Optimal Synthesis of the Zermelo–Markov–Dubins Problem in a Constant Drift Field
- 319 Downloads
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.
KeywordsMarkov–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.
- 2.Isaacs, R.: Games of pursuit. RAND Report P-257, RAND Corporation, Santa Monica, CA (1951) Google Scholar
- 5.Carathéodory, C.: Calculus of Variations and Partial Differential Equations of First Order, 3rd edn. Am. Math. Soc., Washington (1999) Google Scholar
- 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
- 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
- 16.Dolinskaya, I.S.: Optimal path finding in direction, location and time dependent environments. Ph.D. Thesis, The University of Michigan (2009) Google Scholar
- 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
- 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
- 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.Thomaschewski, B.: Dubins’ problem for the free terminal direction, pp. 1–14 (2001). Preprint Google Scholar