Spacecraft Collision Avoidance with Constrained Control via Discrete-Time Generating Functions

This study presents sub-optimal collision-free transfers of spacecraft subject to constraints on control magnitude. In order to mitigate the difficulty in solving an optimal control problem considering directly inequality constraints, the penalty and barrier functions are incorporated into the cost function of optimal tracking problem. Then, the sub-optimal control law is derived by employing the discrete-time generating functions representing the canonical transformation in the discrete-time Hamilton-Jacobi theory. The proposed approach allows us to derive the control law as an algebraic form of the states of spacecraft, reference solution, and obstacles without any iterative process and initial guess. The numerical simulations validate the proposed approach by showing that spacecraft can reach the target point while avoiding obstacles with constrained control.