Two Ways to Control a Pendulum-Type Spherical Robot on a Moving Platform in a Pursuit Problem

Abstract

We consider the problem of controlling a spherical robot with a pendulum actuator rolling on a platform that is capable of moving translationally in the horizontal plane of absolute space. The spherical robot is subject to holonomic and nonholonomic constraints. Some point target moves at the level of the geometric center of the spherical robot and does not touch the moving platform itself. The motion program that allows the spherical robot to pursue a target is specified through two servo-constraints. The robot can follow a target from any position and with any initial conditions. Two ways to control this system in absolute space are proposed: by controlling the forced motion of the platform (the pendulum oscillates freely) and by controlling the torque of the pendulum (the platform is stationary or oscillates inconsistently with the spherical robot). The equations of motion of the system are constructed. In the case of free oscillations of the pendulum, the system of equations of motion has first integrals and, if necessary, can be reduced to a fixed level of these integrals. When a spherical robot moves in a straight line, for a system reduced to the level of integrals, phase curves, graphs of the distance from the geometric center of the spherical robot to the target, the trajectory of the selected platform point when controlling the platform, and the square of the control torque when controlling the pendulum actuator are constructed. When the robot moves along a curved path, integration is carried out in the original variables. Graphs of the squares of the angular velocity of the pendulum and the spherical robot itself are constructed, as well as the trajectory of the robot’s motion in absolute space and on a moving platform. Numerical experiments were performed in the Maple software package.