Robotics Research Technical Report (Classic Reprint)

Excerpt from Robotics Research Technical Report We consider the problem of computing the shortest path between two points in two or three-dimensional space bounded by polyhedral surfaces. In the 2-D case the problem is easily solved in time O(n log n).In the general 3-D case the problem is quite hard to solve, and is not even discrete; we present a doubly exponential procedure for solving the discrete subproblem of determining the sequence of boundary edges through which the shortest path passes. Finally we consider a favorable special case of the 3-D shortest path problem, namely that of finding the shortest path between two points along the surface of a convex polyhedron, and solve it in time O(n log n). The problem of finding the shortest path between two points in Euclidean space bounded by a finite collection of polyhedral obstacles is a special case of the more general problem of planning optimal collision-free paths for a given robot system (here we treat the robot as a single moving point). In two dimensional space the problem is easy to solve, because the shortest path between two given points must be a polygonal line whose vertices are comers of the given polygonal obstacles, so that the problem can be immediately reduced to a discrete graph searching, and can be solved in time 0(n log n), where n is the number of obstacle comers. This two-dimensional problem has been considered by Lozano-Perez and Wesley [Lw], and later also by Lee and Preparata [Lp]. In some special cases, considerably more efficient algorithms exist. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.