Optimizing and Deciphering Design Principles of Robot Gripper

Configurations Using an Evolutionary Multi-Objective Optimization

Method

Rituparna Datta and Kalyanmoy DebDepartment of Mechanical EngineeringIndian Institute of Technology, Kanpur

PIN 208016, India{rdatta,deb}@iitk.ac.in

http://www.iitk.ac.in/kangal/pub.htm

February 10, 2011

KanGAL Report Number 2011002

Abstract

This paper is concerned with the determination of op-timum forces extracted by robot grippers on the sur-face of a grasped rigid object – a matter which is cru-cial to guarantee the stability of the grip without caus-ing defect or damage to the grasped object. A multi-criteria optimization of robot gripper design problemis solved with two different configurations involvingtwo conflicting objectives and a number of constraints.The objectives involve minimization of the differencebetween maximum and minimum gripping forces andsimultaneous minimization of the transmission ratiobetween the applied gripper actuator force and theforce experienced at the gripping ends. Two differentconfigurations of the robot gripper are designed bya state-of-the-art algorithm (NSGA-II) and the ob-tained results are compared with a previous study.Due to presence of geometric constraints, the resultingoptimization problem is highly non-linear and multi-modal. For both gripper configurations, the proposedmethodology outperforms the results of the previousstudy. The Pareto-optimal solutions are thoroughlyinvestigated to establish some meaningful relation-ships between the objective functions and variable val-ues. In addition, it is observed that one of the gripperconfigurations completely outperforms the other onefrom the point of view of both objectives, thereby es-tablishing a complete bias towards the use of one ofthe configurations in practice.

1 Introduction

Evolutionary multi-objective optimization (EMO) isgetting increasing attention to solve real-world prob-

lems among the scientific and engineering commu-nity. This is because, real-world problems are oftenmulti-objective in nature and consist of non-linear,non-convex, and discontinuous objective functions andconstraints. In most of the situations, conventionaloptimization techniques cannot handle such complex-ities well and are largely unsuccessful to produce a sat-isfactory solution. Evolutionary algorithms are flexi-ble, less-structured and are found to provide an alter-native and effective way to tackle these difficult opti-mization problems with the fast increase in computingpower.

The role of a robot gripper mechanism is significantin today’s industrial systems, as it is the interactiondevice between the environment and the object to bepicked and placed to perform grasping and manipula-tion tasks. With the evolution of automation in indus-tries, grasping become an important topic in roboticsresearch community. The motivation of growing re-search in robot gripper design came from human hand[11], [9]. Robot grippers are attached in the wrist ofthe robot to grasp and manipulate an object. Thebasic purpose of robot gripper is to hold, grip, han-dle and release an object the way a human hand cando [3]. The driving mechanism of a gripper can behydraulic, electrical and pneumatic.

There are many previous studies in literature onrobot gripper design. Cutkosky [5] carried out a studyon the choice, model and design of grasp and de-veloped an expert system to resolve grasping issues.Osyczka [10] proposed a multi-objective optimizationbased robot gripper design and solved the grippingproblem with different configurations. Ceccarelli etal. [2], [4] performed dimensional synthesis of grip-per mechanisms using Cartesian coordinates. The

1

formulation was based on practical design require-ments and the aim was to derive an analytical for-mulation using an index of performance (Grasping In-dex) to describe both kinematic and static character-istics. Cabrera et al. [1] developed a strategy usingevolutionary algorithm for optimal synthesis of a two-hand multi-objective constrained planar mechanism.They validated the results by evaluating the mecha-nism on some points on the Pareto-optimal front andshowed satisfactory result. Another study [8] pro-posed an optimum design of two-finger robot grippermechanism using multi-objective formulation, consid-ering the efficiency, dimension, acceleration and veloc-ity of the grasping mechanisms and reported a casestudy by using 8R2P linkage. A recent study [12]proposed kinematic design of spherical serial mech-anisms. The method is a combination between globalmanipulability and uniformity of manipulability overthe workspace. Genetic algorithms were used to solvethe linear problem.

