Three-Dimensional Shortest Path Planning in the Presence of Polyhedral Obstacles

Abstract

A new algorithm is presented for solving the three-dimensional shortest path planning (3DSP) problem for a point object moving among convex polyhedral obstacles. It is the first non-approximate three-dimensional path planing algorithm that can deal with more than two polyhedral obstacles. The algorithm extends the visibility graph concept from two dimensions to three dimensions. The two main problems with 3DSP are identifying the edge sequence the shortest path passes through and the turning points of the shortest path. A technique based on projective relationships is presented for identifying the set of visible boundary edges (VBE) corresponding to a given view point over which the shortest path, from the view point to the goal, will pass. VBE are used to construct an initial reduced visibility graph (RVG). Optimization is used to revise the position of the turning points and hence the three-dimensional RVG (3DRVG) and the global shortest path is then selected from the 3DRVG. The algorithm is of computational complexity O(n3vk), where n is the number of verticles, v is the maximum number of vertices on any one obstacle and k is the number of obstacles. The algorithm is applicable only with polyhedral obstacles, as the theorems developed for searching for the turning points of the three-dimensional shortest path are based on straight edges of the obstacles. It needs to be further developed for dealing with arbitrary-shaped obstacles and this would increase the computational complexity. The algorithm is tested using computer simulations and some results are presented.