The Window Corner Algorithm For Planning Translational Paths Among Polyhedral Subassemblies

In this paper, we present a complete and ex- act algorithm for planning a geometrically feasible path for a polyhedral subassembly (robot) translating amongst poly- hedral obstacles. Our algorithm can be used to determine whether a pair of non-convex subassemblies can be separated by a sequence of fine-motions and, possibly, contact-motions, when the amount of free space tends to be severly limited. We present the three-dimensional Window Corner algorithm to plan feasible paths. The algorithm solves the FeosiblePath problem, in worst-case time O(m5) where m is the product of the number of vertices describing the robot and obstacles respectively. A Feasiblepath solution hierarchy, for assembly sequence planning, is described for incremental, straight-line and multi-step paths. The Polyhedral Cone Representation (PCR) is introduced to efficiently represent the geometrical constraints on trans- lation and as a basis for fast collision avoidance checks. We introduce the concept of Window Corners in the PCR and this results in a finite search space. The Polyhedral Cone Obstacle Representation (PCUR), which is a constructive representation of all boundary contact configurations, trans- forms the problem into that of a point moving amongst a col- lection, O(m), of convex obstacles. In comparison, the worst- case space complexity of the C-space boundary is O(m3). Certain contact-constrained, feasible motions, which lie on surfaces without interior points can be inaccessible in a typ id C-space representation.