In this paper, we borrow two robot gripper configu-rations from Osyczka [10] and perform a bi-objectivestudy to understand the trade-off between chosen ob-jectives. In the following sections, first we describethe gripper mechanism configurations and presentthe corresponding multi-objective optimization prob-lems. The optimization problems are then solved us-ing the Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) [6]. Evaluation of every solution re-quires another single-objective maximization problemwhich we solve using a classical golden section searchmethod. The NSGA-II obtained fronts are comparedwith an earlier study and superiority of obtained re-sults is demonstrated. Thereafter, using the obtainedtrade-off solutions, we perform an innovization study[7] to unveil important design principles related to thegripper dimensions. Finally, we compare optimal so-lutions of both the gripper problems and conclude thesuperiority of one over the other. The paper ends withthe conclusions of this study.

2 Gripper- I Configuration De-sign

The motivation of the present work is to design thestructure of a robot gripper optimally. The origi-nal problem was formulated elsewhere [10]. The goalof the optimization problem is to find the dimen-sions of elements of the grippers and optimize ob-jective functions simultaneously by satisfying the ge-ometric and force constraints. The two-dimensionalstructure of the gripping mechanism is shown in Fig-ure 2. The vector of seven design variables are x =(a, b, c, e, f, l, δ)T , where a, b, c, e, f, l are dimen-sions (link lengths) of the gripper and δ is the angle

between elements b and c. The structure of geometri-cal dependencies of the mechanism can be describe asfollows (Figure 2):

g =√

(l − z)2 + e2,

b2 = a2 + g2 − 2.a.g. cos(α − φ),

α = arccos(a2 + g2 − b2

2.a.g) + φ,

a2 = b2 + g2 − 2.b.g. cos(β + φ),

β = arccos(b2 + g2 − a2

2.b.g) − φ,

φ = arctan(e

l − z).

y

cbz a

f

l

eβα

δ

Fk

Fk

Figure 1: A sketch of robot Gripper-I.

z ab

e

ll−z

f

g

cβ

α

δ

φφ y

2

Fk

Figure 2: Geometrical dependencies of the Gripper-Imechanism.

The free body diagram for the force distribution ofis shown in Figure 3. From the figure, we can writethe following:

R.b. sin(α + β) = Fk.c,

R =P

2. cosα,

Fk =P.b sin(α + β)

2.c cosα,

where R is the reaction force on link a and P is theactuating force applied from the left side to operatethe gripper.

From the above correlations the objective functionscan be formulated as follows:

1. The first objective function can be written as thedifference between the maximum and minimum

2

cb

R

aβ

β

α

α

δFk

P2

Figure 3: Force distribution of mechanism of theGripper-I.

griping forces for the assumed range of gripperends displacement:

f1(x) = maxz

Fk(x, z) − minz

Fk(x, z). (1)

The minimization of this objective ensures thatthere is not much variation in the gripping forceduring the entire range of operation of the grip-per.

2. The second objective function is the force trans-mission ratio, the ratio between the applied actu-ating force P and the resulting minimum grippingforce at the tip of link c:

f2(x) =P

minz Fk(x, z). (2)

The minimization of this objective will ensurethat the gripping force experienced at the tip oflink c has the largest possible value.

In the aforesaid multi-objective optimization prob-lem, both objective functions depend on the vector ofdecision variables and on the displacement z. Thusfor a given solution vector x, the values of the con-flicting objective functions f1(x) and f2(x) requiresthat we find the maximum and minimum value ofgripping force Fk(x, z) for different possible values ofz. The parameter z is the displacement parameterwhich takes a value from zero to Zmax. Taking asmall finite increment in z, recording the correspond-ing Fk value for each z, and then locating the max-imum and minimum values of Fk is a computation-ally time-consuming proposition. Here, we employ thewell-known golden section search algorithm for locat-ing minimum and maximum Fk. It is important tonote that these extreme values may take place at oneof the two boundaries (either z = 0 or z = Zmax) andthe golden section search is capable of locating them.The only drawback of the golden section search is thatit can locate the minimum of a unimodal function ac-curately, but for multi-modal problems the algorithmis not guaranteed to find an optimum. Fortunately, forthis problem, we have checked the nature of Fk varia-tion for a number of different solution vectors (x) andevery time a unique maximum in the specified rangeof z is observed. A typical variation is shown in Fig-ure 4. Although there is usually a single maximum

k

95.4

95.5

95.6

95.7

95.8

95.9

96

96.1

0 5 10 15 20 25 30 35 40 45 50z

F

Figure 4: Variation of force Fk with the displacementz for a typical x for Gripper-I.

point, one of two extremes have been found to cor-respond to the minimum Fk. Thus, for locating theminimum Fk, we simply check the two extreme valuesof z and use the one having the smaller value of Fk.

The gripper problem also has a number of non-linearconstraints:

1. The dimension between ends of gripper for maxi-mum displacement of actuator should be less thanminimal dimension of the gripping object:

g1(x) = Ymin − y(x, Zmax) ≥ 0,

where y(x, z) = 2.[e + f + c. sin(β + δ)] is the dis-placement of gripper ends and Ymin is the minimaldimension of the griping object. The parameterZmax is the maximal displacement of the gripperactuator.

2. The distance between ends of gripper correspond-ing to Z max should be greater than zero:

g2(x) = y(x, Zmax) ≥ 0.

3. The distance between the gripping ends corre-sponding to no displacement of actuator (staticcondition) should be greater than the maximumdimension of gripping object:

g3(x) = y(x, 0) − Ymax ≥ 0,

where Ymax is the maximal dimension of the grip-ing object.

4. Maximal range of the gripper ends displacementshould be greater than or equal to the distancebetween the gripping ends corresponding to nodisplacement of actuator (static condition):

g4(x) = YG − y(x, 0) ≥ 0, (3)

where YG is the maximal range of the gripper endsdisplacement.

3

5. Geometrical properties are preserved by followingtwo constraints:

g5(x) = (a + b)2 − l2 − e2 ≥ 0.

g6(x) = (l − Zmax)2 + (a − e)2 − b2 ≥ 0.

The graphical illustration of constraint g5(x) andg6(x) is shown in Figure 5.

l

a

b

ee

b

a

i) ii)

α

α

l − Zmax

Figure 5: Geometric illustration of constraints i) g5(x)ii) g6(x) for Gripper-I.

6. From the geometry of the gripper the followingconstraint can be derived:

g7(x) = l − Zmax ≥ 0. (4)

7. The minimum gripping force should be greaterthan or equal to chosen limiting gripping force:

g8(x) = minz

Fk(x, z) − FG ≥ 0, (5)

where FG is the assumed minimal griping force.

2.1 NSGA-II-cum-Golden-SectionSearch and Results

We employ the constraint handling strategy of NSGA-II to take care of all of the above constraints. The grip-per dimensions can be such that for some displacementvalue z, the gripper configuration is no more a mecha-nism or there is a locking in the mechanism. We firstcheck for such a scenario by considering a fixed incre-ment of z by 0.1 mm. If such a scenario happens, wepenalize the solution by a large amount to discourageits presence in the NSGA-II population. Otherwise,as discussed above, for every population member, thegolden section search approach is employed to com-pute maximum Fk value and check the extremes of zto compute the minimum value of Fk.

The other fixed parameter values for the problemand NSGA-II parameters are given in the following:

1. The variable bounds of all the link lengths andangle are as follows: 10 ≤ a ≤ 250, 10 ≤ b ≤ 250,100 ≤ c ≤ 300, 0 ≤ e ≤ 50, 10 ≤ f ≤ 250,100 ≤ l ≤ 300, 1.0 ≤ δ ≤ 3.14.

2. The geometric parameters are: Ymin = 50 mm,Ymax = 100 mm, YG = 150 mm, Zmax = 50 mm,P = 100 N and FG = 50 N .

3. NSGA-II parameters are as follows: populationsize = 200, probability of SBX recombination =0.9, probability of polynomial mutation = 0.1,distribution index for real-variable SBX crossover= 20, and distribution index for real-variablepolynomial mutation = 100.

Figure 6 shows the obtained non-dominated frontby NSGA-II-cum-golden-section search procedure. Tocompare our results, we also plot the results reportedin another study [10] (we term these results as ‘Orig-inal’ here). After 1,000 generations of NSGA-II, theobtained trade-off front completely outperforms thatreported in the original study. However, as we keeprunning our procedure, a betterment in the trade-offfrontier is noticed. We then ran our procedure for alarge number of generations and the resulting frontis shown to be substantially better. This front is alsoshown in the figure. An increase in generations did notseem to change the front in any significant manner.

2

1

A

B

C3

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.2 0.4 0.6 0.8 1 1.2f

f

OriginalNSGA−II (Small Computation)

NSGA−II (Large Computation)

Figure 6: Trade-off solutions obtained using NSGA-II run for small and large number of generations forGripper-I. Solutions are compared with the originalstudy [10].

To investigate the changes in configurations of thegripper from one extreme trade-off solution to theother, we take two extreme solutions (A and C) andone intermediate solution (B) and show a diagram foreach in Figure 7.

2.2 Innovization Study on Gripper-I

Interestingly and surprisingly, the grippers change inonly one of the dimensions, whereas the remaining sixdesign variables have an identical value. To investi-gate how one trade-off solutions vary from the other,we now analyze the values of seven design variables asa function of one of the objectives (force transmission

4

(A) (B) (C)

Figure 7: Three configurations taken from the NSGA-II front for Gripper-I.

ratio). Figures 8 to 14 show these plots. It is clear thatdimensions a and b are fixed at their allowable upperlimit value, dimensions e and l are fixed at their al-lowable lower limit value, and dimension f and theangle δ are set to an intermediate value (37 mm and98.6 degrees, respectively). The only way one optimalconfiguration changes to another optimal configura-tion by making a trade-off between two objectives isthat the dimension c (gripper length) changes mono-tonically (Figure 14) with the force transmission ratio(the second objective). For a higher force transmissionratio, the dimension c must be large and vice versa.This can be explained from the fact that for a largelink-length c, the force experienced at the tip will besmall, thereby making a large value of f2. Thus, ifa better grip is desired, it is better to have a smallerlink-length c and keep all the other variables to theirprescribed values as discussed above.

In some sense, this innovized design principles sug-gest that if a designer wants to guarantee an optimalconfiguration the dimensions a, b, e, f and l, and an-gle δ must be set at some specific values, but the di-mension c (gripper length) can be varied to suit therequired trade-off in the two chosen objectives. Whenwe fit the c-variation with the second objective, weobserve the following linear empirical relationship:

c = 243.6f2. (6)

Such design principles are useful to not only design anoptimal configuration, but also to get valuable insightabout the properties of possible optimal solutions un-der a multi-objective scenario.

Before we leave this problem, we would like to dis-cuss an important aspect of the innovization study.Innovization task is performed on a set of Pareto-optimal points to decipher salient principles which arecommon to the optimal points. Such principles arethen expected to provide valuable knowledge about

2

Slope =1.0

100

150

200

250

300

1 1.05 1.1 1.15 1.2 1.25

c

f

Figure 14: Variation of link length c with force trans-formation ratio for Gripper-I.

properties common to optimal solutions and that donot exist in non-optimal solutions. However, if sucha task is performed on a set of non-dominated set ofsolutions away from the Pareto-optimal set, there mayexist some other relationships common to them, butthey may be different from the innovized principlescommon to Pareto-optimal solutions. For this prob-lem, we have two fronts, as shown in Figure 6, oneobtained with a reduced computational effort and onethat was obtained with a large number function eval-uations. When we perform an innovization task onthe front obtained after 1,000 generations, a differentset of principles emerged. Table 1 tabulates the differ-ences in innovization principles of two sets of solutions.

Although both sets make all but one of the vari-ables constant across the front, the constant valuesare somewhat different from each other. The trade-off points in both sets differ only in the link-length c.This study reveals the fact that by keeping all but cvariable fixed may achieve a trade-off among the ob-jectives, but there exist one set of fixed values (shownin the first row of the table) that makes the front close

5

2

10

50

100

150

200

250

1 1.05 1.1 1.15 1.2 1.25

a

f

Figure 8: Variation of link lengtha with force transformation ratiofor Gripper-I.

2

10

50

100

150

200

250

1 1.05 1.1 1.15 1.2 1.25

b

f

Figure 9: Variation of link lengthb with force transformation ratiofor Gripper-I.

2

0

10

20

30

40

50

1 1.05 1.1 1.15 1.2 1.25

e

f

Figure 10: Variation of link lengthe with force transformation ratiofor Gripper-I.

2

10

3750

100

150

200

250

1 1.05 1.1 1.15 1.2 1.25

f

f

Figure 11: Variation of link lengthf with force transformation ratiofor Gripper-I.

2

100

150

200

250

300

1 1.05 1.1 1.15 1.2 1.25

l

f

Figure 12: Variation of link lengthl with force transformation ratiofor Gripper-I.

