Quaternion Solution of the Problem on Optimum Control of the Orientation of a Solid (Spacecraft) with a Combined Quality Criteria

The problem on optimal rotation of a solid (spacecraft) from an arbitrary initial to a prescribed final angular position in the presence of restrictions on the control variables is studied. The turnaround time is set. To optimize the rotation control program, a combined quality criterion that reflects energy costs is used. The minimized functional combines in a given proportion the integral of the rotational energy and the contribution of control forces to the maneuver. Based on the Pontryagin's maximum principle and quaternion models of controlled motion of a solid, an analytical solution of the problem has been obtained. The properties of optimal movement are revealed in analytical form. To construct an optimal rotation program, formalized equations and calculation formulas are written. Analytical equations and relations for finding optimal control are given. The key relations that determine the optimal values of the parameters of the rotation control algorithm are given. In addition, a constructive scheme for solving the boundary value problem of the maximum principle for arbitrary turning conditions (initial and final positions and moments of inertia of a solid) is described. For a dynamically symmetric solid, a closed-form solution for the reorientation problem is obtained. A numerical example and mathematical modeling results that confirm the practical feasibility of the developed method for controlling the orientation of a spacecraft are presented.