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.
KeywordsMotion Planning Time Slice Surface Patch Visibility Graph Active Line
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