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Motion planning among time dependent obstacles

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Summary

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.

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Authors and Affiliations

  1. Stevens Institute of Technology, 07030, Hoboken, NJ, USA

    K. Sutner

  2. University of Illinois at Chicago, 60638, Chicago, IL, USA

    W. Maass

Authors
  1. K. Sutner
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  2. W. Maass
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Additional information

During the preparation of this paper the second author has been supported in part by NSF grant DCR-85-04247

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Sutner, K., Maass, W. Motion planning among time dependent obstacles. Acta Informatica 26, 93–122 (1988). https://doi.org/10.1007/BF02915447

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