Analytic Solution to Optimal Reconfigurations of Satellite Formation Flying in Circular Orbit under $J_2$ Perturbation

Abstract

This paper presents an analytic solution to the optimal reconfiguration problem of satellite formation flying in J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> orbital perturbation. Continuous and variable low-thrust accelerations are represented by the Fourier series, and initial and final boundary conditions are used to establish the constraints on the thrust functions. The thrust functions are implemented by optimal Fourier coefficients that minimize the cost during the maneuver. The analytic solution composed of these Fourier coefficients are simply represented in a closed form, and no approximation is needed. Numerical simulations are conducted to visualize and compare the results obtained in this paper with those of previous papers with no perturbations. The analytic solution developed in this paper is more accurate in that the general behavior of the optimal control history and reconfiguration trajectories are easily calculated even in the presence of the J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> potential disturbance. The analytic solution is useful for designing a reconfiguration controller for satellite formation flying under J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> orbital perturbation.