Optimal Servicing of Geosynchronous Satellites

This paper addresses the problem of how to optimally rendezvous with a set of satellites in geosynchronous orbit that have small inclinations. The method developed has application to two problems, servicing of geosynchronous satellites and the removal of dead satellites in geosynchronous orbit. It is shown that when inclination changes are required the optimal solution reduces to the solution of the traveling salesman problem. The method was applied to a set of non-operational geosynchronous satellites selected from the Space Object catalog. The method also directly identifies those satellites that are very costly (in fuel) to visit so that they can be removed from the list to be visited. NOMENCLATURE Ω right ascension I inclination λ longitude n mean motion h angular momentum v velocity T R g g , geosynchronous radius and orbit period τ jk jk N , time and number of orbits to travel from satellite j to satellite k Tm mission lifetime Ts servicing time for each satellite INTRODUCTION The servicing of satellites in orbit is a concept that is receiving more and more attention. If this could be achieved economically then the system lifetime could be extended and system costs reduced. The Hubble Space Telescope (HST) has been serviced to repair the solar panels, mirror and gyros. The cost of the HST made it worthwhile even though it was a very expensive mission in that it involved the use of astronauts and the Space Shuttle. For servicing to be economical satellites will have to be designed to be serviced. In addition, the servicing will have to be performed by robotic spacecraft; the use of man in space will make the costs prohibitive in most cases. The best economy will be achieved when numerous satellites can be serviced in one servicing mission. Since plane changes require a lot of fuel two scenarios in which numerous satellites could be serviced are constellations with many satellites in each plane, such as Iridium, and geosynchronous satellites. A few satellites could potentially be serviced in low Earth orbit utilizing differential nodal precession if all the satellites had approximately the same inclination. Even though there is a policy now that satellites are to be removed from the geosynchronous belt at end of life there are many non-operational satellites in geosynchronous orbit that are a hazard to other satellites. If there was a way to rendezvous with these satellites and attach a small engine to raise their orbit then over a period of time the geosynchronous belt could be made safer. The method developed in this paper applies to these two problems. The method determines the minimum fuel solution for visiting (rendezvousing with) a set of satellites in geosynchronous orbit with small inclinations. Since the fuel to rendezvous with a satellite in the same orbit is very small if sufficient time is allowed for the maneuver the method finds the order in which the satellites should be visited to minimize the fuel required for the plane changes. It is shown that the minimum fuel solution is proportional to the minimum distance path through the set of points that are the projections of the angular momentum AIAA/AAS Astrodynamics Specialist Conference and Exhibit 5-8 August 2002, Monterey, California AIAA 2002-4905 Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. 2 American Institute of Aeronautics and Astronautics vectors on the equatorial plane. Thus, the minimum fuel (∆v) solution is the solution of the Traveling Salesman Problem (TSP). The fuel required for the in-plane maneuver is not minimized for several reasons. It is much smaller than the fuel required for the plane change and it can be obtained by vectoring the out of plane ∆v to obtain the necessary small in-plane component. Strategies for the rendezvous include equal time for each rendezvous or equal ∆v for the in-plane portion of the maneuver. Parameters in the problem are the total time and the servicing time at each satellite. The required ∆v is relatively insensitive to the servicing time until it becomes so large that only a short time is allowed for the transfer from one satellite to another. Then the in-plane ∆v begins to increase rapidly and dominate. ANALYSIS Consider a set, S, of N geosynchronous satellites where Si represents the ith satellite with orbital elements ei i i i T e e e = ( ) 1 2 6 , , , . K Find the order of visiting the satellites such that the total fuel ∆v ( ) to rendezvous (visit) is minimized. In addition, identify those satellites that should not be serviced because the fuel to service them is prohibitive. In this preliminary study we restrict the set S to consist of satellites in circular orbits and equal periods of 24 hours, small, but not necessarily zero, inclination and arbitrary right ascension. The circular orbit and 24 hour restrictions could be relaxed to consider near circular and near 24 hour satellites, such as the drifters. First we show that the finding the order of visiting each of the satellites that results in the minimum ∆v solution for the plane change portion of the ∆v to rendezvous with each of the N satellites in S is the solution of the Traveling Salesman Problem (TSP). Consider two satellites of the set S with inclination and right ascensions I I 1 1 2 2 , , Ω Ω ( ) ( ) and . Let X Y Z , , ( ) be an inertial coordinate system with Z along the polar axis, and X and Y define the equatorial plane. As shown in Figure 1 the coordinates of the projection of the angular momentum vector in the equatorial plane are h I j j j j j sin sin , cos , , Ω Ω − ( ) = 1 2. With equal angular momentum the distance d between the two points is