On characterizing and computing three- and four-finger force-closure ...

On Characterizing and Computing Three- and Four-Finger Force-Closure Grasps of Polyhedral Objects Jean Ponce and Steven Sullivan Beckman Institute University of Illinois Urbana, IL 61801

Jean-Daniel Boissonnat and Jean-Pierre Merlet INRIA Sophia-Antipolis 2004 Route des Lucioles 06565 Valbonne Cedex, France

Abstract

force-closure. Their method relies on a linear sufficient condition for the existence of planar three-finger forceclosure grasps that allows them to characterize a grasp by a system of linear constraints in the position of the fingers along the polygonal edges. All regions satisfying these constraints are found by a projection algorithm based on linear parameter elimination accelerated by simplex tech-

We address the problem of characterizing the forceclosure grasps of a three-dimensional object by a hand equipped with three or four hard fingers. We prove several necessary and several sufficient conditions for force-closure. For polyhedral objects, we prove sufficient conditions that are linear in the unknown parameters. This reduces the problem of computing force-closure grasps of polyhedral objects to the problem of projecting a polytope onto some linear subspace. We present an efficient, output-sensitive algorithm for computing the projection and an algorithm using linear programming for computing maximal grasp regions.

niques.

In this paper, we extend their approach to the synthesis of three- and four-finger force-closure grasps of 3D polyhedral objects. We prove several necessary and several linear sufficient conditions for force-closure, reducing the problem of computing all force-closure grasp configurations to the problem of projecting a polytope onto some linear sub space. We present a novel and efficient output-sensitive algorithm for computing this projection, and describe an algorithm for computing maximal grasp regions. We are in the process of implementing these algorithms.

1 Introduction We address the problem of computing stable three- and four-finger grasps of polyhedral 3D objects. More precisely, we consider force-closure grasps, such that arbitrary forces and torques exerted on the grasped object can be balanced by the contact forces exerted by the fingers. We assume hard-finger point contacts with friction. The geometry of force-closure grasps is studied in [lo, 16, 171. The existence of force-closure grasps and the minimum number of fingers necessary to achieve these grasps are studied for frictionless contacts [15] as well as for contacts with friction [9,111. Nguyen gives a geometric method for finding maximal independent two-finger grasps of polygons [16]. Pollard and Lozano-PCrez plan three-finger grasps of polyhedral objects by first intersecting the object with some plane and then optimizing the choice of a focus point within this plane [19]. Omata uses fractional programming methods for manipulating polygonal objects with two fingers in the plane [18]. Ferrari and Canny [8] compute optimal force-closure grasps of polygonal objects by computing a maximal ball included in the convex hull of the contact wrenches. Faverjon, Ponce, and Stam [7, 211 and Chen and Burdick [4] use global non-linear optimization methods to compute two-finger grasps of curved objects. In [20], Ponce and Faverjon present a method for constructing maximal segments of a 2D polygon where three fingers can be positioned independently while maintaining

1050-4729193$3.00 0 1993 IEEE

2

Force-Closure Grasps

We characterize the stable grasps of a three-dimensional object by a hand equipped with d hard fingers, assuming point contact with friction. A grasp is defined geometrically by the position X i of the fingers on the boundary of the object, with i = 1, ..,d . We assume that the points X i are all different, that no three of them are aligned, and that no four of them are coplanar. Each finger exerts in X i a force f i and a moment X i x f i with respect to the origin. Force and moment are combined into a six-dimensional zero-pitch wrench W i = ( f i , x i x f i ) [2, 131.

2.1

Equilibrium and Force-Closure

A grasp achieves equilibrium when the sum of the forces and the sum of the moments are zero, i.e.,

cwi d

= 0.

i= 1

Note that, although the value of the wrenches depends on the choice of coordinate frame, the condition for equi-

82 1

Suppose now that we exert an external wrench w =

librium does not. Here, we are interested in force-closure grasps, defined as follows.

(f,x x f + m) on the object. We want to find a set of forces fi,fi,fi lying in the friction cones and balancing

Definition 1 A gmsp is force-closure if any external

this wrench, i.e.,

wrench can be bolanced by the wrenches at the fingertips.

