V.S. Patsko, A.A. Fedotov. The structure of the reachable set for the Dubins car with a strictly one-sided turn ... P. 171-187

We study the structure of a three-dimensional reachable set “at instant” of the nonlinear control system often called the “Dubins car.” A controlled vehicle moves in the plane with constant speed and bounded turning radius. We consider the case where the object can turn to one side only and the rectilinear motion is forbidden by given control constraints. Based on the Pontryagin maximum principle, we obtain variants of controls leading to the boundary of the reachable set. Sections of the three-dimensional reachable set along the angular coordinate are considered. The boundaries of such sections are described analytically in the form of sets of smooth arcs. The paper lists all possible options for the structure of the sections. Each arc is defined by a certain type of piecewise constant control satisfying the maximum principle. The strict convexity of the sections along the angular coordinate is proved, and the smoothness of the boundary of the sections is analyzed.

Keywords: Dubins car, strictly one-sided turn, structure of a three-dimensional reachable set, Pontryagin maximum principle, piecewise constant control, strict convexity of sections of a reachable set along the angular coordinate

Received May 8, 2019

Revised July 22, 2019

Accepted August 5, 2019

Valerii Semenovich Patsko, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: patsko@imm.uran.ru

Andrei Anatol’evich Fedotov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: andreyfedotov@mail.ru

REFERENCES

1.   Dubins L.E. On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. American J. Math., 1957, vol. 79, no. 3, pp. 497–516. doi: 10.2307/2372560 

2.   Laumond J.-P. (ed.) Robot Motion Planning and Control. Lecture Notes in Control and Information Sciences, vol. 229. Berlin; Heidelberg: Springer-Verlag, 1998, 354 p. ISBN: 978-3-540-76219-5 .

3.   Patsko V.S., Pyatko S.G., Fedotov A.A. Three-dimensional reachability set for a nonlinear control system. J. Computer and Systems Sciences International, 2003, vol. 42, no. 3, pp. 320–328.

4.   Fedotov A., Patsko V., Turova V. Reachable sets for simple models of car motion. In A.V. Topalov (ed.) Recent Advances in Mobile Robotics, Rijeka, Croatia: InTech, 2011, pp 147–172. doi: 10.5772/26278 

5.   Patsko V.S., Fedotov A.A. Investigation of reachable set at instant for the Dubins’ car. Proc. 58th Israel Annual Conf. Aerospace Sci., Tel-Aviv&Haifa, Haifa, 2018, рр. 1655–1669. ISBN: 9781510851399.

6.   Choi H. Time-optimal paths for a Dubins car and Dubins airplane with a unidirectional turning constraint, Dissertation for the degree of Doctor of Philosophy, University of Michigan, Michigan, 2014, 134 p.

7.   Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The mathematical theory of optimal processes. N Y; London: John Wiley & Sons, 1962, 360 p. ISBN: 2-88124-077-1 . Original Russian text (2nd ed.) published in Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal’nykh protsessov, Moscow: Nauka Publ., 1969, 384 p.

8.   Lee E.B., Markus L. Foundations of optimal control theory. N Y; London; Sydney: John Wiley & Sons, 1967, 576 p. Translated to Russian under the title Osnovy teorii optimal’nogo upravleniya. Moscow: Nauka Publ., 1972, 576 p. ISBN: 0471522635 .

9.   Patsko V.S., Fedotov A.A. Attainability set at instant for one-side turning Dubins car. Proc. 17th IFAC Workshop Control Appl. Optim., Yekaterinburg, 2018. P. 201–206. doi: 10.1016/j.ifacol.2018.11.381 

Cite this article as: V.S. Patsko, A.A. Fedotov. The structure of the reachable set for the Dubins car with a strictly one-sided turn, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 3, pp. 171–187.