Theoretical and experimental investigations of the controlled motion of a roller racer

Alexander A. Kilin^a, Tatiana B. Ivanova^a, Yury L. Karavaev^b, Kirill S. Yefremov<sup>b

a</sup> Ural Mathematical Center, Udmurt State University, Izhevsk, Russia
^b Kalashnikov Izhevsk State Technical University, Izhevsk, Russia

Abstract: In this paper, we address the problem of the controlled motion of a roller racer on a plane. We assume that the angle between the platforms is a given periodic function of time (control function), and the no-slip conditions (nonholonomic constraint) and viscous friction forces act at the points of contact of the wheels with the plane. In this case, all trajectories of the reduced system tend asymptotically to a periodic solution. In this paper, we show that for a selected periodic control function there exists a motion of the system that is bounded (along a circle) and unbounded (along a straight line). Unbounded motion corresponds to the resonant case which takes place at zero average value of the control function. The theoretical dependence of the trajectory and the velocity of the roller racer on its parameters and the parameters of the selected control function is investigated. These dependences are confirmed experimentally.

Keywords: roller racer, nonholonomic constraint, viscous friction, control, periodic solution.

MSC: 37J60, 70E60

Received: 03.12.2024
Accepted: 23.12.2024

Language: English

DOI: 10.2298/TAM241203010K