Global stabilization of the spinning top with mass imbalance

Abstract

We consider the stabilization of a top with known imbalance to the sleeping motion. We first define the sleeping motion and show that it is a solution of the equations of motion of a balanced top. In the general case where the top is unbalanced, we derive two families of control laws that globally asymptotically stabilize a top with known imbalance to the sleeping motion using torque actuators. The input torque is produced by two body-fixed torque actuators in one case, and is confined to the inertial "XY-plane in the other. The control-design strategy is based on Hamilton-Jacobi-Bellman theory with zero dynamics. The result is global in the sense that the spinning top can be stabilized to the sleeping motion regardless of spin rate, and from an arbitrary initial motion that has a coning angle of up to 90°