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1 A Quality Measure for Compliant Grasps Qiao Lin, Joel Burdick Dept. of Mechanical Engineering California Inst. of Technology Pasadena, CA 925 Elon Rimon Dept. of Mechanical Engineering Technion, Israel Inst. of Technology Haifa, ISRAEL Abstract This paper presents a systematic approach for quantifying the quality of compliant grasps. Appropriate tangent and cotangent subspaces to the object's conguration space are studied, from which frameinvariant characteristic compliance parameters are de- ned. Physical and geometric interpretations are given to these parameters, and a practically meaningful method is proposed to make the parameters comparable. A frameinvariant quality measure is then dened, and grasp optimization using this quality measure is discussed with examples. Introduction This paper presents a quality measure for compliant grasps. Compliance plays a dominant role in passive grasps such as workpiece xturing, and can also be used to model the nger forces in active grasps. To our knowledge, this is the rst systematic approach to quantifying the quality of compliant grasps. The approach is frame-invariant and physically appealing. It applies to the grasping of 2D and 3D objects by any number of ngers, and can be used to determine the optimal nger placement. For the sake of convenience, the term grasping will also apply to xturing. Compliant grasps have received much attention. Hanafusa and Asada [3] used a linear spring model to nd stable 3-ngered planar grasps. Nguyen [] used a linear spring model to compute the stiness matrix of more general grasps. Howard and Kumar [4] also used a linear spring compliance model to study grasp stability, but included the eects of contact geometry. In studying compliance due to friction, Cutkosky and Wright [] noted that stability is inuenced by initial loading as well as local curvature. While the linear spring compliance model has been widely used by roboticists, it is not supported by experiments or results from elasticity theory. Rimon and Burdick [4] used overlap functions to model nonlinear compliance eects. Lin, Burdick and Rimon [7] use these overlap functions to compute and analyze the grasp stiness matrix for various contact models, including the widely veried and theoretically justied Hertz model. While the overlap model is used for illustration, our grasp quality measure can be used with any compliance model. Nearly all prior work on quantifying grasp eectiveness has assumed rigid body mechanics. Let the wrench (i.e. force and torque) due to a unit force applied by a contacting nger be termed a generating wrench. Li and Sastry [6] suggests a quality measure that is the smallest singular value of the grasp matrix, whose columns consist of the generating wrenches. Kirkpatrick, Mishra and Yap [5] dene the radius of the maximal ball inscribed in the convex hull of the generating wrenches as a quality measure. This idea is also followed by Ferrari and Canny [2]. However, these quality criteria are awed by their dependence on the choice of coordinate frame; a grasp which is optimal under one choice of reference frame may fail to be optimal under another. Several authors have devised schemes to avoid this problem. Markensco and Papadimitriou [9] minimize the worstcase nger forces needed to balance any external unit force acting on the object. Mirtich and Canny [] rst compute the grasps that best counteract pure forces. Among these grasps, the one that best resists torques is chosen to be optimal. Teichmann [5] nds the largest inscribed ball (as dened in Ref. [5]) for all choices of coordinate frames, but does not discuss the computation of the optimal grasp. This paper concerns the systematic development of quality measures for compliant grasps. Frame invariance is one of the main attributes of our approach. We consider frame-invariant subspaces of the object's tangent and cotangent spaces, from which frame-invariant characteristic compliance parameters are dened. We give novel geometric interpretations to these parameters, which are also dened by Patterson and Lipkin [2] in a dierent manner. We also propose a practically meaningful method for making these parameters comparable, and dene a frame-invariant quality measure. Examples demonstrate these ideas. 