Indirect optimization of low-thrust collision avoidance trajectories in LEO with bang-bang control laws

The ever-increasing density of objects in low Earth orbit (LEO) has raised concerns about the potential for a future cascade of collisions, a phenomenon known as the Kessler Syndrome. Such collisions could inflate the amount of space debris, posing significant risks to operational satellites and future space missions. In response to this, and in alignment with new de-orbit regulations, there is a shift towards refining collision avoidance strategies to minimize propellant consumption. Indeed, despite the relatively low fuel requirements of individual collision avoidance maneuvers, their growing necessity places a considerable strain on satellite fuel budgets. In this context, the objective of this research is to develop a tool that uses indirect optimization to identify propellant-efficient collision avoidance trajectories between two orbiting objects. To this end, the indirect approach is employed to exploit the theory of optimal control, applied to spacecraft trajectories, and to transform the optimization problem into a boundary value problem, which is subsequently solved by means of shooting procedures. This method ensures precise optimization and offers significant advantages in terms of computational costs, particularly when low-thrust is considered. The study uses a modified version of a collision probability calculation method originally developed by Alfano and Negron. This method assesses the risk of collision between two objects in a high-fidelity LEO dynamic model. Meanwhile, trajectory optimization is performed using a simpler two-body dynamic model. This approach ensures robust convergence of the optimization method and maintains dynamic accuracy, due to the low maneuvering times involved. Moreover, this work offers valuable theoretical insights into the indirect optimization method and illustrative examples that facilitate a more comprehensive grasp of the method's practical implementation.