Valerii Patsko, Andrey Fedotov, Three-dimensional reachable set for the Dubins car: Foundation of analytical description, Vol. 2022 (2022), Article ID 23, pp. 1-42

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DOI: 10.23952/cot.2022.23

Received November 21, 2021; Accepted September 17, 2022; Published November 11, 2022

Abstract. The Dubins car is a model of motion, in which the scalar control $u$ determines the instantaneous angular speed of rotation. The paper considers the symmetric variant of the constraints $u{\in}[u_1,u_2]$ , where $u_1=-u_2$ and $u_2=1$ . We study the three-dimensional reachable set at a given instant $t_f>0$ . An analytical description of two-dimensional sections of the set w.r.t. the angular coordinate $\varphi$ is given. The boundary of each $\varphi$ -section is formed with the help of a certain set of curves obtained using the Pontryagin maximum principle. This set includes circular arcs and circular involutes. The symmetry property of each $\varphi$ -section w.r.t. a certain straight line is established. Classification of possible types of the $\varphi$ -sections is proposed. The greatest difficulty is presented by analysis of the case with non-simply connected $\varphi$ -sections. The range of values $\varphi$ and $t_f$ , at which the $\varphi$ -sections are non-simply connected, is indicated.

How to Cite this Article:
V. Patsko, A. Fedotov, Three-dimensional reachable set for the Dubins car: Foundation of analytical description, Commun. Optim. Theory 2022 (2022) 23.