International Journal of Robotics and Automation, Vol. 24, No. 2, 2009

KINEMATIC ANALYSIS OF A

NOVEL THREE DEGREE-OF-FREEDOM

PLANAR PARALLEL MANIPULATOR

B. Li,∗ J. Zhao,∗ X. Yang,∗ and Y. Hu∗∗

Abstract

In this paper, a novel unsymmetrical five–four bar Planar Parallel

Manipulator (PPM) with three Degree-of-Freedoms (DOFs) is pro-

posed, the new manipulator is composed of a double-layer five-bar

part with a four-bar part located in between. Targeting for the

application of Chinese dish cooking machine, the detailed kinematics

analysis for the new PPM is developed. Different from traditional

three-DOF PPM, the manipulator has only four assembly modes. In

orientation workspace analysis, the workspace map is shown to be

an annular zone. The two-point velocity method is used in dexterity

analysis to solve the inhomogeneous problem, and the five–four bar

PPM exhibits good dexterity. By analysis of both Inverse Kinemat-

ics Singularity (IKS) and Direct Kinematics Singularity (DKS), the

distributions of the singular curves in workspace are obtained, the

DKS analysis also reveals the similarity between the DKS and the

direct displacement analysis: the related analysis can be decomposed

into a four-bar part and a five-bar part respectively. All the analysis

presented in the paper can facilitate the dimensional synthesis and

trajectory planning for the manipulator.

Key Words

Parallel manipulator, kinematics, workspace, dexterity, singularity

1. Introduction

In recent years, there was a trend towards the analysis andsynthesis of the reduced-DOF, i.e. less than six, parallelmanipulators [1]. Although spatial parallel manipulatorsdeserve more research efforts for their complexity in kine-matics, dynamics and control, PPM have also attractedconsiderable attentions for their simple structure and easy

∗ Shenzhen Graduate School, Harbin Institute of Technol-ogy, Shenzhen, 518055, P.R. China; State Key Labora-tory of Robotics and System (HIT), Harbin, 150001, P.R.China; e-mail: libing.sgs@hit.edu.cn, jguozhao@gmail.com,yangxj@hitsz.edu.cn

∗∗ Cognitive Technologies Lab, Shenzhen Institute of AdvancedIntegration Technology, Chinese Academy of Sciences/ChineseUniversity of Hong Kong, Shenzhen, 518067, P.R. China;e-mail: ying.hu@siat.ac.cn

Recommended by Prof. Jingzhou Yang(paper no. 206-3314)

implementation. Many three-DOF PPMs have been pro-posed in the literatures, which can be classified into twocategories: nonredundant and redundant. A nonredundantthree-DOF PPM has only three actuators and is conven-tionally made up of three identical limbs connecting thefixed base to the moving platform. In fact, Merlet [2] enu-merated all the seven feasible structures for them. Some ofthem, especially the 3-RRR, 3-PRR and 3-RPR type, havebeen studied extensively [3, 4]. A redundantly actuatedPPM has more than three actuators. Wu et al. proposeda three-DOF PPM with two symmetrical limbs, each ofwhich contains two prismatic actuators [5]. Ebrahimi et al.invented a new kinematically redundant PPM by addingone prismatic actuator to each limb of traditional 3-RRRmanipulator [6]. All the above mentioned manipulatorsare of symmetrical nature.

In this paper, a new three-DOF PPM with unsymmet-rical structure is proposed. The manipulator evolves fromthe traditional 3-RRR manipulator by relocating the endof one limb to the end of adjacent limb as shown in Fig. 1.In the figure, the end of the second limb is relocated to theend of first limb. Thus the first and second limbs becomeone limb and it constitutes a five-bar part. If we let itbe still, then the remaining part of the manipulator is afour-bar part with the moving platform as one of its link.Therefore, we can call this new manipulator as a five–fourbar manipulator. A similar method has been used to ob-tain parallel kinematics machines, such as 6–6, 3–6 and 3–3connections of a hexapod, but they still have symmetricalstructure. To the best of authors’ knowledge, the proposedunsymmetrical five–four bar manipulator is novel.

