Quantitative Analysis of Inner Force Distribution and Load Capacity of Grasps and Fixtures

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Youlun XiongSchool of Mechanical Engineering,

Huazhong University of Science and Technology,Wuhan 430074, China

Han Ding*School of Mechanical Engineering,

Shanghai Jiaotong University,Shanghai, 200030, China

Michael Yu WangDepartment of Automation & Computer-Aided

Engineering,The Chinese University of Hong Kong,

Shatin, NT, Hong Kong

Quantitative Analysis of InnerForce Distribution and LoadCapacity of Grasps and FixturesThis paper focuses on a quantitative analysis for grasp planning and fixture design bon an analytical description of point contact restraint. In the framework, the analdeals with the fundamental concepts of restraint cone, freedom cone, force-determand relative form closure. A method is presented to quantify the performance of a fi(or grasp) with two major characteristics of inner force distribution and load capacTwo different fixturing (or grasp) models of simplex grasp and elastic grasp are preseIt is shown that the performance of these two types of grasp (or fixturing) couldmeasured with different performance indices. A minimax index (MMI) and a volmeasure are defined for evaluating a simplex grasp, while a measure using the tolerange of differential motion in the twist space or the allowable load polyhedron inwrench space would be suitable for quantifying robustness and load capability oelastic fixture system. Furthermore, for fixture system design a geometric analysireasoning procedure is described for the design of locators, clamps and supplemesupports. The aim of these proposed analysis and design techniques is to provscientific foundation for automated grasping/fixturing system design in the enginepractice. @DOI: 10.1115/1.1459089#

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1 IntroductionGrasping planning and automated fixture design have been

cussed at length in the manufacturing and robotics litera@1–3#. The problem of designing fixtures is closely related to thof grasp analysis. The functions of fixturing and grasping are silar to restraining an object kinematically by means of a suitaset of contacts. Their applications include multifingered robomanipulation@3#, reconfigurable fixtures@4#, and flexible manu-facturing @2#.

Most work in the area of grasping and fixturing can be dividinto two categories, analysis of a given grasp or a fixture aautomatic synthesis of a grasp or fixture configuration. For ovdecade, a popular tool for grasp and fixture analysis has beescrew theory, which was elaborated by several researchers ining Ohwororiole and Roth@5#. Based on the theory, DeMeter@6#studied the surface contact and friction issues in restraint analAsada and By@4# studied kinematic problems of fixture desigincluding deterministic locating, total restraining, accessibility adetachability. Xiong et al.@7# presented the reciprocity betweethe restraint sub-space and freedom sub-space of a kinemapair and showed its application in grasping planning and fixtdesign. Another important issue is how to define quality measuand to optimize the fixturing/grasping forces to balance an exnal wrench. Several grasp quality measures have been propbased on the smallest force necessary to resist applied foFerrari and Canny@8# developed quantitative tests for force clsure using the radius of a maximal ball centered at the originintroduced the convex hull of primitive wrenches as a measurthe goodness of a grasp. An alternative quality measure, propby Trinkle @9# and DeMeter@10#, is the maximum of the minimumvalue of the primitive wrench intensities in equilibrium. Chen@11#developed a procedure for estimating clamping forces andreaction forces of fixtures under the frictionless contact mode

As to computational issues, Ponce et al.@12# gave a geometriccharacterization of equilibrium and force closure and presente

*Corresponding author. Email: [email protected], [email protected] by the Manufacturing Engineering Division for publication in t

JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedApril 2000; Revised August 2001. Associate Editor: E. Demeter.

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efficient algorithm for computing concurrent grasps. Brost aPeters@13# presented an algorithm that automatically designstimal fixtures for a particular class of problems. They also etended a fixture-quality metric to consider 3-D forces, positirepeatability and ease of loading. Xiong et al.@14# studied graspcapability and presented a dynamic stability index for contact cfiguration planning. Liu@15# formalized a qualitative test of 3-Dfrictional form-closure grasps based on the duality of convex hand convex polytopes. Wang@16# proposed an approach to seleoptimal locators from an initial collection of a large numberfixel locations.

Our work described in this paper focuses on issues of fodeterminacy and relative form closure. These issues have notpreviously investigated in systematic fixture design and grplanning. These issues are of practical significance, since thamany cases a form-closed fixture design may be impossible dupossible strong geometric constraints. The concept of the relaform-closure@17# will be illustrated with specific examples ograsps of a sphere and a cylinder. Our work attempts to extendanalysis method of Asada and By@4# by providing a design pro-cedure to incorporate the force determinacy and relative formsure considerations. Furthermore, we will present a quantitaanalysis of inner force distribution and load capacity of graspsfixtures, based on our previous research of optimal synthesipoint contact restraint@7,14,17#. Quantitative indices are described for an optimization approach to fixture design.

The paper is organized as follows. In Section 2 we reviewbasic concepts related to fixture design and grasp planning,as locating and clamping equations, restraint cone and freecone, form closure and relative form closure. In Section 3discuss the force determinacy and inner force distribution osimplex grasp. Performances indices are introduced for evaluadifferent simplex grasps or fixtures. In Section 4 stiffness maof an elastic grasp is defined and the load capacity of the systecharacterized. Considerations for a systematic design of fixsystems, including support stiffness, locating accuracy, prellevel and supplementary supports, are presented in Section 5nally, the results of our work are summarized and conclusionsdrawn in Section 6.

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2 Characterization of Form ClosureIn order to quantify inner force distribution and load capacity

a fixture/grasping system, we first describe a mathematical moof point contact restraint and its various characterizations. Fig1~a! depicts a rigid bodyB grasped~or fixtured! by a set of fingers~or fixels!. Suppose that each fingertip makes a frictionless pocontact on the bodyB; then the grasp~or fixture! system is repre-sented by the set of frictionless contacts as shown in Fig. 1~b!. Ina reference frame (xyz), each contact pointi is described by aline vectorPi of 6-elements:

Pi5F ni

r i3niG i 51,2,̄ ,m

where ni is the unit normal vector of the body surface at thcontact pointi, r i is the position vector of the pointi, andm is thenumber of fingers. In the paper, the grasp/fixture configuration~orsimply grasp! is denoted byG and is defined as follows: GraspGconsists of a set of line vectors

G5$P1 ,P2 ,¯ ,Pm%

Equivalently, graspG is expressed as a matrix consisting of thline vectors as its columns

G5@P1P2¯Pm#

2.1 Locating and Clamping Equations. With the aboverepresentation of a grasp, differential motions of the bodyB canbe mapped from configuration space to the contact space. Supthat the body undergoes a differential motiond in the configura-tion space. Then a corresponding perturbation« i in the position ofthe contact pointi along the normal direction is given as~Fig. 1!:

PiTd5« i ~ i 51,2,̄ ,m!

whered5@hTdT#T, andh andd denote the differential translationand differential rotation of the body respectively. Let«5@«1 ,«2 ,¯ ,«m#T. Then the above equation may be rewrittenthe matrix form:

GTd5« (1)

Equation~1! represents the relationship between the differentmotion d in the configuration space and the motion« in the con-tact space. The equation is often referred to as the locating eqtion.

