A Simple Time Domain Collocation Method to Precisely Search for the Periodic Orbits of Satellite Relative Motion

A numerical approach for obtaining periodic orbits of satellite relative motion is proposed, based on using the time domain collocation (TDC) method to search for the periodic solutions of an exact J2 nonlinear relative model. The initial conditions for periodic relative orbits of the Clohessy-Wiltshire (C-W) equations or Tschauner-Hempel (T-H) equations can be refined with this approach to generate nearly bounded orbits. With these orbits, a method based on the least-squares principle is then proposed to generate projected closed orbit (PCO), which is a reference for the relative motion control. Numerical simulations reveal that the presented TDC searching scheme is effective and simple, and the projected closed orbit is very fuel saving