Semi-Analytical Optimal Formation Reconfiguration via Data-Driven Lyapunov-Floquet Transformation and Differential-Algebra Expansio濢

Time-varying periodic systems are prevalent in orbital dynamics, such as the relative motions in elliptical orbits or near libration point orbits, and their solution is fundamental for spacecraft formation design and control. The Lyapunov-Floquet (LF) theorem is commonly employed to convert such time-varying systems into linear time-invariant ones, thereby revealing intrinsic motion characteristics. However, the associated transformation matrix lacks a generalized analytical form, making it difficult to obtain when the dynamics change. This paper presents a data-driven deep learning framework to uncover the generalized form of the LF transformation, together with an optimization approach for low-thrust formation reconfiguration on elliptical orbits based on the LF transformation and differential algebra (DA) expansion. An encoder-decoder neural network, constrained by periodicity and reconstruction accuracy, is trained on simulated trajectories to identify the transformation matrix without analytical derivations. Once obtained, geometric configuration invariants describe relative motion as linear combinations of essential components. An indirect optimization method is then formulated in the invariant space, where DA expansions transform the boundary-value problem into a linear inverse mapping, eliminating repeated numerical integration. Numerical results demonstrate that the proposed method accurately discovers the LF transformation in the absence of precise dynamics and achieves energy-optimal reconfiguration with high precision. Compared with traditional approaches, the DA-based optimization reduces computational cost by over 90%, highlighting the robustness and efficiency of combining machine learning with analytical techniques for spacecraft formation reconfiguration in complex dynamical regimes.