Path Planning in an Anisotropic Medium

Many of the optimal path finding problems studied to-date are restricted to a direction-independent metric. In this paper we discuss path planning in an anisotropic medium illustrated by the fastest-path problem where speed is direction-dependent. Such problems arise in vessel routing, robotics, and aircraft navigation, where the agent’s speed is affected by the direction of waves, winds or slope of the terrain. The difficulty of optimal-path finding in a direction-dependent medium comes from the fact that our travel-time function is asymmetric, and in general, violates the triangle inequality. We present an analytical form solution for the fastest-path finding problem in an obstacle-free domain without making any assumptions on the structure of the speed function. Subsequently, we merge these results with visibility graph search methods to develop an obstacle-avoiding fastest-path finding algorithm for a direction-dependent speed function. Our results provide computationally fast techniques for finding a closed form solution to a very large class of applied problems.