538 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004

Short Papers_______________________________________________________________________________Optimal Kinematic Design of 2-DOF Parallel ManipulatorsWith Well-Shaped Workspace Bounded by a Specified

Conditioning Index

Tian Huang, Meng Li, Zhanxian Li, Derek G. Chetwynd, andDavid J. Whitehouse

AbstractThis paper presents a hybrid method for the optimum kine-matic design of two-degree-of-freedom (2-DOF) parallel manipulators withmirror symmetrical geometry. By taking advantage of both local and globalapproaches, the proposed method can be implemented in two steps. In thefirst step, the optimal architecture, in terms of isotropy and the behaviorof the direct Jacobian matrix, is achieved, resulting in a set of closed-formparametric relationships that enable the number of design variables to bereduced. In the second step, the workspace bounded by the specified condi-tioning index is generated, which allows only one design parameter to be de-termined by optimizing a comprehensive index in a rectangular workspace.The kinematic optimization of a revolute-jointed 2-DOF parallel robot hasbeen taken as an example to illustrate the effectiveness of this approach.

Index TermsOptimum design, parallel manipulators, performanceindex.

I. INTRODUCTION

Kinematic design or dimensional synthesis is an important issue inthe development of parallel manipulators because the performance of asystem in terms of workspace, velocity, accuracy, and rigidity is dom-inated, to a great extent, by its geometrical characteristics. Kinematicdesign is primarily concerned with the determination of the dimensionsof the geometric parameters and the range of the actuated joint vari-ables, leading to, in general, optimizing a nonlinear cost function, sub-ject to a set of appropriate constraints.The approaches in previous work dealing with the kinematic design

of parallel manipulators may be classified into two categories. One cat-egory is known as the local optimum design [2], [6], [10][12], [18],focusing upon finding the parametric loci necessary to generate a setof isotropy. Another category is referred to as the global optimum de-sign [3], [5], [7], [14], [16], [17], attempting to optimize a global con-ditioning index. The typical work in this phase was proposed by Gos-selin and Angeles [7], in which a global conditioning index representedby the mean value of the reciprocal of conditioning number of the Ja-cobian was used as a cost function for maximization. The reasons forhaving to implement a global approach are: 1) the condition necessaryto generate an isotropy is unable to provide sufficient design constraints

Manuscript received November 22, 2002; revised May 15, 2003. This paperwas recommended for publication by Associate Editor J. Merlet and Editor I.Walker upon evaluation of the reviewers comments. This work was supportedin part by the 863 Development Scheme under Grant 2001AA421220, in partby the National Science Foundation of China under Grant 50075059, in part bythe Royal Society UKChina Joint Research under Grant Q820, and in part bythe Education Ministry of China.T. Huang, M. Li, and Z. Li are with the School of Mechanical Engineering,

Tianjin University, Tianjin 300072, China (e-mail: htiantju@public.tpt.tj.cn;th@eng.warwick.ac.uk).D. G. Chetwynd and D. J. Whitehouse are with the School of Engineering,

The University of Warwick, Coventry CV4 7AL, U.K.Digital Object Identifier 10.1109/TRA.2004.824690

Fig. 1. 2-DOF parallel robots with mirror symmetrical geometry. (a) Proximalarm type. (b) Telescopic leg type. (c) Linear drive type.

to achieve a unique solution; and 2) the global kinematic performancecould not be guaranteed simply by conducting local optimum design.For this reason, the global optimal method has been well accepted andwidely employed for the evaluation of the kinematic performance of ex-isting parallel manipulators and/or for optimization of new machines.Although the global approach is very general and meaningful in prac-tice, it cannot be implemented analytically, especially when a numberof design variables are involved.By taking advantage of both local and global optimal approaches,

this paper presents a hybrid method that enables the optimal kinematicdesign to be implemented by two steps. In the first step, the optimalarchitecture is achieved, resulting in a set of closed-form parametricrelationships. In the second step, the workspace bounded by the spec-ified conditioning index is generated, which allows a well-shaped andconditioned workspace to be obtained via optimizing a comprehen-sive index. A revolute-jointed two-degree-of-freedom (2-DOF) parallelrobot will be taken as an example to demonstrate the effectiveness ofthis approach.

