Rolling of spherical bodies with periodic controls

When constructing various spherical robots, it occurs the problem of a mathematical description of dynamics and prediction of the motion under controls. In particular, the corresponding equations can be derived within the framework of the standard model of nonholonomic rolling (no slipping) and the rubber rolling model (no slipping and twisting). This report is devoted to the study of such models.
The report will be considered mathematical models of rolling on the plane of a balanced and unbalanced spherical bodies with periodically changing moments of inertia and gyrostatic momentum. The purpose of this work is to study the stability of some particular motions, which can be further implemented when constructing some standard maneuvers in full-scale models of spherical robots.
From earlier studies of the inertial motion of a balanced «rubber» sphere, it is known that the motions with rotation around the major or minor axis of inertia are stable, and the motion with rotation around the middle axis of inertia is unstable. The research presented in the report showed that the plane-parallel motion corresponding to the middle axis of inertia can become stable in the case of periodically changing the moments of inertia. In addition, it is shown that there are frequencies (resonant) of changing the moments of inertia at which the stable plane-parallel motions can be destabilized.
A numerical estimation of the stability of plane-parallel motions shows that their stability is neutral. Nevertheless, in the phase space of the system under consideration, various asymptotic regimes of motion can also occur: limit cycles, attracting tori, and strange attractors. The emergence of such regimes of motion in robotics is undesirable; therefore, it is important to understand the mechanism and conditions of their occurrence. In particular, it is shown that for the system considered, strange attractors can occur as a result of a cascade of period doubling bifurcations or a finite number of torus doubling bifurcations.
For an unbalanced «rubber» sphere, it is indicated conditions for the existence of invariant manifolds. The motion on these manifolds is plane-parallel and is described by Hamiltonian equations. Using Melnikov’s method, it is shown that in the case of periodic controls the chaotic dynamics can occur on the invariant manifold. The stability of the equilibrium positions of the system is investigated depending on the parameters of the control actions.