Fuel-optimal impulsive fixed-time trajectories in the linearized circular restricted 3-body-problem

Romain Serra (1) , Denis Arzelier (2) , Florent Bréhard (1, 3, 4) , Mioara Joldes (1)

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
2 LAAS-ROC - Équipe Recherche Opérationnelle, Optimisation Combinatoire et Contraintes
3 ARIC - Arithmetic and Computing
4 PLUME - Preuves et Langages

The problem of fixed-time fuel-optimal trajectories with high-thrust propulsion in the vicinity of a Lagrange point is tackled via the linear version of the primer vector theory. More precisely, the proximity to a Lagrange point i.e. any equilibrium point-stable or not-in the circular restricted three-body problem allows for a linearization of the dynamics. Furthermore, it is assumed that the spacecraft has ungimbaled thrusters, leading to a formulation of the cost function with the 1-norm for space coordinates, even though a generalization exists for steerable thrust and the 2-norm. In this context, the primer vector theory gives necessary and sufficient optimality conditions for admissible solutions to two-value boundary problems. Similarly to the case of rendezvous in the restricted two-body problem, the in-plane and out-of-plane trajectories being uncoupled, they can be treated independently. As a matter of fact, the out-of-plane dynamics is simple enough for the optimal control problem to be solved analytically via this indirect approach. As for the in-plane dynamics, the primer vector solution of the so-called primal problem is derived by solving a hierarchy of linear programs, as proposed recently for the aforementioned rendezvous. The optimal thrusting strategy is then numerically obtained from the necessary and sufficient conditions. Finally, in-plane and out-of-plane control laws are combined to form the complete 3-D fuel-optimal solution. Results are compared to the direct approach that consists in working on a discrete set of times in order to perform optimization in finite dimension. Examples are provided near various Lagrange points in the Sun-Earth and Earth-Moon systems, hinting at the extensive span of possible applications of this technique in station-keeping as well as mission analysis, for instance when connecting manifolds to achieve escape or capture.

Domaines: Optimisation et contrôle [math.OC]; Automatique / Robotique

Romain Serra, Denis Arzelier, Florent Bréhard, Mioara Joldes. Fuel-optimal impulsive fixed-time trajectories in the linearized circular restricted 3-body-problem. IAC 2018 - 69th International Astronautical Congress; IAF Astrodynamics Symposium, International Astronautical Federation, Oct 2018, Bremen, Germany. pp.1-9. ⟨hal-01830253⟩