An Analytical Solution to the Problem of Optimal Control of the Reorientation of a Rigid Body (Spacecraft) Using Quaternions

Abstract

The dynamic problem of the rotation of a rigid body (for example, a spacecraft) from an arbitrary initial to the required final angular position in the presence of control restrictions is considered and solved. The end time of the maneuver is known. To optimize the rotation control program, a quadratic quality criterion is used, the minimized functional characterizes energy costs. The construction of optimal turn control is based on quaternion variables and the L. S. Pontryagin maximum principle. The features of optimal motion are studied in detail. Key properties of the optimal solution are formulated in an analytical form. It is shown that in the case of limited control, the moment of forces in the process of optimal rotation is parallel to a straight line that is stationary in inertial space, and during rotation of a rigid body (spacecraft) the direction of the kinetic moment is constant relative to the inertial coordinate system. Optimal control is presented in the form of synthesis—the synthesizing function is found and the dependence of the control variables on the phase coordinates is given. Formalized equations and calculation expressions are obtained to determine the optimal rotation program. The 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 solid body) is also described. An example and results of mathematical modeling of the motion of a spacecraft as a solid with optimal control are presented, demonstrating the practical feasibility of the developed method for controlling the spatial orientation of the spacecraft. For a dynamically symmetric solid, a complete solution of the reorientation problem in closed form is given, control variables and the optimal trajectory of motion as functions of time are presented in an analytical form.