Approximate Analytical Solutions to Optimal Reconfiguration Problems in Perturbed Satellite Relative Motion

This paper presents new analytical solutions to the optimal reconfiguration problems of formation flying using a power-limited low-thrust propulsion that includes perturbations associated with differential gravity effects, the eccentricity effects of a chief orbit, and J 2 effects. The perturbation approach and the calculus of variations are applied to optimal reconfiguration problems to obtain an approximate analytical solution. Some nonlinear relative dynamics are also presented to account for the nonlinear terms generated by each perturbation. For the sake of validity, numerical simulations are performed for satellite reconfiguration cases in which the distance between the satellites is large and the chief orbit is eccentric. The improved capacity of the new analytical solutions to ensure reasonable fuel usage is confirmed, and the magnitudes of the errors caused by each perturbation are compared and analyzed. It is found that the errors caused by unperturbed analytical solutions based on only the Hill―Clohessy― Wilshire equations are reduced by the new analytical solutions by one-half to one over 500, depending on which perturbation is considered.