Exterior penalty indirect method for collision avoidance among multiple satellites

Abstract

This paper considers the collision avoidance problem among multiple continuous-thrust satellites considering the pursuit-evasion game. First, the problem is formulated as an optimal control problem subject to high-fidelity dynamics and pure state constraints for the collision avoidance between evaders and pursuers. Analytical derivations reveal that the propellant-optimal collision avoidance maneuvers are of bang-bang control structure, and the number and sequence of the constrained arcs must be pre-identified, as each constrained arc introduces at least four more unknowns. To solve this, an exterior penalty indirect method is proposed, incorporating the constraints into the performance index with an activation function. A two-point boundary-value problem is then established and solved by the shooting method, where the constrained arcs are automatically identified without the need for additional shooting variables. By gradually increasing the penalty coefficient, the optimal solution is efficiently obtained starting with the solution neglecting collision. The proposed method is validated through three numerical examples, showing a significant reduction in computational time—cutting it from 2 hours to just 1 minute compared to the widely-used GPOPS. The results demonstrate excellent agreement with GPOPS and previous studies, with relative errors in propellant consumption under 1.0%. Additionally, it is shown that cooperative collision avoidance can reduce fuel consumption by 2.4%, while the presence of pursuers can significantly increase fuel consumption, potentially by several times.