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.