The manipulator is originally designed for an auto-matic Chinese dish cooking machine to imitate skilledchef’s operation, which needs the cooking pan to performa rather complicated planar motion. The typical Chinesecuisine includes a cooking pan and fire for heating up thein-pan material, there are two main motions for a skilledchef to operate on the cooking pan: one is to move thepan with a planar ellipse trajectory to avoid any cookingmaterial sticking to the pan; the second motion is to makethe cooking pan move quickly upward and downward, thusthe in-pan material can be turned over so that fire canevenly heat up the double sides of the material, this is the

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Figure 1. From 3-RRR manipulator to a new manipulator.

Figure 2. Proposed novel three-DOF PPM.

key requirement of Chinese cooking process.

Traditional serial manipulator is used in the develop-ment of the first prototype, which includes a ball screwfor one direction translation, a crank-slider mechanism forthe other direction translation and a motor for rotation.The major drawback for this manipulator is that it cannotimitate the chef’s dexterous motion. Moreover, the actu-ators are rather cumbersome because not all of them aremounted on the base. The proposed PPM in this paper isdesigned to overcome the above shortcomings.

Due to the special architecture of the proposed PPM,the five–four bar manipulator exhibits different character-istics from the traditional 3-RRR manipulator. First ofall, all the three actuators can be installed at the baseand thus reduce their inertia. Furthermore, the controlprocess can be significantly simplified as the manipulatorcan be treated as a combination of five-bar and four-barmechanisms, precise imitation of the chef’s motion can beachieved by trajectory planning. However, the manipula-tor’s stiffness may not be strong enough. For the specificapplication, some modifications are made to improve thecharacteristics of the proposed manipulator. As the as-sembly space is limited, we change the fixed base to forman isosceles triangle; we also use a double-layer five-bar tosolve the problem of stiffness reduction. In addition, themoving platform is also modified to avoid mechanical infer-ences as a cooking container should be attached on it. By

the above modifications, a novel manipulator is obtainedwhose solid model is depicted in Fig. 2(a). The fixed baseis connected to the moving platform by a double-layer five-bar part and with a four-bar part lying in between. Forthis manipulator, a national patent has been applied [7].

Due to the special architecture of proposed manipula-tor, the five–four bar manipulator exhibits different char-acteristics from the traditional 3-RRR manipulator. Thispaper is organized as follows: in Section 2, the displace-ment analysis of the manipulator is performed. Based onthe displacement analysis, the workspace, dexterity andsingularity, analysis are performed in Section 3–5 respec-tively. Finally, a conclusion is drawn based on all theanalysis.

2. Displacement Analysis

2.1 Detailed Description of the Five–Four Bar Ma-nipulator

The kinematic structure of the five–four bar manipulatoris shown in Fig. 2(b). In the manipulator, the five-barpart is O1A1B2A2O2, where B1 also belongs to it. Thefour-bar part is O3A3B3B1. Isosceles O1O2O3 definesthe fixed base, and B3B1P forms the moving platform.ΔA1B1B2 in the figure as a whole is a rigid link connectingto other parts by three revolute joints. By the Grubler-Kutzbach formula, we can easily conclude the DOF of the

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manipulator is three.A base frame OXY with its origin at midpoint of O1O2

is established, its x-axis points along the direction of−−−→O1O2

and y-axis along−−→OO3. Three actuators are all located at

O1, O2 and O3 respectively, and α1, α2, α3 are used todenote the actuating angle with respect to x-axis. Theorientation of moving platform is denoted by β, which is

the angle between−−−→B3B1 and the x-axis. Coordinates of P

in base frame is used to represent the position of the movingplatform. Hence the pose (position and orientation) of themoving platform can be described by p=(x, y, β)T . Abody frame O′X ′Y ′ attached to the moving platform isalso established with it origin at P , x’-axis points along

the direction of−−−→B3B1.

