Definition and Force Distribution of Power Grasps - CiteSeerX

Definition and Force Distribution of Power Grasps Yuru Zhang and William A. Gruver School of Engineering Science Simon Fraser University Burnaby, BC, V5A 1% Canada

Abstract This research treats grasping of an object by a multifingered robot hand. By decomposing the space of contact forces exerted between the jingers and the grasped object into subspaces, we develop a method to determine the dimensions of the subspaces with respect to the connectivity of the grasped object. This approach provides insight into diflerentgrasps basedon a classification into three types.A power grasp is defined when the connectivity of the grasped object is equal to or less than zero. The analysis of contact force distribution is simplifiedfor a power grasp with zero connectivity. Examples for dimensional determination are illustrated.

1 Introduction As multi-fingeredrobot hands are being built in research laboratories, human grasping has been studied as ameans to better understand the planning and control of mechanical hands. Cutkosky [ 13analyzed human grasps in manufacturing and developed a taxonomy of grasps as a hierarchical tree beginning with two basic categories-power grasps and precision grasps. Recently a taxonomy of hand designs has been proposed based on two jaws [9]. This taxonomy shows the functionalcapabilities of each design so that the strengths and weaknessesof a given design can be clearly recognized. Despite their different points of view, both taxonomies include power grasps that frequently occurinmanufacturing. Research involving power grasps has been conducted on modeling, simulation and stability. Analytical models for force distribution in power grasps have been developed [2]. Based on these models, stability analyses were performed and quality measures for stability were proposed [2,51. In addition, the dynamic simulation of power grasps provides a useful tool for stability analysis [3, 81. In a very different approach, Haness, et al. [4], adopted a neural network

technique for force control of power grasps. Very limited research has been conducted on the planning of power grasps, although Trinkle, et al.[7], have developeda system for planning and simulating the envelopinggrasp of convex polygons in the absence of frictional and dynamic effects. The static anallysis by Bicchi and Melchiorri [6,10]focuses on the defectivity of enveloping grasping systems which results in the inability to ensure complete controllability of the intemal ccintact forces. Previously. apower grasp was empiricallyviewed as that having large areas of contactbetween the grasped object and the surfaces of the fingers and palm, and little or no ability to impart motions with fingers [l]. To develop analytical models of grasping, Zhang and Nakamura, et al. [ 5 ] ,defined a power grasp in terms of the ability to passively resist external forces without controling joint torques. With this latter definitioinit is not easy to identify apower grasp. In this paper we fociis on how to analytically characterize and define power grasps. We establish a relationship between the connectiviity of the grasped object and the controlability of the contact forces. Grasps are classified into three types according to their controllabilities. A new definition of powergraspsis proposed with which graspscan be identified by simply determing the connectivityof the grasped object. Furthermore, the value of the connectivity indicates the complexity of the force distribution in power grasps. Cheng and Orin [12] provided an efficient algorithm to solve the force distributilonproblem for fingertipgrasps. When applied to power grasps, Mirza and Orin [21 note that a high degree of static indeterminacy, due to the multiplicity of contactsin the power grasp, results in increased computational complexity. We show in this paper that such complexity does not always exist. F:or power grasps with zero connectivity, the problem can be simplified.

2 Definition of Power Grasps The static force analysis of grasping is based on the following equations

IEEE lnternatlonal C o n f e r e n c e o n Robotics a n d Automation 0-7803-1965-6/95 $4.00 01995 IEEE - 1373 -

where f E R" is the force applied at contacts on the fingers, w E R6 is the external wrench applied to the object, z E R"' is the joint torque of the fingers, W E R6xnis the grasp matrix, and J E Rnxmis the Jacobian matrix of the hand. MeIchiorri [lo] decomposed the space of contact forces into four subspaces as follows: active controllable forces r,, , internal controllable forces Tic,passive contact forces rpnand pre-load contact forces rpl. We shall characterize grasps by determining the dimension of each subspace with respect to the connectivity of the grasped object.