2 Background A grasp or xturing arrangement consists of an object B contacted by k ngers A ; : : : ; A k. We assume that the contacts are frictionless, and that the bodies have a smooth boundary near the contact points. The bodies are assumed to be quasi-rigid, and the ngers A i stationary. In the quasi-rigid assumption, deformations due to compliance eects are assumed to be localized to the vicinity of the contact points, so that the overall motion of B relative to A i can be described using rigid body kinematics. This is an excellent assumption for

2 xturing of mechanical parts. Since the ngers A i are stationary, we can focus on the following conguration space (c-space) of B, which is denoted C. Choose a xed world reference frame, F W, and a frame F B xed to B. A conguration of B is specied by the position, d 2 R 3, and orientation, R 2 SO(3), of F B relative to F W. C-space is given hybrid coordinates q = (d; ) 2 R 3 R 3, which map to (d; R()). The mapping R() is given by R() = exp(b ), where b is a skew-symmetric matrix such that b x = x for x 2 R 3. The tangent space to C at a conguration q, denoted by T q C, is the set of all tangent vectors, or velocities of B, at q. In hybrid coordinates, tangent vectors can be written as vectors _q = (v;!), where v 2 R 3 is the velocity of the origin of F B, and! 2 R 3 is the angular velocity of F B. The wrench space at q, denoted by Tq C, is the set of all wrenches (or covectors) acting on B. A wrench takes the form w = (f; ) in hybrid coordinates, where f 2 R 3 is a force acting at F B 's origin and 2 R 3 is a torque. In the planar case, letting the z-axis be perpendicular to the plane and dropping the identically zero components, we have v; f 2 R 2 and!; 2 R. The hybrid parametrization of c-space depends on the choice of frames. Consider a new world frame, FW, displaced from F W by (d w ; R w ), and a new object frame, F B, displaced from F B by (d b ; R b ). A conguration with coordinates q would now have dierent coordinates q. The tangent and cotangent vectors transform as follows: _q = T? _q; w = T T w; () where the transformation matrix for the 3D and 2D cases are given by T 66 = Rw[R d b R w Rw JR and T 33 = d b ; (2) R w? respectively. Here J =?, and R = R( ). Since d w and R b do not appear in T, a translation of F W or a rotation of F B do not aect the transformation. Rimon and Burdick [4] proposed a model for contact compliance which use overlap functions. These functions allow one to ignore the specic details of deformations when B and A i are quasi-rigid. Rather, the net contact force is modelled as a function of the overlap of the two undeformed rigid body volumes that results from a relative displacement. The overlap approach is briey reviewed here. In the absence of deformation, the two bodies B and A i contact at a single point, and after deformation occurs the bodies inter-penetrate. The overlap between B and A i, denoted i, is the minimum amount of translation separating B from A i. Clearly, i depends on B's conguration: i = i (q). We dene i = when B and A i are disjoint or maintain surface contact. The net contact force is assumed to act on B at the initial contact point, in the direction of the separating translation. The force's magnitude, denoted f i, is assumed to depend on the overlap i (q): f i = f i ( i (q)). The simplest model assumes that f i is a linear function of the overlap: f i ( i ) = k i i ; (3) where k i is determined by the material and surface properties of B and A i. While this model is linear in i, it is typically not linear, since i is nonlinear in q. More sophisticated contact models can be formulated by choosing appropriate functions f i ( i ). See [7] for details. Consider a grasp of B at a conguration q. The arrangement of ngers forms an equilibrium grasp if (in the absence of any external wrench) the nger forces produce a zero net wrench on B. When subjected to an arbitrary external disturbance, B may be displaced from q. The grasp is stable if B returns to q after the external disturbance is removed. A more formal discussion of stability can be found in Ref. [4]. The elastic potential energy of the system consisting of the object B and ngers A ; : : : ; A k is: (q) = kx i= Z i(q) f i ()d: (4) It can be veried that i (q) is dierentiable almost everywhere, hence (q) is dierentiable. In the absence of a disturbing wrench, an equilibrium grasp is characterized by: i= r(q ) = kx i= f i ( i )r i = ~; (5) where i = i (q ) and r i = r i (q ). A suciently small displacement of B can be approximated by a tangent vector. For this reason we will interchangeably use the terms tangent vector and local displacement. The stiness matrix is dened as the Hessian K = D 2 (q ) of the potential. Denoting fi = dfi d i and D 2 i = D 2 i (q ), it follows from (5) that kx K = f i( i )r i r T i + f i ( i )D 2 i : (6) Therefore, the stiness matrix can be computed from the overlaps i and their derivatives. The reader is referred to Ref. [7] for the computation of K. As is well known, at points q = q + _q in the vicinity of q the stiness matrix gives the wrench acting on B, according to the formula w = K _q. We observe that the two summands in Eq. (6) generally depend on the initial deformations i. It is shown in [7] that the second term depends on the surface curvatures at the contacts, while the rst term does not. We say that the rst term accounts for rst order geometrical eects, while the second term accounts for second order (curvature) eects. If the rst term alone is positive denite, the grasp is stable to rst order. Otherwise,

3 if the entire K is positive denite, the grasp is stable to the second order. The relative contributions of rst and second order eects on grasp stability and stiness are analyzed in Ref. [7]. We conclude this section with the following changeof-frame formula for the stiness matrix: K = T T KT; (7) where K is the stiness matrix associated with the new frames FW and FB. This formula can be derived from () and the fact that r(q ) =. 3 Principal Stiness Parameters This section denes the characteristic compliance parameters of a grasp, based on the stiness matrix K, and the compliance matrix C, K?. For clarity, we note that w = K _q while _q = Cw. We use the following partition of K and C into 3 3matrices: K K K = 2 C C ; C = 2 : (8) K 22 C 22 K T 2 C T 2 Note that the diagonal blocks are positive denite, since K and C are. We use _q and _q 2 for the translational and rotational components of _q = (v;!), and use w and w 2 for the force and torque components of w = (f; ). 3. Formal Development The eigenvalues of K, which could provide important insight into the stiness matrix, are not frame invariant. To circumvent this diculty, we look at the tangent subspace dened by V = f _q 2 T q C : f = (K _q) = g: That is, V consists of the small displacements that induce a pure reaction torque on B. Using the partition of K, we obtain V = f(v;!) : v =?K? K 2!g, from which it follows that V can be parametrized as?k? _q = P! where P = K 2 : (9) I Let K V denote the restriction of K to V. Recalling that the stiness matrix represents the symmetric bilinear operator D 2 (q ), we have that! T K V! =! T P T KP! for arbitrary!. Thus under our parametrization of V, K V has the representation K V = P T KP = K 22? K2K T? K 2: Since K maps _q 2 V to pure-torque wrenches, we have that (K _q) 2 = K V!: () Consider now two new frames FW and FB, with overbars denoting objects associated with these frames. The linear operator K V has the following invariance property. Proposition 3. ([8]). Let V and V be the subspaces parametrized by (9) in the q and q coordinates. Let KV be the restriction of K to V. Then K obeys the orthogonal transformation KV = R T w K V R w : Hence the eigenvalues of K V are frame-invariant. Dually, consider the following wrench subspace: W = fw 2 T q C :! = (Cw) 2 = g: In words, W is the subspace of wrenches that induce pure translation, and this subspace can be parametrized as w = Qf Q = I?C? 22 CT 2 : () Using this parametrization, the