Gripping parts at concave vertices

A simple gripper with two vertical cylindrical jaws can make contact with external or internal concavities in polygonal and polyhedral parts to align and grip parts in form closure. This is called a ν-grip. We begin by defining 2D ν-grips, where a pair of frictionless point jaws makes contact with a pair of polygonal part concavities to achieve form-closure. We define a ν-grip quality metric based on the maximum possible change in the part´s orientation when jaw position is relaxed infinitesimally. For a polygonal part with polygonal holes, we give an algorithm for computing and ranking 2D ν-grips. We also extend the definition to jaws with non-zero radii. In 3D, ν-grips are achieved with a pair of frictionless vertical cylinders. We define 3D ν-grips and give a numerical algorithm for computing all 3D ν-grips of a polyhedral part. If n is the number of vertices that describe the part and k is the number of concave vertices, we can compute all 2D ν-grips in O(n+k²) time. Measures of complexity are given for computing offsets for jaws with nonzero radii, a ranked list of 2D ν-grips based on the quality metric, and all 3D ν-grips. A Java implementation of the 2D algorithm is available.