Optimal Control of a Vibratory Robot Defined by a Choice of Angular Acceleration

Abstract

There is considered a vibratory robot, presented by a rigid box and a physical pendulum inside it. The robot moves along a horizontal plane in the gravity field. There is a Coulomb friction between the box and the surface. Control of the robot is defined by the choice of angular acceleration of the pendulum. In the paper there is considered a certain class of motion and a control low that insures sliding of the robot in the desired direction within all given restrictions. The control insures periodic motion of the robot with two phases: sliding phase, where the main body of the robot is moving in the desired direction, and resting phase, where the box of the robot is standing still. On the sliding phase on the maximum interval within the given restrictions there is no friction between the body and the plane. In the paper there is proved that such control law is optimal in terms of maximum velocity for a certain motion class. First, there is shown that for the case when the box is sliding, when the pendulum is in one half of the period it is best to insure its maximum acceleration, and in the second part the angular acceleration should be at its minimum value. Then there is shown that for the motion within all given restrictions when there are set certain starting conditions and ending conditions on phase variables then the considered control law insures maximum average velocity of the robot. After that there is shown that there should be set conditions on phase variables for only one position of pendulum on the period. Lastly there is proved that for a certain angle there is needed to define only the speed of the box, optimum value of angular velocity of the pendulum appears to be as desired.