restriction of C to W, denoted C W, is C W = Q T CQ = C? C 2 C? 22 CT 2 = K?. Moreover, the resulting pure-translation is given by v = (Cw) = C W f, where w 2 W. Proposition 3.2. Let W be the subspace parametrized by () in the q coordinates, and let CW be the restriction of the compliance matrix C to W. Then CW obeys the orthogonal transformation CW = R T w C W R w. Hence, the eigenvalues of C W = K? are frameinvariant. Propositions 3. and 3.2 lead to the following observations. The behavior of K on V characterizes the rotational stiness of the grasp. Regardless of frame location, the same pure-torque is elicited in response to an instantaneous displacement in V. Similarly, the behavior of C on W characterizes the translational compliance of the grasp. A wrench in W generates the same pure-translation when using dierent frames. Since the tangent subspace V and the image of W under C W span T q C, the two subspaces characterize the grasp compliance completely. Summarizing these observations and using the fact that C W = K? ues i (i = ; 2; 3) of K V, we call the eigenval- the principal rotational stinesses, and the eigenvalues i (i = ; 2; 3) of K = C? W the principal translational stinesses of the grasp. In particular, min = minf i g is the smallest principal translational stiness. The associated eigenvectors are called principal rotational and translational stiness directions, respectively. For planar grasps it can be shown that there is a unique location of the origin of FB, given by d b = R T JK? K 2; (2) such that K33 takes the block-diagonal form K = diag(r w T K R w ; ). That is, the translation and rotational eects are decoupled about this special point, called the center of compliance []. The principal translational and rotational stinesses of the grasp are physically the translational and rotational stinesses about the center of compliance. 3.2 Screw Coordinates Interpretation While searching for a 3D analog of the center of compliance, Patterson and Lipkin [2] were the rst to recognize the existence of the principal stiness directions. They used screw coordinates, and now we show that our principal parameters are equivalent to the ones derived

4 by Patterson and Lipkin. First we briey review the notion of screw coordinates. A one-dimensional tangent subspace of the form f _q = (v;!) : 2 Rg with k!k =, is given screw coordinates as follows. The instantaneous screw axis is parallel to! and passes through the point v!. The pitch, h, is equal to v!. For a one-dimensional wrench subspace fw = (f; ) : 2 Rg with kfk =, the screw axis is parallel to f and passes through the point f. The pitch is h = f. Consider now a tangent vector _q i 2 V, where _q i is an eigenvector of K V associated with the eigenvalue i. Using (9), there exists a unit vector! i such that _q i = P! i. Then () gives = (K _q i ) 2 = i! i. That is, the displacement along _q causes a pure-torque about the screw axis associated with _q i. On the other hand, for w = Qf i 2 W where f i is an eigenvector of C W associated with the eigenvalue = i, we have that v = (Cw) = (= i )f i. Hence, the wrench w generates a pure-translation along its screw axis. Patterson and Lipkin [2] call the screw axis associated with these eigenvectors the twist- and wrench-compliant axes, respectively. 3.3 Geometric Interpretation We now present a novel interpretation of the principal stinesses. Consider the quadratic form ( _q) = 2 _qt K _q, where _q 2 T q C. The level set S dened by ( _q) = is a 5-dimensional elliptical surface, and a point on S corresponds to a displacement that produces unit elastic energy. Consider the intersection, denoted S!, of S with the subset of T q C determined by the equation! = const. Letting! (v), (v;!), the points v 2 S! satisfy! (v) = 2 ut K u + 2!T K V! = ; where u = v + K? K 2!. Hence for each xed!, p S! is an ellipsoid with principal semi-axes of length (2?!T K V!)= i (i = ; 2; 3). These lengths are p frame-invariant, and when! = the lengths are 2= i (i = ; 2; 3). Next we consider the collection of points in S, denoted S n, at which the vectors normal to S have zero v-components. For any _q = (v;!) 