Models and motion planning

  • Mark de Berg
  • Matthew J. Katz
  • Mark Overmars
  • A. Frank van der Stappen
  • Jules Vleugels
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1432)


We study the consequences of two of the realistic input models proposed in the literature, namely unclutteredness and small simple-cover complexity, for the motion planning problem. We show that the complexity of the free space of a bounded-reach robot with f degrees of freedom is O(n f/2) in the plane, and O(n 2f/3) in three dimensions, for both uncluttered environments and environments of small simple-cover complexity. We also give an example showing that these bounds are tight in the worst case. Our bounds fit nicely between the θ(nf) bound for the maximum free-space complexity in the general case, and the θ(n) bound for low-density environments.


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  1. [1]
    Pankaj K. Agarwal, M. J. Katz, and M. Sharir. Computing depth orders and related problems. Comput. Geom. Theory Appl., 5:187–206, 1995.MathSciNetGoogle Scholar
  2. [2]
    H. Alt, R. Fleischer, M. Kaufmann, K. Mehlhorn, S. NÄher, S. Schirra, and C. Uhrig. Approximate motion planning and the complexity of the boundary of the union of simple geometric figures. Algorithmica, 8:391–406, 1992.CrossRefMathSciNetGoogle Scholar
  3. [3]
    J. M. Bañon. Implementation and extension of the ladder algorithm. In Proc. IEEE Internat. Conf. Robot. Autom., pages 1548–1553, 1990.Google Scholar
  4. [4]
    M. de Berg, M. J. Katz, A. F. van der Stappen, and J. Vleugels. Realistic input models for geometric algorithms. To appear; a preliminary version appeared in Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 294–303, 1997.Google Scholar
  5. [5]
    Mark de Berg. Linear size binary space partitions for fat objects. In Proc. 3rd Annu. European Sympos. Algorithms, volume 979 of Lecture Notes Comput. Sci., pages 252–263. Springer-Verlag, 1995.Google Scholar
  6. [6]
    A. Efrat and M. J. Katz. On the union of κ-curved objects. In Proc. 14th Annu. ACM Sympos. Comput. Geom., 1998. to appear.Google Scholar
  7. [7]
    A. Efrat, M. J. Katz, F. Nielsen, and M. Sharir. Dynamic data structures for fat objects and their applications. In Proc. 5th Workshop Algorithms Data Struct., pages 297–306, 1997.Google Scholar
  8. [8]
    M. J. Katz. 3-D vertical ray shooting and 2-D point enclosure, range searching, and arc shooting amidst convex fat objects. Comput. Geom. Theory Appl., 8:299–316, 1997.MATHGoogle Scholar
  9. [9]
    M. J. Katz, M. H. Overmars, and M. Sharir. Efficient hidden surface removal for objects with small union size. Comput. Geom. Theory Appl., 2:223–234, 1992.MathSciNetGoogle Scholar
  10. [10]
    J. Matoušek. Epsilon-nets and computational geometry. In J. Pach, editor, New Trends in Discrete and Computational Geometry, volume 10 of Algorithms and Combinatorics, pages 69–89. Springer-Verlag, 1993.Google Scholar
  11. [11]
    J. Matoušek, J. Pach, M. Sharir, S. Sifrony, and E. Welzl. Fat triangles determine linearly many holes. SIAM J. Comput., 23:154–169, 1994.CrossRefMathSciNetGoogle Scholar
  12. [12]
    J. S. B. Mitchell, D. M. Mount, and S. Suri. Query-sensitive ray shooting. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 359–368, 1994.Google Scholar
  13. [13]
    M. H. Overmars and A. F. van der Stappen. Range searching and point location among fat objects. J. Algorithms, 21:629–656, 1996.CrossRefMathSciNetGoogle Scholar
  14. [14]
    J. T. Schwartz and M. Sharir. On the “piano movers” problem I: the case of a two-dimensional rigid polygonal body moving amidst polygonal barriers. Commun. Pure Appl. Math., 36:345–398, 1983.MathSciNetGoogle Scholar
  15. [15]
    Otfried Schwarzkopf and Jules Vleugels. Range searching in low-density environments. Inform. Process. Lett., 60:121–127, 1996.CrossRefMathSciNetGoogle Scholar
  16. [16]
    A. F. van der Stappen, D. Halperin, and M. H. Overmars. The complexity of the free space for a robot moving amidst fat obstacles. Comput. Geom. Theory Appl., 3:353–373, 1993.Google Scholar
  17. [17]
    A. F. van der Stappen and M. H. Overmars. Motion planning amidst fat obstacles. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 31–40, 1994.Google Scholar
  18. [18]
    A. F. van der Stappen, M. H. Overmars, M. de Berg, and J. Vleugels. Motion planning in environments with low obstacle density. Report UU-CS-1997-19, Dept. Comput. Sci., Utrecht Univ., Utrecht, Netherlands, 1997.Google Scholar
  19. [19]
    Jules Vleugels. On Fatness and Fitness—Realistic Input Models for Geometric Algorithms. Ph.d. thesis, Dept. Comput. Sci., Utrecht Univ., Utrecht, the Netherlands, March 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Mark de Berg
    • 1
  • Matthew J. Katz
    • 2
  • Mark Overmars
    • 1
  • A. Frank van der Stappen
    • 1
  • Jules Vleugels
    • 1
  1. 1.Department of Computer ScienceUtrecht UniversityTB UtrechtThe Netherlands
  2. 2.Department of Mathematics and Computer ScienceBen-Gurion University of the NegevBeer-ShevaIsrael

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