Skip to main content
SpringerLink
Log in

Optimal navigation in planar time-varying flow: Zermelo’s problem revisited

  • Special Issue
  • Published:
Intelligent Service Robotics Aims and scope Submit manuscript

Abstract

This paper is concerned with time-optimal navigation for flight vehicles in a planar, time-varying flow-field, where the objective is to find the fastest trajectory between initial and final points. The primary contribution of the paper is the observation that in a point-symmetric flow, such as inside vortices or regions of eddie-driven upwelling/downwelling, the rate of the steering angle has to be equal to one-half of the instantaneous vertical vorticity. Consequently, if the vorticity is zero, then the steering angle is constant. The result can be applied to find the time-optimal trajectories in practical control problems, by reducing the infinite-dimensional continuous problem to numerical optimization involving at most two unknown scalar parameters.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acheson DJ (1990) Elementary fluid dynamics. Oxford University Press, Oxford

    MATH  Google Scholar 

  2. Bell DJ, Jacobson DH (1975) Singular optimal control problems. Academic Press, New York

    MATH  Google Scholar 

  3. Bellingham JG, Willcox JS (1996) Optimizing AUV oceanographic surveys. In: Proceedings of the 1996 symposium on autonomous underwater vehicle technology, 1996 (AUV ’96), June 1996, pp 391–398

  4. Bryson AE, Ho YC (1975) Applied optimal control: optimization, estimation and control. Hemisphere Publishing Corporation, Washington

    Google Scholar 

  5. Cliff EM, Seywald H, Bless RR (1993) Hodograph analysis in aircraft trajectory optimization. In: AIAA guidance, navigation and control conference, pp. 363–371, Monterey, CA, 9–11 August 1993

  6. Crassidis JL, Junkins JL (2004) Optimal estimation of dynamic systems. Chapman & Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

  7. Ebbinghaus HD, Peckhaus V (2007) Ernst Zermelo: an approach to his life and work. Springer, Berlin

    MATH  Google Scholar 

  8. Eichhorn M (2010) Solutions for practice-oriented requirements for optimal path planning for the AUV “Slocum Glider”. In: IEEE oceans 2010, September 2010

  9. Garau B, Alvarez A, Oliver G (2005) Path planning of autonomous underwater vehicles in current fields with complex spatial variability: an A* approach. In: Proceedings of the 2005 IEEE international conference on robotics and automation, 2005 (ICRA 2005). April 2005, pp 194–198

  10. Karamcheti K (1996) Principles of ideal-fluid aerodynamics. Wiley, New York

    Google Scholar 

  11. Kirk DE (1970) Optimal control theory: an introduction. Dover Pulications, Mineola, New York

    Google Scholar 

  12. Knauss JA (2005) Introduction to physical oceanography, 2 edn. Waveland Press, Long Grove

    Google Scholar 

  13. Leitman G (1981) The calculus of variations and optimal control. Plenum Press, New York

    Google Scholar 

  14. Morrow M, Woolsey CA, Hagerman G (2006) Exploring Titan with autonomous, buoyancy-driven gliders. J Br Interplanet Soc 59(1): 27–34

    Google Scholar 

  15. Pamadi B (2004) Performance, stability, dynamics, and control of airplanes, 2nd edn. AIAA Education Series, Reston

    Google Scholar 

  16. Petrich J, Woolsey CA, Stilwell DJ (2009) Planar flow model identification for improved navigation of small AUVs. Ocean Engineering 36(1):119–131. Autonomous Underwater Vehicles

    Google Scholar 

  17. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1963) The mathematical theory of optimal processes. Wiley, New York (translated by K. N. Trirogoff)

  18. Rugh WJ (1996) Linear system theory. Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

  19. Techy L, Woolsey CA (2009) Minimum-time path planning for unmanned aerial vehicles in steady uniform winds. AIAA J Guid Control Dyn 32(6): 1736–1746

    Article  Google Scholar 

  20. Techy L, Woolsey CA, Morgansen KA (2010) Planar path planning for flight vehicles in wind with turn rate and acceleration bounds. In: International conference on robotics & automation, Anchorage, AK, May 2010

  21. Vinh NX (1981) Optimal trajectories in atmospheric flight. Elsevier, New York

    Google Scholar 

  22. Zermelo E (1931) Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung. Zeitschrift für angewandte Mathematik und Mechanik 11(2): 114–124

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aeronautics and Astronautics, University of Washington, Washington, USA

    Laszlo Techy

Authors
  1. Laszlo Techy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laszlo Techy.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Techy, L. Optimal navigation in planar time-varying flow: Zermelo’s problem revisited. Intel Serv Robotics 4, 271–283 (2011). https://doi.org/10.1007/s11370-011-0092-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11370-011-0092-9

Keywords

Search

Navigation