Optimal Steering for North -South Stationkeeping of Geostationary Spacecraft

The problem of north ‐south stationkeeping of geostationary spacecraft using electric thrusters is analyzed. Pure yawing with short-duration low-thrust arcs applied infrequently is assumed, and the dynamics are cast in continuous form to obtain an analytic steering law in the inclination-node (i;X ) space that brings the spacecraft backtotheidealinitialorbitorientationfortheinitiationofafree-driftperiodthatsatise estheinclinationconstraint for the longest possible duration. This problem is posed as a minimum-time navigation problem between two i;X pairs and is similar to the Zermelo problem of navigating a ship in strong variable currents. The simple linear steering law thus obtained is easy to use and fuel optimal compared to other suboptimal strategies for travel between two given (i;X ) pairs. ONTROLstrategiesforthenorth ‐southstationkeepingofgeostationaryspacecraftwithchemicalpropulsionhavebeenthoroughly documented in the literature. 1i7 There exists an ideal drift in the inclination-right ascension of the ascending node (i, \ )space that results in the satisfaction of the inclination deadband constraint for the longest possible duration. This inclination deadband is dee nedby0 · i · imax,withimax themaximumallowableinclination. Once the deadband is consumed, an impulsive maneuver will target tocertainoptimizedinitialconditionsin i and\ tocontinuethesatisfactionofthe deadbandconstraint.Considerable fuel canbesaved if thelow Isp chemicalrockets arereplacedbyhigh Isp electricengines to effect the same stationkeeping maneuvers, thereby extending the operationallifeofthesesatellites.However,thesemaneuverscannot be carried out in an essentially instantaneous manner, but must be implemented in small incremental steps spanning several weeks or more, depending on the level of the thrust acceleration and the frequency of the incremental maneuvering. The inherent long duration of these low-thrust maneuvers must be factored in the design of the maneuver strategy inasmuch as it must account for the natural drift that takes place before the completion of the maneuver sequence. This paper adopts, as an example for illustration purposes, the ideal drift cycle in i, \ as the fundamental cycle to repeat, and casts the dynamics of this problem in a continuous fashion to convert it into a navigation problem of minimum-time travel between given i, \ pairs, not unlike the well-known Zermelo problem in optimal control theory. The control is now a steering angle that dee nes the direction of the change in the i, \ region to be effected by the small incremental D V at that particular moment in the overall sequence. It is assumed that these small D V are large enough to overcome the natural drift in the opposite direction such that the ideal initial conditions are recovered in time. We begin in Sec. II with the description of the idealized (i, \ ) drift dynamics due to Kamel and Tibbitts, 7 which will be used in the design of the optimal steering strategy shown in the subsequent sections. Section III shows how the relevant linearized variation of parameters equations are used to calculate the changes in the orbit plane orientation duetosmallthrustarcs. In Sec.IV,a suboptimalstrategy that does not factor the natural drift in i and \ between successive thrust arcs is depicted. This strategy consists of applying the thrust arc at the current common line of nodes of the current orbit and the e nal target orbit. The more efe cient strategy of Sec. V creates new intermediate target orbits and applies the thrust arc at the common line of nodes of the current orbit and the intermediate target orbit,