A Quality Measure for Compliant Grasps - CiteSeerX
A Quality Measure for Compliant Grasps Qiao Lin, Joel Burdick Dept. of Mechanical Engineering California Inst. of Technology Pasadena, CA 91125 Abstract | This paper presents a systematic approach for quantifying the quality of compliant grasps. Appropriate tangent and cotangent subspaces to the object's con guration 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 de ned, and grasp optimization using this quality measure is discussed with examples.
1 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 [11] used a linear spring model to compute the sti ness matrix of more general grasps. Howard and Kumar [4] also used a linear spring compliance model to study grasp stability, but included the e ects of contact geometry. In studying compliance due to friction, Cutkosky and Wright [1] noted that stability is in uenced 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 [14] used overlap functions to model nonlinear compliance e ects. Lin, Burdick and Rimon [7] use these overlap functions to compute and analyze the grasp sti ness matrix for various contact models, including the widely veri ed and theoretically justi ed 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 e ectiveness has assumed rigid body mechanics. Let the wrench (i.e. force and torque) due to a unit force applied by a
Elon Rimon Dept. of Mechanical Engineering Technion, Israel Inst. of Technology Haifa, ISRAEL 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] de ne 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 [10] rst compute the grasps that best counteract pure forces. Among these grasps, the one that best resists torques is chosen to be optimal. Teichmann [15] nds the largest inscribed ball (as de ned 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 de ned. We give novel geometric interpretations to these parameters, which are also de ned by Patterson and Lipkin [12] in a di erent manner. We also propose a practically meaningful method for making these parameters comparable, and de ne 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 A1 ; : : : ; Ak . 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 Ai stationary. In the quasi-rigid assumption, deformations due to compliance e ects are assumed to be localized to the vicinity of the contact points, so that the overall motion of B relative to Ai can be described using rigid body kinematics. This is an excellent assumption for
xturing of mechanical parts. Since the ngers Ai are stationary, we can focus on the following con guration space (c-space) of B, which is denoted C . Choose a xed world reference frame, FW , and a frame FB xed to B. A con guration of B is speci ed by the position, d 2 R3 , and orientation, R 2 SO(3), of FB relative to FW . C-space is given hybrid coordinates q = (d; ) 2 R3 R3 , 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 R3 . The tangent space to C at a con guration q, denoted by Tq 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 R3 is the velocity of the origin of FB , and ! 2 R3 is the angular velocity of FB . 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 R3 is a force acting at FB 's origin and 2 R3 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 R2 and !; 2 R. The hybrid parametrization of c-space depends on the choice of frames. Consider a new world frame, FW , displaced from FW by (dw ; Rw ), and a new object frame, FB , displaced from FB by (db ; Rb ). A con guration with coordinates q would now have di erent coordinates q. The tangent and cotangent vectors transform as follows: q_ = T ?1q;_ w = T T w; (1) where the transformation matrix for the 3D and 2D cases aregiven by T66 = R0w R0Rdb Rw and T33 = R0w JR10 db ; (2) w ? respectively. Here J = ?01 01 , and R0 = R(0 ). Since dw and Rb do not appear in T , a translation of FW or a rotation of FB do not a ect the transformation. Rimon and Burdick [14] proposed a model for contact compliance which use overlap functions. These functions allow one to ignore the speci c details of deformations when B and Ai 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 brie y reviewed here. In the absence of deformation, the two bodies B and Ai contact at a single point, and after deformation occurs the bodies inter-penetrate. The overlap between B and Ai , denoted i , is the minimum amount of translation separating B from Ai . Clearly, i depends on B's con guration: i = i (q). We de ne i = 0 when B and Ai 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 fi , is assumed to depend on the overlap i (q): fi = fi (i (q)). The simplest model assumes that fi is a linear function of the overlap: fi (i ) = ki i ; (3) where ki is determined by the material and surface properties of B and Ai . 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 fi (i ). See [7] for details. Consider a grasp of B at a con guration q0 . 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 q0 . The grasp is stable if B returns to q0 after the external disturbance is removed. A more formal discussion of stability can be found in Ref. [14]. The elastic potential energy of the system consisting of the object B and ngers A1 ; : : : ; Ak is: (q) =
k Z i (q) X
fi ()d: (4) i=1 0 It can be veri ed that i (q) is di erentiable almost ev-
erywhere, hence (q) is di erentiable. In the absence of a disturbing wrench, an equilibrium grasp is characterized by: k X
fi (i0 )ri0 = ~0; (5) where i0 = i (q0 ) and ri0 = ri (q0 ). A suciently
r(q ) = 0
i=1
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 sti ness matrix is de ned as the Hessian K = D2 (q0 ) of the potential . Denoting fi0 = ddfii and D2 i0 = D2 i (q0 ), it follows from (5) that
K=
k X
fi0(i0 )ri0 ri0 T + fi (i0 )D2 i0 :
(6) Therefore, the sti ness 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 = q0 + q_ in the vicinity of q0 the sti ness 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 i0 . 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 e ects, while the second term accounts for second order (curvature) e ects. If the rst term alone is positive de nite, the grasp is stable to rst order. Otherwise, i=1
if the entire K is positive de nite, the grasp is stable to the second order. The relative contributions of rst and second order e ects on grasp stability and sti ness are analyzed in Ref. [7]. We conclude this section with the following changeof-frame formula for the sti ness matrix: K = T T KT; (7) where K is the sti ness matrix associated with the new frames FW and FB . This formula can be derived from (1) and the fact that r(q0 ) = 0.
