Nonlinear dynamic modeling and optimal control of \(J_2\) perturbed spacecraft formation flying with periodic coefficients

Nonlinear time-varying system, such as a spacecraft formation flying system with chief spacecraft in elliptical orbit and under the effect of perturbation forces, is difficult to analyze, design, and control based on the presence of time-varying parameters. The proper functioning of aerospace systems and their ability to be able to achieve the designed mission objectives depend largely on proper understanding of their nonlinear time-varying nature, dynamics, and ability to keep them in the required mission operation configurations through high-fidelity optimal control strategy. This paper presents nonlinear dynamics and optimal control of \(J_2\) $J_2$ perturbed spacecraft formation flying. Via Euler–Lagrange approach, the nonlinear \(J_2\) $J_2$ perturbed motion dynamics was approximated into a time-varying nonlinear form, having periodic coefficients and time-varying parameters, suitable for designing fuel efficient control strategies, spacecraft formation flying, relative motion, and rendezvous mission analysis. Through the application of State-Dependent Riccati Equation (SDRE) approach, the approximated model was converted into a non-unique, pseudo-linear state-dependent coefficient (SDC) form. The numerical simulations confirmed that the SDRE controllers, developed using SDC parameterized systems, are maximally robust and able to return the system to the desired radial, along-track, and cross-track positions.