In other words, a grasp is force-closure 8, for any external wrench w = (f,x x f m), the fingers can exert a set of zero-pitch wrenches Wi, with i= 1, ..,d, such that:

+

Since this condition does not depend on the choice of origin, we can choose the origin outside of the plane formed

d

w

+ c w i = 0.

by the points xI,xz, x3, and since these points are not aligned, the vectors Xi and Xi x Xi+l are linearly independent. Let us denote by (XI, Xz, AB) the coordinates of the vector (x X I ) x f m in the coordinate system (ax X1,Xl x xz,xz x x3). Consider the vectors V I = AIXI - XZXZ -f, vz = XZXZ X3x3, and va = X3x3 - Xlxl. Suppose we apply the force f,! = afi vi in xi, for i= 1,2,3, with CY = mU(ai), and ai chosen so that aif; +vi lies in the friction cone at that point. Clearly, the resultant of the forces f/ is -f, and their total moment is -x x f - m, which completes the proof. The case d = 4 can be treated similarly, using the four forces fi, with i = 1, ..,4 to achieve equilibrium, and constructing additional forces Vi, with i = 1 , 2 , 3 in the first three friction cones to balance the external force and moment. 0

(2)

is1

-

Again, this definition is independent of the choice of coordinate system. From now on, we assume point contact with Coulomb friction. The force exerted by each finger lies in the internal friction cone of half-angle 8 at the contact point (Fig. l(a)). Note that writing that a force f lies in the friction cone is quadratic in f. This condition can be expressed instead as a set of linear constraints in f by approximating the friction cone by an n-sided pyramid (Fig. l(b)). This will prove important in the sequel.

+

-

+

n

2.2

Zero-Pitch Wrenches and Lines

A zero-pitch wrench w = (f,x x f ) can also be seen as a representation of the line of action of the force f applied at the point x; the six coordinates (w1, ..,w6) of w are c d e d the Pliicker coordinates of the line. More precisely, the Pliicker coordinates are homogeneous coordinates for a projective space of dimension 5: the wrenches w = (f,x x f ) and Xw, with X # 0, both r e p resent the line with direction f passing through the point x (note that w is independent of the choice of x along the line). The set of lines form a quadric, called the Grassmannian, defined by w l w 4 + w ~ w ~ + w 3 w= ~ 0 in this projective space. Pliicker coordinates provide a convenient method for deciding whether two 3D lines intersect: two straight lines associated with wrenches (fi,XI x f1) and (fz,xz x fz) intersect if and only if fi .(xz x fi) fz .(xi x fi) = 0. This condition is valid in any coordinate system. More importantly for us, equilibrium implies that the corresponding lines are linearly dependent. Since Grassmann geometry characterizes the varieties of various dimensions formed by sets of dependent lines [5, 141, it will prove useful to characterize force-closure geometrically. We will use the following results: three linearly dependent lines from a flat pencil; if four lines are linearly dependent then, either they are all coplanar, or they intersect in a single point, or they form two flat pencils of lines having a line in common, or they form a regulus.

Figure 1. Coulomb friction:

(a) continuous friction cone; (b) discrete approximation by an n-sided pyramid.

The following proposition shows that the existence of a grasp achieving equilibrium is necessary and sufficient for force-closure.

Proposition 1 A gmsp is force-closure i#there ezists o force in each open friction wne such that the sum of the corresponding wrenches is zero. Note: to simplify the discussion, we assume in a l l the following that the wrenches at the fingertips are non-zero. PROOF: The condition is clearly necessary. We prove it is sufficient in the cases d = 3 and d = 4. The twodimensional case is a simpler variation of this one. Alternative proofs can be found in [ l l , 151, but they rely on different assumptions. Our proof is rather close in spirit to the one in [ll]. Let us first consider the case d = 3. The key remark is that, if a force f; lies in an open friction cone, then there exists some E, > 0 such that, for any vector Vi in the open ball’ of radius E;, fi Vi also lies in the friction cone. Equivalently, for any vector Vi, there exists ai = IVil/ci > 0 such that, for any (Y 2 ai, afi Vi lies in the friction cone.

+

+

+

822

3

Three-Finger Grasps

the internal friction cones, while others lie in the external ones. Some of these grasps are obtained by squeezing with the three fingers, while others are obtained by moving the fingers away from each other.

Figure 2. A three-finger force-closure grasp.

Proposition 2 A necessary condition for three points to form a force-closure grasp is that there exists a point in the intersection of the plane formed by the three points with the double-sided friction cones a t these points.

Figure 3. Different types of threefinger force-closure grasps.

