Control laws for minimum orbital changes - The satellite retrieval problem, part 2

On a number of occasions, upper stage failures have left satellites stranded in low Earth orbit. In order for a safe retrieval by the space shuttle, firing the engine may be necessary to remove unused propellants. If the satellite is already in a shuttle-type orbit, then the thrust must be controlled to minimize changes in semi-major axis, eccentricity, and inclination. The accepted strategy has been to vary the thrust direction to only modulate the inclination. This paper uses a set of blended extremal controls that minimizes changes in semi-major axis, eccentricity, and inclination. This requires the solution of a nonlinear programming problem using a set of optimization variables to blend the contributions of the extremal controls. A number of fully three-dimensional cases are examined and the solutions found do not include inclination modulation. A set of realistic thrust directions is presented along with the effects of the maneuver on the orbital elements.