LQR Design For Formation Flying Near Halo Orbits Exploiting Quasi-Periodic Symmetry In Toroidal Coordinates

Abstract

This work presents the design of a Linear Quadratic Regulator for relative motion control near periodic orbits in the Circular Restricted Three-Body Problem (CR3BP), formulated in non-singular toroidal coordinates. The key result exploits the rotational quasi-periodicity of the toroidal coordinate transformation. A uniqueness argument on the stabilizing solution of the associated difference Riccati equation proves the optimal control gains need only be computed for a single orbital period. The controller is validated against the full non-linear CR3BP dynamics on an L1 Northern halo orbit in the Earth-Moon system, and further against a high-fidelity ephemeris model, demonstrating successful reconfiguration maneuvers on the invariant torus with low control effort. Compared with an impulsive targeting and station-keeping baseline, the proposed controller attains a comparable tracking accuracy while reducing the total control effort by approximately 40%.