PROOF: Consider three points

XI,x2,x3 and three forces inside the corresponding open friction cones such that the sum of the corresponding wrenches WI, w2,w3 is zero. Since the sum of the three wrenches is zero, they are linearly dependent, but a set of three linearly dependent lines form a flat pencil of lines, so the three lines intersect in a point XO, trivially lying in the intersection of the plane formed by XI,x2,x3 with the open double-sided friction cones. The above necessary condition is not sufficient. We now prove several sufficient conditions for three-finger forceclosure.

We now prove a linear sufficient condition which does not require the intersection points to lie in the internal friction cones. We first need a definition and some lemmas.

f 1 , f 2 , f3

Definition 2 Three vectors are said to positively span a plane if zero is a positive linear combination of these vectors. Lemma 1 If the angle between two vectors is less than - 28, then the cones of half-angle B centered at these vectors do not contain a pair of opposite vectors.

A

PROOF: Immediate. 0 Proposition 3 A suficient condition for three contact points to form a force-closure grasp is that there ezists a point in the intersection of the three open internal friction cones with the triangle formed by these points.

PROOF : Let ( ( ~ 1a2, , a3) denote the barycentric coordinates of the intersection point xo in the basis XI,XZ,X~. Since xo is in the triangle, these coordinates are strictly positive. Let f, = ai(x0 - xi), with i = 1, ..,3. Since xo belongs to the open internal friction cone in X i , so does f,. Clearly, the sum of the corresponding wrenches is zero, which shows that the grasp is force-closure. 0 Other forms of Propositions 2 and 3 appear in [lo, 231. As noted before, for polyhedra, the condition that a point xo belongs to the internd friction cones at the contact points xi, i = 1 , 2 , 3 , can be expressed by linear constraints in xo and xi by approximating the cones by n-sided pyramids. Unfortunately, writing that xo belongs to the triangle formed by XI, x2,x3 is not linear. In addition, only considering the case of contact points inside the internal friction cones may be overly restrictive: as demonstrated by Fig. 3, there are many types of three-finger grasps, and some contact points may lie in

823

Lemma 2 A suficient condition for three contact points XI,x2,x3 to form a force-closure grasp is that the pairwise angles between the internal normals n; a t these points are less than A - 28, there exist three uectors hi making angles less than 813 with these normals and positively spanning some plane, and there ezists a point xo in the intersection of the plane formed by XI,x2,x3 with the double-sided friction cones of half-angle 813 at these points. Before proving this lemma, note that it does not require that the normals themselves be coplanar. Also, the point xo may lie in the internal or the external friction cones at the contact points. PROOF: As before, let ( ( ~ 1a2, , aa) denote the barycenlet fi = tric coordinates of xo in the basis (xI,x~,x~), ai(x0 - xi), and let wi = ( f i , xi x fi) denote the wrenches associated with the three lines going through xo and xi. Clearly, w1 + w2 + w3 = 0. To prove the proposition, it is sufficient to prove that the three vectors f i either all lie in the internal friction cones at the points xi, or all lie in the external friction cones at these points, since, in the latter case, inverting the vectors fi will achieve force-closure.

4

Suppose this is not the case, we can assume without loss of generality that f1 lies in the external friction cone in XI, and that f 2 (resp. f 3 ) lies in the internal cone in x 2 (resp. x3). Let i i l , b 2 , & denote three unit vectors positively spanning some plane such that, for i = 1,2,3, ni .hi 2 cos(8/3), let b denote a unit normal to this plane, and ti = b x &, we write each vector fi in the basis (hi,f i , fi) as fi = X i i i i piti via. Since, for i = 1,2,3, the angle between n; and bi is less than 8/3, and the angle between ni and the line s u p ported by fi is less than 8/3, we conclude that the angle between iii and this line ia less than 28/3, so (p? U:) < AT tanz(28/3), X i < 0, and X2, x3 > 0. Since the vectors bi positively span a plane, we can write b1 = -ap& - a3fi3, with az,a3 > 0. Since f1 f2 f 3 = 0, it follo~athat VI g = 0, and if Xli = X i aiX1, pli = pi ( ~ i ~ (and 1 , f1i = Alibi plifi, with i = 2,3, we have f12 +f13 = 0. In addition, we have, for i = 2,3, X1i > 0, and lplil 5 IPiI IaillPlI S IXiItan(28/3) Iaillhl tan(28/3) = (Xi a i X 1 ) tan(28/3) = X l i tan(28/3). Since the angle between n 2 (resp. n 3 ) and 5 2 (resp. &) is lesa than 8/3, it follows that f12 and f13 respectively belong to the internal friction cones in x 2 and x g , with f12 f13 = 0, but, from lemma 1, this contradicts the hypothesis that the angle between nz and ns is less than r - 28, which impliea that there ia no such vector. 0 The above condition is not linear because the direction of the plane may vary. We now give a linear sufficient condition for threefinger forceclosure using a fixed plane direction (see[lo] for a similar result). As before, the three normals are not required to lie in this plane.

