Bounded and Linear Quadratic Optimal Low-Thrust Collision Avoidance in Circular Orbits

This paper proposes two optimal-control-based algorithms for low-thrust collision avoidance. The maneuvering object’s nominal orbit is assumed to be circular, and only the position of the passive object relative to the nominal orbit at the conjunction is used. Therefore, the Clohessy-Wiltshire equations are used to model the relative dynamics. The first algorithm maximizes the weighted miss distance with a bounded thrust, whereas the second also minimizes the propellant consumption via a quadratic cost function on the weighted miss distance and the control effort. The two guidance laws are given in closed form and have only minor numerical components. The bounded guidance law only requires a numerical gradient-based optimization of two parameters, and the linear quadratic guidance law only requires solving for the roots of a 6-th-order polynomial. The two guidance laws require very low computational effort, making them suitable for the onboard implementation of small satellites. The guidance laws were evaluated in simulation and showed excellent evasion performance. Comparison between the two proposed guidance laws shows that for the same miss distance and maximum thrust, the minimum propellant law saves a significant amount of propellant, compensating for a slightly longer maneuver duration. Therefore, choosing which algorithm to use in a specific scenario depends on time and propellant considerations.