II. KINEMATIC EQUATIONS

As shown in Fig. 1(a), the example object of study is a 2-DOF par-allel robot with mirror symmetrical geometry. The robot is revolutejointed and driven by two proximal arms connected with the frameby the fixed revolute joints. Although discussion here will concentrateon this type of robot, similar robots, either internally driven by twotelescopic legs or externally driven by two linear drives, as shown inFig. 1(b) and (c), can be dealt with in the same way.In order to evaluate the kinematic performance, the inverse position

and velocity analyses should be carried out. In the O xy coordinate

1042-296X/04$20.00 2004 IEEE

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004 539

system shown in Fig. 1(a), the closed-loop constraint equation can bewritten as

rrr sgn(i)eeee

1

l

1

uuu

i

= l

2

www

i

; i = 1; 2 (1)

where rrr = (x y )T is the position vector of the reference pointO0 ofthe end-effector; l1

, l2

, uuui

, and wwwi

are the lengths and unit vectors ofthe proximal and distal links; e is the distance between O and the axisof the actuated joint; anduuu

i

= ( cos

1i

sin

1i

)

T

; www

i

= ( cos

2i

sin

2i

)

T

eee

1

= (1 0 )

T

; sgn(i) =

1; i = 1

1; i = 2

where 1i

(or qi

) and 2i

are the position angles of the proximal anddistal links, respectively.Taking dot product on both sides of (1) and adding the assembly

mode gives the solution to

1i

= 2arctan

A

i

+ sgn(i) A

2

i

C

2

i

+B

2

i

C

i

B

i

(2)

where

A

i

= 2l

1

y

B

i

= 2l

1

(x sgn(i)e)

C

i

=x

2

+ y

2

+ e

2

+ l

2

1

l

2

2

2sgn(i)ex:

Thus, uuui

can be determined, and from (1)

www

i

=

(rrr sgn(i)eeee

1

l

1

uuu

i

)

l

2

:

Differentiating (1) with respect to time yields

vvv = l

1

_

1i

QuQuQu

i

+ l

2

_

2i

QwQwQw

i

(3)

where QQQ = 0 11 0

.

Without loss of generality, let the proximal link have a unit length,i.e., l1

= 1 and replace _1i

by _qi

. Multiplying both sides of (3) by wwwTi

and rewriting in matrix form yields the general structure of the velocitymapping function of the 2-DOF parallel robots shown in Fig. 1

_qqq = AAA

1

BvBvBv = JvJvJv; _qqq = ( _q

1

_q

2

)

T (4)

where JJJ is the Jacobian andAAA,BBB are known as the direct and indirectJacobian matrices [3], respectively

AAA =

diag www

T

i

QuQuQu

i

; proximal arm typeEEE

2

; telescopic leg typediag www

T

i

uuu

i

; linear drive type; BBB = [www

1

www

2

]

T

:

EEE

2

is a unit matrix.

III. OPTIMUM DESIGN

A. Proposed IdeaIt has been well accepted that the most suitable local performance

index is the condition number of the Jacobian [15]. Isotropy has alsobeen used as a design criterion to find the optimal configuration [2],[6][10], [18]. That is a configuration at which the condition numberof the Jacobian reaches its minimum value in the overall workspace.Therefore, a parallel robot must behave well at the optimal configu-ration, in terms of isotropy. This will allow a hybrid approach to bedeveloped that can be implemented in two steps. The first step of thisapproach aims at formulating a set of closed-form parametric relation-ships necessary to obtain an optimal configuration, in terms of the

Fig. 2. Singular configurations of the example robot.

Fig. 3. Isotropic configurations.

isotropy and the force-transmission behavior. Then, the second stepinvolves determining the independent kinematic parameters by opti-mizing a global index in a well-shapedworkspace bounded by the spec-ified conditioning index.