+

Let us now switch to the four-finger case. Thiswill actually allow us to derive a stronger linear sufficient condition for threefinger forceclosure in the case where gravity acts as a fourth finger,

+

Proposition 6 A necessary condition for four points to form a force-closure gmsp is that there en’st four lines in the open internal fiction that either intersect in a single point, or form two pat pencils having a line in common but lying in different planes, or form a ngulus (Fig. 4).

+

+ -

+

-

+ +

Four-Finger Grasps

+

+

+

Figure 4. four-finger grasps: (a) four intersecting lines, (b) two flat pencils of lines having a line in common, (c) a regulus.

+

PROOF: Consider four points Xi, with i = 1,..,4 and four forces f; inside the corresponding open friction cones such that the sum of the corresponding wrenches W i is zero. Since the sum of the four wrenches is zero, they are linearly dependent, and it is known from Grassmann geometry [5, 141 that a set of four linearly dependent lines either lie in a single plane, or intersect in a single point, or form two flat pencils having a line in common but lying in different planes, or form a regulus. The first case is not possible since the four contact points are not coplanar, and the result follows. 0 Note that a regulus is the set of lines intersecting three given skew lines. The lines in the regulus lie on a doubly ruled hyperboloid of one sheet (Fig. 4(c)). So far, we have not been able to derive simple sufficient conditions for grasps corresponding to two flat pencils of lines or to a regulus. From now on, we concentrate on grasps where the four lines of action of the contact forces intersect in a single point. We first give a simple sufficient condition

Proposition 4 A suficient condition for three contact points x 1 , x p I x 3 to form a force-closure grasp is that the pairwise angles between the internal normals at these points are less than r - 28, there ezist three vectors making angles less than 8/3 with these normals and positively spanning some fixed vector plane 11, and there ezists a point xo such that, for i = 1,2,3, the vector xo - Xi lies in the intersection of the plane II with the double-sided friction cone8 of half-angle 8/3 in xi.

(Fig. 5 ) .

Proposition 6 A sufficient condition for four contact points to form a force-closure gmsp is that there ezists a point in the intersection of the four open internal friction cones with the tetrahedron formed by these points,

PROOF : This is a corollary of lemma 2: note that if the vector xo - X i lies in the plane II for all i,then the points and xo are all coplanar. For a fixed vector plane 11, the above condition can be decomposed into eight linear sufficient conditions depending on whether xo lies in the internal or external friction cone at each contact point X i . Note that the plane containing the three contact points is not fixed but merely parallel to II. Finally, note that for any triple of faces, a set of three vectors lii can be found (if it exists) through linear programming. Xi

PROOF : Similar to proof of Proposition 3. 0 Now we are going to prove a linear sufficient condition, which will ensure that the intersection of the friction cones lies in the tetrahedron. Notation: The plane defined by three points xi, i = 1,2,3 is denoted by [ X ~ X Z X S ] .A point xo and a vector ui, i = 1 , 2 , 3 , define a half-line x = xo aiui, with ai 2

+

824

Proposition 7 A suficient condition f o r four-finger force-closure is t h a t the i n t e m a l normals a t the f o u r contact points 0-positively span @ and there exists a p o i n t i n the intersection of the i n t e r n a l f r i c t i o n cones a t the contact points. PROOF : Immediate from 6 and lemma 5. Note that the above proposition provides a linear sufficient condition for forceclosure. On the other hand, like Proposition 3, it only characterizes grasps for which all the fingers are moving toward each other, squeezing the object. Fnrther work is required to characterize more general grasps as well as grasps involving wrenches that form two flat pencils of planes or a regulus. To close this section , we are finally going to derive a linear sufficient condition for three-finger force-closure in the case where gravity is present, and acts as a fourth finger.