To simplify subsequent analysis, we denote the

vector of each link as follows: r1 =−−→OO1, r2 =

−−→OO2,

r3 =−−→OO3, a1 =

−−−→O1A1, a2 =

−−−→O2A2, a3 =

−−−→O3A3, b1 =

−−−→A1B1,

b2 =−−−→A2B2, b3 =

−−−→A3B3, c1 =

−−→B1P , c2 =

−−−→B2B1, c3 =

−−→B3P ,

d1 =−−−→A1B2, d2 =

−−−→B3B1. The bold face letter is used to

represent vectors. Let λ=π−∠B3B1P , ∠B2B1A1 = γ1,∠A1B2B1 = γ2. In the rest of this paper, the normallowercase letters are scalars and can be used to representthe Euclidian norm of vectors, e.g. a1 = ||a1|| is the lengthof link O1A1. We use subscript x and y to denote thecoordinates of any point. For example, (B1x, B1y)

T repre-sents the coordinates of point B1 in base frame. The samenotation applies to the two components of a vector. Forinstance, a1 =(a1x′ , a1y′)T means the vector componentsin the body frame.

2.2 Inverse Displacement Analysis

For the Inverse Displacement Analysis (IDA), p=(x, y, β)T

is given, the actuation angles q=(α1, α2, α3)T are to be

found. Although the manipulator evolves from the 3-RRRPPM, we cannot solve the IDA by substitution of specificvalue to existing solutions because of the modifications wemade. Therefore, the conventional vector loop method isused to solve the problem. In fact, the manipulator can bedecomposed into three subchains as shown in Fig. 3.

We can formulate three vector loop equations accord-ing to the three subchains as follows:

Figure 3. Three sub chains in the five–four bar manipulator.

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−−→OP = r1 + a1 + b1 + c1−−→OP = r2 + a2 + b2 + c2 + c1−−→OP = r3 + a3 + b3 + c3

(1)

From which we can obtain the solution to IDA asfollows:

αi = arctan

(figi − δiei

√e2i + f2

i − g2ieigi + δifi

√e2i + f2

i − g2i

)(i = 1, 2, 3)

(2)where:

e1 = 2a1[x+ r1 − c1 cos(λ+ β)]

f1 = 2a1[y − c1 sin(λ+ β)]

g1 = [x+ r1 − c1 cos(λ+ β)]2 + [y − c1 sin(λ+ β)]2

+ a21 − b

e2 = 2a2[x− r2 − c1 cos(λ+ β)− c2x]

f2 = 2a2[y − c1 sin(λ+ β)− c2y]

g2 = [x− r1 − c1 cos(λ+ β)− c2x]2 + [y − c1 sin(λ+ β)

− c2y]2 + a22 − b22

e3 = 2a3[x− d2 cosβ − c1 cos(λ+ β)]

f3 = 2a2[y − r3 − d2 sinβ − c1 sin(λ+ β)]

g3 = [x− d2 cosβ − c1 cos(λ+ β)]2 + [y − r3

− d2 sinβ − c1 sin(λ+ β)]2 + a23 − b23

and δi =±1 is branch index, which determines the workingmodes of the manipulator. Note that c2x and c2y in theabove equations can be obtained only after α1 is solved.Therefore, the solution for the second subchain depends onthe first subchain. This is the reason that the IDA cannotbe obtained from existing 3-RRR formulas.

The manipulator has a total of eight working modesas shown in Fig. 4. In the figure ‘+’ means the value ofbranch index being 1, while ‘−’ means −1. For example,mode (a) with ‘+ + +’ means δ1, δ2 and δ3 all taking the

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values of 1. For automatic cooking application, the cookingpan connected in the point P of Fig. 2(b) will swing backand forth to avoid any material sticking to the pan duringthe cooking process, the required workspace is almost arectangular area located in the right-side of the mechanism,so the configuration of the manipulator is selected to workin the mode of Fig. 4(e).

