Advertisement

Journal of Optimization Theory and Applications

, Volume 156, Issue 2, pp 469–492 | Cite as

Optimal Synthesis of the Zermelo–Markov–Dubins Problem in a Constant Drift Field

  • Efstathios Bakolas
  • Panagiotis Tsiotras
Article

Abstract

We consider the optimal synthesis of the Zermelo–Markov–Dubins problem, that is, the problem of steering a vehicle with the kinematics of the Isaacs–Dubins car in minimum time in the presence of a drift field. By using standard optimal control tools, we characterize the family of control sequences that are sufficient for complete controllability and necessary for optimality for the special case of a constant field. Furthermore, we present a semianalytic scheme for the characterization of an optimal synthesis of the minimum-time problem. Finally, we establish a direct correspondence between the optimal syntheses of the Markov–Dubins and the Zermelo–Markov–Dubins problems by means of a discontinuous mapping.

Keywords

Markov–Dubins problem Optimal synthesis Zermelo’s navigation problem Non-holonomic systems 

Notes

Acknowledgements

This work has been supported in part by NASA (award no. NNX08AB94A). E. Bakolas also acknowledges support from the A. Onassis Public Benefit Foundation.

References

  1. 1.
    Dubins, L.E.: On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Am. J. Math. 79(3), 497–516 (1957) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Isaacs, R.: Games of pursuit. RAND Report P-257, RAND Corporation, Santa Monica, CA (1951) Google Scholar
  3. 3.
    Patsko, V.S., Turova, V.L.: Numerical study of the “homicidal chauffeur” differential game with the reinforced pursuer. Game Theory Appl. 12(8), 123–152 (2007) MathSciNetGoogle Scholar
  4. 4.
    Zermelo, E.: Über das Navigationproble bei ruhender oder veränderlicher Windverteilung. Z. Angew. Math. Mech. 11(2), 114–124 (1931) CrossRefGoogle Scholar
  5. 5.
    Carathéodory, C.: Calculus of Variations and Partial Differential Equations of First Order, 3rd edn. Am. Math. Soc., Washington (1999) Google Scholar
  6. 6.
    Sussmann, H.J.: The Markov–Dubins problem with angular acceleration control. In: Proceedings of 36th IEEE Conference on Decision and Control, San Diego, CA, pp. 2639–2643 (1997) CrossRefGoogle Scholar
  7. 7.
    Bakolas, E., Tsiotras, P.: Optimal synthesis of the asymmetric sinistral/dextral Markov–Dubins problem. J. Optim. Theory Appl. 150(2), 233–250 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Reeds, A.J., Shepp, R.A.: Optimal paths for a car that goes both forward and backwards. Pac. J. Math. 145(2), 367–393 (1990) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sussmann, H.J.: Shortest 3-dimensional path with a prescribed curvature bound. In: Proceedings of 34th IEEE Conference on Decision and Control, New Orleans, LA, pp. 3306–3312 (1995) Google Scholar
  10. 10.
    Vendittelli, M., Laumond, J.-P., Nissoux, C.: Obstacle distance for car-like robots. IEEE Trans. Robot. Autom. 15(4), 678–691 (1999) CrossRefGoogle Scholar
  11. 11.
    Chitour, Y., Sigalotti, M.: Dubins’ problem on surfaces. I. Nonnegative curvature. J. Geom. Anal. 15(4), 565–587 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Sigalotti, M., Chitour, Y.: On the controllability of the Dubins’ problem for surfaces. II. Negative curvature. SIAM J. Control Optim. 45(2), 457–482 (2006) MathSciNetCrossRefGoogle Scholar
  13. 13.
    McGee, T.G., Hedrick, J.K.: Optimal path planning with a kinematic airplane model. J. Guid. Control Dyn. 30(2), 629–633 (2007) CrossRefGoogle Scholar
  14. 14.
    Bakolas, E., Tsiotras, P.: On the generation of nearly optimal, planar paths of bounded curvature. In: Proceedings of the 2009 American Control Conference, St. Louis, MO, June 10–12, 2010, pp. 385–390 (2010) Google Scholar
  15. 15.
    Giordano, P.L., Vendittelli, M.: Shortest paths to obstacles for a polygonal Dubins car. IEEE Trans. Robot. 25(5), 1184–1191 (2009) CrossRefGoogle Scholar
  16. 16.
    Dolinskaya, I.S.: Optimal path finding in direction, location and time dependent environments. Ph.D. Thesis, The University of Michigan (2009) Google Scholar
  17. 17.
    Techy, L., Woolsey, C.A.: Minimum-time path-planning for unmanned aerial vehicles in steady uniform winds. J. Guid. Control Dyn. 32(6), 1736–1746 (2009) CrossRefGoogle Scholar
  18. 18.
    Bakolas, E., Tsiotras, P.: Time-optimal synthesis for the Zermelo–Markov–Dubins problem: the constant wind case. In: Proceedings of the 2010 American Control Conference, Baltimore, MD, June 30–July 2, 2010, p. 6163 (2010) Google Scholar
  19. 19.
    Glizer, V.Y.: Optimal planar interception with fixed end conditions: a closed-form solution. J. Optim. Theory Appl. 88(3), 503–539 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Glizer, V.Y.: Optimal planar interception with fixed end conditions: approximate solutions. J. Optim. Theory Appl. 93(1), 1–25 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Pecsvaradi, T.: Optimal horizontal guidance law for aircraft in the terminal area. IEEE Trans. Autom. Control 17(6), 763–772 (1972) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Bui, X.-N., Boissonnat, J.D.: The shortest paths synthesis for nonholonomic robots moving forwards. Research Note 2153. Institut National de Recherche en Informatique et en Automatique, Sophia-Antipolis, France (1993) Google Scholar
  23. 23.
    Patsko, V.S., Pyatko, S.G., Fedotov, A.A.: Three-dimensional reachability set for a nonlinear control system. J. Comput. Syst. Sci. Int. 42(3), 320–328 (2003) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Cesari, M.: Optimization—Theory and Applications. Problems with Ordinary Differential Equations. Springer, New York (1983) zbMATHGoogle Scholar
  25. 25.
    Sussmann, H.J., Tang, G.: Shortest paths for the Reeds-Shepp car: a worked out example of the use of geometric tecnhiques in nonlinear optimal control. Research Note SYCON-91-10, Rutgers University, New Brunswick, NJ (1991) Google Scholar
  26. 26.
    Thomaschewski, B.: Dubins’ problem for the free terminal direction, pp. 1–14 (2001). Preprint Google Scholar
  27. 27.
    Jurdjevic, V.: Geometric Control Theory. Cambridge University Press, New York (1997) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations