Movement planning in the presence of flows

This paper investigates the problem of time-optimum movement planning in d = 2, 3 dimensions for a point robot which has bounded control velocity through a set of n polygonal regions of given translational flow velocities. This intriguing geometric problem has immediate applications to macro-scale motion planning for ships, submarines and airplanes in the presence of significant flows of water or air. Also, it is a central motion planning problem for many of the mesoscale and micro-scale robots that recently have been constructed, that have environments with significant flows that affect their movement. In spite of these applications, there is very little literature on this problem, and prior work provided neither an upper bound on its computational complexity nor even a decision algorithm. It can easily be seen that optimum path for the d = 2 dimensional version of this problem can consist of at least an exponential number of distinct segments through flow regions. We provide the first known computational complexity hardness result for the d = 3 dimensional version of this problem; we show the problem is PSPACE hard. We give the first known decision algorithm for the d = 2 dimensional problem, but this decision algorithm has very high complexity. We also give the first known efficient approximation algorithms with bounded error.

Duke Scholars

Author John H. Reif Computer Science