Application of Lyapunov–Floquet transformation to the nonlinear spacecraft relative motion with periodic-coefficients

Abstract

The Lyapunov–Floquet (L–F) theory, a powerful result of Floquet theory, is one of the main tools for the stability analysis and study of nonlinear periodic systems. In this paper, Lyapunov–Floquet and modal transformations of the nonlinear approximation model of spacecraft relative motion, containing periodic state dependent coefficients (SDC), are developed. The transformations, advantageously, retain the stability characteristics of the original nonlinear relative motion dynamics. To provide a form to which averaging method can be applied, the nonlinear approximated equation of motion is scaled using the chief’s eccentricity. Through the application of L–F transformation, the linear part with periodic coefficients is reduced to time-invariant form and the transformed periodic nonlinear part has coefficients with true anomaly as the independent variable. The modal transformation reduced LF transformed equation into a form containing Jordan canonical form. Afterward, the averaging method is applied to the L–F transformed equation to describe the evolution of the motion in terms of the average values of the dynamical variables. Both the LF and modal transformed relative motion equations and the averaged equations, with chief in elliptical orbit, present forms that are useful and applicable for spacecraft guidance, navigation and control, formation flying, rendezvous and proximity operations.