High-order analytical solutions of the equations of relative motion with elliptic reference orbit

Abstract

The study of the dynamics of relative motion plays an important role in the application of formation flying. For a more accurate configuration, much less fuel consumption is required for station-keeping. Traditionally, the configuration is designed based on the periodic solutions of the linear equation of relative motion called C–W equation for circular reference orbit or Lawden equation for elliptic reference orbit. However, the linear solutions are suitable for the configuration with small amplitudes. In this paper, based on the nonlinear equations of relative motion with elliptic reference orbit, the solution is expanded as formal series of the eccentricity of the reference orbit, in-plane amplitude and out-of-plane amplitude, then,taking the Lawden periodic solution as starting point, the high-order analytical solution is constructed by means of the Lindstedt–Poincar′e method. In particular, when the eccentricity is zero, the analytical solution constructed in this paper could be reduced to express the relative motion with circular reference orbit. At last, the practical convergence of the analytical solution is considered in order to check its validity and applicability.