Easily computable optimum grasps in 2-D and 3-D

We consider the problem of finding 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 fingers, with friction at the contacts. Assuming frictional contact and rounded finger tips-very mild assumptions in practice-we give new upper (and lower) bounds on the number of fingers 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 finger grasps of 2-D objects, and optimum three finger grasps for 2-D objects. We present a simple O(n) algorithm for computing these optimum grasps for convex polygons, a O(n log n) algorithm for nonconvex polygons, and an O(n³) algorithm for polyhedra. In studying these optimum grasps, we derive several interesting theoretical results concerning grasp geometry