Optimal Reorientation of a Rigid Body (Spacecraft) with Limited Control Based on a Combined Quality Functional

A quaternion solution to the problem of optimal rotation of a rigid body (spacecraft) from an arbitrary initial to a designated angular position in the presence of constraints on the control variables is presented. To optimize the control process, a combined quality functional was used, which combines in a given proportion the sum of time and control efforts spent on the turn, and the integral of the kinetic energy of rotation during the turn. Based on the maximum principle of L.S. Pontryagin and quaternion models of controlled motion of a rigid body, a solution to the problem was obtained. The properties of optimal motion are disclosed in analytical form. Formalized equations and calculation formulas are recorded to build an optimal rotation program. Analytical equations and relations for finding optimal control are provided. Key relations determining the optimal values of the parameters of the rotation control algorithm are given. Also, a constructive scheme for solving the boundary-value problem of the maximum principle for arbitrary rotation conditions (initial and final positions and moments of inertia of a rigid body) is presented. For a dynamically symmetric rigid body, a solution to the reorientation problem in closed form is obtained. Numerical examples and the results of mathematical modeling are presented, confirming the practical feasibility of the developed method for controlling the attitude of a spacecraft.