2.3 Direct Displacement Analysis

Direct displacement analysis is to solve the moving plat-form’s pose p=(x, y, β)T given the actuation anglesq=(α1, α2, α3)

T . The vector loop method is used to solvethe problem. Obviously, if we have the coordinates of B1

and B3, then the pose of the moving platform can be easilyobtained. Therefore, the direct displacement analysisis converted to solve B1 and B3. In the five-bar part,from loop OO1A1B2O and OO2A2B2O, we can solve for−−→OB2, and then coordinates of B1 can be easily obtained.Similarly, in the four-bar part, from loop OO3A3B3O and

OB1B3O, we can solve for−−→OB3, thus coordinates of B3

can be obtained. Note that there exists two solutions forboth B1 and B3, for a given set of actuator angles, themanipulator has four assembly modes.

3. Workspace Analysis

The workspace of parallel manipulators can be dividedinto different categories [2]: the constant orientationworkspace(COW), orientation workspace, inclusive orien-tation workspace, reachable or maximum workspace, to-tal orientation workspace(TOW), dexterous workspace etc.For automatic cooking application, only the COW andTOW are studied in this paper.

Merlet [8] used efficient geometrical algorithms to de-termine various workspace for 3-RPR manipulator whichcould be easily extended to other symmetrical PPMs. How-ever, these algorithms cannot be applied to five–four barPPM with unsymmetrical structure. Thus, in this paper,the usual numerical polar search method will be used toobtain the boundary of different workspace [9].

To facilitate the analysis, without otherwise stated, themanipulator’s dimensions are as follows: r1 = r2 =0.075,r3 =0.1, a1 =0.22, a2 =0.2, a3 =0.3, b1 =0.6, b2 =0.4,b3 =0.44, c1 =0.2, c2 =0.22, d1 =0.4, d2 =0.12, λ= π

4 .COW is defined as the set of positions of a reference

point on the moving platform when its orientation is fixed.For the five–four bar manipulator, we choose P as thereference point, then COW is the translation workspaceof P when β is a constant. TOW is defined as the setof positions that the moving platform can be reached forany orientation in a specified range. In this paper, thenumerical search method is used to obtain the boundaryfor TOW.

Generally speaking, the rotational range for a rev-olute joint is [0, 2π], but in practical applications dueto the working environment limitation, the actuator canonly rotate in a specified range. Thus, it is importantto study the COW under actuator constraints. Let usspecify the ranges of the actuator angles as: π

3 ≤α1 ≤ 4π3 ,

−π3 ≤α2 ≤ 2π

3 , −π6 ≤α3 ≤ π

2 and let the manipulator workas mode (e) as shown in Fig. 4. The resulting COW withβ=π/3 and TOW with β ∈ [0, 2π] are illustrated in Fig. 5.In the figure, the annular area circled by the solid line isthe COW, while inside the COW the annular area circledby the dashed line is the TOW. As a matter of fact, theTOW is the intersection of COW for all the β in a specifiedinterval, hence the TOW is always a subset of COW.

Dexterous workspace is defined as the positions ofa reference point on the moving platform which can bereached for any orientation. It is a special case of TOWas the range of orientation is [0, 2π], hence the dexterousworkspace can be obtained by the same searching methodfor TOW.

