A fast approach for relative orbital determination in spacecraft formations

The design and operation of spacecraft formation-flying missions require a suitable knowledge of the relative dynamics of the platforms belonging to the formations. The perturbed, non-Keplerian, orbital environment can be easily faced by numerical approaches, like Cowell integration. However, that solution is accurate but quite slow to obtain, and indeed more appealing in the design phase. Instead, in a number of applications, above all from the operation point of view, a faster propagation would have a strong interest. This paper proposes a special writing of the equations of motion which does provide a closed form solution for orbits including the oblateness, i.e. a more realistic representation than the Keplerian one for higher LEO and medium altitude formations environment. The idea, which originated from previous literature, is to express the variables of interest (radius, node, inclination, etc) as a series, which can be limited to the desired accuracy level in terms of eccentricity. The adoption of symbolic mathematics currently allows for exact computation of the parameters of interest at every time. More important, the paper offers this formulation in terms of relative dynamics, i.e. in terms of differences in the orbital parameters of the platforms. Such a writing, which is the novelty of the paper, is actually what is required in the spacecraft formation case. The strong advantage of the proposed formulation is the quick availability of the solution, orders of magnitude faster than numerical integrators of similar accuracy. In such a way the future behavior of the platforms belonging to the formation can be easily computed, and these data are available for on-line control strategy evaluation as well as for payload operation schedule. The paper details this special writing of the equation of motion, provides the solutions for eccentricities up to 0.01 while validating the solutions with respect to standard, accurate numerical propagators.