2

1

1.5

1.72

2

2.5

33.14

1 1.05 1.1 1.15 1.2 1.25f

δ

Figure 13: Variation of angle δwith force transformation ratio forGripper-I.

to the Pareto-optimal and is of interest to designers.It can observed from the table that while the larger

computational results make four of the seven variablesto lie on their allowable bounds (lower or upper), whilethe smaller computational results make only one of thevariables lie on its upper bound. Due to the highlynon-linear and trigonometric terms in the constraints,it took a large number of generations before the near-optimal solutions could be found.

3 Gripper- II Configuration De-sign

An overall sketch of the Gripper-II is shown in Fig-ure 15 and was discussed in [10]. This configuration

bc

l

a

d

z

β

α

Figure 15: A sketch of robot Gripper-II.

is somewhat simpler from Gripper-I. The design vari-able vector is x = (a, b, c, d, l)T , where the elementsof this vector are five link-lengths of the gripper. Byusing geometrical relationships, we can write the fol-lowing:

g =√

(l + z)2 + d2,

α = arccos(a2 + g2 − b2

2.a.g) + φ,

a2 = b2 + g2 − 2.b.g. cos(β + φ),

β = arccos(b2 + g2 − a2

2.b.g) − φ,

φ = arctan(d

l + z).

The distribution of the forces is presented in Fig-ure 16. Based on the figure, we can write the follow-

l

d

P

g

c

b

RF

z

β

αφ

φ

Figure 16: Distribution of forces on Gripper-II.

6

Table 1: Comparison of innovized principles of two sets of trade-off solutions for Gripper-I.

a b c e f l δ[10,250] [10,250] [100,300] [0,50] [10,250] [100,300] [1.00,3.14]

Large computations 250 250 243.6f2 0 37 100 98.6o

Small computations 250 233 210.3f2 14 15 180 106.0o

ing:

R.b. sin(α + β) = F.(b + c),

R =P

2. cos(α),

Fk =P.b sin(α + β)2.(b + c). cosα

.

Two objectives used in this study are identical tothat in Gripper-I. However, the constraints are some-what different. The first four constraints are exactlythe same as discussed in Gripper-I. The other con-straints are as follows:

g5(x) = c − Ymax − a ≥ 0,

g6(x) = (a + b)2 − d2 − (l + z)2 ≥ 0,

g7(x) = α ≤ π

2,

where, y(x, z) = 2(d + (b + c)) sin(β) is the displace-ment of gripper ends, Ymin is the minimal dimensionof the griping object, Ymax is the maximal dimensionof the griping object, YG is the maximal range of thegripper ends displacement, and Zmax is the maximaldisplacement of the gripper actuator.

3.1 NSGA-II-cum-Golden-SectionSearch and Results

We use the identical algorithm as that used in the caseof Gripper-I. The following parameter values are used:

1. Variable bounds are as follows: 10 ≤ a ≤ 250,10 ≤ b ≤ 250, 100 ≤ c ≤ 250, 10 ≤ d ≤ 250,10 ≤ l ≤ 250

2. Other parameters are as follows: Ymin = 50 mm,Ymax = 100 mm, YG = 150 mm, Zmax = 50 mm,P = 100 mm,

3. NSGA-II Parameters are exactly the same as inthe case of Gripper-I.

Another study on a number of solutions, it wasfound that the gripping force Fk has a single maxi-mum point, which is found using the golden sectionmethod. The minimum of Fk lies usually on one ofthe extreme values of z. Figure 17 shows a typicalvariation of Fk on the entire range of z values used inthis study.

k

40

50

60

70

80

90

100

110

120

130

140

150

0 5 10 15 20 25 30 35 40 45 50z

FFigure 17: A typical variation of gripping force Fk

with the displacement z for Gripper-II.

Figure 18 shows the trade-off frontiers obtainedby the NSGA-II procedure after 500 generations. A

1

A

B C

2

1

2

3

4

5

6

7

8

9

0 100 200 300 400 500 600 700 800 900f

f

NSGA−II

Original

Figure 18: Trade-off frontiers obtained by NSGA-IIand the original study are compared for Gripper- II.

comparison with the original study [10] reveals thatthe frontier obtained using NSGA-II completely dom-inates the front reported in the original study. The useof efficient a constraint handling strategy, an accuratemethodology for identifying minimum and maximumof gripping force, and modular operations of NSGA-IIallowed better non-inferior solutions to be found. Fig-ure 19 shows three widely distributed configurations,including two extreme trade-off solutions (A and C).

7

(A) (B) (C)

Figure 19: Three NSGA-II solutions selected from trade-off frontier for Gripper-II.