3 Principal Sti ness Parameters
This section de nes the characteristic compliance parameters of a grasp, based on the sti ness matrix K , and the compliance matrix C , K ?1 . For clarity, we note that w = K q_ while q_ = C w. We use the following partition of K and C into 3 3matrices: K11 K12 ; C = C11 C12 : K= K (8) T K22 C12T C22 12 Note that the diagonal blocks are positive de nite, since K and C are. We use q_1 and q_2 for the translational and rotational components of q_ = (v; !), and use w1 and w2 for the force and torque components of w = (f; ).
3.1 Formal Development
The eigenvalues of K , which could provide important insight into the sti ness matrix, are not frame invariant. To circumvent this diculty, we look at the tangent subspace de ned by V = fq_ 2 Tq0 C : f = (K q_)1 = 0g: That is, V consists of the small displacements that induce a pure reaction torque on B. Using the partition ?1K12 !g, from of K , we obtain V = f(v; !) : v = ?K11 which it follows that V can be parametrized as ?1K12 ? K 11 q_ = P! where P = : (9) I Let KV denote the restriction of K to V . Recalling that the sti ness matrix represents the symmetric bilinear operator D2 (q0 ), we have that !T KV ! = !T P T KP! for arbitrary !. Thus under our parametrization of V , KV has the representation KV = P T KP = K22 ? K12T K11?1 K12 : Since K maps q_ 2 V to pure-torque wrenches, we have that (K q_)2 = KV !: (10) Consider now two new frames FW and FB , with overbars denoting objects associated with these frames. The linear operator KV has the following invariance property. Proposition 3.1 ([8]). Let V and V be the subspaces parametrized by (9) in the q and q coordinates. Let K V be the restriction of K to V . Then K obeys the orthogonal transformation K V = Rw T KV Rw : Hence the eigenvalues of KV are frame-invariant.
Dually, consider the following wrench subspace: W = fw 2 Tq0 C : ! = (C w)2 = 0g: In words, W is the subspace of wrenches that induce pure translation, and this subspace can be parametrized as w = Qf Q = ?C ?I1 C T : (11) 12 22 Using this parametrization, the restriction of C to W , ?1 C T = denoted CW , is CW = QT CQ = C11 ? C12 C22 12 ? 1 K11 . Moreover, the resulting pure-translation is given by v = (C w)1 = CW f , where w 2 W . Proposition 3.2. Let W be the subspace parametrized by (11) in the q coordinates, and let CW be the restriction of the compliance matrix C to W . Then CW obeys the orthogonal transformation CW = Rw T CW Rw . ?1 are frameHence, the eigenvalues of CW = K11 invariant. Propositions 3.1 and 3.2 lead to the following observations. The behavior of K on V characterizes the rotational sti ness 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 di erent frames. Since the tangent subspace V and the image of W under CW span Tq0 C , the two subspaces characterize the grasp compliance completely. Summarizing these observations and ?1, we call the eigenvalusing the fact that CW = K11 ues i (i = 1; 2; 3) of KV the principal rotational sti nesses, and the eigenvalues i (i = 1; 2; 3) of K11 = CW?1 the principal translational sti nesses of the grasp. In particular, min = minfi g is the smallest principal translational sti ness. The associated eigenvectors are called principal rotational and translational sti ness directions, respectively. For planar grasps it can be shown that there is a unique location of the origin of FB , given by db = R0T JK11?1 K12; (12) such that K33 takes the block-diagonal form K = diag(Rw T K11 Rw ; ). That is, the translation and rotational e ects are decoupled about this special point, called the center of compliance [11]. The principal translational and rotational sti nesses of the grasp are physically the translational and rotational sti nesses about the center of compliance.
3.2 Screw Coordinates Interpretation
While searching for a 3D analog of the center of compliance, Patterson and Lipkin [12] were the rst to recognize the existence of the principal sti ness directions. They used screw coordinates, and now we show that our principal parameters are equivalent to the ones derived
by Patterson and Lipkin. First we brie y review the notion of screw coordinates. A one-dimensional tangent subspace of the form fq_ = (v; !) : 2 Rg with k!k = 1, 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 kf k = 1, 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 KV associated with the eigenvalue i . Using (9), there exists a unit vector !i such that q_i = P!i . Then (10) 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 = Qfi 2 W where fi is an eigenvector of CW associated with the eigenvalue 1=i , we have that v = (C w)1 = (1=i )fi . Hence, the wrench w generates a pure-translation along its screw axis. Patterson and Lipkin [12] call the screw axis associated with these eigenvectors the twist- and wrench-compliant axes, respectively.
3.3 Geometric Interpretation
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 (6:15; 5:54), while the slanted ellipsoid corresponds to the origin located at (0; 0). The lengths of the principal semi-axes of each horizontal cross section of S are frame invariant. Similarly, the projection of S ispbounded 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=. v1
v2
-0.5 0.5 0.5 -0.5 2
w0
-2
Fig. 1. The elastic energy ellipsoid in Tq0 C
For the wrench space on which the quadratic form (w) = 12 wT Cw is de ned, we have the following analogous interpretation, shown in Fig. 2. The level set T given by (w) = 1 is a 5-dimensional elliptical surface corresponding to wrenches that induce unit elastic energy. The intersection Tf of T with the set f = const isq an ellipsoid whose principal semi-axes are equal to i (2 ? f T K11?1f ) (i = 1; 2; 3) and are frame-invariant. When principal semi-axes of Tf are given by p2 . fLet= 0,T the i h be the subset of T such that the normal vector to T at a point w 2 Th has zero -component. The projection of Th to the subspace = 0 is given by ?1 f = 1g: (Th ) =0 = f(f; ) : = 0 and 12 f T K11 ?1 = CW is frame invariant, the principal semiSince K11 p axes of (Th ) =0, given by 2i , are frame invariant. In the planar case, the elliptical surface T intersects p the -axis at two points whose coordinates are 2 (Fig. 2). If T is vertically oriented, the horizontal pro?1 f = 1. Any jection of T is the planar ellipse 21 f T K11 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. 1. The upright and slanted ellipsoids correspond to the same frames as their counterparts in Fig. 1.