For an m-fingered grasp, the equilibrium condition for thegrasped body under the action of an external force~wrench! w canbe represented as:

Gl5w, l>0 (2)

where l5@l1 ,l2 ,¯ ,ln#T with l i denoting the magnitude ofnormal contact force at the pointi. Note that it must be nonnega-tive, i.e.,l i>0. The equation is often referred to as the clampinequation in fixture analysis since it defines locators’ reactiforcesl for any given clamping wrenchw. These two equations

Fig. 1 Fixture „Grasp … system

Journal of Manufacturing Science and Engineering

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are fundamental for grasp planning and for fixture design.example, with a given graspG, any feasible differential motiondof the object is defined by the following linear constraint contion:

GTd<0 (3)

On the other hand, clamping Eq.~2! denotes the possible disturbance wrenchw which can be resisted by the grasp configuratiG.

2.2 Freedom Cone and Restraint Cone.Definition 1: Freedom Cone

The freedom cone of a graspG is defined as all twists whichare reciprocal or contrary to all line vectorsPi ( i 51,2,̄ ,m) ofG, i.e.,

R* @G#5$d:GTd<0% (4)

Obviously, R* @G# is the intersection ofm closed half spacesformed by the planes reciprocal to line vectorsPi , and it is apolyhedral cone in the twist space.Definition 2: Restraint Cone

The restraint cone of a graspG is defined as the aggregate of apossible applied disturbance wrenchesw that can be resisted bythe graspG, i.e.,

R@G#5$w:w5Gl,l>0% (5)

Obviously, the restraint coneR@G# is a polyhedral cone in thewrench space. Using the Farkas theorem@18#, The restraint coneR@G# and the freedom coneR* @G# are dual to each other withfollowing relationships:

~1! ;dPR* @G# and;wPR@G#, we havewTd<0. This meansthat w and d are contrary or reciprocal. In other words,R* @G#andR@G# are the negative-work dual polyhedral cones in the twand wrench spaces respectively@5#.

~2! It is equivalent thatR* @G#5R6 and thatR@G#5$0%.~3! It is equivalent thatR* @G#5$0% and thatR@G#5R6

It is convenient to use these geometric properties of the restrconeR@G# and the freedom coneR* @G# for characterizing thefixture systemG. For example, the third dual relationship impliethat if the set of feasible motions of the grasped object is n~$0%!, then the object is fully immobilized. This also indicates ththe set of all external disturbance wrenches which can be resby fixtureG is the entire wrench spaceR6. This is known as formclosure and is further discussed as follows.

2.3 Relative Form Closure. Form closure is an importanconcept in grasping and fixturing. For completeness, we spresent a formal definition for it. Our emphasis here is given tconcept of relative form closure, which will be defined and illutrated with examples.Definition 3: Form Closure

A graspG is said to be form closed, if and only if the restraiconeR@G# spans the whole wrench spaceR6, i.e.,

R@G#5R6 (6)

Based on the third dual relationship above, we can obtainequivalence definition of form closure asR* @G#5$0%. Further-more, an equivalence theorem can be derived to make use oconcept of convex hull, whose proof appears in the appendixTheorem 1: Equivalence Theorem for Form Closure

A graspG is form closed, if and only if the origin of the wrencspaceR6 lies inside the interior of the convex hull ofG. In otherwords, the conditionOP int(C@G#) is equivalent to the conditionR@G#5R6, where the convex hullC@G# of the set of line vectorsG5$P1 ,P2 , . . . ,Pm% is defined as

C@G#5H w:w5(i 51

m

l i Pi ,(i 51

m

l i51,l i>0J (7)

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As a corollary of Theorem 1, the following necessary conditiofor form closure hold. These necessary conditions are usefufixture design and grasp planning.Theorem 2: Necessary Conditions for Form Closure

If a grasp configurationG is form closed, then

~1! the interior ofC@G# is nonempty, or int(C@G#)ÞF.~2! the number of its line vectorsm must satisfy thatm>6

11 ~or 7!.

In practice, a familiar principle in design of fixtures to useleast 6 locators and one clamp is consistent with these nececonditions. However, there exist a set of special surfaces sucfinite surfaces of revolution, infinite planes, cylinders and helisurfaces for which no form-closed grasp exists@19#. It is highlyuseful to extend the concept of form closure to be able to accmodate these situations.Definition 4: Relative Form Closure

A graspG is said to be relatively form closed, if and only if itrestraint coneR@G# is equal to the linear subspaceL@G# spannedby G, i.e.,

R@G#5L@G# (8)

where linear subspaceL@G# of G is defined as follows:

L@G#5H w:w5(i 51

m

l i Pi ,l iPRJor L@G#5$w:w5Gl,lPRm% (9)

Furthermore, ifG is affine independent, then the convex huC@G# is a simplex, andG is said to be a simplex grasp witrelative form closure.Theorem 3: Equivalence Theorem for Relative Form Closure

A grasp configurationG is of relatively form closure, if andonly if the origin of the wrench space is inside the relative interof the convex hullC@G#. The relative interior of a convex hull isa well defined concept in convex analysis@20#, and it is denotedas rint(C@G#).

Applying the definition of relative form closure~Eq. ~8!!, wehave a relationship known as the topological equivalence betwthe relative form closure and the convex hull. In other words,following two statements are equivalent:

~a! 0Pr int~C@G# !.

~b! R@G#5L@G#. (10)

Furthermore, using the dual relations between the restraint cand the freedom cone defined in Definition 2, the following twstatements are also equivalent for relative form closure:

~a! R@G#5L@G#.

~b! R* @G#5L* @G#.

where the linear subspaceL* @G# ~also referred to as the reciprocal subspace ofL@G#! is defined by all the reciprocal screwsatisfying

L* @G#5$d:GTd50%

It should be noted that, for a relative form closed graspG, itsrestraint coneR@G# is reciprocal to its freedom coneR* @G#.Therefore, for any wrenchw and twistd satisfying;wPR@G#and;dPR* @G#, thenw andd are reciprocal to each other, i.ewTd50. Thus, the relative form closure is closely related toconcept of reciprocity in the extended screw theory@5#.

2.4 Examples. Three planar examples are used here to illtrate the concepts of restraint cone, freedom cone and relaform closure defined above. As shown in Fig. 2~a! for the firstexample, when an object is restrained by 3 point contacts in3-D wrench space (f x , f y ,mz), the contact line vectors are giveas follows:

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P15F 010G P25F 0

11G P35F 1

00G G5@P1P2P3#5F 0 0 1

1 1 0

0 1 0G

The clamping equation of the configurationG is given as:

F 0 0 1

1 1 0

0 1 0G F l1

l2

l3

G5F f x

f y

mz

GIn the 3-D twist space (dx ,dy ,dz), the locating equation is described as

F 0 1 0

0 1 1

1 0 0G F dx

dy

dz

G5F «1

«2

«3

GAccording to Definition 2, the restraint coneR@G# spanned by

P1 , P2 and P3 is a polyhedral cone in the wrench space. Obously,R@G# has 3 extreme directionsP1 , P2 andP3 ; the convexhull C@G# is a triangle with 3 vertices defined byP1 , P2 andP3in the wrench space; and the linear subspaceL@G# is the 3-Dwrench spaceR35( f x , f y ,mz). These are shown in Fig. 2~b!.Based on Definition 1, the freedom coneR* @G# is the intersectionof the following 3 closed half spaces in the twist space:

F 0 1 0

0 1 1

1 0 0G F dx

dy

dz

G<F 000G

Similarly, as shown in Fig. 2~c!, R* @G# is a polyhedral cone withfollowing 3 extreme directions in the twist space:

F 00

21G , F21

00

G , F 0211

GAs illustrated in Figs. 2~b! and 2~c!, it is obvious that the con-

figurationG is not form closed, since

~1! R@G#ÞR3

~2! 0¹ int~C@G# !

~3! R* @G#Þ$0%

Fig. 2 Restraint cone R†G‡ and freedom cone R* †G‡ of a 2-Dobject restrained by 3 point contacts

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As shown in@21#, form closure of any 2-D bounded object witpiecewise smooth boundary~except for a circle! can be achievedby 4 fingers. In order to complete the form closure, we mustanother point contactP4 on the object. In the case of fixturdesign, this point contactP4 is usually referred to as a clampFurthermore, the necessary and sufficient conditions to compleform closure with the last~fourth! point contact are as follows:

~a! dim~R@G# !53, and

~b! P4P int~2R@G# !

Suppose that the fourth point contact yields, as shown in3~a!,

P45@21,22,21#T

This will complete the form closure. It is clear that the originthe 3-D wrench space is now in the interior of the tetrahedC@G# with verticesP1 , P2 , P3 and P4 , as shown in Fig. 3~b!.However, in the most general sense with respect to the entirewrench space, this form closure is a relative form closure.

Two more examples of grasp are shown in Figs. 4~a! and 4~b! toillustrate relative form closure in 2-D wrench space and in 1wrench space respectively. In Fig. 4~a!, graspG5$P1 ,P2 ,P5% isform closed inR25( f y ,mz)

T, but it is relatively form closed inR3. In Fig. 4~b!, G5$P3 ,P6% is form closed inR85( f x), but it isalso a relative form closure in a wrench space of higher dimsion, e.g.,R6 or R3.

In general, we may state that if a graspG is of relative formclosure in a wrench space, it must be a complete form closura lower dimensional space, and vice versa. Moreover, ifG isrelatively form closed, then the whole wrench spaceR6 can beconsidered as the Cartesian product

R65R1r 3R2

s

WhereR1r is the subspace whereG is relatively form closed and

R1s is the complementary space ofR1

r (r 1s56). As pointed outby DeMeter @6#, if a workpiece is totally restrained by contaregion geometry, it is said to be form closed; if it is restrained wthe aid of friction, it is said to be force closed. WhenG is relativeform closed, the workpiece is restrained in the subspaceR1

r by thecontact region geometry, while it may be restrained in the sspaceR2

s by the actions of frictional forces.

Fig. 3 Relative form closure and C†G‡

Fig. 4 Relative form closure


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3 Inner Force Distribution of a Simplex GraspIn fixture design or grasp planning, an important requiremen

on the distribution of the contact forces among the fixels or rofingers. In general these forces should be distributed as uniforas possible. This is often referred to as a balanced design@22#. Inthis section, we first discuss the inner force distribution and fodeterminacy of a simplex grasp. We then introduce performaindices for evaluating the force distribution of a graspG.

Several common aspects of qualitative analysis of fixturing agrasping have been previously investigated@17,23#. In fixture de-sign, one must consider the necessary and sufficient conditR* @G#ÞF and int(R@G#)ÞF for respective accessibility and determinacy of a configurationG. It is convenient for the fixturedesign to use 6 contact points with 6 independent line vectThese are commonly referred to as 6 locators. Such a set ofvectors span the whole screw spaceR6. In order to fulfill thecondition of form closure, it is necessary to use one more pocontact such thatScP2 int(R@G#). This point is referred to asclamping point. Obviously, these 7 point contact together formfixture configurationG0: $Si ,i 51,2,...,7% and satisfy the follow-ing conditions: 1! 0P intC@G0#; 2! G0 is affine independent; and3! Any arbitrary 6 points among them are linearly independ~i.e., Haar condition!. This configurationG0 is referred to as asimplex grasp. In practice, most fixture systems can be descras a simplex grasp mathematically, whose algebraic representcan be described in the following theorem which has been proin the appendix.Theorem 4: Algebraic Equivalence Theorem for Relative ForClosure@17#

The (r 11) point graspG5$p1 ,p2 ,...,pr 11% in wrench spaceRr is relatively form closed if, and only if, the linear equations

(i 51

r 11

l ipi50 (11)

(i 51

r 11

l i51

have a strictly positive solutionl i.0 (i 51,2, . . . ,r 11). Physi-cally, l i represents the distribution of the inner forces of tgrasped object at the point contacti.

In fact, whenw50, the clamping Eq.~2! becomes a homogeneous oneGl50 ~as shown in Eq.~11!. In the case thatG issimplex grasp with relative form closure, the associated homoneous equationGl50 has a non-zero solution (lÞ0). Thenal(a.0) is also a non-zero solution. The strictly positive soltion l i.0 (i 51,2, . . . ,r 11) of Eq. ~11! would denote the mag-nitudes of reaction forces on the fixels, when the sum of allmagnitudes equals the unit, i.e.,( i 51

r 11l i51. If ( i 51r 11l i5a.0,

then Eq.G l50 has solutionl i* 5al i ( i 51,2, . . . ,r 11) whichis also strictly positive.

3.1 Force Determinacy. Since a simplex grasp G5@p1 ,p2 , . . . ,pr 11# is affine independent, the augmented matof G

GO 5Fp1 p2 ¯ pr 11

1 1 . . . 1 G (12)

has full rank, i.e., rank (GO )5(r 11). The solution l i (I51,2, . . . ,r 11) of the system~11! is unique. Therefore, for asimplex graspG with relative form closure, we have an uniqusolution:l i.0 (i 51,2, . . . ,r 11) and( i 51

r 11l i51. Obviously,l iis similar to the distribution density of a discrete probability. Fthe simplicity, we rearrangel i in the order:l1>l2>l3>¯

>l r 11.0. Then, we have the following proposition for forcdeterminacy.Proposition 1: Force Determinacy

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For a simplex graspG with relative form closure, the inneforce distributionl i of the fixtured object at fixeli is deterministicwith a unique solutionl i . For example, suppose that the foramplitudel r 11* on the fixel for clamping is specified by an actutor, then the reaction forces acting on the locators can be calated as

l i* 5l r 11*

l r 11l i5al i , i 51,2, . . . ,r

Proposition 1 asserts that the simplex fixturing system wrelative form closure possesses determinacy of inner force dibution, shortly referred to as force determinacy~orF-determinacy! which is different from the configuration determnacy proposed by Asada and By@4# in concept. The latter is concerned with the properties of all locators, while the former dewith the attributes of forces on the whole fixture system includall locators and clamps. As we know, the configuration deternacy~or C-determinacy! of a graspG is characterized by its rankrank(G)56, but it is neither necessary nor sufficient forF-deterministic graspG to have rank(G)56. However, if the fix-ture systemG05Gø$p0% belongs to a simplex grasp with dimensionality of 6~F-deterministic!, then theG0 is F-deterministic andform closed, andG is C-deterministic.

3.2 Minimax and Maximin Measures. As an importantcharacterization for evaluating a grasp configuration, the inforces on the fixels of a fixturing system are often required toof a uniform distribution. In order to reach this goal, the perfomance index for judging inner force distribution of a given graG should be defined. For a simplex and relatively form closgrasp, we define two performance indices: minimax and maxiindices~MMI !.