Figure 5. A 3 D four-finger force-closure grasp. 0. This half-line is denoted by (xo,ui). A point xo and three directions ui, with i = 1 , 2 , 3 define a trihedron x = xo C i ~ i ~ with i ,ai > 0. This trihedron is denoted

Proposition 8

I n the presence of gravity, a suficient condition f o r three contact points t o form a force-closuw grasp is t h a t the i n t e r n a l normals a t these points and the and there .en'sts a weight of the object 8-positively span p o i n t i n the intersection of the three open i n t e r n a l f r i c t i o n cones w i t h the half-line whose o r i g i n is the center of mass of the object and whose direction is i t s weight.

+

< XO,Ul,U2,U3 > Lemma 3 Let < XO, u1, u 2 , ~3 > be some trihedron, and, = 1 , 2 , 3 , let xi be some p o i n t o n W e half-line

for i

(=,Ui).

hedron

T h e plane

[ X ~ X Z X does ~]

< XO,-u1, - u 2 ,

-ug

n o t intersect the tri-

>

PROOF : This is almost a corollary of the previous proposition. The only change in the proof is that the point xo is the intersection of the half-line with the three internal friction cones instead of being the intersection of four friction cones. 0

PROOF : Immediate. Lemma 4 kt xo be some point and let Ci, f o r i = 1 , 2 , 3 , denote a single-sided cone centered in xo w i t h direction Ui. L e t T I 2 3 denote the intersection of the trihedra formed by all triples of half-lines belonging t o the cones -Cl, 4 2 , -C3. If,for i = 1 , 2 , 3 , X i denotes a p o i n t i n Ci , then the plane [ x ~ x zdoes ~ ] n o t intersect T123.

5

The algorithm

The sufficient conditions for force-dosure derived in the previous two sections can be written as a set of linear inequalities in the position of a point xo and the position of three or four contact points X i . For a polyhedral object, each contact can be parameterized linearly by two variables ( U , , vi) specifying its position in the plane of the corresponding face. Note that the fact that xi lies inside a convex face can be characterized by a set of linear constraint in (tti,vi)TOobtain all the solutions (xi, vi) yielding force-closure grasps, we must first eliminate the point XO.

PROOF: Immediate from lemma 3. 0 Deflnition 3 Four vectors are said t o @-positively span when, f o r any t r i p l e U), u2,~3 of these vectors, the cone C4 of half-angle B centered in the f o u r t h vector u 4 lies in the the intersection of the trihedra formed by a l l triples of half-lines belonging t o the cones -Cl, -C2,-C3 of half-angle @ centered in -u1, -u2, -U.

5.1

Lemma 6 If the i n t e r n a l normals a t f o u r xi, w i t h i = 1,..,4, @-positively span J?

contact points a n d the p o i n t xo lies in the intersection of the i n t e r n a l f r i c t i o n cones a t the points x;, i = 1,..,4, then the p o i n t xo lies i n the tetrahedron formed b y the points x i .

The Projection Algorithm

The projection algorithm takes as input a &polytope of a d-dimensional Euclidean space E, defined as the intersection of n half-spaces

P

Hi : A i X S b i , i = l , ..., n.

PROOF : From lemma 4 and the hypotheses, the points xo and x4 lie on the same side of the plane [XI, x2, a]. The same reasoning applies to the three other planes bounding the tetrahedron and the lemma follows. 0

Here Ai is a 1 x d real matrix, X = (21,. . . ,zd) the vector of coordinates of a point of E and b; a real number. The problem is to construct the orthogonal projection P'

825

the queries performed so far exceeds the preprocessing time and the memory storage. At that point, we can allow more memory. Thus we restart a new preprocessing with a bigger value of m. And repeat that process until all the faces have been constructed. We obtain the following result.

of P onto a k-dimensional subspace (k-flat) E’ of E in an output sensitive way. In the worst-case, P and P‘ have size O(nL41). The whole facial structure of P can be constructed in time O(n14J) [l, 31 and then, P’ can be computed in time proportional to its size by a simple traversal of the faces of the silhouette of P (i.e. the faces that project onto faces of P’ ). However, the worst-case situation is rarely encountered and moreover it is worth constructing P’ (a k-dimensional object) without constructing all the faces of P (a d-dimensional object of a possibly much bigger size than P’). We restrict the discussion to the case k > 1. The case k = 1 can be easily solved using linear programming. Without loss of generality, we will assume that E’ is defined by x i = 0, for k < i 5 d. Moreover we will assume that the hyperplmes AiX = b i , i = 1,. .. , n , are in general position, i.e. no (d l)-uplets of hyperplanes have a common intersection and no two vertices of P project onto the same vertex of P’. General techniques such as the simulation of simplicity of [6] can be used to make this hypothesis valid. The general outline of the algorithm is the following : 1. Find a first vertex of P which projects onto a vertex of P’ , e.g., an extremum in some direction orthogonal to the projection direction (this is a linear program). Put it in a stack S . 2. Repeat the following steps until S is empty