We now present a novel interpretation of the principal sti nesses. Consider the quadratic form (q_) = 21 q_T K q_, where q_ 2 Tq0 C . The level set S de ned by (q_) = 1 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 Tq0 C determined by the equation ! = const. Letting ! (v) , (v; !), the points v 2 S! satisfy ! (v) = 21 uT K11 u + 12 !T KV ! = 1; ?1 K12 !. Hence for each xed !, where u = v + K11 S! is an ellipsoid with principal semi-axes of length p (2 ? !T KV !)=i (i = 1; 2; 3). These lengths are p frame-invariant, and when ! = 0 the lengths are 2=i (i = 1; 2; 3). Next we consider the collection of points in S , denoted Sn , at which the vectors normal to S have zero v-components. For any q_ = (v; !) 2 Sn , the condi?1K12 ! and tion (r(q_))1 = 0 implies that v = ?K11 1 T consequently (q_) = 2 ! KV ! = 1. By setting the vcoordinates of the points in Sn to zero, we obtain the projection of Sn to the subspace v = 0 as follows. (Sn )v=0 = f(v; !) : v = 0 and 12 !T KV ! = 1g: This p is an ellipsoid with principal semi-axes of lengths 2=i , where i for i = 1; 2; 3 are the eigenvalues of 4 A Frame-Invariant Quality Measure KV . Guaranteeing that the displacement of a grasped obFor planar grasps S is 2-dimensional. Fig. 1 shows ject will not exceed a speci ed tolerance is one of the
f1 0
-1
1
0.5
τ 0.250
-0.25 -0.5 1
0
f2
-1
Fig. 2. The elastic energy ellipsoid in Tq0 C
most important concerns in xture design [13]. Hence we wish to develop a grasp quality measure which is related to the de ection of the object under the action of disturbing forces. In particular, we wish to relate the principal translational and rotational sti nesses to the object's de ection, and use this relation to evaluate alternative grasps. Let q_ = (v; !) be a displacement of B, where kvk = 1 if ! = 0, and k!k = 1 if ! 6= 0. We de ne the de ection 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 ! = 0, the de ection is simply jj. If ! 6= 0, let max (q_) be the maximal distance from the screw p axis of q_ to B 's boundary points. The de ection is jj max(q_)2 + (v !)2 , where v ! is the pitch of q_. For planar grasps v ! = 0 and B's de ection is jjmax (q_). First we present our quality measure in the context of planar grasps. For planar grasps, we wish to compare the principal rotational sti ness with the smallest principal translational sti ness min . As previously discussed, and min are associated with pure rotation and translation of B with respect to the center of compliance. The de ection 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 de ection of B due to this rotation is jjmax , where max is the maximal distance from the center of compliance to B's boundary. The equivalent sti ness associated with , denoted eq , is de ned by the relationship: 1 ( )2 = 1 2 ; 2 eq max 2 where the right hand side is the elastic energy generated by the rotation . This relationship yields eq = ( )2 : (13) max The parameters eq and min are now comparable. We de ne the grasp quality measure as: Q = minfmin ; eq g: (14) The scalar Q measures the worst-case characteristic sti ness based on B's de ection. Moreover, Q is frame invariant. We now de ne the quality measure for a 3D grasps. For 3D objects, we must scale the principal rotational sti nesses i so that they become comparable with the
translational sti nesses. Let q_i = (vi ; !i ) 2 V be the eigenvector of KV associated with i , such that k!i k = 1. Then the elastic energy generated by the displacement q_i is given by 21 pi 2 , while the de ection of the object due to q_i is (max i )2 + (vi !i )2 , where max i = max (q_i ). Analogously to the 2D case, we de ne eq i by the following energy equivalence relationship 1 p( )2 + (v ! )2 2 = 1 2 ; i i max i 2 eq i 2 i which yields eq i = ( )2 +i (v ! )2 for i = 1; 2; 3. (15) max i i i We de ne the following 3D grasp quality measure : (16) Q = minfmin ; eq 1 ; eq 2 ; eq 3 g: Again, Q is a frame-invariant scalar which measures the worst-case characteristic sti ness as determined by B's de ection.
5 Optimal Grasping of Polygons
To illustrate our methodology and its possible utility, we apply the quality measure (14) 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 sti ness coecient ki is assumed to be the same for all nger locations on a given edge. We exclude nger placements at vertices and choose coincident frames FW and FB . Let the contact con guration 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 di erent 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 fv1 ; : : : ; vn g be the vertices of B. For vertex vi , let Hj be the closed half plane that does not contain vi and is bounded by the bisector between vi and vj (j 6= i). Let Ri be the intersection of these half planes. Then max(p) = kp ? vi k for p 2 Ri . Clearly, Ri is a convex polygonal region, and [ni=1 Ri = R2 .
5.1 Optimal Three-Finger Grasping
For 3- ngered planar equilibrium grasps, the sti ness 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, ni are the unit contact normals pointing into B. Also, P the total initial nger force is fT = ki=1 fi (i0 ).
