Skip to main content

Movement Planning in the Presence of Flows

  • Conference paper
  • First Online:
Algorithms and Data Structures (WADS 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2125))

Included in the following conference series:

  • 875 Accesses

Abstract

This paper investigates the problem of time-optimum movement planning in d = 2,3 dimensions for a point robot which has bounded control velocity through a set of n polygonal regions of given translational flow velocities. This intriguing geometric problem has immediate applications to macro-scale motion planning for ships, submarines and airplanes in the presence of significant flows of water or air. Also, it is a central motion planning problem for many of the mesoscale and micro-scale robots that recently have been constructed, that have environments with significant flows that affect their movement. In spite of these applications, there is very little literature on this problem, and prior work provided neither an upper bound on its computational complexity nor even a decision algorithm. It can easily be seen that optimum path for the d = 2 dimensional version of this problem can consist of at least an exponential number of distinct segments through flow regions. We provide the first known computational complexity hardness result for the d = 3 dimensional version of this problem; we show the problem is PSPACE hard. We give the first known decision algorithm for the d = 2 dimensional problem, but this decision algorithm has very high complexity. We also give the first known efficient approximation algorithms with bounded error.

Supported by NSF ITR EIA-0086015, NSF-IRI-9619647, NSF CCR-9725021, SEGR Award NSF-11S-01-94604, Office of Naval Research Contract N00014-99-1-0406.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. L. Aleksandrov, M. Lanthier, A. Maheshwari, and J.-R. Sack. An ε-approximation algorithm for weighted shortest paths on polyhedral surfaces. Lecture Notes in Computer Science, 1432:11–22, 1998.

    Google Scholar 

  2. L. Aleksandrov, A. Maheshwari, and J.-R. Sack. Approximation algorithms for geometric shortest path problems. In Proceedings of the 32nd ACM Symposium on Theory of Computing, 2000.

    Google Scholar 

  3. J. Canny. Some algebraic and geometric computations in PSPACE. In Proceedings of the 20th Annual ACM Symposium on the Theory of Computing, 1988.

    Google Scholar 

  4. J. Canny and J. Reif. New lower bound techniques for robot motion planning problems. In Proceedings of the 28th IEEE Annual Symposium on Foundations of Computer Science, 1987.

    Google Scholar 

  5. G. E. Collins. Quantifier elimination for real closed fields by cylindric algebraic decomposition. In Proc. Second GI Conference on Automata Theory and Formal Languages, volume 33 of Lecture Notes in Computer Science, 1975.

    Google Scholar 

  6. M. Lanthier, A. Maheshwari, and J. Sack. Approximating weighted shortest paths on polyhedral surfaces. In 6th Annual Video Review of Computational Geometry, Proc. 13th ACM Symp. Computational Geometry, 1997.

    Google Scholar 

  7. C. Mata and J. Mitchell. A new algorithm for computing shortest paths in weighted planar subdivisions. In Proceedings of the 13th ACM International Annual Symposium on Computational Geometry (SCG-97), 1997.

    Google Scholar 

  8. J. S. B. Mitchell. Geometric shortest paths and network optimization. In J.R. Sack and J. Urrutia, editors, Handbook of Computational Geometry. Elsevier Science Publishers B.V. North-Holland, Amsterdam, 1998.

    Google Scholar 

  9. J. S. B. Mitchell and C. H. Papadimitriou. The weighted region problem: Finding shortest paths through a weighted planar subdivision. Journal of the ACM, 38(1):18–73, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  10. N. Papadakis and A. Perakis. Minimal time vessel routing in a time-dependent environment. Transportation Science, 23(4):266–276, 1989.

    MATH  MathSciNet  Google Scholar 

  11. N. Papadakis and A. Perakis. Deterministic minimal time vessel routing. Operations Research, 38(3):426–438, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. Reif. Complexity of the mover’s problem and generalizations. In Proceedings of the 20th IEEE Symposium on Foundations of Computer Science, 1979.

    Google Scholar 

  13. J. Reif and M. Sharir. Motion planning in the presence of moving obstacles. In 26th Annual Symposium on Foundations of Computer Science, 1985.

    Google Scholar 

  14. J. H. Reif and J. A. Storer. A single-exponential upper bound for finding shortest paths in three dimensions. Journal of the ACM, 41(5):1013–1019, Sept. 1994.

    Google Scholar 

  15. J. Reif and Z. Sun. BUSHWHACK: An Approximation Algorithm for Minimal Paths Through Pseudo-Euclidean Spaces. Submitted for publication.

    Google Scholar 

  16. J. Reif and Z. Sun. An efficient approximation algorithm for weighted region shortest path problem. In Proceedings of the 4th Workshop on Algorithmic Foundations of Robotics, 2000.

    Google Scholar 

  17. J. T. Schwartz and M. Sharir. On the piano movers problem: II. general techniques for computing topological properties of real algebraic manifolds. Advances in applied mathematics, 4:298–351, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Sellen. Direction weighted shortest path plannin. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA-95), 1995.

    Google Scholar 

  19. G. Wilfong. Motion planning in the presence of movable obstacles. In Proceedings of the Fourth ACM Annual Symposium on Computational Geometry, 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Reif, J., Sun, Z. (2001). Movement Planning in the Presence of Flows. In: Dehne, F., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 2001. Lecture Notes in Computer Science, vol 2125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44634-6_41

Download citation

  • DOI: https://doi.org/10.1007/3-540-44634-6_41

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42423-9

  • Online ISBN: 978-3-540-44634-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics