Efficient Techniques for Relative Motion Analysis between Eccentric Orbits under J2 Effect

Accurate orbital propagation is required in order to correctly estimate (in design phase) and carry out (during operations) the control actions needed to maintain relative geometry among formation's spacecraft. Such an accurate propagation could be easily obtained by numerical methods, but their relevant computation cost would neither allow for general trade-offs during design nor match the onboard capabilities once spacecraft are in space. The interest for analytic, closed form solution is clear, as it would save on computation resources and allows for both speed and portability advantages. This paper proposes a special writing of the equations of motion which does provide a closed form solution for orbits including the oblateness effect. Such a representation is far more realistic than the approximate Keplerian one for LEO and medium altitude formations environment, and has indeed a remarkable appeal. The approach, originated from previous literature, is to express the variables of interest (radius, node, inclination, anomaly) as a series, which can be limited to the desired accuracy level in terms of eccentricity. The authors worked on this approach for several years, including currently available symbolic mathematics to allow for exact computation of the parameters of interest at every desired time. The more important contribution is a correct writing of the formulation in terms of relative dynamics, i.e. in terms of differences in the orbital parameters of the platforms, which is actually what is required in the spacecraft formation case. The paper details this special writing of the equation of motion and provides the analytical solution for eccentricities up to 0.2; i.e., remarkably extending the range of orbits previously considered in literature. These solutions are validated with respect to standard numerical propagators that end up with taking orders of magnitude longer to provide the same accuracy. Their quite high efficiency in terms of computational resources needed make them a suitable solution for inclusion in the onboard software, or a performing option for trade-off analysis during design phase.