Omniwheel implementation of the Suslov problem with a rheonomic constraint: dynamic model and control

The classical Suslov problem of the motion of a rigid body with a fixed point is well known and has been studied in detail. In this paper, an omniwheel implementation of the Suslov problem is proposed. The controlled motion of a rigid body with a fixed point in the presence of scleronomic nonholonomic constraints and rheonomic artificial kinematic constraint is considered. The rigid body rotates around a fixed point, rolls around a spherical shell from the inside and contacts it by means of omniwheels with a differential actuator. We believe that the omniwheels are in contact with the spherical shell only at one point. In order to subordinate the motion of the rigid body to an artificial rheonomic constraint, a differential actuator creates control torques on omniwheels. Based on the d’Alembert–Lagrange principle, equations of motion of the mechanical system with indeterminate multipliers specifying constraint reactions are constructed. The problem is reduced to the study of a non-autonomous two-dimensional dynamical system. Using the generalized Poincar.e transformation, the study of a two-dimensional dynamical system is reduced to the study of the stability of a one-parameter family of fixed points for a system of differential equations with a degenerate linear part. We determine numerical parameters for which phase trajectories of the system are bounded and for which phase trajectories of the system are unbounded. The results of the study are illustrated graphically. Based on numerical integration, maps for the period (Poincar.e sections) and a map of dynamic regimes are constructed to confirm the Feigenbaum scenario of transition to chaotic dynamics.