The problem of controlling the rolling of a spherical robot with a pendulum actuator pursuing a moving target by the pursuit method, but with a minimal control, is considered. The mathematical model assumes the presence of a number of holonomic and nonholonomic constraints, as well as the presence of two servo-constraints containing a control function. The control function is defined in accordance with the features of the simulated scenario. Servo-constraints set the motion program. To implement the motion program, the pendulum actuator generates a control torque which is obtained from the joint solution of the equations of motion and derivatives of servo-constraints. The first and second components of the control torque vector are determined in a unique way, and the third component is determined from the condition of minimizing the square of the control torque. The system of equations of motion after reduction for a given control function is reduced to a nonautonomous system of six equations. A rigorous proof of the boundedness of the distance function between a spherical robot and a target moving at a bounded velocity is given. The cases where objects move in a straight line and along a curved trajectory are considered. Based on numerical integration, solutions are obtained, graphs of the desired mechanical parameters are plotted, and the trajectory of the target and the trajectory of the spherical robot are constructed.

The dynamics of a body with a fixed point, variable moments of inertia and internal rotors are considered. A stability analysis of permanent rotations and periodic solutions of the system is carried out. In some simplest cases the stability analysis is reduced to investigating the stability of the zero solution of Hill's equation. It is shown that by periodically changing the moments of inertia it is possible to stabilize unstable permanent rotations of the system. In addition, stable dynamical regimes can lose stability due to a parametric resonance. It is shown that, as the oscillation frequency of the moments of inertia increases, the dynamics of the system becomes close to an integrable one.

No abstract