Proposition 5.1 ([8]). If the origin of FW coincides itive when the geometric center belongs to S . In this 1
with the point of concurrency, the sti ness matrix for a grasp employing a given triplet of edges takes the form P3 T K = i=1 k0i ni ni 0 ; with the principal rotational sti ness given by fT a sin 1 sin 2 sin 3 ; = 2sin 1 + 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. α3
n2
n3
case the optimal concurrency point location coincides with the geometric center, and the optimal quality measure is given by (see footnote 1 for the radius r0 ). fT a sin 1 sin 2 sin 3 : Qopt = r22(sin 1 + sin 2 + sin 3 ) 0 Example 5.1. Consider grasping a quadrilateral by three identical ngers (Fig 4). We take ki = 1 (i = 1; 2; 3) without loss of generality. The radius of the object is 6:7315 and its geometric center is at (6:5; 1:75). 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 = 0:8609 and Q=fT = 0:081 for grasp I, while min = 1:2764 and Q=fT = 0:0955 for grasp II, which is globally optimal. D(2,6)
α1
S n1
C(9,8)
α2
Fig. 3. 3- nger grasp on a particular edge triplet It follows P that min is the smallest eigenvalue of the 2 2 matrix 3i=1 ki ni ni T , and is constant for the given
edge triplet. It is even more interesting to observe that , resulting from curvature e ects and depending on the total nger force fT , is also constant for all grasp arrangements on the same edge triplet. For the grasp to be stable, fT must assume a positive value (i.e., initial deformations are nonzero). Since the rst order e ects are dominant, eq min and therefore T a sin 1 sin 2 sin 3 : Q = eq = 2 2f(sin (17) 1 + sin 2 + sin 3 ) max In practice, fT 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 R2 , 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 fT , the quality measure (17) is maximized over a given edge triplet as 2max is minimized. This agrees with the intuition that the de ection of B about the concurrency point due to a unit torque is minimized for the optimal grasp. For a given edge triplet, we maximize 2max, a positive de nite 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 intu-
- Grasp I (6.5,1.75) A(0,0)
- Grasp II
B(13,0)
Fig. 4. 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 ni top the ith edge. Use s = (s1 ; s2 ; s3 ; s4 )T , where si = ki 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 placedpat in nity. Let Ni = ki ni , and hi (si ) = (NiT ; sTi )T . Then the sti ness matrix is givenby (see [7] fora proof) T Ns K (s) = (NN (18) Ns)T sT s ; where N = (N1 ; N2 ; N3; N4 ) and the second order e ects 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 sti ness is a constant. Using (18), we nd the dependence of the principal rotational sti ness on contact con guration: (s) = sT s, where = I ? N T (NN T )?1 N . From (12) and (18) the center of compliance as a function of s is given by db (s) = ?s, where ? = J (NN T )?1 N . Thus in the polygonal region Ri we can write eq as T eq (s) = (?s ? vs)T(?s s ? v ) : i i Here the geometric center is the point p0 at which minp2R2 max (p) = max (p0 ), and r0 is called the radius. 1
r0
=
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. De ne di (s) = det (hi+1 ; hi+2 ; hi+3 ) (mod 4), for i = 1; 2; 3; 4. Ref. [8] shows that the sti ness matrix K (s) is positive de nite if and only if d1 (s), ?d2 (s), d3 (s) and ?d4 (s) are all nonzero and have the same sign. Therefore the collection of stable grasps is S1 [ S2 , 4 where S1 and S2 4are bounded convex polytopes in R . S1 = G \ s 2 R : d1 (s); ?d2 (s); d3 (s); ?d4 (s) < 0 ; S2 = G \ s 2 R4 : d1 (s); ?d2 (s); d3 (s); ?d4 (s) > 0 : We can maximize eq over convex polyhedral regions of the form P = Si \ Dj , where Dj = fs 2 R4 : ?s 2 Rj g. For t 2 R, de ne (t) = maxs2P (t; s) where (t; s) = sT s ? t (?s ? vi )T (?s ? vi ): Proposition 5.2. If the function has a zero, it is unique. Moreover, t = eq (s ) = maxs2P eq (s) if and only if (t ) = (t ; s ) = 0. This proposition is proved in Ref. [8]. It follows that maximizing eq over P is equivalent to solving the scalar equation (t) = 0. The evaluation of the function is an inde nite quadratic programming problem. While inde nite quadratic programming is NP-hard, there are many ecient approximate algorithms. In fact, with our 4-dimensional problems, an exhaustive search scheme is quite a ordable. The remarkable fact is that global optimality is guaranteed at reasonable cost despite the nonconvex and strongly nonlinear nature of the quality measure. (1.175,3.525)
(0,0)
(11,4)
(13,0)
Fig. 5. Global optimal grasp of a quadrilateral
Example 5.2. Let us look at the quadrilateral used in
Example 5.1 and assume ki = 1. 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 = 1:684 < eq = 1:865.