If fpl = 0 E r,, , then r,, is not a linear space. In this latter case, dimensional relations between subspaces are not applicable to r,, . To solve the problem, we augment it with the nullspace { 0) so that r,, becomes a subspace. Similarly, the other subspaces can be defined as

r;,= [fhl I wfh1 = 0, JTfhl = z2, 7 2 # 0) {o} rpn= if,, 1 Wfp2= w2, JTfp2= 0, w2 # 0) U [O} rpl= (fh2 I wfh2 = 0, JTfhZ = 0) where w1+ w2 = w, r can be defined as

r = { f I Wf = w, JTf = 7)

2.1 Mobility Equation A single link of a finger may contact an object with 1 through 5 degrees of freedom (DOF) . To simplify the problem, we assume that all contacts allow the same DOF. Considering point contact with or without friction, the mobility M of a hand-object system can be formulated as [141,

where m is the number of joints, p is the number of contact points, g is the DOF of motion at contact points, L is the number of independent loops in the system, M ' is the mobility of the system with the fingerjoints locked. Because each loop contains two contact points, we have (3)

p=L+1 Substituting Eq. (3) into Eq. (2), we obtain M' = 6 - n

M=m+6-n,

(4)

where n = ( 6 - g) p. The special case that the inequality in Eq. (2) holds will be discussed in Section 4. We use N, to denote the connectivity between the palm and the grasped object. The relation between N, and M is

M = N,

+ N,

(5)

where Nr is the number of redundancies [ 11I.

2.2 Dimensions of the Subspaces The subspace of active controllable forces can be defined as

r,, = ifp, I Wfpl = ~

z1 + z2 = z. The contact force space

1 JTfpl , =

zl, w1 f 0, T~ f O}U (0)

We can derermine the orthogonal complement r* so that R" = r 0r', where r1 = (X I fTx = O}. The n-dimensional velocity vector x at the contacts satisfies

in which U and q are the 6-vector of object velocities and them-vectorofjointvelocities. Since fTx = wTU = zTq ,if wTU=Ofor w # 0 and U # 0 , U E N(WT) which implies x = WTU = O.Ontheotherhand,if zTq = 0 for z f 0 and 4 # 0 , q E: N(J) , then x = Jq = 0. Therefore I" = (0), dimr=dimR" = n

(7)

The space of external wrenches is the range space R(W). Its orthogonal complement R I (W) contains all object velocity vectors that satisfy wTli = O.We have R~ = R(W) + R' (w) . Since U E N ( W ), ~ R'W) = N(WT).We observe that N(WT) is the indeterminacy subspace of the object velocities that remain free by contact constraints. Hence dimN(WT)=Ni, dimR(W)=6-Ni

(8)

where Ni is the number of indeterminacies. 6111.The space of joint torques is the range space R ( J ~ )~. t orthogonal s

complement R' ( J )~contains alljoint velocity vectors that satisfy ~~q = 0. We have Rm = R(JT)O R*(JT). Because q E N(J), R*(JT) = N(J). We observe that N(J) is the redundancy subspace of the joint velocities that do not affect the object velocity,but only modify the configuration of the fingers. Hence

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dimN(J) = N,

dimR(JT) = m - N,

(9)

Fig. 1 shows the relationships between all the subspaces defined above, from which it can be seen that dim r,, + dim rpn = dim R(W) dim I?,, + dim ri,= dim R(JT) dimr,, + dimTi, = di"(W) + dim rpn= dim N(JT) dim rpl

2.3 Classificationof Grasps It can be seen from the dimensional equations derived above that the connectivity is closely related to the dimensions of the contact force subspaces. The following dimensional analysis for each value of connectivity will characterize different grasps. To obtain a stable grasp, the case Ni > 0 must be avoided. Therefore, we assume Ni = 0. (1)When N ,= 6, dim Fa, = 6, dim rpn = dim rpl =O and

dimTi, =

Figure 1. Contact force subspaces From the physical meanings of Nc, Ni and ,?I determine dimr,, =

N,-Ni {0

we can

N,>O N,