Linear-Programming Solutions for Orbital-Transfer Trajectories

This paper describes how the relatively efficient computational techniques of linear programming can be used to obtain near-minimum fuel solutions to the problem of controlling spacecraft midcourse or terminal homing trajectories. These “orbital-transfer” maneuvers are required (a) to bring a spacecraft to a prespecified position, velocity, and time state so that its subsequent free-fall trajectory follows a desired path, or (b) to match the position and velocity of another orbiting spacecraft in order to achieve a rendezvous or station-keeping condition. Modern optimal control theory has provided means by which the necessary (and in some cases, sufficient) conditions for the optimal control function for this problem can be derived. However, each method leads to subsidiary computational requirements that have proved troublesome in practice. In view of the computational demands of optimal control theory, it appears worthwhile to study how the computational power of linear programming can be brought to bear. Therefore, this paper applies linear programming (LP) methods to the terminal guidance of an orbiting spacecraft. Typical trajectories based on linearized equations of motion are calculated. Under the assumptions made, the LP solutions minimize fuel consumption. Attention is also called to current research that is beginning to provide a more profound interconnection between optimal control theory and the mathematical programming techniques familiar to operations research and management science.