There are two extreme cases of the inner force: the maximinner force, and the minimum inner force, i.e.,

l̄@G#5max$l1 , l2 , . . . , l r 11% (13)

lI @G#5min$l1 , l2 , . . . , l r 11%

Usually, a graspG1 is better than a graspG2 if

l̄@G1#,l̄@G2# and lI @G1#.lI @G2# (14)

Thus, it is natural to select a graspG to minimize l̄@G# ~theminimax criterion! and to maximizelI @G# ~the maximin crite-rion!, particularly if one wishes to result in a uniform force ditribution.

When considering task requirements and structural featuresfixture system, it may be desirable to assign the inner forces aa given distributionn i* .0 (i 51,2, . . . ,r 11) and ( i 51

r 11n i* 5a.Fortunately, the minimax and maximim criteria~l̄@G# andlI @G#!can be modified to accommodate this requirement by introducweighing coefficients n i5n i* /a.0 (i 51,2, . . . ,r 11) and( i 51

r 11n i51. Letpi5n i pi8 for i 51,2, . . . ,r 11, wherepi is a screwwith amplitude ~or intensity! n i . In terms of pi8 , the uniformdistribution solution will ensure that the intensities of all internforces reach their upper bounds at the same time. These remay be summarized in the following proposition.Proposition 2: Minimax ~Maximin! Measures

For a simplex graspG with relative form closure, the followingis true:

~1! l i is similar to the density distribution of a discrete proability.

~2! 0<lI @G#<l̄@G#,1.~3! If lI @G#5l̄@G#51/r 11 ~i.e., uniform distribution!, thenG

is an optimal grasp in the sense of the minimax and maximcriteria.

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~4! For a given distributionn i>0 (i 51,2, . . . ,r 11) and( i 51

r 11n i51, the uniform distribution ofpi8 will ensure that theinner forces of the graspG5$p1 ,p2 , . . . ,pr 11% have the givendistribution.

Notes: If the graspG85$p18 ,p28 , . . . ,pr 118 ,% has the uniformdistribution of inner forcesl8, i.e., the system of equations

(i 51

r 11

l i8pi850 ~* !

(i 51

r 11

l i851

has the solution

l185l285l5 . . . 5l r 118

Then the inner forcesn i of the graspG have the given distri-bution. In fact, from (* ) we have

(i 51

r 11

l i8n ipi851

r 11Sn ipi850

(i 51

r 11

l i8n i51

r 11Sn i51

It is shown that letpi85n ipi , whenG8 complies with the uni-form distribution ~l185l285l5 . . . 5l r 118 , and ( i 51

r 11l i851!,thenG has the given distribution.

3.3 Volume Measure. Eitherl̄@G# andlI @G# gives only theworst case analysis for upper or lower extreme respectively. Tdo not represent the ability ofG to withstand external forcesTherefore, another quality measurem@G#, called the volume mea-sure ofG, is introduced as follows

m@G#5uGI u/~r ! ! (15)

where uGI u denotes the absolute value of the determinant ofaugmented matrixGI of G. WhenG is a simplex grasp, its convexhull C@G# is anr dimensional simplex. Thus, indexm@G# definesthe volume of the simplexC@G#.

Suppose that fixture configurationG is affine independent inr -D vector spaceRr . Then, we have the following:

~1! The coefficient matrix~12! of the system of simultaneoulinear equation~11!

GO 5FP1 P2 ¯ Pr 11

1 1 ¯ 1 Ghas full rank, i.e., rankGO 5r 11.~2! The determinant of the augmented matrixGO

det@GI #5UP1 P2 ¯ Pr 11

1 1 ¯ 1UÞ0

~3! The convex hullC@G# is a r -D simplex inRr .~4! The volume of the simplexC@G# is proportional to the

absolute value of the determinantuGI u, which is represented byV(C@G#)5uGI u/(r !).

Now, let us show an example of unit circle~Fig. 5!. A fixture Gfor the circle is composed of 3 point contacts. Hence,G5$P1 ,P2 ,P3%, and

P15F10G P25Fcosu2

sinu2G P35Fcosu3

sinu3G

Obviously, the convex hullC@G# is the triangleDP1P2P3 ~sim-plex!, and the augmented matrix


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al

GI 5F 1 cosu2 cosu3

0 sinu2 sinu3

1 1 1G

The absolute value of the determinant det@GI # is found to be

uGI u5usin~u32u2!1sinu22sinu3u

On the other hand, as shown in Fig. 5, the area of the trianDP1P2P3 can be partitioned into 3 parts:DOP1P2 , DOP2P3 ,andDOP3P1 . The total area is given as

V~C@G# !5r2 sinu2/21r2 sin~u32u2!/21r2 sin~360°2u3!/2

5~sinu21sin~u32u2!2sinu3!/2

with r in unit length. Therefore, we have

m@G#5uGI u/2!5V~C@G# ! (16)

The general results are summarized as follows:Proposition 3: Volume Measurem@G#

For a simplex graspG with relative form closure, the volumemeasurem@G# possesses the following properties:

~1! m@G#5V(C@G#)5uGI u/(r !)~2! m@G# could be used to represent the capacity of graspG to

withstand external forces, in the case that the originO is at thecenter ofC@G#. Geometrically, the capacity could be denotedthe radius of the inscribed supersphere of simplexC@G#.

3.4 Grasp of a Sphere. Sphere is a good example for relative form closure. Obviously, it is impossible to achieve complform closure on a smooth sphere. However, it is easy to achierelatively form closed grasp. A set of 4 point contacts would costruct a relatively form closed grasp if the center of the spherin the interior of the tetrahedron with the 4 point contactsvertices, as shown in Fig. 6~a!. Furthermore, a set of 3 pointcontact on a great circle of the sphere~Fig. 6~b!! could alsoachieve relatively form closure, if the center of the sphere is inrelative interior of the triangle formed by the 3 contact points. B

Fig. 5 Volume of simplex C†G‡

Fig. 6 Optimal grasp configurations on a sphere


gle

as

-tee an-is

as

theut

the latter is a relative form closure in 2-dimension, while tformer is in 3-dimension. Figure 6~c! depicts a set of 2 antipodapoints on the sphere, which can achieve a relatively form clograsp in 1-dimension.

The normal vector of a point (u i ,f i) on a sphere is given as

ni5F cosu i cosf i

cosu i sinf i

sinu i

GIn the case of 4 point contacts, the linear equation~11! becomes

5 (i 51

4

l ini50

(i 51

4

l i51

Since that the 4 contact points are the vertices of a regular tehedron as shown in Fig. 6~a!, the system has a solutionl15l25l35l451/4. This is an optimal grasp configuration in termthe minimax measurel@G#.

Similarly, in the case of 3 point contacts whose convex hulan equilateral triangle, the system of linear equations

5 (i 51

3

l ini50

(i 51

3

l i51

have a solutionl15l25l351/3, which yields an optimal graspconfiguration with uniform distribution of inner forces~Fig. 6~b!!.In the case of 2 point contacts, whose convex hull is a diameline, then Eq.~11! has a solutionl15l251/2, which is also op-timal as shown in Fig. 6~c!. The optimal configurations would bethe same if the volume measurem@G# is used.

