Hill's equations, mean orbital elements, and formation flying of satellites
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This paper demonstrates the use of Hill's equations to design relative motion orbits between two satellites. Hill's equations provide initial conditions that will result in periodic relative motion under ideal conditions and the Chief being in a circular orbit. In this paper, the effect of J2 perturbation is accounted for through the use of the mean orbital elements. An in-plane period matching constraint between the orbits is imposed. However, it is not possible to match differential nodal precession, if an inclination difference is desired, without using thrust. The required mean orbital element differences between the Chief and the Deputy are obtained numerically, to satisfy the relative motion initial conditions obtained from Hill's equations. Nonlinear simulation results indicate that actual relative orbits are very close to the desired Hill's orbits in the short-term. However, over the long-term, the orbits slowly stretch in the out-of-plane direction. This drift can be corrected and at the same time exploited, to reposition the satellite on the Hill's orbit with a change in the initial relative inclination and ascending node difference. Modified Hill's equations are also derived using a mean, precessing, circular, ghost orbit as a reference. The results of numerical integration of these linear periodic differential equations, with the initial conditions obtained using the new technique, agree sufficiently well with the nonlinear simulations for the purpose of control design.
