D-space and Deform Closure Grasps of Deformable Parts

Abstract

Building on the well-established form closure framework for holding rigid parts, in this paper we propose a new framework for holding deformable parts. We consider the class of deformable parts that can be modeled as linearly elastic polygons with a triangular finite element mesh and given stiffness matrix. We define the D-space (deformation-space) of a part as the C-space of all its mesh nodes. Free space is the intersection of the set of topology-preserving mesh configurations with the complement of the union of all D-obstacles that represent collisions of the part with finger bodies. Consider a given set of finger bodies in frictionless contact with a part. When positive work is needed to release the part, we say that it is in deform closure. We present numerical examples and prove two results: (1) if a contact set holds a rigid part in form closure, it will hold the equivalent deformable part in deform closure and (2) deform closure is frame invariant. We then consider frictionless deform closure grasps with two contact points. We define a measure of grasp quality based on balancing the potential energy needed to release the part against the potential energy that would result in plastic deformation. Given two jaw contacts at the perimeter nodes, we develop numerical algorithms to determine the optimal jaw separation based on this metric. For a part with n mesh nodes and p perimeter nodes, we give an algorithm that computes the optimal separation in time O(n3 p2 +p6 log p) and an approximation algorithm that runs in time O(n3 p2 +(p2 /ε)log p) .