Nonlinear and linear local cartesian relative motion state models for J2 pertubed elliptical orbits

Formulating a relative motion model between artificial satellites keeps a very actual interest in order to achieve devices adapted to autonomous rendezvous operating or formation flying control. Scanning the existing models field leads to distinguish them according to their nature -- linear or nonlinear -- the relative state variables -- local cartesian or curvilinear variables, relative orbital elements, classical or not -- the reference state variables -- inertial cartesian or spherical variables, orbital elements -- the pertubations taken into account -- drag, J2, ... -- and other assumptions as eccentricity of the reference satellite. The historical Clohessy-Wiltshire and Tschauner-Hempel models have been outclassed by improved linear models that include J2 pertubation [1, 2, 3] or drag [4] but do not take rigorously into account the perturbed dynamics of the reference local frame. As far as this fundamental point is concerned, Kechichian's nonlinear model [5] provides an interesting but complex formulation because of a non optimal derivation method. More over, it does not take full advantage of the Lagrange conditions [6] implied by orbital elements definition which allows simplifications without loss of generality. These elements are presented in this article to reach an improved relative motion nonlinear model under J2 perturbation assumption which results are validated by comparison with those produced by a nonlinear equinoctial propagator. A linear model is also proposed.