Dynamics of Rubber Chaplygin Sphere under Periodic Control

Abstract

This paper examines the motion of a balanced spherical robot under the action of periodically changing moments of inertia and gyrostatic momentum. The system of equations of motion is constructed using the model of the rolling of a rubber body (without slipping and twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the equations of motion admit three invariant submanifolds corresponding to plane-parallel motion of the sphere with rotation about the minor, middle and major axes of inertia. The above-mentioned motions are quasi-periodic, and for the numerical estimate of their stability charts of the largest Lyapunov exponent and charts of stability are plotted versus the frequency and amplitude of the moments of inertia. It is shown that rotations about the minor and major axes of inertia can become unstable at sufficiently small amplitudes of the moments of inertia. In this case, the so-called “Arnol’d tongues” arise in the stability chart. Stabilization of the middle unstable axis of inertia turns out to be possible at sufficiently large amplitudes of the moments of inertia, when the middle axis of inertia becomes the minor axis for a part of a period. It is shown that the nonconservativeness of the system manifests itself in the occurrence of limit cycles, attracting tori and strange attractors in phase space. Numerical calculations show that strange attractors may arise through a cascade of period-doubling bifurcations or after a finite number of torus-doubling bifurcations.