Models and motion planning

  • Authors:

  • Mark de Berg;Matthew J. Katz;Mark H. Overmars;A. Frank van der Stappen;Jules Vleugels

  • Affiliations:

  • Institute of Information and Computing Sciences, Utrecht University, P.O. Box 80089, 3508 TB Utrecht, The Netherlands;Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel;Institute of Information and Computing Sciences, Utrecht University, P.O. Box 80089, 3508 TB Utrecht, The Netherlands;Institute of Information and Computing Sciences, Utrecht University, P.O. Box 80089, 3508 TB Utrecht, The Netherlands;Institute of Information and Computing Sciences, Utrecht University, P.O. Box 80089, 3508 TB Utrecht, The Netherlands

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2002

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Abstract

We study the complexity of the motion planning problem for a bounded-reach robot in the situation where the n obstacles in its workspace satisfy two of the realistic models proposed in the literature, namely unclutteredness and small simple-cover complexity. We show that the maximum complexity of the free space of a robot with f degrees of freedom in the plane is Θ (nf/2 + n) for uncluttered environments as well as environments with small simple-cover complexity. The maximum complexity of the free space of a robot moving in a three-dimensional uncluttered environment is Θ(n2f/3 + n). All these bounds fit nicely between the Θ(n) bound for the maximum free-space complexity for low-density environments and the Θ(nf) bound for unrestricted environments. Surprisingly--because contrary to the situation in the plane---the maximum free-space complexity is Θ(nf) for a three-dimensional environment with small simple-cover complexity.