B. Local Optimum Design1) Conditioning Index: Following the notation used by [7], the con-

ditioning index is defined by

0

1

=

1

2

1 (5)

where , 1

, and 2

are the condition number, and the minimum andmaximum singular values of the Jacobian. For the example parallelrobot, we have

1;2

=

1

p

2jv

1

jjv

2

j

v

2

1

+v

2

2

v

2

1

v

2

2

2

+(2v

1

v

2

w)

2

1=2

1=2

(6)

1

=

1

2

=

v

2

1

+ v

2

2

v

2

1

v

2

2

2

+ (2v

1

v

2

w)

2

1=2

(v

2

1

+ v

2

2

) + (v

2

1

v

2

2

)

2

+ (2v

1

v

2

w)

2

1=2

1=2

(7)

where

v

2

i

= www

T

i

QuQuQu

i

2

= 1 www

T

i

uuu

i

2

= 1 cos

2

i

jwj = www

T

1

www

2

(8)

and i

= arccos www

T

i

uuu

i

is the acute angle (pressure angle) betweenthe proximal and distal links in the ith kinematic chain [see Fig. 1(a)],representing the force-transmission behavior of the robot.2) Singularity and Optimal Configuration: By observing (7) and

Fig. 2, it is easy to see in the following cases where the singularityoccurs.

540 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004

Fig. 4. Contours of 1= versus in workspace bounded by 1= = 0:5. (a) = 1:5. (b) = 2:0. (c) = 2:5.

Fig. 5. Contours of versus in workspace bounded by 1= = 0:5. (a) = 1:5. (b) = 2:0. (c) = 2:5.

Case 1) If jwj ! 1(www1

==www

2

), then 1= ! 0.Case 2) If either jv1

j ! 0(www

1

==uuu

1

) or jv2

j ! 0(www

2

==uuu

2

) but notboth, then (1= ! 0).

Case 3) If jv1

j = jv

2

j = v

c

, (7) gives

1

=

1 jwj

1 + jwj

1=2

: (9)

Thus, 1= 6= 0 unless jwj ! 1. In this case, however, if jv1

j ! 0 andjv

2

j ! 0, then (6) gives 1

=

2

!1, leading to the singularity dueto the degeneration of the direct Jacobian.The necessary condition for achieving an isotropy is that (7) has two

identical roots, such that

jwj = 0; jv

1

j = jv

2

j: (10)

In particular, when jv1

j = jv

2

j = 1, (7) gives 1= = 1, and thecorresponding isotropic configuration is a pentagon with three rightcorner angles (see Fig. 3). However, when jv1

j = jv

2

j = 0, (9) alsogives 1= = 1, and the isotropic configuration turns out to be a righttriangle (see Fig. 3). Referring back to (6), it is clear that the former casegives1

=

2

= 1, while the latter leads to 1

=

2

!1. Therefore,the so-called optimal configuration should be that at which 1

takesthe maximum and 2

takes the minimum values simultaneously in theoverall workspace.According to the above analysis, the conditions for the example robot

to achieve an optimal configuration are

jwj = 0; jv

1

j = jv

2

j = 1: (11)

Obviously, the conditions given in (11) are also suitable for the othertwo parallel robots shown in Fig. 1(b) and (c), depending upon thecorresponding jvi

j.

For the example robot, (11) will produce a set of parametric relation-ships as follows:

H =

p

2

2

(l

1

+ l

2

); e =

p

2

2

(l

2

l

1

) (12)

whereH is the distance from O to O0 when the robot is at the optimalconfiguration (see Fig. 3).

C. Global Optimum Design1) Observations and Discussions: (1) Let be the ratio of l2

to l1

.

This means that the parametric relationships given by (12) leave onlyone independent geometric parameter, , to be determined as l1

= 1. Inorder to achieve a robot having a relatively large yet well-conditionedworkspace, it is necessary in the first place to gain an insight into the in-fluence of on the location, shape, and size of the workspace boundedby a specified conditioning index 1=0

. Here, this set of workspacesis defined as W0

with each element W0

() 2 W

0

being associ-ated with a given . Given 1=0

and , the boundary ofW0

() can befound using the 1-D root search algorithm, the golden-section method,for example. It is worth pointing out, however, that for the systems withhigher DOFs, it is necessary to develop an effective algorithm to findthe boundary bounded by the specified 1=0

.

