Autonomous Small Satellite Trajectory Correction under Third Body Perturbation using Pseudo-Spectral Method

This paper presents an integrated methodology for autonomous orbit maintenance of small satellites in highly elliptical orbits subject to strong third body gravitational perturbations. A pseudo-spectral collocation scheme with Legendre–Gauss–Lobatto nodes transcribes the nonlinear equations of motion into algebraic form, solved using a Newton–Raphson procedure with Broyden rank-one updates to generate optimal low-thrust profiles with onboard computation times in the order of 0.01 s per orbit. Thrust execution is realized through Pulse Amplitude Modulation (PAM) of a Pulsed Plasma Thruster (PPT), enabling continuouscontrol- like behaviour while respecting the discrete firing nature of electric propulsion hardware. The methodology is demonstrated on the Venus–Sun–satellite system, where solar gravity dominates the dynamics of a 200 × 66,000 km highly elliptical orbit around Venus. Periapsis altitude is maintained within 0.25 km of the nominal 200 km over 365 orbits and sustained over a 10-year, 3,650-orbit simulation, with a fuel consumption rate of 2.1541 milligrams per orbit, giving a 25-fold propellant saving relative to conventional discrete impulsive maintenance. Sensitivity analysis quantifies the impact of collocation point count, PAM pulse width, and thrust quantization, while Monte Carlo simulations confirm robustness under parameter and model uncertainties. The proposed framework offers a fuel-efficient, computationally tractable, and practically implementable solution for unmanned small satellite orbit maintenance in complex multi-body gravitational environments.