6 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 e ort along this direction. A frame-invariant quality measure was de ned based on characteristic compliance parameters of the sti ness 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
[1] M. Cutkosky and P. Wright. Friction, stability and the design of robotic ngers. Int. J. of Robotic Research, 5(4):20{37, 1986. [2] C. Ferrari and J. Canny. Planning optimal grasps. In Proc. Int. Conf. on Robotics and Automation, pages 2290{2295, Nice, France, May 1992. [3] H. Hanafusa and H. Asada. Stable prehension by a robot hand with elastic ngers. In Proc. 7th Int. Symp. on Industrial Robots, pages 361{368, Tokyo, October 1977. [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 1994. [5] D. G. Kirkpatrick, B. Mishra, and C. K. Yap. Quantitative steinitz's theorems with applications to multi ngered grasping. In Proc. 20th ACM Symp. on Theory of Computing, pages 341{351, Baltimore, MD, May 1990. [6] Z. Li and S. Sastry. Task oriented optimal grasping by multi ngered robot hands. IEEE Trans. on Robotics and Automation, 6(2):32{44, 1987. [7] Q. Lin, J. W. Burdick, and E. Rimon. Computing sti ness matrices for grasping and xturing. Submitted, IEEE Int. Conf. on Robotics and Automation, 1996. [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, 1996. [9] X. Markensco and C. H. Papadimitriou. Optimum grip of a polygon. Int. J. of Robotic Research, 8(2):17{29, 1989. [10] 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 1994. [11] V.-D. Nguyen. Constructing stable grasps. Int. J. of Robotic Research, 8(1):26{37, 1989. [12] T. Patterson and H. Lipkin. Structure of robot compliance. ASME J. of Mechanical Design, 115(3):576{580, 1993. [13] P. C. Pong, R. R. Barton, and P. H. Cohen. Optimum xture design. In Proc. 2nd Industrial Eng. Research Conf., pages 6{10, Los Angeles, CA, May 1993. [14] E. Rimon and J. W. Burdick. Mobility of bodies in contact| ii: How forces are generated by curvature e ects. In Proc. Int. Conf. on Robotics and Automation, pages 2336{2341, San Diego, CA, May 1994. [15] M. Teichmann. The power of friction: Quantifying the \goodness" of frictional grasps. In Proc. Int. Conf. on Robotics and Automation, Minneapolis, MN, April 1996.
1 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 [11] used a linear spring model to compute the sti ness matrix of more general grasps. Howard and Kumar [4] also used a linear spring compliance model to study grasp stability, but included the e ects of contact geometry. In studying compliance due to friction, Cutkosky and Wright [1] noted that stability is in uenced 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 [14] used overlap functions to model nonlinear compliance e ects. Lin, Burdick and Rimon [7] use these overlap functions to compute and analyze the grasp sti ness matrix for various contact models, including the widely veri ed and theoretically justi ed 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 e ectiveness has assumed rigid body mechanics. Let the wrench (i.e. force and torque) due to a unit force applied by a
Elon Rimon Dept. of Mechanical Engineering Technion, Israel Inst. of Technology Haifa, ISRAEL 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] de ne 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 [10] rst compute the grasps that best counteract pure forces. Among these grasps, the one that best resists torques is chosen to be optimal. Teichmann [15] nds the largest inscribed ball (as de ned 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 de ned. We give novel geometric interpretations to these parameters, which are also de ned by Patterson and Lipkin [12] in a di erent manner. We also propose a practically meaningful method for making these parameters comparable, and de ne 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 A1 ; : : : ; Ak . 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 Ai stationary. In the quasi-rigid assumption, deformations due to compliance e ects are assumed to be localized to the vicinity of the contact points, so that the overall motion of B relative to Ai can be described using rigid body kinematics. This is an excellent assumption for
xturing of mechanical parts. Since the ngers Ai are stationary, we can focus on the following con guration space (c-space) of B, which is denoted C . Choose a xed world reference frame, FW , and a frame FB xed to B. A con guration of B is speci ed by the position, d 2 R3 , and orientation, R 2 SO(3), of FB relative to FW . C-space is given hybrid coordinates q = (d; ) 2 R3 R3 , 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 R3 . The tangent space to C at a con guration q, denoted by Tq 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 R3 is the velocity of the origin of FB , and ! 2 R3 is the angular velocity of FB . 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 R3 is a force acting at FB 's origin and 2 R3 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 R2 and !; 2 R. The hybrid parametrization of c-space depends on the choice of frames. Consider a new world frame, FW , displaced from FW by (dw ; Rw ), and a new object frame, FB , displaced from FB by (db ; Rb ). A con guration with coordinates q would now have di erent coordinates q. The tangent and cotangent vectors transform as follows: q_ = T ?1q;_ w = T T w; (1) where the transformation matrix for the 3D and 2D cases aregiven by T66 = R0w R0Rdb Rw and T33 = R0w JR10 db ; (2) w ? respectively. Here J = ?01 01 , and R0 = R(0 ). Since dw and Rb do not appear in T , a translation of FW or a rotation of FB do not a ect the transformation. Rimon and Burdick [14] proposed a model for contact compliance which use overlap functions. These functions allow one to ignore the speci c details of deformations when B and Ai 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 brie y reviewed here. In the absence of deformation, the two bodies B and Ai contact at a single point, and after deformation occurs the bodies inter-penetrate. The overlap between B and Ai , denoted i , is the minimum amount of translation separating B from Ai . Clearly, i depends on B's con guration: i = i (q). We de ne i = 0 when B and Ai 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 fi , is assumed to depend on the overlap i (q): fi = fi (i (q)). The simplest model assumes that fi is a linear function of the overlap: fi (i ) = ki i ; (3) where ki is determined by the material and surface properties of B and Ai . 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 fi (i ). See [7] for details. Consider a grasp of B at a con guration q0 . 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 q0 . The grasp is stable if B returns to q0 after the external disturbance is removed. A more formal discussion of stability can be found in Ref. [14]. The elastic potential energy of the system consisting of the object B and ngers A1 ; : : : ; Ak is: (q) =
k Z i (q) X
fi ()d: (4) i=1 0 It can be veri ed that i (q) is di erentiable almost ev-
erywhere, hence (q) is di erentiable. In the absence of a disturbing wrench, an equilibrium grasp is characterized by: k X
fi (i0 )ri0 = ~0; (5) where i0 = i (q0 ) and ri0 = ri (q0 ). A suciently
r(q ) = 0
i=1
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 sti ness matrix is de ned as the Hessian K = D2 (q0 ) of the potential . Denoting fi0 = ddfii and D2 i0 = D2 i (q0 ), it follows from (5) that
K=
k X
fi0(i0 )ri0 ri0 T + fi (i0 )D2 i0 :
(6) Therefore, the sti ness 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 = q0 + q_ in the vicinity of q0 the sti ness 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 i0 . 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 e ects, while the second term accounts for second order (curvature) e ects. If the rst term alone is positive de nite, the grasp is stable to rst order. Otherwise, i=1
if the entire K is positive de nite, the grasp is stable to the second order. The relative contributions of rst and second order e ects on grasp stability and sti ness are analyzed in Ref. [7]. We conclude this section with the following changeof-frame formula for the sti ness matrix: K = T T KT; (7) where K is the sti ness matrix associated with the new frames FW and FB . This formula can be derived from (1) and the fact that r(q0 ) = 0.
3 Principal Sti ness Parameters
This section de nes the characteristic compliance parameters of a grasp, based on the sti ness matrix K , and the compliance matrix C , K ?1 . For clarity, we note that w = K q_ while q_ = C w. We use the following partition of K and C into 3 3matrices: K11 K12 ; C = C11 C12 : K= K (8) T K22 C12T C22 12 Note that the diagonal blocks are positive de nite, since K and C are. We use q_1 and q_2 for the translational and rotational components of q_ = (v; !), and use w1 and w2 for the force and torque components of w = (f; ).
3.1 Formal Development
The eigenvalues of K , which could provide important insight into the sti ness matrix, are not frame invariant. To circumvent this diculty, we look at the tangent subspace de ned by V = fq_ 2 Tq0 C : f = (K q_)1 = 0g: That is, V consists of the small displacements that induce a pure reaction torque on B. Using the partition ?1K12 !g, from of K , we obtain V = f(v; !) : v = ?K11 which it follows that V can be parametrized as ?1K12 ? K 11 q_ = P! where P = : (9) I Let KV denote the restriction of K to V . Recalling that the sti ness matrix represents the symmetric bilinear operator D2 (q0 ), we have that !T KV ! = !T P T KP! for arbitrary !. Thus under our parametrization of V , KV has the representation KV = P T KP = K22 ? K12T K11?1 K12 : Since K maps q_ 2 V to pure-torque wrenches, we have that (K q_)2 = KV !: (10) Consider now two new frames FW and FB , with overbars denoting objects associated with these frames. The linear operator KV has the following invariance property. Proposition 3.1 ([8]). Let V and V be the subspaces parametrized by (9) in the q and q coordinates. Let K V be the restriction of K to V . Then K obeys the orthogonal transformation K V = Rw T KV Rw : Hence the eigenvalues of KV are frame-invariant.
Dually, consider the following wrench subspace: W = fw 2 Tq0 C : ! = (C w)2 = 0g: In words, W is the subspace of wrenches that induce pure translation, and this subspace can be parametrized as w = Qf Q = ?C ?I1 C T : (11) 12 22 Using this parametrization, the restriction of C to W , ?1 C T = denoted CW , is CW = QT CQ = C11 ? C12 C22 12 ? 1 K11 . Moreover, the resulting pure-translation is given by v = (C w)1 = CW f , where w 2 W . Proposition 3.2. Let W be the subspace parametrized by (11) in the q coordinates, and let CW be the restriction of the compliance matrix C to W . Then CW obeys the orthogonal transformation CW = Rw T CW Rw . ?1 are frameHence, the eigenvalues of CW = K11 invariant. Propositions 3.1 and 3.2 lead to the following observations. The behavior of K on V characterizes the rotational sti ness 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 di erent frames. Since the tangent subspace V and the image of W under CW span Tq0 C , the two subspaces characterize the grasp compliance completely. Summarizing these observations and ?1, we call the eigenvalusing the fact that CW = K11 ues i (i = 1; 2; 3) of KV the principal rotational sti nesses, and the eigenvalues i (i = 1; 2; 3) of K11 = CW?1 the principal translational sti nesses of the grasp. In particular, min = minfi g is the smallest principal translational sti ness. The associated eigenvectors are called principal rotational and translational sti ness directions, respectively. For planar grasps it can be shown that there is a unique location of the origin of FB , given by db = R0T JK11?1 K12; (12) such that K33 takes the block-diagonal form K = diag(Rw T K11 Rw ; ). That is, the translation and rotational e ects are decoupled about this special point, called the center of compliance [11]. The principal translational and rotational sti nesses of the grasp are physically the translational and rotational sti nesses about the center of compliance.
3.2 Screw Coordinates Interpretation
While searching for a 3D analog of the center of compliance, Patterson and Lipkin [12] were the rst to recognize the existence of the principal sti ness directions. They used screw coordinates, and now we show that our principal parameters are equivalent to the ones derived
by Patterson and Lipkin. First we brie y review the notion of screw coordinates. A one-dimensional tangent subspace of the form fq_ = (v; !) : 2 Rg with k!k = 1, 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 kf k = 1, 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 KV associated with the eigenvalue i . Using (9), there exists a unit vector !i such that q_i = P!i . Then (10) 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 = Qfi 2 W where fi is an eigenvector of CW associated with the eigenvalue 1=i , we have that v = (C w)1 = (1=i )fi . Hence, the wrench w generates a pure-translation along its screw axis. Patterson and Lipkin [12] call the screw axis associated with these eigenvectors the twist- and wrench-compliant axes, respectively.
