# Restricted Quadratic Optimal Control of a Spacecraft Turning in a Fixed Time Period

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### Abstract

A dynamic problem of a spacecraft (SC) turning from an arbitrary initial position to the required terminal angular position with restricted control is considered and solved. The termination time of the maneuver is known. The control program is optimized using the quadratic criterion of quality; the minimized functional characterizes the energy consumed in the turn. The construction of the optimal control of the turn is based on quaternion variables and Pontryagin’s maximum principle. It is shown that during the optimal turn, the moment of force is parallel to the straight line fixed in the inertial space, and the direction of the angular momentum in the process of the SC’s rotation is constant relative to the inertial coordinate system. The formalized equations and computational expressions for determining the optimal turning program are obtained. A special mode of control is studied in detail, and the conditions of the impossibility of occurrence of this mode are formulated. The control algorithms allow performing turns over a fixed time period with the minimum angular kinetic energy. A comprehensive solution to the control problem is given for a dynamically symmetric SC: the dependences as explicit functions of time for the control variables and relations for calculating the key parameters of the turn maneuver’s control law are obtained. A numerical example and the results of the mathematical modeling of the SC’s motion with the optimal control are given, which demonstrate the practical feasibility of the method for controlling the attitude of the SC.

## Notes

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