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

Romain Serra 1 Denis Arzelier 2 Florent Bréhard 1, 3 Mioara Joldes 1
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes [Toulouse]
2 LAAS-ROC - Équipe Recherche Opérationnelle, Optimisation Combinatoire et Contraintes
LAAS - Laboratoire d'analyse et d'architecture des systèmes [Toulouse]
3 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : 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.
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IAC 2018 - 69th International Astronautical Congress; IAF Astrodynamics Symposium, Oct 2018, Bremen, Germany. pp.1-9, 2018, CSA/IAF Special issue IAF Astrodynamics Symposium (69TH international astronautical congress). 〈https://www.iac2018.org/〉
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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, Oct 2018, Bremen, Germany. pp.1-9, 2018, CSA/IAF Special issue IAF Astrodynamics Symposium (69TH international astronautical congress). 〈https://www.iac2018.org/〉. 〈hal-01830253〉

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