Partial closure grasps: Metrics and computation

We extend the notion of grasp metrics to partial force-closure grasps. We describe two metrics which measure the maximum and sum, respectively, of the forces that need to be applied at the contacts involved in a grasp, in order to exert some given unit wrench on the grasped object. For a given object P of complexity O(n) and a pure force T, we describe efficient algorithms which compute all combinations of m features (edges of polygons or facets of polyhedra) that admit grasps capable of exerting T such that the value of our metric is greater than some threshold. In particular, we show that if P is a polygon, all pairs of edges that admit frictionless two-finger grasps capable of exerting T can be computed in O(n log² n+K) time, where K is the number of pairs of edges in the output. Also, all two-finger grasps with friction of a polyhedral object can be computed in O(n^3/2+ε + K´) time and all frictionless three-finger grasps of a polyhedron can be computed in O(n^5/2+ε + K´) time, where K⁰ is the number of pairs or triples of facets that satisfy some slightly weaker condition and ε is some arbitrarily small, positive constant.