Modal Solutions-based Approach to Relative Dynamics in the Cislunar Environment

The raising interest in the cislunar space as a strategic environment to facilitate the access to the surface of the Moon and as an outpost to begin Solar System exploration has fostered the development of several prospective missions in the Earth-Moon system. In particular, close-proximity operations (such as those involving the relative motion of a spacecraft with respect to a lunar station) require the development of specific tools for trajectory design and path-planning. This paper introduces an original relative motion representation with respect to a periodic chief in the circular restricted three-body problem and an original procedure to design proximity trajectories in the cislunar environment based on fundamental modal solutions decomposition. The relative motion model is first developed in a velocity-based orbiting frame, in which the velocity (i.e., flight-path) direction is one of the axes of the frame. Then, modal decomposition is applied to separate the fundamental modes of motion, showing that the modal coefficients, eigenvectors, and eigenvalues can be employed to characterize the geometry of the relative motion. In this way, the coefficients act as geometrically insightful relative orbital elements for trajectory design and path planning purposes.