4. Dexterity Analysis

Condition number of the Jacobian matrix is a prevalentkinematic performance index to evaluate the manipulator’sdexterity. Let the actuated joint variables be denoted bya vector q and the center point of the moving platform bedescribed by a vector p by differentiating the displacementconstraint equations with respect to time, two Jacobianmatrices Jp and Jq can be obtained as follows:

Jp =

⎛⎜⎜⎜⎝

b1x b1y

b2x− (c2xb2y − c2yb2x)a1xb1xa1y − b1ya1x

b2y− (c2xb2y − c2yb2x)a1yb1xa1y − b1ya1x

b3x b3y

b1xc1y − b1yc1x(c1xa1y − c1ya1x)(c2xb2y − c2yb2x)

b1xa1y − b1ya1x− (c1xb2y − c1yb2x)

b3xc3y − b3yc3x

⎞⎟⎟⎟⎠(3)

Jq =

⎛⎜⎜⎜⎝

a1xb1y − a1yb1x 0 0

0 a2xb2y − a2yb2x 0

0 0 a3xb3y − a3yb3x

⎞⎟⎟⎟⎠

Then the overall Jacobian matrix for the parallel ma-nipulator can be written as J=J−1

p Jq.The condition number of traditional Jacobian can only

be applied to those manipulators with both only one typeof actuation joints and for either positioning or orientingtask [10]. If these conditions are not satisfied, the condi-tion number may vary if different units are chosen in theJacobian. For the five–four bar manipulator, although itsactuation joints are all revolute joints, its moving platformhas mixed translational and rotational motion. Thus, thelast row in the direct Jacobian is dimensionless while thefirst two rows have units of length. To solve the unit inho-mogeneous problem, a two points’ linear velocities methodproposed by Gosselin [11] is utilized in this paper.

The Jacobian matrix is posture dependent, thus thecondition number also varies with posture. To get a per-formance index that only depends on the manipulator’s di-mensions, a global conditioning index based on the averagecondition number over the whole workspace was proposed

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Figure 4. Eight working modes of five–four bar manipulator.

[12–13]. For the five–four bar PPM, the global condition-

ing index can be defined as C =λ2∫λ1

C(β)dβ/λ2 −λ1, where

C(β) is the global condition number with β ∈ [λ1, λ2]. LetW be the workspace and c(J′) be the condition number,C(β) can be defined as C(β)=

∫W c(J′)dW/

∫W dW.

For the application of automatic cooking, we needthe five–four bar PPM to work as mode Fig. 4(e) in aspecified workspace. As mentioned in Section 2.2 thetarget workspace area is a rectangular workspace which ismarked with a black square in Fig. 5 with vertices’ planarcoordinates at [−0.1, 0.7], [−0.1, 0.8], [0.2, 0.7] and [0.2,0.8]. Based on the equation of global condition number,we can get C(π4 )= 1.76. Given β ∈ [0, 4π

9 ], by calculatingglobal condition numbers for every β in this interval, theminimum global condition number happens at β=0.83776,which means the manipulator’s average dexterity is thebest in the specified workspace when β=0.83776. By theequation of global conditioning index and using the samedata, we can get C =2.61 indicating the good dexterity ofthe five–four bar PPM.

5. Singularity Analysis

There are a variety of classifications for singularities, themost common classification is based on the two Jaco-

Figure 5. COW for β= π3 , TOW for β ∈ [0, π

2

].

bian matrices. For this classification, three types ofsingularities are: Inverse Kinematic Singularity (IKS)when det(Jq)= 0, Direct Kinematic Singularity (DKS)when det(Jp)= 0 and combined singularity when bothdet(Jp)= 0 and det(Jq)= 0. Since the combined singular-ity can occur only in special architecture, only the IKS andDKS are studied in this paper. For simplicity, we assumethe orientation of the moving platform is fixed, i.e. β is aconstant.

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Figure 6. IKS curves.

Figure 7. DKS configurations.

5.1 IKS Curve

In (3), we have obtained the expression for Jq. Fromdet(Jq)= 0, we know that the first two links in each

subchain are collinear, i.e.,−−−→OiAi and

−−−→AiBi are collinear.

