1 Ground-Based Orbit Determination for Spacecraft Formations 1,2 Silvano Sgubini and Giovanni B. Palmerini DIAA – Dipartimento di Ingegneria Aerospaziale e Astronautica Università di Roma La Sapienza Via Salaria 851/881 – 00138 Roma (Italy) silvano.sgubini@uniroma1.it , giovanni.palmerini@uniroma1.it 1 978-1-4244-3888-4/10/$25.00 ©2010 IEEE. 2 IEEEAC paper #1480, Version 2, Updated January 4, 2010 Abstract— Spacecraft formations offer interesting challenges to orbit determination, especially for ground- based tracking. In fact, the limited distances between spacecraft and the possible ambiguity of the observables gathered from the ground have an impact on the solution process. The paper aims to apply the filtering techniques on a refined dynamical model, which can include the main perturbation effects - due to the oblateness of the Earth and, at the lower altitudes, the air drag - on spacecraft trajectories, representing them in series with a remarkably limited number of terms even in eccentric case. The idea is to focus on theoretically expected behavior rather than dealing with an enriched but heavier state including parameters directly related to the perturbing effects. In such a way, it could be possible to obtain a good estimate even with limited spacecraft tracking information. This is an important asset in navigating a formation from the Earth, due to the needed partition of ground station resources among different platforms belonging to the formation, and to the possible ambiguity among the measurements, which further reduce the available data. The specific nature of the dynamic model calls for an estimator with a flexible and ”open” architecture, easily allowing for changes and additions in the model itself. Therefore, the estimator selected for testing the approach has been the Unscented Kalman Filter, versatile enough to allow for increasing model accuracy without the need for tedious computation of the Jacobian. This approach is also intended to offer a different way to investigate special perturbed configurations, via the semi- analytical and almost exact representation of the trajectories. In such a perspective, one of the first application, which is shortly outlined in the paper, will be the analysis of spacecraft formations under the J2 effect. In fact, recent studies identified a set of almost periodic relative configurations among the spacecraft. This set (sometimes referred as the special or magical inclination’s one) has been recently identified by means of numeric search, and has also received some (partial) explanation. Due to the interest in control effort reduction, it is deemed that a better understanding of this special dynamics, possibly provided by means of a selected modeling approach, can be of some interest. TABLE OF CONTENTS 1. INTRODUCTION ................................................................ 1 2. THE APPROACH ARCHITECTURE.................................... 2 3. DYNAMICS MODEL .......................................................... 2 4. UKF ESTIMATOR............................................................. 4 5. APPLICATION OF THE TECHNIQUE TO THE SPECIAL INCLINATION PROBLEM...................................................... 6 6. FINAL REMARKS.............................................................. 6 REFERENCES........................................................................ 7 1. INTRODUCTION Autonomous orbit determination is clearly assuming an increasing relevance, due above all to the availability of space-rated GPS systems for kinematic state determination. The benefit is especially remarkable when the mission is operated by several co-operating spacecraft, as the savings with respect to tracking from ground the overall formation become quite appealing. However, such an approach requires the availability of GPS receivers on board all the spacecraft, as well as bi-directional inter-satellite data links and a computation capability for at least one of the elements. Therefore, non-autonomous (i.e. ground-based) techniques still maintain a remarkable interest, both as a back-up or as a basic asset for loose formation, where continuous accurate navigation is not an issue. The paper moves from the concept that a better estimation, even with a small number of available measurements, can be obtained if the data are matched to a more accurate model. Typically, the influence of the perturbations is considered by introducing process noise in the filter or, at a more accurate level, by introducing perturbation models in Newton equations. Moreover, these additions are usually performed on the linearized formulations, which in the formation case are the largely used Hill-Clohessy-Wiltshire equations. Here, the perturbation effects are seen from a slightly different point of view. Kinematics description is adapted to the perturbed field, by using an approach originally suggested by Broglio [1]. The main interest of this approach, which deals with a series expansion, is to allow for a gradual, hierarchical introduction of the effects. In this paper