2 S n, the condition (r( _q)) = implies that v =?K? K 2! and consequently ( _q) = 2!T K V! =. By setting the v- coordinates of the points in S n to zero, we obtain the projection of S n to the subspace v = as follows. (S n ) v= = f(v;!) : v = and 2!T K V! = g: This p is an ellipsoid with principal semi-axes of lengths 2=i, where i for i = ; 2; 3 are the eigenvalues of K V. For planar grasps S is 2-dimensional. Fig. shows two such ellipsoids for a 4-ngered grasp of the quadrilateral given in Fig. 4 with ngers placed on the edges AB (at the vertices A and B), BC (at C) and DA (at D), respectively. The upright ellipsoid in the gure corresponds to the origin located at the center of compliance with coordinates (6:5; 5:54), while the slanted ellipsoid corresponds to the origin located at (; ). The lengths of the principal semi-axes of each horizontal cross section of S are frame invariant. Similarly, the projection of S is p bounded by two points, whose!- coordinates are 2=. These two points are frame invariant, and S p is always bounded by the two horizontal planes! = 2=. Fig.. 2 w -2 v v The elastic energy ellipsoid in Tq C For the wrench space on which the quadratic form (w) = 2 wt Cw is dened, we have the following analogous interpretation, shown in Fig. 2. The level set T given by (w) = is a 5-dimensional elliptical surface corresponding to wrenches that induce unit elastic energy. The intersection T f of T with the set f = const is an ellipsoid whose principal semi-axes are equal to q i (2? f T K? f) (i = ; 2; 3) and are frame-invariant. When p f =, the principal semi-axes of T f are given by 2i. Let T h be the subset of T such that the normal vector to T at a point w 2 T h has zero -component. The projection of T h to the subspace = is given by (T h ) = = f(f; ) : = and 2 f T K? f = g: Since K? = C W is frame invariant, the principal semiaxes of (T h ) =, given by p 2 i, are frame invariant. In the planar case, the elliptical surface T intersects the -axis at two points whose coordinates are p 2 (Fig. 2). If T is vertically oriented, the horizontal projection of T is the planar ellipse f T K? 2 f =. Any other T is inscribed in the vertical cylinder whose base set is this ellipse. These features can be observed in Fig. 2, for the same grasp as used for Fig.. The upright and slanted ellipsoids correspond to the same frames as their counterparts in Fig.. 4 A Frame-Invariant Quality Measure Guaranteeing that the displacement of a grasped object will not exceed a specied tolerance is one of the

5 τ Fig. 2. f f2 - The elastic energy ellipsoid in Tq C most important concerns in xture design [3]. Hence we wish to develop a grasp quality measure which is related to the deection of the object under the action of disturbing forces. In particular, we wish to relate the principal translational and rotational stinesses to the object's deection, and use this relation to evaluate alternative grasps. Let _q = (v;!) be a displacement of B, where kvk = if! =, and k!k = if! 6=. We dene the deection of B due to a displacement _q as the maximal displacement of any point in B. Since B is bounded, such a maximal displacement always exists and is independent of frame choice. If! =, the deection is simply jj. If! 6=, let max ( _q) be the maximal distance from the screw p axis of _q to B's boundary points. The deection is jj max ( _q) 2 + (v!) 2, where v! is the pitch of _q. For planar grasps v! = and B's deection is jj max ( _q). First we present our quality measure in the context of planar grasps. For planar grasps, we wish to compare the principal rotational stiness with the smallest principal translational stiness min. As previously discussed, and min are associated with pure rotation and translation of B with respect to the center of compliance. The deection of the object can be used to compare these two parameters as follows. Consider a rotation of B of magnitude about the center of compliance. Then the deection of B due to this rotation is jj max, where max is the maximal distance from the center of compliance to B's boundary. The equivalent stiness associated with, denoted eq, is dened by the relationship: 2 eq( max ) 2 = 2 2 ; where the right hand side is the elastic energy generated by the rotation. This relationship yields eq = ( max ) 2 : (3) The parameters eq and min are now comparable. We dene the grasp quality measure as: Q = minf min ; eq g: (4) The scalar Q measures the worst-case characteristic stiness based on B's deection. Moreover, Q is frame invariant. We now dene the quality measure for a 3D grasps. For 3D objects, we must scale the principal rotational stinesses i so that they become comparable with the translational stinesses. Let _q i = (v i ;! i ) 2 V be the eigenvector of K V associated with i, such that k! i k =. Then the elastic energy generated by the displacement _q i is given by p 2 i 2, while the deection of the object due to _q i is ( max i )2 + (v i! i ) 2, where max i = max ( _q i ). Analogously to the 2D case, we dene eq i by the following energy equivalence relationship p 2 2 eq i ( maxi ) 2 + (v i! i ) 2 = 2 i 2 ; which yields i eq i = for i = ; 2; 3. (5) ( max i) 2 + (v i! i ) 2 We dene the following 3D grasp quality measure: Q = minf min ; eq ; eq 2 ; eq 3 g: (6) Again, Q is a frame-invariant scalar which measures the worst-case characteristic stiness as determined by B's deection. 5 Optimal Grasping of Polygons To illustrate our methodology and its possible utility, we apply the quality measure (4) to the planar polygonal objects grasped by three or four disc ngers. For simplicity, we employ the overlap model of Eq. (3). Since each nger boundary has constant curvature and B's edges are straight, the stiness coecient k i is assumed to be the same for all nger locations on a given edge. We exclude nger placements at vertices and choose coincident frames F W and F B. Let the contact conguration space (contact c-space) be the set of all possible contact arrangements (each contact can be parametrized by a scalar). For polygonal objects, the contact c-space can be decomposed into subspaces corresponding to dierent combinations of edges. Consider the computation of max for polygons. If the center of compliance is at p, max (p) is the distance from p to the farthest vertex of B. For ecient computation, we may presort the plane into regions whose points correspond to the same farthest vertex. Let fv ; : : : ; v n g be the vertices of B. For vertex v i, let H j be the closed half plane that does not contain v i and is bounded by the bisector between v i and v j (j 6= i). Let R i be the intersection of these half planes. Then max (p) = kp?v i k for p 2 R i. Clearly, R i is a convex polygonal region, and [ n R i= i = R Optimal Three-Finger Grasping For 3-ngered planar equilibrium grasps, the stiness matrix corresponding to a particular edge triplet (Fig. 3), can be computed according to (6). A formula is given in the following proposition. In the proposition, n i are the unit contact normals pointing P into B. Also, k the total initial nger force is f T = f i= i( i ).

6 Proposition 5. ([8]). If the origin of F W coincides with the point of concurrency, the stiness matrix for a grasp employing a given triplet of edges takes the form P 3 K = k T i= in i n i ; with the principal rotational stiness given by = 2f T a sin sin 2 sin 3 sin + sin 2 + sin 3 ; where i are the triangle's three angles and a is the radius of its circumscribed circle. Moreover, the center of compliance coincides with the concurrency point. Fig. 3. n 3 α 3 n 2 S α n α 2 3-nger grasp on a particular edge triplet It follows P that min is the smallest eigenvalue of the 3 22 matrix k i= in i n it, and is constant for the given edge triplet. It is even more interesting to observe that, resulting from curvature eects and depending on the total nger force f T, is also constant for all grasp arrangements on the same edge triplet. For the grasp to be stable, f T must assume a positive value (i.e., initial deformations are nonzero). Since the rst order eects are dominant, eq min and therefore 2f T a sin sin 2 sin 3 Q = eq = 2 max(sin + sin 2 + sin 3 ) : (7) In practice, f T is xed at a value which is the same for all edge triplets, and a threshold value can be chosen for min such that a triplet with min < is rejected. For an edge triplet whose inward normals positively span R 2, the collection of stable equilibrium grasps is parametrized by the location of the concurrency point. Consider the three strips in Fig. 3. The two lines bounding each strip are perpendicular to an edge, and pass through the edge's endpoints. For each point in the region S formed by intersecting the three strips, there exists a nger placement such that this point is the concurrency point of the contact normals. For a given xed preloading f T, the quality measure (7) is maximized over a given edge triplet as 2 max is minimized. This agrees with the intuition that the deection of B about the concurrency point due to a unit torque is minimized for the optimal grasp. For a given edge triplet, we maximize 2 max, a positive denite quadratic function, over a collection of convex polygonal regions described in [8]. While these convex quadratic programming problems can be solved by many ecient algorithms, the optimal grasp arrangement is very intuitive when the geometric center belongs to S. In this case the optimal concurrency point location coincides with the geometric center, and the optimal quality measure is given by (see footnote for the radius r ). Q opt = 2f T a sin sin 2 sin 3 r 2 (sin + sin 2 + sin 3 ) : Example 5.. Consider grasping a quadrilateral by three identical ngers (Fig 4). We take k i = (i = ; 2; 3) without loss of generality. The radius of the object is 6:735 and its geometric center is at (6:5; :75). For the edge triplets (AB; BC; CD) and (AB; BC; DA) stable grasps exist, with the optimal grasps given by arrangements I and II, respectively. We have min = :869 and Q=f T = :8 for grasp I, while min = :2764 and Q=f T = :955 for grasp II, which is globally optimal. D(2,6) C(9,8) A(,) Fig. 4. (6.5,.75) B(3,) - Grasp I - Grasp II 3-nger quadrilateral grasps 5.2 Optimal Four-Finger Grasping A 4-ngered polygonal grasp involves three or four edges. Thus edge combinations of interest include all triplets and quadruplets of edges. For a given edge combination, let i be the moment about the origin of the inward unit normal n i to the i th edge. Use s = (s ; s 2 ; s 3 ; s 4 ) T, where s i = p k i i, to parametrize G, the contact c-space. Since the geometric constraints on contact locations are all linear inequalities, G is a convex polytope. Moreover it is bounded since no nger can be placed at innity. Let N i = p k i n i, and h i (s i ) = (Ni T ; st i. Then the )T stiness matrix is given by (see [7] for a proof) NN T Ns K(s) = (Ns) T s T ; (8) s where N = (N ; N 2 ; N 3 ; N 4 ) and the second order eects have been neglected [7]. For a given edge combination, the contact normals do not change directions and the matrix N is constant. Hence the smallest principal translational stiness is a constant. Using (8), we nd the dependence of the principal rotational stiness on contact conguration: (s) = s T s, where = I?N T (NN T )? N. From (2) and (8) the center of compliance as a function of s is given by d b (s) =?s, where? = J(NN T )? N. Thus in the polygonal region R i we can write eq as eq (s) = s T s (?s? v i ) T (?s? v i ) : Here the geometric center is the point p at which r = min p2r 2 max(p) = max(p), and r is called the radius.

7 If there is some feasible s such that eq (s) > min, then the corresponding grasp is optimal for this edge combination. Otherwise eq can be maximized, which is considered below. Dene d i (s) = det (h i+ ; h i+2 ; h i+3 ) (mod 4), for i = ; 2; 3; 4. Ref. [8] shows that the stiness matrix K(s) is positive denite if and only if d (s),?d 2 (s), d 3 (s) and?d 4 (s) are all nonzero and have the same sign. Therefore the collection of stable grasps is S [S 2, where S and S 2 are bounded convex polytopes in R 4. S = G \ s 2 R 4 : d (s);?d 2 (s); d 3 (s);?d 4 (s) < ; S 2 = G \ s 2 R 4 : d (s);?d 2 (s); d 3 (s);?d 4 (s) > : We can maximize eq over convex polyhedral regions of the form P = S i \ D j, where D j = fs 2 R 4 :?s 2 R j g. For t 2 R, dene (t) = max s2p (t; s) where (t; s) = s