Application of Fundamental Modal Solutions to Relative Dynamics in the Cislunar Environment

The rising interest in cislunar space as a strategic environment for easier access to the Moon and the solar system has fostered the development of several future missions. In such an environment, close-proximity operations will require innovative trajectory design and path-planning techniques. Moving toward this goal, this paper introduces an original relative motion representation with respect to a periodic chief in the circular restricted three-body problem and an original relative trajectories design methodology based on fundamental modal solutions decomposition. Specifically, a relative motion model is developed in a velocity-based orbiting frame, in which the flight-path direction is considered to define one of its primary axes. Then, modal decomposition is applied to separate the fundamental modes of motion, showing that modal coefficients, eigenvectors, and eigenvalues can be employed for a geometrical characterization of relative dynamics. The resulting relative motion model is applied to describe the relative dynamics between a chaser spacecraft and a target satellite moving onto an L2 halo orbit, and its accuracy is assessed by comparisons against numerical integration of the three-body problem equations. In addition, the use of modal decomposition coefficients as geometrically insightful relative orbital elements for trajectory design and path planning is illustrated through various applicative examples.