Fig. 4 shows the contours representing the variations in shape, size,and location of W0

() versus changes in = 1.52.5 and 1=0

=

0.51.0. It can be seen that the shape ofW0

() is similar to the reach-able workspace of the robot of this type [1]. Meanwhile, the distancefrom the center ofW0

() toO increases with the increase of , and

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004 541

Fig. 6. 1-D search for the maximum rectangular workspace inW ().

Fig. 7. Variation of 1= versus [] and .

increasing 1=0

reduces the size ofW0

(), meaning that to achieve abetter-conditioned robot must be at the cost of size of workspace. Fig. 4also shows that the area ofW0

() bounded by a given 1=0

deceaseswith the increase in , but this tendency becomesweaker when > 2:0.Fig. 5 shows the contours representing the variations of the minimumpressure angle min

= min (

1

2

) versus changes in = 1.52.5in W0

() bounded by 1=0

= 0.51.0. It can be seen that the dis-tribution pattern of min

appears to be closely similar to that of 1=.For example, the contour associated with min

= 30

approximatelymatches that associated with 1=0

= 0:5. Meanwhile, min

alwaystakes the minimum value on the upper bound of W0

(), regardlessof what value may take. These observations indeed provide us withsome important information for the global optimum design in what fol-lows.

2) Global Performance Index: From an application point of view,it is more meaningful for the example robot to possess a well-shapedworkspace that can be tailored out of W0

() for a given 1=0

. Arectangular workspaceW

() with fixed width/height ratio = b=hmight be typically the first priority. As shown in Fig. 6,W

() is de-fined in such a way that it is tangential to the upper bound ofW0

() atthe points P1

and P2

, and intersects with the lower bound ofW0

()

at the cornersQ1

andQ2

, respectively. In this case, a global and com-prehensive performance index for determining is proposed as an ob-jective function for maximization. This index can be represented by

=

s

! max (13)

where 1= is the mean value of the conditioning index 1= evaluated inW

();s

= S=S

max

, and S is the area ofW

(), Smax

is the max-imum value of S when varies in the range of 1 max

. Here,

max

= 2:5 is suggested for a practical design. The index proposedhere is a modified version of the global conditioning index proposed

Fig. 8. Performance indexes versus . (a) 1= = 0:256. (b) 1= = 0:581with = 4.

TABLE IRESULTS OF OPTIMIZATION WITH = 4

in [7]. In computer implementation, 1= can be numerically calculatedby

1

=

1

M N

M

m=1

N

n=1

1

mn

(14)

where mn

0

is the value of at node (m n ) of (M 1) (N 1) equally meshed W0

(). In order to choose the right valueof the conditioning limit 1=0

, the force-transmission behavior shouldbe considered. 1=0

can be determined such that min(min

) [] in

542 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004

Fig. 9. 1= and distributions in a normalized rectangular workspace. (a) 1= = 0:526, = 1:7. (b) 1= = 0:581, = 1:85 with = 4.

W

(), with [] being a prescribed allowable pressure angle. In prac-tice, [] = 3035 would be a reasonable choice. It can be seen fromFig. 1(a) that the pressure angle i

can be expressed by

i

= arccos

1

2

1 +

2

(x sgn(i)e)

2

y

2 (15)

where

1

2

b x

1

2

b;

1

2

hH

0

y

1

2

hH

0

; =

b

h

and H 0 denotes the distance from the center of W

() to the pointO. Observation of (15) indicates that i

takes the minimum value inW

() when O0 is located at

x

0

=

e; i = 1

e; i = 2

y

0

=

h

2

H

0

: (16)

Substituting (16) into (15) and replacing i

in (15) by [], leads to

x

0

= e; y

0

= (1 +

2

2 cos[])

1=2

: (17)

This allows 1=0

to be determined using (7). Fig. 7 shows the variationof 1=0

versus changes in [] and . Therefore, max(1=0

) can beused as the conditioning limit for 1 max

. For example, if[] = 30, then max(1=0

) = 0:526, as shown in Fig. 7.In addition, referring back to Fig. 6, a numerical algorithm can be

developed for searching S as follows.

