Cislunar Relative Dynamics in the Elliptic Restricted Three-Body Problem Using Fundamental Modal Solutions

In the framework of the newfound interest in the cislunar environment and the related growing interest in formation flying and proximity operations, this paper focuses on the application of the method of fundamental modal solutions to represent relative dynamics with respect to periodic orbits in the elliptic restricted three-body problem (ER3BP). Based on linearized relative motion equations, the method allows representing relative dynamics using the coefficients of the modal decomposition as constants of motion. The validity of the model is first assessed by comparison with the numerical integration of the non-linear dynamics in the ER3BP. Since the coefficients can be related to the geometrical properties of motion, they can be employed to ease trajectory design; therefore, applicative examples are provided for trajectories that are relevant to proximity operations. The proposed design methodologies are ultimately applied to define initial and final states in the state-space of the coefficients for finite-burn maneuvers. The modal decomposition representation of relative dynamics is employed as dynamics constraint in the formulation of a non-linear optimization problem, involving piece-wise constant control action and including the time of flight of the maneuver among optimization variables.