Minimum-Fuel Finite-Thrust Relative Orbit Maneuvers via Indirect Heuristic Method

Fuel-optimal space trajectories in the Euler–Hill frame represent a subject of great relevance in astrodynamics, in consideration of the related applications to formation flying and proximity maneuvers involving two or more spacecraft. This research is based upon employing a Hamiltonian approach to determining minimum-fuel trajectories of specified duration. The necessary conditions for optimality (that is, the Pontryagin minimum principle and the Euler–Lagrange equations) are derived for the problem at hand. A switching function is also defined, and it determines the optimal sequence and durations of thrust and coast arcs. The analytical nature of the adjoint variables conjugate to the dynamics equations leads to establishing useful properties of these trajectories, such as the maximum number of thrust arcs in a single orbital period and a remarkable symmetry property, which holds in the presence of certain boundary conditions. Furthermore, the necessary conditions allow translating the optimal control problem into a parameter optimization problem with a fairly small parameter set composed of the unknown initial values of the adjoint variables. A simple swarming algorithm is chosen among the different available heuristic techniques as the numerical solving algorithm, with the intent of finding the optimal values of the unknown parameters. Five examples illustrate the effectiveness and numerical accuracy of the indirect heuristic method applied to optimizing orbital maneuvers in the Euler–Hill frame.