Grasping and fixturing: a geometric study and an implementation

Abstract

The problem of immobilizing an object by placing fingers (or points) on its boundary occurs in the field of dexterous manipulation, manufacturing and geometry. In this dissertation, we consider the purely static problems of good grasp and fixture set synthesis, and explore their connection to problems in computational and combinatorial geometry. Two efficient randomized approximation algorithms are proposed for finding the smallest cover for a given convex set and for finding the largest scale by which a convex set can be scaled and still be covered by a cover of a given size. These algorithms have applications to finding a good grasp or fixture set. An $O(n\sp2$ log n) algorithm for finding optimal 3 finger grasps for n sided polygons is also given. We introduce a new grasp efficiency measure based on a certain class of ellipsoids, invariant under rigid motions of the object coordinate system. To our knowledge, this is the first measure having this property. We also introduce a new reactive grasping paradigm which does not require a priori knowledge of the object. This paradigm leads to several reactive algorithms for finding a grasp for parallel jaw grippers and three and four finger robot hands equipped with simple sensors. We show their correctness and discuss our implementation of one such algorithm: a parallel jaw gripper with light-beam sensors which we have built.