3.5 Grasp of a Cylinder. Another special case is to graspcylinder. Similarly, it is impossible to achieve form closed graon a smooth cylinder. Therefore, we are concerned with the cponents of differential motion (dx ,dy ,dx ,dy), without consider-ing prismatic motiondz and rotationdz ~Fig. 7!. The unit wrenchpi of line vector at any point (u i ,zi) on the surface of cylinder isgiven by

pi5F cosu i

sinu i

2zi sinu i

zi cosu i

G5F f x

f y

mx

my

GSuch a set of 5 point contactspi ( i 51,2,...,5) will construct anoptimal grasp, if and only if the system of linear equation~11!

5 (i 51

5

l ipi50

(i 51

5

l i51

have the solution:l15l25...5l551/5. It should be noted thathe optimal grasp satisfying conditionl̄@G#5lI @G# is not unique.For example, ifu i is uniformly distributed along the circle, i.eu150°, u2572°, u35144°, u45216°, u55288°, andZ150,then the following 3 sets of point contacts achieve the optimgrasp configuration:

G1 :Z25Z5 ,Z35Z4 ,Z3 /Z250.62

G2 :Z450,Z25Z350.62Z5 (17)

G3 :Z250,Z35Z5 ,Z4 /Z5520.62

MAY 2002, Vol. 124 Õ 449

450 Õ Vol. 124, MAY

Fig. 7 Optimal grasp configurations on a cylinder

o

f

i

e

t

,

o

Eq.nde-rt to

x-then

one

gu-

Figure 7~a! shows optimal grasp configurationG1 . Figure 7~b!shows another optimal grasp configurationG4 . It is noted that allthese optimal grasp configurations form a symmetric group wthe rotational transformation aboutz axis and transnational transformation alongz axis.

In general, there may exist a lot of optimal grasp configuratiothat satisfy the minimax measurel@G#. Therefore, we may usethe volume measurem@G# as an additional criterion to select thbest. General speaking, there are three possible cases.

1 The optimal criterial@G# and m@G# yield the same solu-tion~s!. For a sphere of radiusr, it is shown that when 4 points onthe sphere comprise a regular tetrahedron, thenl15l25l35l451/4r . The volume measurem@G# achieves its maximumm@G#58)r 3/27.

2 Among the set of the optimum solutions found for criteril@G#, there is a subset of the solutions which are also optimalm@G#. For the cylinder example, 4 sets of point contacts,G1 ,G2 , G3 and G4, are obtained such thatl15l25l35l45l551/5. They are optimal in the sense that they can generateform inner forces among the contacts. However, they have difent capacities to withstand external forces. For example, foz51, their corresponding values of the volume measure are gasm@G1#5m@G2#50.38m@G3#.

3 The optimal grasps obtained forl@G# and m@G# respec-tively may be not consistent. A trade-off method has to be madachieve a balance between the two performance indices. A meof multi-objective optimization was suggested in@7#.

4 Load Capacity of an Elastic GraspThe inner force distribution of simplex grasp mentioned abo

uses the ratio-of-amplitude concept without considering the acof external forces. It is noted that the concept for a simplex safies the criterion ofF-determinacy. If a relatively form closedgrasp is not simplex, then there exists a strictly positive solutl i.0 (i 51,2,...,r 11) for Eq.~11!, but the solution is not uniquesince that the matrixG has a rank,(r 11), the graspG5$p1 ,p2 ,...,pr 11% is affine dependent. Such a graspG is called acomplex grasp, because that the convex hullC@G# is a complex inthe screw space. A fixturing system with supplementary suppis always a complex grasp. This situation prompts two problethat need consideration in fixture design or grasp planning:

~1! to determine the ratio-of-amplitudes of inner forces forcomplex grasp~without F-determinacy!.

~2! to compute effective forces of point contacts under extence of an external wrenchw.

In this case, it is well known that the solution to Eq.~2! consists oftwo parts

l5l01ls (18)

2002

ith-

ns

e

nfor

uni-er-rven

tothod

veiontis-

ion

rtsms

a

is-

with ls being a special solution of Eq.~2! and l0 being thegeneral solution of the homogeneous system corresponding to~2!. The related mechanical systems are called as statically utermined. In order to obtain a unique solution, one has to resouse an additional principle such as the least squares@16,24# or toincorporate the elasticity of the grasp system.

4.1 Stiffness Matrix Ks of an Elastic Grasp System„G, K…We shall consider the effect of elasticity of contact point in fituring and grasping, as modeled in Fig. 8. It is assumed thatelastic deformations are so small such the grasp configuratioGremains unchanged under the action of an external wrenchw.Consequently, the differential motiond of the object underw isrelated tow as

w5K sd (19)

whereK s is the stiffness matrix of the fixture system.The stiffness matrixKs of graspG with relative form closure

depends upon the grasp configurationG, and the normal stiffnesski of the point contacts. The normal deformation« i at a contactpoint is linearly related to the normal forcel i

l i5ki« i , or l5K« (20)

whereK5diag (k1 ,k2 ,...,kr 11). By the principle of virtual work,the work done in the configuration space is equal to the one din the contact space, i.e.,

dTw5lT«5(i 51

r 11

l i« i (21)

As shown in Fig. 9, the differential motionsd in the configurationspace and« in the contact space are related by the grasp confiration as

GTd5« (22)

Substituting Eqs.~19!, ~20! and ~22! into Eq. ~21!, we obtain

Fig. 8 Elastic grasp modeling „G,K …


m

t

i

h

act

blece,

ble

t

e

en

n

istingnal

it

n,ws:

g-r.

dTK sd5dTGKGTd

Therefore,

KsÄGKGT (23)

Since thatK is positive definite andG is relatively form closed,K s is also positive definite; therefore, the inverse matrix (Ks

21)exists. From Eq.~19!, we have

d5Ks21w (24)

Proposition 1 asserts that a simplex fixture with relative forclosure possessesF-determinacy. Such a fixture is fully describeby its graspG. For an elastic fixture system, it is characterizedits graspG as well as its contact stiffness matrixK . The elasticsystem is modeled by~G, K !.

For fixture design and analysis, there are two descriptivepects from the viewpoints of motion and force. The locating E~1! maps the differential motion from the configuration spacethe contact space, while the equilibrium Eq.~2! maps forces fromthe contact space to the configuration space. These two equaare dual as illustrated in Fig. 9. The elastic model~G, K ! for afixture system interconnects motion and force in the configuraspace and the contact space. In the configuration space, the dential motiond and forcew are linked by the system stiffnesmatrix Ks, while in the contact space the contact deformation«and the contact forcel are linked by the contact stiffness matrK .

