PERFORMANCE GUARANTEES FOR MOTION PLANNING WITH TEMPORAL UNCERTAINTY

Recently, Computational Geometry has contributed many new algorithms for planning the motion of a point mobile robot from its start to its target in a field of obstacles. A typical aim of such an algorithm is to find a shortest path using minimal computation time. Most algorithms assume that the motion planner has complete information, in other words, that the set of obstacles is known to the robot before it leaves the start point. The advantage of the Computational Geometry approach is that the performance of the motion planner may be rigorously analysed using mathematical methods. As such, this approach is popular with theorists who generally believe in the Grand Ideal of Computer Science, that software should be guaranteed by mathematical proof In practice, the physical constraints of robotics are myriad and some seem impossible to define precisely in mathematical terms; thus the methods of Computational Geometry have had limited applicability. From this point of view, it seems that the performance of motion planners must be measured by experience, and the Grand Ideal must be abandoned. In this paper we present recent research which represents a small step toward resolving this dilemma, by considering a robot with the physical constraint of temporal uncertainty. The inputs to the motion planner arrive sequentially, and the robot must act on one input before it receives the next. Specifically, the robot can only perceive an obstacle when it bumps into it. Of course such a robot cannot find a shortest path from its start to its target, and the usual shortest path algorithms from Computational Geometry are of little significance. We show that, however, this constraint does not preclude a rigorous mathematical analysis. Specifically, we present ''performance guarantees'' for such a robot in a simple environment: we measure the ratio p of the length of the robot's path to the length of the shortest path. The main results are upper and lower bounds for p. The results fall far short of considering all the constraints of a real robot. However, we believe that they show that the gap between the theoretical approach of Computational Geometry and realistic robotics can be reduced.