3.3 Geometric Interpretation
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 (6:15; 5:54), while the slanted ellipsoid corresponds to the origin located at (0; 0). The lengths of the principal semi-axes of each horizontal cross section of S are frame invariant. Similarly, the projection of S ispbounded 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=. v1
v2
-0.5 0.5 0.5 -0.5 2
w0
-2
Fig. 1. The elastic energy ellipsoid in Tq0 C
For the wrench space on which the quadratic form (w) = 12 wT Cw is de ned, we have the following analogous interpretation, shown in Fig. 2. The level set T given by (w) = 1 is a 5-dimensional elliptical surface corresponding to wrenches that induce unit elastic energy. The intersection Tf of T with the set f = const isq an ellipsoid whose principal semi-axes are equal to i (2 ? f T K11?1f ) (i = 1; 2; 3) and are frame-invariant. When principal semi-axes of Tf are given by p2 . fLet= 0,T the i h be the subset of T such that the normal vector to T at a point w 2 Th has zero -component. The projection of Th to the subspace = 0 is given by ?1 f = 1g: (Th ) =0 = f(f; ) : = 0 and 12 f T K11 ?1 = CW is frame invariant, the principal semiSince K11 p axes of (Th ) =0, given by 2i , are frame invariant. In the planar case, the elliptical surface T intersects p the -axis at two points whose coordinates are 2 (Fig. 2). If T is vertically oriented, the horizontal pro?1 f = 1. Any jection of T is the planar ellipse 21 f T K11 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. 1. The upright and slanted ellipsoids correspond to the same frames as their counterparts in Fig. 1.
We now present a novel interpretation of the principal sti nesses. Consider the quadratic form (q_) = 21 q_T K q_, where q_ 2 Tq0 C . The level set S de ned by (q_) = 1 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 Tq0 C determined by the equation ! = const. Letting ! (v) , (v; !), the points v 2 S! satisfy ! (v) = 21 uT K11 u + 12 !T KV ! = 1; ?1 K12 !. Hence for each xed !, where u = v + K11 S! is an ellipsoid with principal semi-axes of length p (2 ? !T KV !)=i (i = 1; 2; 3). These lengths are p frame-invariant, and when ! = 0 the lengths are 2=i (i = 1; 2; 3). Next we consider the collection of points in S , denoted Sn , at which the vectors normal to S have zero v-components. For any q_ = (v; !) 2 Sn , the condi?1K12 ! and tion (r(q_))1 = 0 implies that v = ?K11 1 T consequently (q_) = 2 ! KV ! = 1. By setting the vcoordinates of the points in Sn to zero, we obtain the projection of Sn to the subspace v = 0 as follows. (Sn )v=0 = f(v; !) : v = 0 and 12 !T KV ! = 1g: This p is an ellipsoid with principal semi-axes of lengths 2=i , where i for i = 1; 2; 3 are the eigenvalues of 4 A Frame-Invariant Quality Measure KV . Guaranteeing that the displacement of a grasped obFor planar grasps S is 2-dimensional. Fig. 1 shows ject will not exceed a speci ed tolerance is one of the
f1 0
-1
1
0.5
τ 0.250
-0.25 -0.5 1
0
f2
-1
Fig. 2. The elastic energy ellipsoid in Tq0 C
most important concerns in xture design [13]. Hence we wish to develop a grasp quality measure which is related to the de ection of the object under the action of disturbing forces. In particular, we wish to relate the principal translational and rotational sti nesses to the object's de ection, and use this relation to evaluate alternative grasps. Let q_ = (v; !) be a displacement of B, where kvk = 1 if ! = 0, and k!k = 1 if ! 6= 0. We de ne the de ection 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 ! = 0, the de ection is simply jj. If ! 6= 0, let max (q_) be the maximal distance from the screw p axis of q_ to B 's boundary points. The de ection is jj max(q_)2 + (v !)2 , where v ! is the pitch of q_. For planar grasps v ! = 0 and B's de ection is jjmax (q_). First we present our quality measure in the context of planar grasps. For planar grasps, we wish to compare the principal rotational sti ness with the smallest principal translational sti ness min . As previously discussed, and min are associated with pure rotation and translation of B with respect to the center of compliance. The de ection 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 de ection of B due to this rotation is jjmax , where max is the maximal distance from the center of compliance to B's boundary. The equivalent sti ness associated with , denoted eq , is de ned by the relationship: 1 ( )2 = 1 2 ; 2 eq max 2 where the right hand side is the elastic energy generated by the rotation . This relationship yields eq = ( )2 : (13) max The parameters eq and min are now comparable. We de ne the grasp quality measure as: Q = minfmin ; eq g: (14) The scalar Q measures the worst-case characteristic sti ness based on B's de ection. Moreover, Q is frame invariant. We now de ne the quality measure for a 3D grasps. For 3D objects, we must scale the principal rotational sti nesses i so that they become comparable with the
translational sti nesses. Let q_i = (vi ; !i ) 2 V be the eigenvector of KV associated with i , such that k!i k = 1. Then the elastic energy generated by the displacement q_i is given by 21 pi 2 , while the de ection of the object due to q_i is (max i )2 + (vi !i )2 , where max i = max (q_i ). Analogously to the 2D case, we de ne eq i by the following energy equivalence relationship 1 p( )2 + (v ! )2 2 = 1 2 ; i i max i 2 eq i 2 i which yields eq i = ( )2 +i (v ! )2 for i = 1; 2; 3. (15) max i i i We de ne the following 3D grasp quality measure : (16) Q = minfmin ; eq 1 ; eq 2 ; eq 3 g: Again, Q is a frame-invariant scalar which measures the worst-case characteristic sti ness as determined by B's de ection.
