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


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.


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



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


  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

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA

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