Probabilistic algorithms for efficient grasping and fixturing

Given an object with n points on its boundary where fingers can be placed, we give algorithms to select a "strong" grasp with the least number c of fingers (up to a logarithmic factor) using several measures of goodness. Along similar lines, given an integer c, we find the "best" κc log c finger grasp for a small constant κ. In addition, we generalize existing measures for the case of frictionless assemblies of many objects in contact. We also give an approximation scheme which guarantees a grasp quality close to the overall optimal value where fingers are not restricted to preselected points. These problems translate into a collection of convex set covering problems where we either minimize the cover size or maximize the scaling factor of an inscribed geometric object L. We present an algorithmic framework which handles these problems in a uniform way and give approximation algorithms for specific instances of L including convex polytopes and balls. The framework generalizes an algorithm for polytope covering and approximation by Clarkson [Cla] in two different ways. Let γ = 1/[d/2], where d is the dimension of the Euclidean space containing L. For both types of problems, when L is a polytope, we get the same expected time bounds (with a minor improvement), and for a ball, the expected running time is O(n^1+δ + (nc)^{1/(1+γ/(1+δ))} + c log(n/c)(c log c)^[d/2] for fixed d, and arbitrary positive δ. We improve this bound if we allow in addition a different kind of approximation for the optimal radius. We also give bounds when d is not a constant.