Motion planning among time dependent obstacles

Abstract

In this paper we study the problem of motion planning in the presence of time dependent, i.e. moving, obstacles. More specifically, we will consider the problem: given a bodyB and a collection of moving obstacles inD-dimensional space decide whether there is a continuous, collision-free movement ofB from a given initial position to a target position subject to the condition thatB cannot move any faster than some fixed top-speedc. As a discrete, combinatorial model for the continuous, geometric motion planning problem we introduce time-dependent graphs. It is shown that a path existence problem in time-dependent graphs is PSPACE-complete. Using this result we will demonstrate that a version of the motion planning problem (where the obstacles are allowed to move periodically) is PSPACE-hard, even ifD=2, B is a square and the obstacles have only translational movement. ForD=1 it is shown that motion planning is NP-hard. Furthermore we introduce the notion of thec-hull of an obstacle: thec-hull is the collection of all positions in space-time at which a future collision with an obstacle cannot be avoided. In the low-dimensional situationD=1 andD=2 we develop polynomial-time algorithms for the computation of thec-hull as well as for the motion planning problem in the special case where the obstacles are polyhedral. In particular forD=1 there is aO(n lgn) time algorithm for the motion planning problem wheren is the number of edges of the obstacle.