Closed-loop control of the motion of a sphere rolling on a moving horizontal plane
The motion of a sphere rolling on a plane without slipping is a typical example of a mechanical system with nonholonomic constraints. It can be modeled on the tangent bundle of the state space \(\mathbb{R}^2\times\text{SO}(3)\) corresponding to the position of the point of contact relative to some fixed origin on the plane and to the orientation of the sphere. Control problems associated to this system have been studied, for example, in a paper of \textit{V. Jurdjevic} [|Arch. Ration. Mech. Anal. 124, 305-328 (1993; Zbl 0809.70005)]. There the motion of the sphere was controlled through the motion of the plane, and the objective was to minimize a certain integral cost functional. In the paper at hand the authors study a prolongation of that control system: the motion of the sphere is controlled through the acceleration of the motion of the plane. And the objectives are different, too. The present authors consider a global tracking problem. The goal is to find a closed loop control law driving the system from an arbitrary initial configuration into a prescribed neighborhood of a particular type of equilibrium: the point of contact is the origin, the plane at rest, and the sphere is only spinning around the vertical axis. Such a control law is constructed as a full-information feedback law (all states and velocities available), and its performance is illustrated by two examples.
scientific article; zbMATH DE number 1069791
scientific article