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Quantitative Steinitz's Theorems with Applications to Multi Ngered

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Quantitative Steinitz's Theorems with Applications to Multi Ngered Quantitative Steinitzs Theorems with Applications to Multingered Grasping David Kirkpatrick Bhubaneswar Mishra and CheeKeng Yap Courant Institute of Mathematical Sciences New York University New York NY Abstract We prove the following quantitative form of a classical theorem of Steinitz Let m b e suciently large If the convex hull of a subset S of Euclidean dspace contains a unit ball centered on the origin then there is a subset of S with at most m p oints whose convex hull contains a solid ball also centered on the origin and having residual radius 2 d1 d d m The case m d was rst considered by Barany Katchalski and Pach We also show an upp er b ound on the achievable radius the residual radius must b e less than 2 d1 d m These results have applications in the problem of computing the socalled closure grasps by an mngered rob ot hand The ab ove quantitative form of Steinitzs the orem gives a notion of eciency for closure grasps The theorem also gives rise to some new problems in computational geometry We present some ecient algorithms for these problems esp ecially in the two dimensional case Septemb er 1 Supp orted by NSF Grants DCR and DCR ONR Grant NJ NSF Grant Sub contract CMU Chee Yap is supp orted in part by the German Research Foundation DFG and by the ESPRIT II Basic Research Actions Program of the EC under contract No pro ject ALCOM 2 Department of Computer Science University of British Columbia Vancouver Canada Section Intro duction d Caratheo dorys theorem states that given a subset S of Euclidean dspace E any p oint inside its convex hull is also inside the convex hull of some subset of S with at most d p oints d Steinitzs theorem states that given a subset S E any p oint in the interior of its convex hull is also in the interior of the convex hull of some subset of S with at most d p oints Barany Katchalski and Pach showed that the following quantitative version of Steinitzs theorem holds Theorem Quantitative Steinitzs Theorem d d For any positive d there is a constant r r d d such that given any set S E of points in dspace whose convex hul l contains the unit bal l centered at the origin o there is a subset X S with at most d points whose convex hul l contains a bal l centered at o with radius r d bdc In fact notes that r d ced d for some constant c In this pap er we generalize this quantitative Steinitzs theorem and study various algorithmic questions related to it d We intro duce the following terminology For any set S E let the residual bal l of S refer to the largest closed ball BS centered at the origin o such that the interior of BS is either fully outside or fully contained inside the convex hull of S The residual radius of S denoted r S is the signed radius of this residual ball where the sign is zero if BS is a p oint otherwise the sign is p ositive or negative dep ending resp ectively on whether BS lies inside or outside the convex hull Let r m S or r m S if d is clear from the context denote the largest d residual radius of a subset X of S with at most m p oints Let r m denote the minimum value d d of r m S as S ranges over all subsets S E with r S Hence for m d the result d d of Barany Katchalski and Pach shows that d r m Here we derive tighter upp er d and lower b ounds for r m Note that the notation r d in the Quantitative Steinitzs theorem d ab ove is simply the case of r d d Application in Rob otics Our interest in these theorems comes from the study of rob ot hand grasps We are interested in hands with m frictionless p ointngers A grasp in this mo del consists of m p oints on the b oundary of the b o dy that we want to grasp To actually grasp the b o dy we must then sp ecify forces to b e applied at these m grasp p oints A desirable notion of grasping is that of a closure grasp see for example Intuitively a closure grasp has the ability to resp ond to any external force or torque by applying appropriate forces at the grasp p oints The quantitative Steinitz theorem gives us a measure of the eciency of such closure grasps Roughly sp eaking the eciency of the grasp is given by the ratio of largest external forcetorque that can b e resisted by applying at most unit forces at each of the grasp p oints so a ratio of one corresp onds to the most ecient grasp The quantity r m in the d quantitative Steinitzs theorems gives this eciency directly Computational problems These theorems naturally lead to new problems in computa tional geometry For instance given a nite set S of p oints and a numb er m we want to nd an msubset of S that achieves the residual radius r m S We will present algorithms for such problems Here our strongest