Modeling and analysis of nonlinear spacecraft relative motion via harmonic balance and Lyapunov function

摘要

The harmonic balance method, an approximate method for the study of nonlinear oscillating systems described by ordinary nonlinear differential equations, is applied to the cubic polynomial approximation of the relative motion to develop approximate periodic solutions of the motion. Firstly, the cubic nonlinear terms neglected in the formulation of linear time invariant Clohessy-Wilshire (CW) equations are obtained from the original nonlinear equation of motion. Secondly, using the CW solutions as the generating solutions, harmonic balance technique is applied to the cubic nonlinear terms and correction terms are obtained. The new and improved harmonically linearized model contains CW equations and the correction terms. Using total energy as a Lyapunov function, the stability of the new model is determined. The additional terms in the new harmonically linearized spacecraft relative motion enabled us to obtain excellent approximate periodic solutions of the nonlinear equations as evidenced from the numerical simulation results. (C) 2020 Elsevier Masson SAS. All rights reserved.