This is the case when different branches of the inversedisplacement meet. Hence, this type of singularity oftenoccurs at the workspace boundary. The singularity curvesfor β=π/3 are shown in Fig. 6. All the curves for the threesubchains are plotted in Fig. 6(a), where the dash dottedline is the curve for the first chain, the dashed line is for thethird and the solid line is the coupler curve for the second.The COW for β=π/3 is added to all the IKS curves inFig. 6(b), where the boundary of COW is the bold solidline. From the figure, the boundary of COW is made up ofportions of IKS curves. Thus, we can alternatively get theCOW from IKS analysis.

5.2 DKS Curve

The DKS is much more difficult than IKS, as one can seefrom the complex structure of Jp in (3). Many workson DKS just list a few special DKS configurations which

can be viewed geometrically [14]. To obtain the DKScurve for the five–four bar PPM, the analytical methodis very complicated and sometimes is not available. Infact, numerical method is sufficient to get the DKS curves.Therefore, by using Matlab simulation, the DKS curves forall the eight working modes are obtained.

Actually in the expressions of DKS, when δ3 takesthe same sign it means that different branches of directdisplacement meet for the four-bar part, as shown inFig. 7(a), the singularity happens when A3B3 and B3B1

are collinear. If we disregard the DKS curve for the four-bar part, then there will be the cases of the first two branchindices δ1 and δ2 taking the same sign, in this case thecorresponding configuration for the manipulator is shownin Fig. 7(b). In the figure, A1 and A2 are coincident,this case can happen only when b2 = d1. Note that inthe configuration shown in Fig. 7(b) the manipulator canperform self-motion, i.e. the manipulator can undergocontinuous motion if all the actuators are locked. Indeed,under this condition, it is a four-bar mechanism with A3

and A1 or A2 as its joints attach to the fixed base.

Based on the simulation results, we can find that theDKS curves can be divided into two parts: the four-bar

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part and the five-bar part. For the working modes havingthe same sign of δ3, the four-bar part of the DKS curvesis the same. Similarly, if δ1 and δ2 have the same sign,then the five-bar part of the DKS curves is the same. Thisresult corresponds to the four-bar part and five-bar part indirect displacement analysis.

Apart from the two singular configurations in Fig. 7,there is another type of singularity. The singular con-figuration happens when A1B2 and A2B2 are collinear,which means two branches of direct displacement forthe five-bar part meet. This can not happen ifd1 + b2 >a1 + a2 + r1 + r2 is satisfied.

6. Conclusion

In this paper, a novel unsymmetrical five–four bar PPMwith three-DOFs is proposed, the manipulator is composeda double-layer five-bar part with a four-bar part in between.Due to the special architecture, all the three subchains areconsidered in solving the kinematics of the manipulator.The vector loopmethod is utilized to solve the displacementproblem, different from traditional three-DOF PPM, themanipulator has only four assembly modes. In workspaceanalysis, the COW is shown to be an annular zone, thenumerical search method is used to analyse the TOWof the manipulator. The two points’ velocity methodwhich can better solve the inhomogeneous problem is usedin dexterity analysis, in this way homogeneous Jacobianmatrix is generated. The five–four bar PPM also has gooddexterity as can be seen from the resulting dexterity plot.In singularity analysis, both the IKS and DKS curves aredepicted. The results for the IKS show that the boundaryof COW is a part of IKS curve. Examination of the DKScurve reveals the similarity between the DKS and the directdisplacement problem: they can all be decomposed into afour-bar part and a five-bar part. All the analysis presentedin this paper can facilitate the dimensional synthesis andtrajectory planning for the new manipulator.

The results of the kinematics analysis for the novelPPM in this paper show that the proposed five–four barPPM can better meet the requirements of the Chinesedish cooking machine. The selected work mode for thisparticular application can provide a desired workspace withgood dexterity throughout the effective workspace zone.

Acknowledgements

This work is supported by Natural Science Foundation ofChina (Project No: 60875060) and State Key Laboratory ofRobotics and System (HIT) (Project No: SKLRS200719).The work is also partly supported by 863 Hi-Tech Re-search and Development Program of China (Project No:2006AA040205).

