Optimal control, stability and numerical integration analysis of unicycle
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Optimal control, stability and numerical integration analysis of unicycle Archana Tiwari1 · K. C. Pati1 Received: 3 July 2020 / Revised: 28 October 2020 / Accepted: 30 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper addresses the problem of the study of controllability and stability analysis of an elementary but useful example of unicycle. We have considered a three dimensional control system of a unicycle. The control system on unicycle is related to a group, which has a Lie group structure. Here we show how concepts of differential geometry and Lie algebra can be elegantly applied to explain the behavior of such a system. Controllability and a minimum effort problem for the system are studied using well known theorems and necessary optimality conditions. The two unconventional integrators, namely Kahan and Lie–Trotter integrators have been implemented for numerical integration and the results are compared with the conventional Runge-Kutta integrator. Keywords Lie group · Lie algebra · Unicycle · Optimal control · Stability
1 Introduction A unicycle type robot is generally a robot moving in a 2 dimensional world, with some forward speed and zero instantaneous lateral motion. Contrary to its name, unicycle describes in general, carts or cars having two parallel driven wheels, one mounted of each side of their center. Unicycle model includes many known differential drive robots and approximate in several situations, even the four wheeled cars. Kinematic modeling describes the trajectories that the unicycle follows when subjected to commanding speeds and controls. The kinematic unicycle model is frequently used in path planning for ground vehicles, designing and modeling robots. For these reasons the unicycle dynamics with controls has received a great deal of interest in recent times [4,23]. The kinematic model of a unicycle can be represented by elements of the Special euclidean group S E(2), which has a Lie group structure. Optimal control problems on Lie groups were introduced by Roger Brocket [2,3]. Jurdjevic and Sussman [10,20] also studied the controllability on Lie groups. They emphasized on the use of Lie group structures to characterize controllability and existence of optimal controls.
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K. C. Pati [email protected] Department of Mathematics, National Institute of Technology, Rourkela, Odisha 769008, India
They also obtained many analytical results for the solution of certain types of optimal control problems which was not possible by ordinary state space method. Control system on matrix Lie group with applications to many physical problems were studied in [13,19]. Pop [17,18] discussed optimal control problem on the matrix Lie group S O(4). Lazureanu and Binzer [14] discussed the dynamical and geometrical properties of controls dynamic for a drift-free left invariant control system on G 4 , along with the integrability of the system. From a theoretical perspective, a specific class of optimal control problems [1] has a rich geometric structure and giv
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