PROPERTIES OF SOLUTIONS IN THE DUBINS CAR CONTROL PROBLEM

Artem A. Zimovets     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)

Abstract


TThis paper addresses the time-optimal control problem of the Dubins car, which is closely related to the problem of constructing the shortest curve with bounded curvature between two points in a plane. This connection allows researchers to apply both geometric methods and control theory techniques during their investigations. It is established that the time-optimal control for the Dubins car is a piecewise constant function with no more than two switchings. This characteristic enables the categorization of all such controls into several types, facilitating the examination of the solutions to the control problem for each type individually. The paper derives explicit formulas for determining the switching times of the control signal. In each case, necessary and sufficient conditions for the existence of solutions are obtained. For certain control types, the uniqueness of optimal solutions is established. Additionally, the dependence of the movement time on the initial and terminal conditions is studied.

Keywords


Dubins car, Dubins problem, Time-optimal control, Curve with bounded curvature

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References


  1. Ayala J., Rubinstein H. The classification of homotopy classes of bounded curvature paths. Isr. J. Math., 2016. Vol. 213. P. 79–107. DOI: 10.1007/s11856-016-1321-x
  2. Balkcom D., Furtuna A., Wang W. The Dubins car and other arm-like mobile robots. In: Proc. 2018 IEEE Int. Conf. on Robot. and Aut. (ICRA), Brisbane, QLD, Australia, 2018. P. 380–386. DOI: 10.1109/ICRA.2018.8461017
  3. Berdyshev Iu.I. Time-optimal control synthesis for a fourth-order nonlinear system. J. Appl. Math. Mech., 1975. Vol. 39, No. 6. P. 948–956. DOI: 10.1016/0021-8928(76)90081-2
  4. Boissonnat J.-D., Cerezo A., Leblond J. A note on shortest paths in the plane subject to a constraint on the derivative of the curvature. In: INRIA Research Report. RR-2160. 1994. URL: inria-00074512
  5. Boissonnat J.-D., Cérézo A., Leblond J. Shortest paths of bounded curvature in the plane. J. Intell. Robot. Syst., 1994. Vol. 11. P. 5–20. DOI: 10.1007/BF01258291
  6. Buzikov M.E., Galyaev A.A. Minimum-time lateral interception of a moving target by a Dubins car. Automatica, 2022. Vol. 135. Art. no. 109968. DOI: 10.1016/j.automatica.2021.109968
  7. Dubins L.E. On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. Amer. J. Math., 1957. Vol. 79, No. 3. P. 497–516. DOI: 10.2307/2372560
  8. Johnson H.H. An application of the maximum principle to the geometry of plane curves. Proc. Amer. Math. Soc., 1974. Vol. 44, No. 2. P. 432–435. DOI: 10.1090/S0002-9939-1974-0348631-6
  9. Kaya C.Y. Markov–Dubins path via optimal control theory. Comput. Optim. Appl., 2017. Vol. 68. P. 719–747. DOI: 10.1007/s10589-017-9923-8
  10. Khabarov S.P., Shilkina M.L. Geometric approach to the solution of the Dubins car problem in the formation of program trajectories. Nauch.-Tehn. Vest. Inf. Tekh., Mekh. i Opt., 2021. Vol. 21, No. 5. P. 653–663. DOI: 10.17586/2226-1494-2021-21-5-653-663 (in Russian)
  11. Markov A.A. A few examples of solving special problems on the largest and smallest values. Soobshch. Kharkov. Mat. Obshch., 1889. Ser. 2, vol. 1, No. 2. P. 250–276. (in Russian)
  12. McGee T.G., Hedrick J.K. Optimal path planning with a kinematic airplane model. J. Guid. Cont. Dyn., 2007. Vol. 30, No. 2. P. 629–633. DOI: 10.2514/1.25042
  13. Patsko V.S., Pyatko S.G., Fedotov A.A. Three-dimensional reachability set for a nonlinear control system. J. Comp. Syst. Sci. Int., 2003. Vol. 42, No. 3. P. 320–328.
  14. Patsko V.S., Fedotov A.A. Three-dimensional reachable set for the Dubins car: Foundation of analytical description. Commun. Optim. Theory, 2022. Vol. 2022. P. 1–42. DOI: 10.23952/cot.2022.23
  15. Pecsvaradi T. Optimal horizontal guidance law for aircraft in the terminal area. IEEE Trans. Automatic Control, 1972. Vol. 17, No. 6. P. 763–772. DOI: 10.1109/TAC.1972.1100160
  16. Reeds J.A., Shepp L.A. Optimal paths for a car that goes both forwards and backwards. Pacific J. Math., 1990. Vol. 145, No. 2. P. 367–393. DOI: 10.2140/pjm.1990.145.367
  17. Shkel A.M., Lumelsky V.J. Classification of the Dubins set. Robotics Auton. Syst., 2001. Vol. 34, No. 4. P. 179–202. DOI: 10.1016/S0921-8890(00)00127-5
  18. Sussmann H.J. Shortest 3-dimensional paths with a prescribed curvature bound. In: Proc. of 1995 34th IEEE Conf. on Decision and Control, 1995. Vol. 4. P. 3306–3312. DOI: 10.1109/CDC.1995.478997
  19. Sussmann H.J., Tang G. Shortest paths for the Reeds–Shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control. In: Rutgers Center for Syst. and Cont. Technical Report 91–10, 1991. URL: SYCON-91-10




DOI: http://dx.doi.org/10.15826/umj.2024.2.014

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