5 Optimal Grasping of Polygons
To illustrate our methodology and its possible utility, we apply the quality measure (14) 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 sti ness coecient ki is assumed to be the same for all nger locations on a given edge. We exclude nger placements at vertices and choose coincident frames FW and FB . Let the contact con guration 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 di erent 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 fv1 ; : : : ; vn g be the vertices of B. For vertex vi , let Hj be the closed half plane that does not contain vi and is bounded by the bisector between vi and vj (j 6= i). Let Ri be the intersection of these half planes. Then max(p) = kp ? vi k for p 2 Ri . Clearly, Ri is a convex polygonal region, and [ni=1 Ri = R2 .
5.1 Optimal Three-Finger Grasping
For 3- ngered planar equilibrium grasps, the sti ness 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, ni are the unit contact normals pointing into B. Also, P the total initial nger force is fT = ki=1 fi (i0 ).
Proposition 5.1 ([8]). If the origin of FW coincides itive when the geometric center belongs to S . In this 1
with the point of concurrency, the sti ness matrix for a grasp employing a given triplet of edges takes the form P3 T K = i=1 k0i ni ni 0 ; with the principal rotational sti ness given by fT a sin 1 sin 2 sin 3 ; = 2sin 1 + 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. α3
n2
n3
case the optimal concurrency point location coincides with the geometric center, and the optimal quality measure is given by (see footnote 1 for the radius r0 ). fT a sin 1 sin 2 sin 3 : Qopt = r22(sin 1 + sin 2 + sin 3 ) 0 Example 5.1. Consider grasping a quadrilateral by three identical ngers (Fig 4). We take ki = 1 (i = 1; 2; 3) without loss of generality. The radius of the object is 6:7315 and its geometric center is at (6:5; 1:75). 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 = 0:8609 and Q=fT = 0:081 for grasp I, while min = 1:2764 and Q=fT = 0:0955 for grasp II, which is globally optimal. D(2,6)
α1
S n1
C(9,8)
α2
Fig. 3. 3- nger grasp on a particular edge triplet It follows P that min is the smallest eigenvalue of the 2 2 matrix 3i=1 ki ni ni T , and is constant for the given
edge triplet. It is even more interesting to observe that , resulting from curvature e ects and depending on the total nger force fT , is also constant for all grasp arrangements on the same edge triplet. For the grasp to be stable, fT must assume a positive value (i.e., initial deformations are nonzero). Since the rst order e ects are dominant, eq min and therefore T a sin 1 sin 2 sin 3 : Q = eq = 2 2f(sin (17) 1 + sin 2 + sin 3 ) max In practice, fT 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 R2 , 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 fT , the quality measure (17) is maximized over a given edge triplet as 2max is minimized. This agrees with the intuition that the de ection of B about the concurrency point due to a unit torque is minimized for the optimal grasp. For a given edge triplet, we maximize 2max, a positive de nite 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 intu-
- Grasp I (6.5,1.75) A(0,0)
- Grasp II
B(13,0)
Fig. 4. 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 ni top the ith edge. Use s = (s1 ; s2 ; s3 ; s4 )T , where si = ki 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 placedpat in nity. Let Ni = ki ni , and hi (si ) = (NiT ; sTi )T . Then the sti ness matrix is givenby (see [7] fora proof) T Ns K (s) = (NN (18) Ns)T sT s ; where N = (N1 ; N2 ; N3; N4 ) and the second order e ects 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 sti ness is a constant. Using (18), we nd the dependence of the principal rotational sti ness on contact con guration: (s) = sT s, where = I ? N T (NN T )?1 N . From (12) and (18) the center of compliance as a function of s is given by db (s) = ?s, where ? = J (NN T )?1 N . Thus in the polygonal region Ri we can write eq as T eq (s) = (?s ? vs)T(?s s ? v ) : i i Here the geometric center is the point p0 at which minp2R2 max (p) = max (p0 ), and r0 is called the radius. 1
r0
=
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. De ne di (s) = det (hi+1 ; hi+2 ; hi+3 ) (mod 4), for i = 1; 2; 3; 4. Ref. [8] shows that the sti ness matrix K (s) is positive de nite if and only if d1 (s), ?d2 (s), d3 (s) and ?d4 (s) are all nonzero and have the same sign. Therefore the collection of stable grasps is S1 [ S2 , 4 where S1 and S2 4are bounded convex polytopes in R . S1 = G \ s 2 R : d1 (s); ?d2 (s); d3 (s); ?d4 (s) < 0 ; S2 = G \ s 2 R4 : d1 (s); ?d2 (s); d3 (s); ?d4 (s) > 0 : We can maximize eq over convex polyhedral regions of the form P = Si \ Dj , where Dj = fs 2 R4 : ?s 2 Rj g. For t 2 R, de ne (t) = maxs2P (t; s) where (t; s) = sT s ? t (?s ? vi )T (?s ? vi ): Proposition 5.2. If the function has a zero, it is unique. Moreover, t = eq (s ) = maxs2P eq (s) if and only if (t ) = (t ; s ) = 0. This proposition is proved in Ref. [8]. It follows that maximizing eq over P is equivalent to solving the scalar equation (t) = 0. The evaluation of the function is an inde nite quadratic programming problem. While inde nite quadratic programming is NP-hard, there are many ecient approximate algorithms. In fact, with our 4-dimensional problems, an exhaustive search scheme is quite a ordable. The remarkable fact is that global optimality is guaranteed at reasonable cost despite the nonconvex and strongly nonlinear nature of the quality measure. (1.175,3.525)
(0,0)
(11,4)
(13,0)
Fig. 5. Global optimal grasp of a quadrilateral
Example 5.2. Let us look at the quadrilateral used in
Example 5.1 and assume ki = 1. 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 = 1:684 < eq = 1:865.
6 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 e ort along this direction. A frame-invariant quality measure was de ned based on characteristic compliance parameters of the sti ness 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
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