,

∆Qκ =[∆q1

κ · · ·∆qNκ

]T,

(13)

were Cκ is the identification matrix of the κth leg. Theparameters of the κth leg can be updated using a lin-ear least-squares solution of Eq. 13, (Hollerbach andWampler, 1996):

∆ϕκ = (CTκ Cκ)

−1CTκ ∆Qκ. (14)

10. An alternative traditional kinematic identificationby inverse kinematic modeling is calculated and usedas a comparison with respect to the proposed kine-matic identification protocol. The traditional identifi-cation is performed by means of a set of 18 optimized

postures selected in order to maximize the observabil-ity of the total identification matrix. The observabilitywas defined as the Eq. 7. The designed set of identi-fication postures is presented in Fig. 4a.

The results of the kinematic identification underthese conditions are presented in Fig. 3 (residual er-rors in kinematic parameters before and after calibra-tion). The residual errors are calculated as the dif-ference between the real (virtually disturbed) param-eters and the estimated parameters. Finally, Fig. 4presents the estimated local position errors for theMIW after calibration and the selected postures forkinematic identification. Additionally the computa-tional and measurement identification costs are esti-mated for the identification of the planar 5R parallelmechanism, Table 3. The measurement costs of theDivide and Conquer strategy are incremented with re-spect to a traditional identification method. The incre-ment of the measurements is required because eachleg requires an independent set of end-effector mea-surements. In the case of a traditional identificationthe set of end-effector measurements is common to allthe legs. In despite of the measurement increment theDivide and Conquer identification results in a supe-rior estimation with respect to a traditional kinematicidentification methods (Renaud et al., 2006; Zhuanget al., 1998).

Table 3: Planar 5R symmetrical mechanism. Computa-tional and measurement costs of kinematic identification.

Traditional kinematic Divide and conqueridentification identification

Regressor CTC(36×36) CTκ Cκ(18× 18)

Computationalcost (Matrix ∝ 183 ·23 ∝ 183 ·2inversion)Measurement 2 ·18 ·2 = 72 18 ·2(2+1) = 108cost

6 CONCLUSION

This article presents a new (Divide and Conquer)strategy for the kinematic identification of parallelsymmetrical mechanisms. The new strategy devel-ops a formalization of the inverse calibration methodproposed by (Zhuang et al., 1998). The identificationstrategy (section 3) is based on the independent identi-fication of the kinematic parameters of each leg of theparallel mechanism by minimizing an error betweenthe measured active joint variable of the identified legand their corresponding value, estimated through aninverse kinematic model. With respect to traditional

identification methods the Divide and Conquer strat-egy presents the following advantages:1. Reduction of the kinematic identification computa-tional costs,2. Improvement of the numerical efficiency of thekinematic identification algorithm and,3. Improvement of the kinematic identification re-sults.

Based on the Divide and Conquer strategy, a newprotocol for the kinematic identification of parallelsymmetrical mechanisms is proposed (section 4, Fig.1). For the selection of optimal identification posturesthe protocol adopts the active robot calibration algo-rithm of (Sun and Hollerbach, 2008). The main ad-vantage of the active robot calibration algorithm is thereduction of the complexity of computing an observ-ability index for the kinematic identification, allowingto afford more candidate poses in the optimal pose se-lection search. The kinematic identification protocolsummarizes the advantages of the Divide and Con-quer identification strategy and the advantages of theactive robot calibration algorithm.

The kinematic identification protocol is demon-strated with the simulated identification of a planar5R symmetrical mechanism (section 5). The perfor-mance of our identification protocol is compared witha traditional identification method obtaining an im-provement of the identification results (Figs. 3, 4).

ACKNOWLEDGMENTS

The authors wish to acknowledge the financial sup-port for this research by the Colombian Administra-tive Department of Sciences, Technology and Innova-tion (COLCIENCIAS), grant 1216-479-22001.

REFERENCES

Besnard, S. and Khalil, W. (2001). Identifiable parametersfor parallel robots kinematic calibration. In IEEE In-ternational Conference on Robotics and Automation,2001. Proceedings 2001 ICRA, volume 3.

Cervantes-Sanchez, J. J., Hernandez-Rodrıguez, J. C., andAngeles, J. (2001). On the kinematic design of the 5rplanar, symmetric manipulator. Mechanism and Ma-chine Theory, 36(11-12):1301 – 1313.