results are in two dimensions The rest of this pap er is organized as follows In section we explain more precisely the connection b etween quantitative Steinitzs theorem and grasping Sections and prove the generalized Steinitzs theorems Sections and present algorithms for computational versions of the generalized Steinitzs theorems We conclude in section Application to Multingered Positive Grasps We refer to for a general survey of the eld and to for the theory of rob ot grasping as used in this work Consider an idealized dextrous hand consisting of m indep endently movable forcesensing ngers These ngers can only contact ob jects at their tips and can thus b e represented as p oints in threedimensional space The goal is to grasp a closed b ounded and connected rigid ob ject K A nger can only apply a force on the ob ject K at the p oint of contact with K We assume that the p oints of contact are nonsingular ie the surface of K has a unique surface normal at each such p oint and frictionless and hence the force can only b e applied along the surfacenormal at the p oint of contact directed inward into the ob ject K An interesting task for such a hand is that of grasp selection for a given ob ject K by a grasp we mean a set of m p oints on the b oundary of K We also describ e such grasps in our mo del as positive b ecause the ngers can only push into the b o dy but not pull at the b o dy as might happ en if we p ostulate sticky ngers The ob ject K to b e grasp ed is assumed to have a piecewise smo oth b oundary Assume that the grasp p oints are to b e chosen from a given subset S of the nonsingular p oints of For example S may consist of all nonsingular p oints of by denition the surface normals at nonsingular p oints are uniquely dened Or again S may b e a set of nitely many preselected p oints For any p oint r in S let nr denote the unit surface normal directed inward at r Dene the function mapping S into the sixdimensional forcetorque space as follows S R r nr r nr where here denotes the vector crosspro duct of dimensional vectors Essentially maps r to the p oint r in the forcetorque space that represents the eects of applying a unit force at r in the direction nr If X S is a set of mp oints we call X an mnger closure grasp if the interior of the convex hull of X fr r X g contains the origin o It is shown in that for some m an mnger closure grasp of K exists if and only if o interiorconvexhull S Moreover if is not a surface of revolution the ab ove condition is always satised with S equal to the set of nonsingular p oints of For p olyhedral ob jects Mishra Schwartz and Sharir also gave an algorithm to nd a nger closure grasp in linear time However in the absence of any measure of go o dness for closures grasps the synthesized grasp may not b e very robust The motivation for our work is to quantify the go o dness of closure grasps and to synthesize provably go o d closure grasps One criterion for go o dness is the eciency of a grasp which measures the amount of external force and torque that can b e resisted by applying at most a unit of force at each grasp p oint This is precisely the value r m S To see this note that if we cho ose m p oints in Section S with residual radius r then any forcetorque vector v whose Euclidean norm is at most r can b e written as a convex combination of the m chosen p oints So if v is any external forcetorque that is applied to the b o dy K and v lies in the residual ball of radius r we can counter this external forcetorque by applying suitable forces of magnitude at most at the grasp p oints such that these forces sum to v hence we maintain the b o dy in equilibriu m These concepts can b e sp ecialized to the case where K is a planar b o dy in which case the forcetorque space is dimensional The numb er of ngers for closure grasps can b e reduced to in this case Quantitative Steinitzs Theorem in Dimensions Let S E b e a subset of the Euclidean plane and P its convex hull Without loss of generality we may assume that P has at least four vertices Also it is assumed that a unit disc B centered ab out the origin o is contained inside P In general let B denote the closed disc of radius centered at the origin Our goal is to develop techniques given S and m for cho osing a set X of at most m p oints from S so that the residual radius of X is maximized Lemma Given S as above for any integer k we can nd a subset X of at most k points of S such that the convex hul l of X contains B with cos k proof Take k equally spaced rays from o making sure that one of them pass through a vertex of P the convex hull of S Let these rays intersect the unit circle centered at o at the p oints v v For each ray if it intersects a vertex of P then we cho ose that vertex and if it intersects k an edge we cho ose the two vertices
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