Easily Computable Optimum Grasps in 2-D and 3-D

We consider the problem of nding optimum force closure grasps of two and three-dimensional objects. Our focus is on grasps which are useful in practice, namely grasps with a small number of ngers, with friction at the contacts. Assuming frictional contact and rounded nger tips|very mild assumptions in practice|we give new upper (and lower) bounds on the number of ngers necessary to achieve force closure grasps of 2-D and 3-D objects. We develop an optimality criterion based on the notion of decoupled wrenches, and use this criterion to derive optimum two and three nger grasps of 2-D objects, and optimum three nger grasps for 3-D objects. We present a simple O(n) algorithm for computing these optimum grasps for convex polygons, a O(n logn) algorithm for non-convex polygons, and an O(n) algorithm for polyhedra. In studying these optimum grasps, we derive several interesting theoretical results concerning grasp geometry.