Cervantes-Sanchez, J. J., Hernandez-Rodrıguez, J. C., andRendon-Sanchez, J. G. (2000). On the workspace, as-sembly configurations and singularity curves of therrrrr-type planar manipulator. Mechanism and Ma-chine Theory, 35(8):1117 – 1139.

Daney, D., Lorraine, I., and LORIA, V. (2002). Op-timal measurement configurations for Gough plat-

|φreal

−φnominal

|

|φreal

−φtraditional Identif.

|

|φreal

−φD&C Identif.

|

0123456

l1 l2 L 1 L 2

∆φ

[mm

]a. Link lengths

01234567

A 1x A 1y A 2x A 2y

∆φ

[mm

]

b. Position of fixed joints1 2

01

2

3

4

5

k k γ1

γ2

∆φ

c. Active joint parametersFigure 3: Planar 5R mechanism. Residual errors in the kinematic parameters before and after calibration.

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

0.70.80.91

1.11.21.31.41.5

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

0.70.80.91

1.11.21.31.4

x [m]

y [m

]

x [m]

y [m

]

1.19

0.89

6.68e-03

0.30

0.60

Position Error [mm]

A1

P(x,y)

X

Yψ1

l1 l2A2

L1 L2

ψ2

a) Calibration with traditional identification method.

b) Calibration with divide and conqueridentification strategy.

Poses for traditional identification.Poses for D&C identification of left leg.Poses for D&C identification of rigth leg.

Root mean square error = 0.575 [mm] Root mean square error = 0.251 [mm]

Figure 4: Planar 5R mechanism. Estimated end-effector local root mean square error for the maximal inscribed workspace(MIW) after calibration.

form calibration. In IEEE International Confer-ence on Robotics and Automation, 2002. Proceedings.ICRA’02, volume 1.

Daney, D., Papegay, Y., and Madeline, B. (2005). Choos-ing measurement poses for robot calibration with thelocal convergence method and Tabu search. The Inter-national Journal of Robotics Research, 24(6):501.

Hollerbach, J., Khalil, W., and Gautier, M. (2008). SpringerHandbook of Robotics, Model Identification. Springer.

Hollerbach, J. and Wampler, C. (1996). The calibrationindex and taxonomy for robot kinematic calibrationmethods. The International Journal of Robotics Re-search.

Jang, J., Kim, S., and Kwak, Y. (2001). Calibration of geo-metric and non-geometric errors of an industrial robot.Robotica, 19(03):311–321.

Liu, X., Wang, J., and Pritschow, G. (2006a). Kinematics,singularity and workspace of planar 5r symmetricalparallel mechanisms. Mechanism and Machine The-ory, 41(2):145 – 169.

Liu, X., Wang, J., and Pritschow, G. (2006b). Performanceatlases and optimum design of planar 5r symmetricalparallel mechanisms. Mechanism and Machine The-ory, 41(2):119 – 144.

Liu, X., Wang, J., and Zheng, H. (2006c). Optimum designof the 5r symmetrical parallel manipulator with a sur-

rounded and good-condition workspace. Robotics andAutonomous Systems, 54(3):221 – 233.

Merlet, J. (2006). Parallel Robots. Springer Netherlands,second edition.

Renaud, P., Vivas, A., Andreff, N., Poignet, P., Martinet, P.,Pierrot, F., and Company, O. (2006). Kinematic anddynamic identification of parallel mechanisms. Con-trol Engineering Practice, 14(9):1099–1109.

Ryu, J. and Rauf, A. (2001). A new method for fullyautonomous calibration of parallel manipulators us-ing a constraint link. In 2001 IEEE/ASME Inter-national Conference on Advanced Intelligent Mecha-tronics, 2001. Proceedings, volume 1.

Sun, Y. and Hollerbach, J. (2008). Active robot calibrationalgorithm. In Robotics and Automation, 2008. ICRA2008. IEEE International Conference on, pages 1276–1281.

Tsai, L. (1999). Robot analysis: the mechanics of serialand parallel manipulators. Wiley-Interscience.

Zhuang, H., Yan, J., and Masory, O. (1998). Calibrationof Stewart platforms and other parallel manipulatorsby minimizing inverse kinematic residuals. Journal ofRobotic Systems, 15(7).