Optimal Spacecraft Engagements with Terminal Velocity Constraints

Satellites play a significant role in various daily activities, such as communication, navigation, scientific research, and intelligence gathering. As the number of satellites in space and the complexity of their missions continue to increase rapidly, it becomes necessary to refuel, repair, and protect them. These missions involve on-orbit engagements, in which one spacecraft maneuvers into close proximity with another. The effectiveness of these missions highly depends on the terminal relative velocity of the engagement, and therefore, it should be taken into consideration.
We propose two analytical, closed-loop, minimum-fuel-consumption, linear-quadratic spacecraft engagement guidance laws with terminal velocity constraints, based on the Clohessy-Wiltshire equations. The first is an optimal guidance law, in which it is assumed that the target spacecraft does not maneuver, and the terminal relative velocity is optimally selected to be either speed-constrained or direction-constrained. The second is a differential games guidance law, in which the target spacecraft optimally evades, and the terminal speed is constrained by the selection of the cost function’s weighting matrix. A conjugate point analysis of the game reveals that both positive and non-positive definite weighting matrix solutions can be selected, depending on the required terminal speed.
The performance of the two guidance laws is evaluated and compared in simulations. It will be shown that both laws achieve accurate terminal conditions, and that the choice of guidance law depends on the engagement scenario: when the target follows a predictable trajectory, the optimal guidance law is more appropriate, whereas the differential games guidance law is better suited against evading targets.