Translation Dynamical Nonlinear Models in Perturbed Keplerian Conditions For a Geostationary Satellite

Abstract

The dynamics of a GEO satellite will be studied in this work to obtain a dynamical model as accurate as possible. This model will be obtained in terms of Gauss' variation of osculating parameter (VOP) equations containing the environmental perturbing accelerations, which are traditionally used to plan the station keeping maneuvers. The idea is to implement a controller for geostationary station keeping purposes based on a model written in terms of osculating orbital elements instead of averaged elements. Such a controller plans in an automatic way the station keeping (SK) maneuvers and it could be integrated on board in view of autonomous station keeping control loop. The Geostationary Earth Orbit (GEO) satellites maintain an essentially fixed position with respect to the surface of the Earth. This is made possible by inserting the spacecraft into a circular, equatorial orbit at an altitude of roughly 36000 km. At this altitude, if the main environmental disturbing forces (the Earth's non-spherical gravity attraction, the Moon's and Sun's gravity attraction and the solar radiation pressure) are neglected except the Kepler attraction of the Earth, the geostationary motion is ideal. The satellite remains fixed with respect to the surface of the Earth. The mean motion n matches the Earth's rotation rate  of one revolution per 23 hours and 56 minutes. In presence of perturbations, it is a common practice to control a GEO satellite actively via station keeping maneuvers such that it stays confined in a box of 100-150 km width around a nominal geostationary longitude and latitude (1). Traditionally this is done with an open loop control technique based on a dynamical model of the satellite state vector subject to the only environmental perturbing forces and on a separate dynamical model taking into account the only thrust effects. Moreover, this latter model is derived supposing to use a chemical propulsion system characterized by high thrusts and very short thrust durations relative to the orbital period. These impulsive thrust hypotheses lead to assume that maneuvers induce jumps in the velocity part of the state vector but not in the position part. The GEO station keeping problem is thus dealt with in a discrete way, without considering spacecraft acceleration but only velocity and position vectors. With a view to substitute chemical propulsion systems with electrical ones, the solution approaches of the GEO station keeping problem should become continuous, in order to gain benefit from the technology change. A GEO satellite dynamics model has to be obtained taking into account all the perturbing forces (environmental or not) acting on the spacecraft. In this paper, we explain in detail the nonlinear dynamical model used to design the station keeping controller and the analytical approximations done to implement this