3.2 Innovization Study on Gripper-II

Like in Gripper-I, here we study the obtained NSGA-IIsolutions for any hidden design principles. Figures 20,21, 22, 23, and 24 show the variation of variables fordifferent values of the second objective function (forcetransformation ratio). The dimension a takes an iden-tical intermediate value for all trade-off solutions. Thedimensions c and l take their upper bound value forall trade-off solutions. The dimensions b and d re-duce with force transmission ratio (f2) in the followingmanner:

b = 270.426f−0.2132 , (7)

d = 138.380f−0.1492 . (8)

These variations in the range of their allowable rangeis rather small, but are enough to cause a trade-offbetween the two objectives considered in this study.The reduction in link-length b with increasing f2 canbe explained as follows. Since b and c act like a balancewith end forces proportional to P and Fk, respectively,for a fixed c, f2 which is the ratio of P/Fk is inverselyproportional to b.

4 Combined Consideration ofGrippers I and II

After finding the trade-off solutions for both grip-pers independently, one may wonder which of the twogrippers are better from the point of view of twoobjectives used in this study (minimum variation ingripping force and minimum force transmission ra-tio). Although they look similar, both grippers arefundamentally different in their operations. Gripper-I closes its grip when the actuation takes place to-wards right, whereas in Gripper-II it happens whenit is pushed left. Moreover, Gripper-II has fewer linksthan Gripper-I. Trade-off plots (Figures 6 and 18) sug-gest that Gripper-II causes a larger variation in grip-ping force Fk compared to that in Gripper-I. Also, forGripper-I, the force transmission ratio is better. Thus,in general, Gripper-I can be considered to be a betterdesign than Gripper-II.

Figure 25 shows the trade-off frontiers fromGripper-I and Gripper-II studies. It is clear from the

1

2

1

1.5

2

2.5

3

3.5

3.8

0.01 0.1 0.6 1 10 100 900f

f

Gripper−II

Gripper−I

Figure 25: Trade-off frontiers of the two gripper stud-ies are shown.

figure that Gripper-I solutions outperform Gripper-IIsolutions, hence Gripper-I can be considered as a bet-ter design solution than Gripper-II.

5 Conclusions

In this paper, we have considered the design optimiza-tion of two robot gripper configurations, each having anumber of non-linear and geometrical constraints. An-other difficulty of these problems is that the objectivesand constraints involve a couple of single-variable op-timization tasks, thereby making the problem a nestedoptimization problem. Here, we have used the goldensection search method for solving the inner optimiza-tion task and the constrained NSGA-II procedure forthe bi-objective optimization task.

First, the obtained trade-off frontier for each con-figuration has been found to outperform the front re-ported in another study on the same problems. Sec-ond, the trade-off solutions have been analyzed to de-cipher salient design principles associated in them. In-terestingly, all the trade-off solutions have been foundto vary in only one of the link-lengths and other vari-ables have been found to remain constant in all trade-off solutions. Such knowledge about trade-off solu-

8

2

50

100

150

200

250

2 2.5 3 3.5 3.75

a

1.5 10

f

Figure 20: Variation of link lengtha with force transformation ratiofor Gripper-II.

Slope = −0.213

2

50

100

150

200 250

2 2.5 3 3.5 3.75

b

f1.5

10

Figure 21: Variation of link lengthb with force transformation ratiofor Gripper-II.

2

150

200

250

2 2.5 3 3.5 3.75

c

1.5 100

f

Figure 22: Variation of link lengthc with force transformation ratiofor Gripper-II.

Slope = −0.149

2

10

50

100

130

200 250

1.5 2 2.5 3 3.5 3.75

d

f

Figure 23: Variation of link lengthd with force transformation ratiofor Gripper-II.

2

150

200

250

2 2.5 3 3.5 3.75

l

1.5 100

f

Figure 24: Variation of link lengthl with force transformation ratiofor Gripper-II.

tions at large are valuable to the designers.From the two sets of trade-off solutions obtained by

our proposed approach, it has also become clear thatGripper-I configuration is better than Gripper-II froma perspective of both objectives. Thus, our optimiza-tion task has finally suggested the use of Gripper-I fora possible implementation.

Acknowledgments

The study is funded by Department of Sci-ence and Technology, Government of Indiaunder SERC-Engineering Sciences scheme (No.SR/S3/MERC/091/2009).

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