Formation Control in CR3BP Using Toroidal State Elements from an Atlas of Invariant Tori

RAMIREZ P. 1, CAPANNOLO A. 1

1 Purdue University, West Lafayette, United States

Formation flight in the Circular Restricted Three–Body Problem (CR3BP) presents significant engineering challenges due to the chaotic nature of the dynamical environment and the absence of analytical orbital elements capable of fully characterizing each agent’s motion. Quasi–periodic tori act as invariant sets that define bounded regions of motion, providing a natural geometric structure within which spacecraft can remain confined and evolve coherently over long durations. However, long–duration formation maintenance and reconfiguration remain challenging when performed directly in Cartesian coordinates, where geometric drift, inaccurate phase evolution, and the inability to constrain trajectories to quasi–periodic motion lead to inconsistent performance. These challenges motivate the use of coordinate systems that directly reflect the geometry of the invariant sets governing the local dynamics. In this context, several geometric parameterizations have been explored. Linearized local toroidal coordinates provide an approximate, oscillatory representation of motion near a periodic orbit and have been applied to relative–motion modeling and control, but their validity is restricted to small neighborhoods and cannot maintain accuracy over large excursions. Action–angle coordinates constructed from high–order normal forms offer a canonical, integrable approximation of the dynamics in the libration point vicinity and preserve phase information over long horizons, albeit with limited domain of convergence. More broadly, existing control approaches do not ensure that controlled trajectories remain on, or sufficiently near, a chosen family of quasi–periodic tori, even though the presence of a secondary frequency enables organized oscillatory motion around a baseline periodic structure. In light of the above, this work introduces an optimal control framework formulated directly in a regional toroidal state element set built from a precomputed atlas of quasi–halo invariant tori. The atlas is generated through invariant–tori family continuation, in which each torus is parameterized by two angles and a secondary frequency while the primary frequency matches the reference periodic orbit. The database spans the full two–dimensional angle domain and a finite but dense range of rotation numbers, supporting two oscillatory modes and enabling robust representation of continuous quasi–periodic motion. A spline–based interpolation method maps the regional toroidal coordinates to the six–dimensional Cartesian state, providing efficient state reconstruction during optimization. The optimal control framework can explicitly enforce that trajectories remain within the domain of the generated tori atlas, where the interpolation holds. Within this framework, an optimal trajectory–optimization formulation is posed directly in toroidal state elements, where the dynamics evolve linearly in time. The resulting problem minimizes low–thrust actuation while enforcing initial and terminal conditions, formation–geometry requirements, thrust bounds, dynamic feasibility, and consistency with the admissible atlas domain. For comparison, parallel trajectory–optimization formulations are also implemented using linearized local toroidal coordinates and action–angle coordinates derived from a Lie–transform normal form. Preliminary results indicate that the three–dimensional toroidal framework enables reliable optimal control for long–duration formation maintenance and reconfiguration when trajectories remain within the frequency range spanned by the atlas. In this coordinate system, the dynamics are linear and time–invariant, which simplifies the trajectory–optimization problem while accurately capturing the two independent oscillatory motions that define quasi–periodic behavior. The optimized toroidal trajectories map back to Cartesian space with high fidelity, closely matching the true nonlinear motion. Unlike Cartesian formulations, the toroidal state directly parameterizes motion on invariant tori, preserving phase coherence and preventing long–term geometric drift, which makes it a natural and effective framework for formation–flying trajectory optimization in the CR3BP.