T s? t (?s? v i ) T (?s? v i ): Proposition 5.2. If the function has a zero, it is unique. Moreover, t = eq (s ) = max s2p eq (s) if and only if (t ) = (t ; s ) =. This proposition is proved in Ref. [8]. It follows that maximizing eq over P is equivalent to solving the scalar equation (t) =. The evaluation of the function is an indenite quadratic programming problem. While indenite quadratic programming is NP-hard, there are many ecient approximate algorithms. In fact, with our 4-dimensional problems, an exhaustive search scheme is quite aordable. The remarkable fact is that global optimality is guaranteed at reasonable cost despite the nonconvex and strongly nonlinear nature of the quality measure. Fig. 5. (.75,3.525) (,4) (,) (3,) Global optimal grasp of a quadrilateral Example 5.2. Let us look at the quadrilateral used in Example 5. and assume k i =. By considering all feasible edge combinations we can nd the optimal grasp associated with each combination, and then determine the global optimal grasp arrangement. The global optimal grasp is the one in Fig. 5, with optimal quality measure equal to min = :684 < eq = : Conclusion While compliance plays an important role in grasping and xturing, systematic approaches to assessing the quality of compliant grasps have been lacking. In this paper we presented an eort along this direction. A frame-invariant quality measure was dened based on characteristic compliance parameters of the stiness matrix. It applies to the grasping of 2D and 3D objects by any number of ngers, and can be used to determine the optimal nger placement. The promise of this quality measure is shown by examples applying it to polygonal grasps. We believe that this quality criterion will allow the development of more ecient and accurate algorithms for optimal planning of compliant grasps or xtures. References [] M. Cutkosky and P. Wright. Friction, stability and the design of robotic ngers. Int. J. of Robotic Research, 5(4):2{37, 986. [2] C. Ferrari and J. Canny. Planning optimal grasps. In Proc. Int. Conf. on Robotics and Automation, pages 229{2295, Nice, France, May 992. [3] H. Hanafusa and H. Asada. Stable prehension by a robot hand with elastic ngers. In Proc. 7 th Int. Symp. on Industrial Robots, pages 36{368, Tokyo, October 977. [4] W. S. Howard and V. Kumar. Stability of planar grasps. In Proc. Int. Conf. on Robotics and Automation, pages 2822{ 2827, San Diego, CA, May 994. [5] D. G. Kirkpatrick, B. Mishra, and C. K. Yap. Quantitative steinitz's theorems with applications to multingered grasping. In Proc. 2 th ACM Symp. on Theory of Computing, pages 34{35, Baltimore, MD, May 99. [6] Z. Li and S. Sastry. Task oriented optimal grasping by multingered robot hands. IEEE Trans. on Robotics and Automation, 6(2):32{44, 987. [7] Q. Lin, J. W. Burdick, and E. Rimon. Computing stiness matrices for grasping and xturing. Submitted, IEEE Int. Conf. on Robotics and Automation, 996. [8] Q. Lin, J. W. Burdick, and E. Rimon. Issues in optimal grasping and xturing of workpieces. Technical report, Dept. of Mechanical Engineering, California Inst. of Technology, Pasadena, CA, 996. [9] X. Markensco and C. H. Papadimitriou. Optimum grip of a polygon. Int. J. of Robotic Research, 8(2):7{29, 989. [] B. Mirtich and J. Canny. Easily computable optimum grasps in 2-D and 3-D. In Proc. Int. Conf. on Robotics and Automation, pages 739{747, San Diego, CA, May 994. [] V.-D. Nguyen. Constructing stable grasps. Int. J. of Robotic Research, 8():26{37, 989. [2] T. Patterson and H. Lipkin. Structure of robot compliance. ASME J. of Mechanical Design, 5(3):576{58, 993. [3] P. C. Pong, R. R. Barton, and P. H. Cohen. Optimum xture design. In Proc. 2nd Industrial Eng. Research Conf., pages 6{, Los Angeles, CA, May 993. [4] E. Rimon and J. W. Burdick. Mobility of bodies in contact ii: How forces are generated by curvature eects. In Proc. Int. Conf. on Robotics and Automation, pages 2336{234, San Diego, CA, May 994. [5] M. Teichmann. The power of friction: Quantifying the \goodness" of frictional grasps. In Proc. Int. Conf. on Robotics and Automation, Minneapolis, MN, April 996.