Particle swarm optimization of cooperative and competitive relative orbit maneuvers

Minimum-time relative trajectories between two orbiting spacecraft represent a subject of great relevance in astrodynamics, also in consideration of the related applications to formation flying and proximity maneuvers involving two or more space vehicles. This work uses the Hill-Clohessy-Wiltshire linear equations of motion to model the spacecraft dynamics, and employs a Hamiltonian approach for the determination of optimal relative trajectories. Specifically, the orbital interception problem involving two maneuvering vehicles is considered. If both of them intend to minimize the time until interception, then the problem can be formulated as an optimal control problem. In the presence of a pursuing vehicle and an evading target, which aims at delaying interception indefinitely, then the problem is formulated as a zero-sum differential game. In the latter case, the optimal trajectories of the two space vehicles correspond to a saddle point equilibrium solution. The necessary conditions either for an optimal solution or for an equilibrium solution are derived for the two cases of cooperating and competitive spacecraft, respectively. These conditions allow translating the original problems into parameter optimization problems with a fairly reduced parameter set, mainly composed of the unknown initial values of the adjoint variables. A basic version of particle swarm algorithm is then employed, and proves effective and accurate in finding the optimal values of the unknown parameters, thus identifying the optimal trajectories and the related control laws, either when the two spacecraft cooperate or when they are involved in a competitive scenario