High-order analytical solutions of bounded relative motions for Coulomb formation flying

Close-proximity Coulomb formation flying offers attractive prospects in multiple astronautical missions. A semi-analytical method of constructing the series solution up to an arbitrary order for the relative motion near the equilibrium of Coulomb formation systems is proposed to facilitate the design of Coulomb formations based on a Lindstedt–Poincaré method. The details of the series expansion and coefficient solution for arbitrary m:n -period orbits are discussed. To verify the effectiveness of the proposed method, the practical convergence domain of the high-order series solution is computed by comparing it with corresponding numerical solutions that satisfy the specified boundary conditions. Simulation results demonstrate the efficacy of the Lindstedt–Poincaré method in constructing series solutions for the relative motion of Coulomb formation systems.