A Direct Minimization Approach for Obtaining the Distance between Convex Polyhedra

Abstract

Computational methods used for the automatic generation of robot paths must be fully developed if truly automated manu facturing systems are to become a reality. An important requirement for determining feasible robot paths is the ability to compute the distance between the various elements of the robot and the workspace fixtures, jigs, and machinery. In this research, it is assumed that the robot and workspace solid geometry are represented as a collection of convex polyhe drons, and an efficient numerical algorithm for determining the minimum distance between two such polyhedrons is presented. In addition to determining the minimum distance between solids, the algorithm can also be used to efficiently ascertain whether a collision has occurred. The numerical technique presented uses a sequence of constrained minimizations to obtain the closest three-dimen sional points on any two solid objects. Computational effi ciency is achieved with the algorithm presented because only the collection of planes (points and normals) which define the solids are used for the analysis. The bounding lines and vertices do not need to be explicitly calculated during the minimization procedure. The algorithm uses a direct ap proach for minimizing the nonlinear distance function which generates a sequence of search directions along the surfaces of the objects to obtain the global minimum. An extensive set of numerical tests are used to demonstrate the performance of the algorithm.