Application of Complex Functional of Quality in Optimal Control of Spacecraft Motion

Abstract

The problem of optimal control of the reorientation of a spacecraft as a solid body from an arbitrary initial ‎position into a prescribed final angular position is considered and solved. The case is ‎studied in detail when the ‎minimized index combines, in a given proportion, the integral of modulus of angular momentum and duration of ‎maneuver. It is proved that the accepted optimality ‎criterion guarantees the motion of a spacecraft with modulus of ‎angular momentum not exceeding the required value. Formalized equations and expressions for the synthesis of ‎the optimal rotation program are obtained using quaternion models. It is shown that the optimal solution corresponds to the strategy “acceleration - rotation with constant modulus of angular momentum-‎braking”, the ‎angular momentum and the controling moment are perpendicular during optimal rotation between acceleration and ‎braking. On the basis of necessary optimality conditions, the ‎main properties, laws, and key characteristics ‎‎(parameters, constants, integrals of motion) of the ‎optimal solution of the control problem, including the turn time ‎and the maximum angular momentum for the optimal motion, are determined. An estimation of the influence of the ‎bounded ‎controling moment on the character of the optimal motion and on the indicators of quality is ‎made. The ‎construction of an optimal control program of rotation is based on the quaternion variables and Pontryagin’s ‎maximum principle. The value of maximal angular momentum magnitude is calculated by condition of ‎transversality. The designed method is universal and invariant ‎relative to the moments of inertia. For dynamically ‎symmetric spacecraft, a complete solution of ‎the reorientation problem in closed form is presented. An example ‎and results of mathematical ‎modeling of the motion of a spacecraft under optimal control are presented, ‎demonstrating the ‎practical feasibility of the method for controlling spacecraft's spatial orientation.