4.2 Computation of Effective Force and Initial NormalDeformation. For an elastic grasp system~G, K !, Eq. ~2! has aspecial solution

ls5KGT~GKGT!21w5KGTd (25)

This special solution is the solution of the elastic grasp withminimum potential energyE:

E51

2«TK«5

1

2ls

TK21ls51

2wTKs

21w51

2dTKsd (26)

Proposition 4: Effective ForcelsSuppose thatG is relatively form closed and the initial inne

forcel0 is given. Then the reacting forcel in the contact space isuniquely determined for an external forcew such that

l5l01ls , l>0 (27)

where effective forcels is is given in Eq.~25!, while the innerforcesl0 is given by the initial fixturing and satisfiesGl050.Proposition 5: Preloadl0 and Initial Deformation«0

For a relatively form closed graspG5$p1 ,p2 ,...,pi 11%, theamplitudesl i0 of inner forces at all point contacts must satisfy tfollowing

(i 51

r 11

l i0pi50 l i0>0~ i 51,2,...,r 11! (28)

Fig. 9 Dual relationship and its links of elastic grasp modeling„G,K …


-dby

as-q.to

tions

ioniffer-s

x

a

r

e

Correspondingly, the initial normal deformations at the contpoints are given as follows

« i05l i0 /ki (29)

4.3 Tolerable Limit of Differential Motion „TLDM …. It isdesirable that all fixels of the grasp system will maintain stacontact with the workpiece, even in the course of external fori.e.,

l5l01ls.0 (30)

which is equivalent to

«01«s.0, or «01GTd.0 (31)

Therefore, tolerable limit of differential motiond ~TLDM ! ofthe grasped object is defined as

D05$dPRr :«01GTd>0% (32)

Physically,D0 represents the limit of differential motiond for thegrasped object in the twist spaceRr while the object remains incontact with all fixels. The following proposition characterizesD0defined above, whose proof appears in the appendix.Proposition 6: TLDM D0

The following statements can be made with respect to toleralimit of differential motion.

~1! If «0.0, thenD0ÞF. That is, the range of tolerable limiof differential motion~TLDM ! is nonempty. In fact, 0P intD0 .

~2! If «01>«0

2, then D01$D0

2, which means that the more thinitial normal deformation«0 is larger if the rangeD0 of TLDM islarger.

~3! D0 is a polyhedron made of the intersection of (r 11) half-spaces in the twist spaceRr .

~4! If G is a relatively form-closed grasp and is simplex, thD0 is a simplex in the twist space.

From Proposition 6, the larger the initial deformation«0 ~or l0!is, the largerD0 is. Therefore, increasing«0 ~or l0! is beneficialto maintaining stable contacts for the grasp system~G, K!. How-ever, the value of«0 ~or l0! is bounded by physical limits of thecontacts. Let«̄ be the upper limit of supporting deformation. ThepolyhedronD1 defined in twist spaceRr

D15$dPRr :«01GTd<«̄%

represents the tolerable limit of differential motion~TLDM ! forthe grasped object in the twist spaceRr for the given limit«̄. Eachface of the polyhedronD1 is parallel to one of that ofD0 . Thus,D0 and D1 are polar to each other as shown in Figs. 10~a-b!.When considering bothD0 andD1 , the tolerable range of differ-ential motion~TRDM! should be specified as

D25$dPRr :«01GTd>0,«01GTd<«̄%5D0ùD1 (33)

The positioning accuracy of the grasped object in the twspaceRr depends upon the geometric accuracy of the locatfixels and the deformation of the contacts owing to the exterforces and clamping forces. As usual, it is suggested thatr dimen-sional sphereDs in twist space is defined for specifying the limof positioning error,

Ds5$dPRr :udu<«s%

where udu is the dual norm of differential motiond and«s is theequivalent maximum error.

If it is required to have higher accuracy in a specific directiothen an error ellipsoid in the twist space can be defined as follo

De5$dPRr :dTEd<«e2% (34)

whereE is a weighting matrix and must be positive definite. Fiure 10~c! shows a 2-D error ellipsoid with the origin at its cente

MAY 2002, Vol. 124 Õ 451

452 Õ Vol. 124, MAY 2

Fig. 10 Tolerable Limit of Differential Motion „TLDM… in 2-D twist space

di

e

r

aagr

e

f

ility

the

--e

trib-ends

lop-an-

are

With the above discussions, it is concluded that the toleralimit differential motion ~TLDM ! of a fixture system is boundeby D0 , D1 andDe simultaneously. Therefore, the TLDM is theintersection,

D5D0ùD1ùDe (35)

This is illustrated in Fig. 10~d!. When taking into accounts of thmultiple requirements, it is necessary to verify the differentmotion d of the grasped object under the action of external fow to satisfy the following:Proposition 7: Verification of Elastic Fixture System in the TwisSpace

~1! dP int D0 , indicating the locating stability~Fig. 10~a!!.~2! dP int D1 for the physical limit in strength and stress~Fig.

10~b!!.~3! dP int De , indicating locating accuracy~Fig. 10~c!!.~4! dP int(D0ùD1ùDe), indicating a comprehensive verifica

tion ~Fig. 10~d!!.

4.4 Load Capacity. Strictly speaking, load capacity ofmachine or a mechanism is referred to as its ability to resist mmum external forces without any adverse effects or damaLoad capacity of a fixture is considered as the maximum exteforces allowed to be exerted on the fixture without losing aconditions set for the tolerable limit of differential motion. It isdual relation expressed in the wrench space in terms of a seallowable load polyhedrons

F5F0ùF1ùFe (36)

WhereF0 , F1 and Fe are related withD0 , D1 and De , respec-tively, based on the Eq.~19!,

F05K sD05$w5K sd:dPD0%

F15K sD15$w5K sd:dPD1%

Fe5K sDe5$w5K sd:dPDe%

Proposition 8: Verification of Elastic Fixture System in thWrench Space

Equivalently, we define the allowable load polyhedrons aslows:

002

ble

r

ialce

t

-

xi-es.nalnyat of

ol-

~1! F05K sD05$wPRr :l01ls>0% (37)

~2! F15K sD15$wPRr :l01ls<l̄% (38)

~3! Fe5K sDe5$wPRr :~l01ls!TE~l01ls!<le

2% (39)

~4! dP int D or equivalently wP int F

Thus, either the allowable load polyhedronF ~ALP! in the wrenchspace or the tolerable range of differential motionD ~TRDM! inthe twist space can be used for evaluating robustness, reliaband load capability of an elastic fixture system~G, K !.

4.5 Load Capacity of Grasped Cylinder. We shall studythe cylinder shown in Fig. 7 again as an example. We considerinfluences of lengthz, grasp matrixG, and supporting stiffnessKon the load capacityF, for two cases:z51 andz52. AssumeK5diag(1,1,...,1), then the volume measuresm@G1# andm@G2# forthe two cases are found from Eqs.~15! and ~16! to be m@G1#50.7524/4! for case 1 andm@G2#53.0781/4! for case 2. For apreload levell051 which is uniformly distributed on each contact point and an allowable loadl̄54, we can construct the allowable load polyhedronF. For simplicity, we assume that thexternal forces are acting in the planez50. The allowable loadpolyhedron~ALP! for graspG1 and graspG2 are obtained andshown in Fig. 11. It is clear that the graspG2 is superior to graspG1 in two respects:

~1! m(G1),m(G2)~2! F(G1),F(G2)

In these case, it is clear that a larger valuez results in a larger loadcapacityF. It is also true that a larger system stiffnessKs results ina higher locating accuracy. Therefore, locators should be disuted as widely as possible, for example, located as near theof the cylinder as possible in these cases.

5 Considerations of Fixture System DesignThe framework presented above could be useful for deve

ment of a computer-aided fixture design system, including qutitative analysis, performance evaluation, and verification@25# asillustrated in Fig. 12. In such a system, decision variables


-i

g

t

a

rm

-ure

ror

of

er-

-

en

n,

ve

eimu-

n

t

mp-ncescu-se-dedion,

iditylls,

defined by the system~G, K !. The goal of fixture design is todetermine the fixture configurationG that satisfies several fixturing requirements and constraints, such as accessibC-determinacy,F-determinacy, form closure~or relative form clo-sure! and others. In general, an optimal fixture system cancharacterized by the following requirements:

~1! Rational distribution of inner forces~e.g., uniform distribu-tion or specified distribution!.