1) Given and [], generate the boundary ofW0

() bounded bymax(1=

0

).

2) Move O0 along the x axis to the tangential point P (xP

; y

P

) ofW

0

(), with the point P1

being the starting point. Then, moveO

0 along they axis in an iterative manner to the pointQ(xQ

=

x

P

; y

Q

) located on the lower bound ofW0

() until

2x

P

(y

P

y

Q

)

< " (18)

where " is the accuracy of convergence. In this way, the width b,height h, ofW

(), and distanceH 0 from its center to the pointO can be found by

b = 2x

P

; h = y

P

y

Q

; H

0

= y

P

+

h

2

: (19)

The golden-section method can also be employed for this purpose.

IV. EXAMPLES

Given = 4, two cases associated with 1=0

= 0:526 ([] = 30

)

and 1=0

= 0:581 ([] = 35

) have been considered as examples forthe optimum kinematic design of the robot. Fig. 8 shows the variationsof s

, 1=, and versus . It can be seen from Fig. 8(a) that in thefirst case, 1= varies slightly between 0.74 and 0.80, while s

takesthe maximum value at = 1:7, resulting in the maximum value of atthe same . The same rule is followed by the second case [see Fig. 8(b)]where 1= varies between 0.77 and 0.83, while takes the maximum

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004 543

value at = 1:85. An interesting observation is that for a given [], the for 1=0

to take the maximum value is identical to that for to dothe same. The results of optimization of both cases are given in Table I.By taking the coordinate transformation

x =

x

b

; y =

(y +H

0

)

b

:

Fig. 9 shows the distributions of 1= and min

in the normalizedworkspace W

(). It can be seen that for both cases, the boundaryof W

is strictly bounded by the specified 1=0

at four points, andthe minimum value of min

in the entire workspace equals [], asexpected. This means that bounded by the conditioning index, arelatively large, yet well-behaved, workspace of the robot can beachieved by using this approach. Meanwhile, the computation resultsalso show that has little bearing on the optimized , meaning thatany rectangular workspace having guaranteed performance can betailed out of the W0

().

V. CONCLUSIONS

This paper presents a hybrid method for the optimum design of a2-DOF parallel manipulator. The conclusions are drawn as follows.

1) A set of closed-form parametric relationships has been found forgenerating an optimal architecture, using the isotropic conditionand a maximization of the coefficient appearing in the direct Ja-cobian matrix. This allows only one parameter to be determinedin the global optimum design.

2) A modified global index is proposed. Maximization of this indexenables one to achieve a relatively large, yet well-conditioned,rectangular workspace bounded by the conditioning limit 1=0

,

which can be determined by a specified allowable pressure angle[].

3) The width/height ratio has little bearing on the optimized , al-lowing any rectangular workspace having guaranteed good per-formance to be tailed out of theW=0

().

4) For the right use of the proposed method to the optimal design ofother robots that might be a mix between translational and rota-tional DOFs, the normalization technique should be consideredso as to allow the entries of the Jacobian matrix to have the samedimensions.

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proach for obtaining the workspace of a class of 2-DOF planar parallelrobots, Mech. Mach. Theory, vol. 34, no. 7, pp. 10571073, 1999.

[2] D. Chablat, P. Wenger, and J. Angeles, The iso-conditioning loci of aclass of closed-chain manipulators, in Proc. IEEE Int. Conf. Roboticsand Automation, Leuven, Belgium, 1998, pp. 19701975.

[3] J. Duffy, Static and Kinematics With Application toRobotics. Cambridge, U.K.: Cambridge Univ. Press, 1996.

[4] F. Gao, A physical model of the solution space and the atlas of thereachable workspace for 2-DOF parallel planar manipulators, Mech.Mach. Theory, vol. 31, no. 2, pp. 173184, 1996.

[5] F. Gao and X. Liu, Performance evaluation of 2-DOF planar parallelrobots, Mech. Mach. Theory, vol. 33, no. 6, pp. 661668, 1998.