References

[1] Q.C. Li, Z. Huang, & J.M. Herve, Type synthesis of 3R2T 5-DOF parallel mechanisms using the Lie group of displacements,IEEE Transaction of Robotics and Automation, 20(2), 2004,173–180.

[2] J.P. Merlet, Parallel robots, Second Edition (Netherlands:Springer, 2006), 54–68.

[3] I.A. Bonev, D. Zlatanov, & C.M. Gosselin, Singularity analysisof 3-DOF planar parallel mechanisms via screw theory, ASMEJournal of Mechanical Design, 125(3), 2003, 573–581.

[4] C.M. Gosselin, S. Lemieux, & J.P. Merlet, A new architectureof planar three-degree-of-freedom parallel manipulator, Proc.IEEE Int. Conf. Robotics and Automation, Minneapolis, MN,1996, 3738–3743.

[5] J. Wu, J.S. Wang, & L.P. Wang, Optimal kinematic design andapplication of a redundantly actuated 3DOF planar parallelmanipulator, ASME Journal of Mechanical Design, 130(5),2008, 054503.

[6] I. Ebrahimi, J.A. Carretero, & R. Boudreau, 3-PRRR re-dundant planar parallel manipulator: inverse displacement,workspace, and singularity analyses, Mechanism and MachineTheory, 42(8), 2007, 1007–1016.

[7] B. Li, H.J. Yu, & J.G. Zhao, A Novel Three DOF PlanarParallel Manipulator, P. R. China Patent Pending, 2007,Registration No. 2007100730733.

[8] J.P. Merlet, C.M. Gosselin, & N. Mouly, Workspace of pla-nar parallel manipulators, Mechanism and Machine Theory,33(1/2), 1998, 7–20.

[9] Z. Huang, Y.S. Zhao, & T.S. Zhao, Advanced Spatial Mech-anism Chinese Edition, (China: Advanced Education Press,2006).

[10] L.W. Tsai, Robot Analysis: The Mechanics of Serial andParallel Manipulators (USA: John Wiley & Sons, 1999).

[11] C.M. Gosselin, Dexterity indices for planar and spatial roboticmanipulator, Proc. IEEE Int. Conf. Robotics and Automation,Cincinnati, USA, 1990, 650–655.

[12] C.M. Gosselin & J. Angeles, A global performance index forthe kinematic optimization of robotic manipulators, ASMEJournal of Mechanical Design, 113(3), 1991, 220–226.

[13] K.H. Hunt, Kinematic Geometry of Mechanisms (New York:Oxford University Press, 1978).

[14] I.A. Bonev, Geometric analysis of parallel mechanisms, doc-toral disseration, Laval University, Quebec, Canada, 2002.

Biographies

Bing Li is a Professor and cur-rently Head of the Departmentof Mechanical Engineering andAutomation, Shenzhen GraduateSchool, Harbin Institute of Tech-nology of China. He received hisPh.D. from Department of Me-chanical Engineering of the HongKong Polytechnic University. Hisresearch interests include parallelmanipulators, parallel kinematicsmachine, sheet metal assembly,

robust design, etc.

Jianguo Zhao is a graduate stu-dent from the Department ofMechanical Engineering and Au-tomation, Shenzhen GraduateSchool, Harbin Institute of Tech-nology of China. His researchinterests focus on parallel manip-ulators, etc.

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Xiaojun Yang is an AssistantProfessor of Department of Me-chanical Engineering and Au-tomation, Shenzhen GraduateSchool, Harbin Institute of Tech-nology of China. His researchinterests focus on the design andcontrol of the parallel mecha-nisms.

Ying Hu is an Associate ResearchProfessor of Cognitive Technolo-gies Lab, Shenzhen Institute ofAdvanced Technology, ChineseAcademy of Sciences. Her re-search interests focus on parallelrobot, medical assistant robot,etc.

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