Trajectory Optimization for Low-Thrust Multi-Platform Servicing Missions in Elliptical Orbits: A Fourier-Based Approach

This paper presents a framework to optimize low-thrust missions involving a fleet of servicing spacecraft assigned to visit multiple client satellites in elliptical orbits. The problem of maximizing the number of visited client satellites is addressed under fuel capacity and mission duration constraints. To manage computational complexity, the overall problem is decomposed into two subproblems: trajectory cost assessment and sequence optimization. Trajectory costs are computed by deriving a distance metric that leverages a Fourier series representation of the control. Each possible orbit-to-orbit transfer cost—expressed in terms of ∆V and time of flight—is determined through a Fourier series expansion of the control accelerations. The optimal transfer costs are then obtained by solving a minimum-time trajectory optimization problem, where the Fourier coefficients are optimized via nonlinear programming. The trajectory cost assessment results are subsequently incorporated into a Mixed-Integer Linear Programming formulation to solve the combinatorial sequencing problem. The proposed methodology is validated through numerical simulations involving 42 client satellites from a high-eccentricity Molniya constellation. Various mission scenarios are analyzed, considering multiple servicing spacecraft equipped with different low-thrust propulsion systems. The novelty of this work lies in the introduction of a Fourier-based trajectory optimization framework integrated into a Mixed-Integer Linear Programming formulation, ensuring convergence to at least one globally optimal solution. Numerical results demonstrate the scalability and computational efficiency of the approach, highlighting its effectiveness in optimizing multi-platform servicing missions within practical mission constraints.