[6] C. M. Gosselin and J. Angeles, The optimum kinematic design of aspherical 3-DOF parallel manipulator, ASME J. Mech., Transm., Au-tomat. Des., vol. 111, no. 2, pp. 202207, 1989.

[7] C.M. Gosselin and J. Angeles, A globe performance index for the kine-matic optimization of robotic manipulators, ASME J. Mech. Des., vol.113, no. 3, pp. 220226, 1991.

[8] C. M. Gosselin, The optimum design of robotic manipulators usingdexterity indexes, J. Robot. Auton. Syst., vol. 9, no. 4, pp. 213226,1993.

[9] C. M. Gosselin and J. Wang, Singularity loci of planar parallel manip-ulators with revolute actuators, Robot. Auton. Syst., vol. 21, no. 4, pp.377398, 1997.

[10] T. Huang, D. J. Whitehouse, and J. S. Wang, Local dexterity, optimumarchitecture and design criteria of parallel machine tools, CIRP Ann.,vol. 47, no. 1, pp. 347351, 1998.

[11] T. Huang, J. S. Wang, and D. J. Whitehouse, Theory and methodologyfor kinematic design of GoughStewart platforms, Sci. China, ser. (E),vol. 42, no. 4, pp. 425436, 1999.

[12] T. Huang, J. S. Wang, C.M. Gosselin, and D. J. Whitehouse, Kinematicsynthesis of hexapods with prescribed orientation capability and well-conditioned dexterity, SME J. Manufact. Processes, vol. 2, no. 1, pp.3647, 1999.

[13] M. G. Mohamed and J. Duffy, A direct determination of the instanta-neous kinematics of fully parallel robot manipulators, ASME J. Mech.Des., vol. 107, no. 2, pp. 226273, 1985.

[14] J. R. Singh and J. Rastegar, Optimal synthesis of robot manipulatorsbased on global kinematic parameters, Mech. Mach. Theory, vol. 30,no. 4, pp. 569580, 1995.

[15] J. K. Salisbury and J. J. Craig, Artificial hands: Force control and kine-matic issues, Int. J. Robot. Res., vol. 1, no. 1, pp. 417, 1983.

[16] L. Stocco, S. E. Salcudean, and F. Sassani, Fast constrained global min-imax optimization of robot parameters, Robotica, vol. 16, no. 6, pp.595605, 1998.

[17] L. W. Tsai and S. Joshi, Kinematics and optimization of a 3UPU par-allel manipulator, ASME J. Mech. Des., vol. 122, no. 4, pp. 439446,2002.

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A Numerical Test for the Closure Properties of 3-D Grasps

Xiangyang Zhu, Han Ding, Senior Member, IEEE, andMichael Y. Wang, Member, IEEE

AbstractThis paper presents a numerical test for the closure proper-ties (force closure and form closure) of multifingered grasps. For three-di-mensional (3-D) grasps with frictional point contacts or soft contacts, thenumerical test is formulated as a convex constrained optimization problemwithout linearization of the friction cone. For 3-D frictionless grasps, it canbe calculated by solving a single linear program. The proposed numericaltest (along with the rank of the grasp matrix) provides an efficient toolfor the analysis of the force-closure property and the relative force-closureproperty.

Index TermsMultifingered grasp, numerical test, (relative) force clo-sure.

I. INTRODUCTION

As one of the fundamental issues of multifingered grasping anddexterous manipulation, form-closure and force-closure analysiswas extensively investigated in the context of robotics [1][30]. The

Manuscript received May 19, 2003. This paper was recommended for publi-cation by Associate Editor I. Kao and Editor I. Walker upon evaluation of thereviewers comments. This work was supported by the National Natural ScienceFoundation of China under Grants 50175014 and 50390063.X. Zhu and H. Ding are with the Robotics Institute, School of Mechanical

Engineering, Shanghai Jiaotong University, Shanghai 200030, China (e-mail:mexyzhu@sjtu.edu.cn; hding@sjtu.edu.cn).M. Y. Wang is with the Department of Automation and Computer-Aided

Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong(e-mail: yuwang@acae.cuhk.edu.hk).Digital Object Identifier 10.1109/TRA.2004.825514

1042-296X/04$20.00 2004 IEEE