Stable configurations for bounded satellite formation flying in the presence of gravitational perturbations

A formation of satellites operating in low Earth orbits is subject to erturbations associated to the nonuniform gravitational field of the Earth, whose entities depend on the position, resulting in a different effect on each satellite of the formation. As a result, any bounded configuration designed using the traditional methods, based on the Hill - Clohessy - Wiltshire (HCW) equations, is not immutable, but instead degrades over time, and periodic reconfiguration maneuvers are required. We propose a method to characterize bounded configurations for satellite formations which are stable with respect to the gravitational perturbation associated to the zonal harmonics. The problem is modeled using he Hamiltonian formalism and canonical transformations are introduced to set the Hamiltonian function of the perturbed problem to a form equivalent to that associated to the HCW equations plus some negligible terms. This method allows selecting the desired configuration for bounded formation flying from the well-known analytical solutions of the HCW equations for the unperturbed case and determining the corresponding ones for the perturbed case by applying the inverse of the canonical transformation. The suitability of the problem is verified y means of numerical analysis on satellites formations in sun-synchronous orbits and medium inclination low Earth orbits.