Proposition 9 The projection of a d-polytope of E onto a k-dimensionalflat can be computed in time and space

T = 0 (kn7trtYtr) where E is any positive constant, t is the size of the P’,and

’+*

-f=-

1

w i t h y = $ forp=3,4; y = t forp=5,6.

+

5.2

Computing Maximal Grasps

Eliminating XO, we obtain the convex set of all forceclosure grasps for each triple or four-tuple of faces. Because of the uncertainty in the robotics system, we would like to minimize the sensitivity of a grasp to positioning errors. A way of achieving this is to seek triples of independent contuct regions [16]. These regions are such that for any tuple of contact points chosen in them, the corresponding grasp is force-closure. In the grasp parameter space, these regions are obviously represented by parallelepipeds with sides aligned with coordinate axes and contained in the convex set of force-closure grasps. Among all the solutions to this problem, we define the maximal independent contact regions as those maximizing a criterion depending on their size and location. A reasonable criterion is to maximize the minimum of the lengths of the edges of the parallelepiped. Because of convexity, we only need to write that the vertices of the parallelepiped are contained in the set of force-closure grasps in order to guarantee that the entire parallelepiped is also contained in it, and finding the maximal independent contact regions reduces once again to solving a linear programming problem.

(a) Take a vertex V out of S and put it in a dictionary D of already considered vertices. Construct the facial structure of the cone incident to V and find all the edges (rays) of the cone that contain an edge of the silhouette of P, i.e. that project onto edges of P’. (This is again a linear program [2O].) (b) Each such ray r is considered in turn. We compute the vertex W distinct from V of the edge e contained in the ray. Let W‘ be its projection onto E’. If W has not been already considered (it does not belong to D),add W to S, and report the corresponding edge of p’, V’W’, where V’ is the projection of V onto E’.

6

P’ from the set of its edges. A simple implementation of the above algorithm takes O(tn) time where 2 is the size of P’. This algorithm is quite efficient in practice since the only costly construction is done in E‘, not in E. Moreover, the time complexity can be improved by preprocessing the hyperplanes Hi so as to answer the ray shooting queries in sublinear time. In the full version of this paper [22], we present a solution to that problem, based on a recent result by MatouBek and Schwarzkopf [12]. The idea of our algorithm is to start with little memory and do ray shootings until the cumulated time of all 3. Compute the complete facial structure of

Discussion

We have used Grassmann geometry to derive necessary condition for three- and four-finger force-closure grasps of three-dimensional objects. For polyhedral objects, we have also proven a set of linear sufficient conditions reducing the problem of computing force-closure grasps of polyhedral objects to the problem of projecting a polytope onto some linear subspace. We have presented an efficient, outputsensitive algorithm for computing the projection and an algorithm using linear programming for computing maximal grasp regions. We are in the process of implementing these algorithms in simulation. We also plan to address the issues of feasibility and reachability [19], and to eventually implement our approach on a real robot equipped

826

[14] J.P. Merlet. Singular configurations of parallel manipulators and grassmann geometry. In J.D. Boissonnat and J.P. Laumont, editors, Geometry and Robotics, volume 391 of Lecture Notes in Computer Science, pages 194-212. Springer-Verlag, 1988.

with a dextrous hand. Future work will address the problem of computing a more general set of four-finger grasps, in particular in the case where the fingertips wrenches form either two flat pencils of lines or a regulus. We will also explore the case of soft fingers that can exert pure torques in addition to pure forces, a more difficult setting in which the results of Grassmann geometry do not apply.

[15] B. Mishra, J.T. Schwartz, and M. Sharir. On the existence and synthesis of multifinger positive grips. Algorithmica, Special Issue: Robotics, 2(4):541-558, November 1987.

Acknowledgements: We wish to thank the anonymous referees for their suggestions and comments.

[16] V-D. Nguyen. Constructing force-closure grasps. International Journal of Robotics Research, 7(3):3-16, June 1988.

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