~2! Great ability of loading capacity.~3! High locating accuracy and low deformation.

While the distribution of inner force depends on grasp confirationG for a simplex grasp, the load capacityF depends not onlyon G, but also on support stiffnessK and pre-fixturing forcel0 .Usually in fixture design, the first step is to determine a suitagrasp configurationG, then to select support stiffnessK , and fi-nally to specify pre-fixturing forcel0 and supplementary supporif required. We shall discuss each of these steps in the framewof our analysis presented above.

5.1 Grasp Configuration. Any fixture configurationG canbe put into two parts: locatorsGL and clampsGC , i.e.,

G5GLøGC

where GL5@p1 ,p2 ,...,pr # is the collection of locators andGC5@pr 11 ,pr 12 ,...,pr 1s# is the collection of point contact clampsThe types of locators and clamps with line and surface contactalways be represented by a set of point contacts@6#. It is knownthat the relativeC-determinacy of a set of locatorsGL in Rn isequivalent to rank (GL)5n (0,n<6). In particular, locating a3-D workpiece is deterministic if, and only if, rank (GL)56. Ap-plying the results of our analysis in previous sections, we hthat the necessary and sufficient conditions for achieving relaC-determinacy and relative form closure inRn can be describedby the restraint conesR(GL) andR(Gc) as follows:

Fig. 11 Allowable Load Polyhedral „ALP … for G1 and G.2

Fig. 12 The simulation of decision variables


lity,

be

u-

ble

sork

.can

vetive

~1! For a fixture, a set of locatorsGL should achieve relativeC-determinacy inRn, it requires that dim (R(GL))5n, so that thesolution of locating Eq.~1! is deterministic.

~2! In order to achieve a relative form closure inRn, it requiresthat ~i!. dim (R(GL))5n and ~ii !. 0Pr int(R@GL#1R@Gc#)

Therefore, for a set of relatively deterministic locatorsGL inRn, dim (R(GL))5n. Then, a clamppr 11 can make the locatedworkpiece to contact with the locators to achieve relative foclosure, if and only if-pr 11Pr int(R(GL)). A set of clampsGL5$pr 11 ,pr 12 ,...,pr 1s% (s.0) can maintain the located workpiece contact with the locators to achieve relative form closR@GLøGC#5L@GLøGC#, if and only if

0Pr int~R@GL#1R@Gc# !

When G5$p1 ,p2 ,...,pr 11% is a simplex grasp, then all ther11 point contacts form a simplex graspG which satisfies thecondition ofF-determinacy regardless of the normal contact er«L5@«1 ,«2 ,...,« r # and contact stiffness K . When G5@p1 ,p2 ,...pr ,pr 11 ,...pr 1s# is a complex group, orGL5@p1 ,p2 ,...pr # is not linearly independent, then the elasticitythe grasp system must be taken into consideration.

5.2 Locating Accuracy. In fixture development, it is inevi-table that there exist locator positioning errors and geometricrors of workpiece. These errors at contacts« i ( i 51,2,...,r ) willresult in workpiece locating errord as follows~Eq. ~1!!:

piTd5« i , i 51,2,...,r

Obviously,d is dependent on both« andG. Consequently, matrixGL ~locating matrix! plays an important role for the locating accuracy. For a simplex graspG, matrix GL can be measured byusingAdet(GLGL

T), the condition number ofGL , or the eigenval-ues values ofGL . This aspect of accuracy quantification has bewell discussed recently in@16#.

5.3 Support StiffnessK and Preload Level l0. Contactsupport stiffnessK is related to mechanical structure, dimensiolocking state and contact positions of the fixture~or fingertips!.The locators have usually high rigidity~may be considered asK5` in some cases!. On the other hand, the clamp elements havery low rigidity, and it even may be regarded asK50 when it isnot locked. For a given grasp configurationG, the decision vari-ablesK , l0(«0) and l̄( «̄) may be determined with a procedursuggested here based on a mathematical programming and slation technique, as follows:

~1! Specify the set of generalized external forcesw according totask model.

~2! Specify the upper limit of support deformation«̄, accordingto the fixture mechanical structure and positioning accuracy.

~3! Specify the preloadl0 , according to the load ability of theactive elements.

~4! Calculate«05K s21l0 , Ks5GKGT, and determineD andF.

~5! Verify if wPF or dPD. If the conditions are satisfied, thethe grasp is feasible; otherwise, proceed to a redesign ofG, K andl0 to obtain a satisfactory grasp.

5.4 Supplementary Supporting. In the above discussion, iis supposed that the grasped body is rigid. A simplex graspG issufficient to meet the needs ofC-determinacy andF-determinacy,accessibility, relative form closure, etc. However, such an assution is not valid for the workpiece with thin wall. The deformatioof the workpiece under the actions of the inner and external forshould not be neglected, for it may deteriorate the locating acracy and bring about significant manufacturing error. Conquently, in this situation, supplementary supports are often adto increase system rigidity, to decrease workpiece deformatand to raise load ability.

Supplementary supports are used in increasing system rigand locating accuracy. Especially for workpieces with thin wa

MAY 2002, Vol. 124 Õ 453

a

s

a

r

iu

fia

l

n

n

e,

er

m

he

ble

t

e

they are necessary to minimize the workpiece deformation unapplied loads. In fact, the system stiffness matrixKs can be ex-pressed as a sum for a diagonal matrixK ,

K s5GKGT5(i 51

r 11

kipipiT

Clearly, a supplementary supportpa will change the system stiff-ness matrix fromKs into K s

1,

K s15K s1kapapa

T

The rules for placing supplementary supporting are to ensthe locating condition unchanged, to maintain the determinand accessibility, and to maximize the rigidity of the fixturinsystem. The following considerations ought to be taken whening supplementary supports.

~1! The graspG is no longer a simplex, but a complex. Accesibility and F-determinacy may be lost.

~2! The selection of positioning locators and supplementsupportsGa5$p1 ,p2 ,...,pr ,pa% must satisfy intR* @Ga#ÞF, inorder to achieveC-determinacy. As intR* @Ga#ÞF is equivalentto dimR* @Ga#5n ~definition of C-determinacy forGa!.

~3! Support stiffness must match the system stiffness andpreload level. The supplementary supportpa should be locatedwhere the fixel is highly sensitive to any deformation of the wopiece.

~4! Contact force caused by the supplementary supportpashould be bounded by a specified value. The normal error«a ofthe supplementary support should be negative in general.

6 ConclusionIn this paper we focus qualitative characteristics of a fixtur

~grasping! system and present a quantitative analysis of fixt~grasp! configurations with point contacts. Two major charactertics of relative form closure and force-determinacy are speciand discussed. A method is presented to quantify the performof a fixture ~or grasp! with inner force distribution and load capacity requirements. Two performance indices of minimaxl@G#and volume measurem@G# are defined for simplex graspingfixturing configurations. For an elastic graspG, it is proposed toapply the tolerable range of differential motion~TRDM! and theallowable load polyhedron~ALP! criteria to analyze the feasibilityunder a given upper limit of preload and with an elastic modethe fixturing system (G,K ). Furthermore, for fixture system design a geometric analysis and reasoning procedure is describethe design of locators, clamps and supplementary supports. Tproposed analysis and design techniques may provide a sciefoundation for automated grasping/fixturing system design inengineering practice.

