Fuel-Optimal In-Track Satellite Formation Trajectory with J2 Perturbation Using Pontryagin Neural Networks

Satellite formation flying faces significant challenges in maintaining its desired configurations due to various orbital perturbations, particularly in low-Earth-orbit environments. This paper presents a novel approach to generating fuel-optimal reference trajectories for in-track satellite formations by incorporating both the Earth’s oblateness (J2 perturbation) and the inherent nonlinearity of the two-body problem. The resulting indirect optimal control problem is solved using Pontryagin Neural Networks (PoNNs). The proposed method transforms the conventional two-point boundary value problem into a mathematical programming problem, enabling the efficient computation of optimal trajectories. The effectiveness of our approach is validated through extensive numerical simulations at different inclinations of the chief satellite (0–90°) and cross-track separation distances (1–400 km), demonstrating significant reductions in annual fuel consumption compared to conventional approaches. The feasibility of these optimal trajectories is verified through closed-loop simulations using a PD controller, confirming their practical applicability in realistic mission scenarios. This research contributes to enhancing the long-term sustainability of satellite formation flying missions by optimizing fuel efficiency while maintaining precise formations.