Preliminary study on relative motion and rendezvous between spacecraft in the restricted three-body problem

The focus of this paper is to present a preliminary study concerning relative motion and rendezvous in the restricted three-body problem. This paper presents full numerical simulations compared with linearized results of relative motion around well-known Lagrangian orbits such as halo orbits around Earth-Moon L2. Additionally, an initial linearization study is performed and presented to understand the general dynamics of such relative motion. Previous work on this topic relies on simplifications and assumptions that constrain the results to specific spatial domains and geometries. The reason to analyze such motion in linearized form as opposed to purely numerically integrate the equations of motion is to being able to study rendezvous and formation flying maneuvers around multiple families of Lagrangian orbits at once. Additionally, analytical and linearized analyses can provide important physical insight and help to quickly determine optimal solutions when searching through a large tradespace of orbital transfers and rendezvous maneuvers for both control-free and controlled dynamics. Future work is aimed to develop algorithms that, given a nominal Lagrangian orbit of interest, can describe the relative motion of two spacecraft that are operating "close" to each. Thus a more streamlined analytical work will be developed to compute which maneuvers are optimal to reduce Δv consumption, time of flight, or other parameters of interest.