A Constellation-Volume-Preserved Approach for Formation Flying Design near Triangular Libration Points

This paper investigates spacecraft formation flight dynamics near Earth-Moon libration points within the Circular Restricted Three-Body Problem, focusing on phase-space conservation and contraction. A Hamiltonian formulation combined with Liouville’s theorem is employed to analyze the structure-preserving nature of natural motion. For triangular libration points, linearized equations analogous to the Clohessy-Wiltshire model are derived, allowing relative trajectories to be explicitly parameterized by initial conditions. This facilitates systematic formation design through adjustments in amplitude and phase. The analysis shows that the triangular points exhibit stable center-center dynamics, and rigid formations can be achieved by appropriately selecting short- and long-period modal components. Short-period orbits are identified as particularly advantageous, as they minimize distance variability, a result supported by optimization and long-term simulations. Furthermore, Liouville’s theorem ensures that deviations evolve on a bounded hypersphere in phase space, providing a theoretical guarantee of long-term bounded motion. For collinear libration points, where the dynamics possess a saddle-center structure, a passivity-based control framework using interconnection and damping assignment is proposed. By introducing damping selectively into the hyperbolic modes while preserving the oscillatory ones, the closed-loop system is reshaped into a port-Hamiltonian form that guarantees contraction of phase-space volume. Numerical simulations confirm that this method suppresses divergence, stabilizes unstable directions, and maintains conservative oscillatory behavior, thereby enabling bounded relative motion in formations. This paper combines structure-preserving analysis with energy-based control, offering both theoretical insights and practical guidelines for applications in long-term observation constellations, interferometry, and cooperative deep-space missions.