AppendixThe appendix contains of Theorems 1, 3, 4, and propositio

Theorem 1: Equivalence theorem for form closure, i.e.,

R@G#5R6⇔OP int~C@G# !

Proof SinceR@G# is a convex polyhedral cone inR6, andC@G#is a convex polytope inR6. Both of them are convex sets~poly-hedra! in R6, furthermore,C@G#,R@G#. If OP int(C@G#), thenexists an«-neighborhood around originO, denoted byN«(O),such thatN«(O),C@G#. Any point pPR6, the line segmentOPmust contain a pointp«ÞO, and p«PN«(o), Therefore, p«PN«(o),C@G#,R@G#, so pPR@G#, from the definition ofpositive cone, henceR@G#5R6, and we have provedOP int(C@G#)⇒R@G#5R6. To show thatO¹ int(C@G#)⇒R@G#ÞR6, supposeO¹ int(C@G#), according to the separation theorem ~@18#!, there exists a nonzero vectornPR6 such thatn•p<0, for all pPC@G#, and for allpPG. SinceR@G# is the set of

454 Õ Vol. 124, MAY 2002

der

urecy

gus-

-

ry

the

k-

ngre

is-ednce

-

/

of-d forhesetific

the

6.

-

all positive linear combinations ofG, defined by~5!. So n•p<0, for all pPR@G#, we haveR@G#ÞR6. This completes theproof.Theorem 3: Equivalence Theorem for Relative Form Closuri.e.,

R@G#5L@G#⇔OP r int~C@G# !

Proof It is noted that if a graspG is of relative form closure in awrench space, it must be a complete form closure in a lowdimensional space, and vice versa. In fact, a graspG is said to berelatively form closed, if and only ifR@G#5L@G# ~Definition 4!,that is, G is form closed in subspaceRr5L@G#. Therefore, theproof of Theorem 3 is following that of Theorem 1.Theorem 4: Algebraic Equivalence Theorem for Relative forclosure.

The (r 11) point graspG5$p1 ,p2 , . . . ,pr 11% in wrench spaceRr is relatively form closed if, and only if, the linear equations

(i 51

r 11

l ipi50 (11)

(i 51

r 11

l i51

have a strictly positive solutionl i.0 (i 51,2, . . . ,r 11). Physi-cally, l i represents the distribution of the inner forces of tgrasped object at the point contacti.Proof Since the (r 11) point graspG5$p1 ,p2 , . . . ,pr 11% inwrench spaceRr is relative form closed, thenR@G#5L@G#, andOPr int(C@G#), furthermore L@G#5Rr(G). It is shown thatC@G# is r-dimensional simplex. According to the definition~7! ofC@G#, any pointpPC@G# can be denoted by

p5(i 51

r 11

l i pi , (i 51

r 11

l i51, l i>0

Based on the construction of simplexC@G#, we have:

~1! Any point pPC@G# is vertex of C@G#. if p5pi , i51,2, . . .r 11, i.e.l i51.0, l j50, j Þ i , j 51,2, . . . ,r 11.

~2! Any point pPC@G# is on an edge ofC@G#. if there existtwo positive coefficientsl i , l i.0, and otherlk50, kÞ i , j .

~3! Any point pPC@G# is on s(s,r ) facet ofC@G#, if thereexist s11 positive coefficients and others equal to zero.

~4! If all coefficientsl i.0, i 51,2, . . .r 11, ( i 51r 11l i51. Then

p5( i 51r 11l i piPr int(C@G#)8.

Therefore, the expression:

(i 51

r 11

l i pi50

(i 51

r 11

l i51, and l i.0, i 51,2, . . . ,r 11

implies the originO is on the relative interior ofC@G#, i.e., OPr int(C@G#).This completes the proof.Proposition 6: TLDM D0

The following statements can be made with respect to toleralimit of differential motion.

~1! If «0.0, thenD0ÞF. That is, the range of tolerable limiof differential motion~TLDM ! is nonempty. In fact, 0P intD0 .

~2! If « 01>« 0

2, thenD 01$D 0

2, which means that the more thinitial normal deformation«0 is larger if the rangeD0 of TLDM islarger.

~3! D0 is a polyhedron made of the intersection of (r 11) half-spaces in twist spaceRr .


e

5f

le

ex-n-

J.

gJ.

J.

ce

int

ed

-

rk-,

ce-

nd

-

lm.,

nt

d-

-f

.

of

ust

y-

lei-

~4! If G is a relatively form-closed grasp and is simplex, thD0 is a simplex in twist space.

Proof ~1! SinceOP intD0,D0 , henceD0Þf~2! Based on the definition of~32!, for any dPD0

2, we have«0

11GTd>0, owing to «01>«0

2, then «011GTd>0, therefored

PD01. This completes the proof ofD0

1$D02

~3! It is obvious from~32!~4! If G is a relatively from closed grasp, thenO

Pr int(C@G#). SinceG is a simplex grasp, i.e.,C@G# is a simplexin wrench space, hence its dual

D* @G#5$dPRr :I 1GTd>O%

is a simplex in twist space. So, for«0.0, theD0 is a simplex inthe twist space.

AcknowledgmentThe work is supported by the National Science Foundation

China under Grants No. 59990470, No. 59985004 and 50128The authors would like to thank the reviewers for their helpcomments and suggestions.

Nomenclature

pi 5 the i th line vectorn 5 an outer normal vector~unit vector!r 5 a position vector

G 5 @P1 ,P2 , . . . ,Pm# or @P1 ,P2 , . . . ,Pr 11# agrasp~fixture! configuration

GI 5 the augmented matrix ofGGL 5 locating configuration,GC5clamping configura-

tionw 5 @ f T,mT#T5a wrench vectord 5 @hT,dT#T5differential motion

R@G# 5 the restraint cone ofGR* @G# 5 the freedom cone ofGC@G# 5 the convex hull of setGL@G# 5 the subspace spanned by matrixGint(S) 5 the internior of setS

r int(S) 5 the relative internior of setGrank(G) 5 the rank of matrixG

det(G) 5 the determinant of matrixGuGu 5 udet(G)u5the absolute value of determinant ofG

l̄@G#,lI @G# 5 minimum measure, maximum measurem@G# 5 volume measure

l 5 l01ls5contact force vector,l0 5 the inner force vector,ls5the effect force vec-

torl i 5 the i th contact force~scalar!« 5 contact deformation vector

« i 5 the i th normal deformation~scalar!K 5 contact rigidity matrix

KS 5 system rigidity matrixD1 ,D2 5 TLDM5Tolerable Limit of Differential MotionDe ,DS 5 error ellipsoid


n

of03.ul

F 5 ALP5Allowable Load PloyhedronR6 5 ( f x , f y , f z ,mx ,my ,mz)

T the whole wrench spaceR1

r5 the subspace ofR6, whereG is relatively form

closedR2

s5 the complementary space ofR1

r

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MAY 2002, Vol. 124 Õ 455

Documents

Quantitative Analysis of Inner Force Distribution and Load Capacity of Grasps and Fixtures