Optimal Control of Rotation of a Rigid Body by a Movable Internal Mass

Abstract

The two-dimensional problem on the fastest turn of a rigid body by moving an internal mass is considered. It is assumed that the rigid body with an internal movable mass is a closed mechanical system. In the general case, equations for the trajectory of the motion of mass, optimal control, and the Bellman function are represented in terms of elliptic integrals that include two unknown variables. To determine these variables, two nonlinear scalar equations based on boundary conditions must be solved numerically. If the location of the mass at the terminal time is not specified, then the problem is reduced to solving a single scalar equation on the known interval, and the corresponding root always exists and is unique for arbitrary boundary conditions. In the special case when the internal mass is small compared to the mass of the rigid body, optimal trajectories in the form of circular arcs are obtained. The relation between approximate and exact solutions is investigated, and numerical examples of using these relations are discussed.