A precise analytic solution to the relative motion of formation flying satellites.

摘要

A precise analytic solution that includes the effects of the reference orbit eccentricity and the perturbations is needed for the relative motion of formation flying satellites. As a result of the assumptions of the spherically symmetric Earth and the circular reference orbit, Hill's equations have large errors and are insufficient for the long-term prediction of the relative motion that is needed to minimize the fuel consumption and maximize the lifetime. The new approach, called the geometric method, is proposed to obtain the state transition matrix (STM) with the effects of the reference orbit eccentricity and the primary gravitational perturbation J₂. Based on the geometric transformation and the STM for the relative mean variables, a simple form of a STM is obtained in closed form for the mean non-singular variables under the existence of the gravitational perturbation J ₂. Using the closed form transformation matrix between the mean and the osculating non-singular variables, the closed form STM of the relative motion is derived for the osculating non-singular variables without singularity in eccentricity. Using the same processes but with the equinoctial variables, the closed form STMs for the mean and the osculating equinoctial variables are derived without the singularities at zero eccentricity and at zero inclination. Using numerical evaluations, the geometric method is shown to provide a very precise analytic solution for the near-circular relative orbit, highly eccentric reference orbit, various inclinations, various eccentricities, various initial radii of relative orbit, the reference orbit near the critical inclination, and the geostationary reference orbit. Since the geometric method contains all the properties of the reference orbit and the J ₂ effects, it is possible to determine the effects of orbit parameters on the errors by evaluations and qualitative analyses of the relative motion. The method can be extended easily to include other perturbing forces. Finally, the geometric method provides a precise analytic solution to the relative motion of formation flying satellites in closed form for any kind of reference orbit without singularity and without solving directly the differential equations.