Optimal Control of the Angular Momentum for a Solid (Spacecraft) Performing a Spatial Turn

A quaternion solution of the dynamic problem on the optimal turn of a solid (for example, a spacecraft) from a known initial to a given final angular position is presented. Optimization of the control program is carried out by using a combined indicator that combines a quadratic performance criterion and turn time; the minimized functional combines energy costs and maneuver duration in a given proportion. Based on the maximum principle, quaternion models, and methods for studying the controlled motion of a solid (spacecraft), a solution of the problem has been obtained. The construction of the optimal rotation is based on a differential equation relating the angular momentum and the orientation quaternion of a solid. The conditions of optimality are written in analytical form and the properties of the optimal motion are studied. Analytical equations and calculation formulas for finding the optimal control are presented. The control law is formulated as an explicit dependence of the control variables on the phase coordinates. Key relations that determine the optimal values of the parameters of the angular momentum control algorithm are given. In the case of a dynamically symmetric body, a complete solution of the turn problem in a closed form is obtained: analytical dependences as explicit functions of time for control variables and relations for calculating the parameters of the control law are given. A numerical example and the results of numeric simulation of the rotation of a spacecraft as a solid under optimal control that demonstrate the practical feasibility of the proposed control method are given.