Competitive Complexity of Mobile Robot On Line Motion Planning Problems

Abstract

This paper is concerned with on-line problems where a mobile robot of size D has to achieve a task in an unknown planar environment whose geometry is acquired during task execution. The critical parameter in such problems is physical motion time which corresponds to length or cost of the path traveled by the robot. The competitiveness of an on-line algorithm measures its performance relative to the optimal off line solution to the problem. While competitiveness usually means constant relative performance, this paper generalizes competitiveness to any functional relationship between on-line performance and optimal off-line solution. Given an on-line task, its competitive complexity class is a pair of lower and upper bounds on the competitive performance of all on-line algorithms for the task, such that the two bounds satisfy the same functional relationship. We classify some common online motion planning problems into competitive classes. In particular, navigation to a target whose position is either a priori known or recognized only upon arrival belongs to a quadratic competitive class. The hardest on-line problems belong to exponential and even non-boundable competitive classes. We present examples of such problems, which involve navigation in unknown variable traversibility environments.