Rolling motion dynamics of a spherical robot with a pendulum actuator controlled by the Bilimovich servo-constraint

We discuss the problem of rolling without slipping for a spherical shell with a pendulum actuator (spherical robot) installed in the geometric center of the sphere. The motion of the spherical robot is controlled by the Bilimovich servo-constraint. To implement the servo-constraint, the pendulum actuator creates a control torque. because the physical implementation of the Bilimovich constraint as a nonholonomic constraint is somewhat difficult, it can be implemented as a servo-constraint. Based on the general equations of motion, the kinematic constraints, and the servo-constraint, the equations of motion for this mechanical system are obtained. When the pendulum moves in the vertical plane at fixed levels of the first integrals, the resulting system of the equations of motion reduces to a non-Hamiltonian system with one degree of freedom. We find the conditions for the implementation of the motion program specified by the servo-constraint. The dynamics analysis is based on the study of phase portraits of the system, period maps, and plots of the desired mechanical parameters.