Hybrid (Ion and Chemical) GEO Stationkeeping Maneuver Planning Software

Transcription

1 Hybrid (Ion and Chemical) GEO Stationkeeping Maneuver Planning Software J. K. Skipper, D. Racicot, S. Li, R. Provencher and J. Palimaka Telesat Canada, Ottawa, Ontario, Canada. Abstract In the geochronous domain, a flight dynamics system is a set of software applications used to support all aspects of satellite operations from injection into transfer orbit, through arrival on station, equatorial or inclined orbit stationkeeping, station relocation and end-of-life disposal. The support provided can be divided into three areas: prediction, orbit (and possibly attitude) estimation, and maneuver planning (and evaluation). A satellite placed in geostationary orbit will not remain within its optimal ±0.05 latitude and longitude constraints as a result of orbit perturbations. Perturbations are forces acting on the satellite that cause it to deviate from its geostationary orbit, and are due to such factors as oblateness and triaxiality (nonspherical shape) of the earth, solar radiation force and the gravitational effects of the sun and moon. Maintaining the satellite orbit with efficient use of fuel is the primary goal of flight dynamics operations. This is accomplished through high-fidelity propulsion models and effective tools for maneuver planning, reconstruction and evaluation. In recent years, manufacturers of GEO satellites (Boeing Satellite Systems and Space Systems/Loral in particular) have been adding ion-electric propulsion systems to their satellite bus designs. The higher specific impulse of ion thrusters compared to chemical thrusters results in a lower beginning-of-life fuel load requirement which can be traded off for a larger payload or lower launch costs. Depending on the satellite design, orbit control may be performed using either ion-only or ion and chemical (hybrid) maneuvers. These changes in propulsion technology have prompted Telesat to initiate development of a new flight dynamics system product line known as the OnOrbit FDS. This system handles both ion and chemical maneuver planning. This paper describes the maneuver planning strategies for hybrid maneuver planning as implemented in the OnOrbit FDS. Introduction The concept of communications satellites encircling the globe in geostationary orbits was first proposed by Sir Arthur C. Clarke in Published that year in the Wireless World magazine, his vision of the future became a reality 20 years later with the launch of the Intelsat I Early Bird satellite on April 6, Clarke s concept involved placing a satellite in a circular orbit in the earth s equatorial plane at an altitude 1 of 1

2 of approximately 35,787 km. At this height, a satellite orbits the earth once per sidereal day, thus remaining above the same longitudinal position on the equator. In practice, a satellite initially placed in geostationary orbit will not remain at a fixed location relative to the earth s surface because of orbit perturbations (forces acting on the satellite that cause it to deviate from its geostationary orbit). The dominant perturbing force is due to solar and lunar gravity. This effect, coupled with the oblateness of the earth, causes the orbit plane to precess relative to the equatorial plane, resulting in daily oscillations in the satellite latitude. The earth s triaxiality (longitudinal harmonics in the geopotential) causes the mean satellite longitude to drift either eastward or westward depending on the satellite longitude. (Stable triaxiality nulls are found at approximately 75.1 E and W longitude.) Lastly, the solar radiation force and sun/moon gravitational effects cause the eccentricity (circularity) of the orbit to change over time, resulting in daily oscillations of the satellite longitude about the mean longitude. The effects of perturbations on the satellite orbit are offset by performing maneuvers (thruster firings) to adjust the orbital velocity vector. A North-South maneuver applies the velocity change ( v) primarily in a north or south direction to adjust the orbit plane (inclination and right ascension of ascending node). An East-West maneuver applies the v primarily in an east or west direction to increase or decrease the satellite orbital speed, which simultaneously affects the eccentricity and longitudinal drift. For older satellite designs, thrusters are fueled by all-chemical propulsion systems, producing thrust by means of a chemical reaction of the propellant(s). For such satellites, the North-South and East-West maneuvers tend to be largely uncoupled (independent orbital effects), and are typically performed once every one or two weeks. More recent satellite designs include ion-electric propulsion systems, in which high-efficiency, low-thrust ion thrusters are used to provide some (or even all) of the orbit control. Ion thrusters are usually positioned and oriented such that they produce both north-south and nadir (earth-directed) components of v, while nominally directing the thrust vector through the satellite center-of-mass. (In some configurations, the ion thrusters provide east-west v as well.) Due to their low thrust levels, ion thrusters must be fired much more frequently than weekly or biweekly to maintain the same degree of orbit control; for some satellites, the nominal firing strategy consists of two burns per day (one north burn and one south burn roughly 12 hours apart), seven days a week. The nadir component of v from each ion maneuver affects both the eccentricity and mean longitude of the satellite. The eccentricity effect allows ion thrusters to be used to control eccentricity in addition to the orbit inclination. Because the mean longitude shift is the same regardless of where in the orbit the burn is performed, there is a cumulative effect from each firing that must be offset to keep the satellite from drifting eastward away from its desired longitudinal position. The effect is offset by simply increasing the nominal orbit altitude, such that the westward drift due to the larger orbital period matches the eastward drift due to the daily ion thruster firings. 2 of 2

3 For those satellites in which the ion thrusters are oriented to provide minimal tangential (in-track) v, ionthruster maneuvers are supplemented with chemical maneuvers to provide drift-only or drift-andeccentricity control. The frequency of the chemical maneuvers may range from as often as twice per day to as infrequently as once every two weeks, depending on the satellite design and ion-thruster firing strategy. With the much higher frequency of maneuvers, and the interdependence between ion and chemical maneuvers, the classical approach of planning and executing one maneuver at a time is not feasible with ion-thruster satellites. In what follows, a summary of the planning algorithms used in Telesat Canada s OnOrbit FDS hybrid maneuver planning software is presented. This software allows satellite operators to plan up to 56 ion maneuvers and 56 chemical maneuvers spanning a 28-day period at once. Non-Singular Orbital Elements In formulating the equations that describe the orbital effects of maneuvers, it is useful to define a set of non-singular elements derived from mean values of the classical Keplerian elements. The non-singular elements are defined as follows: d = mean longitude drift rate ( /day E) l = mean right ascension of the satellite (= mean longitude + Greenwich Hour Angle) ( E) h 1 = e sin (ω+ω) k 1 = e cos (ω+ω) h 2 = sin i sin Ω i sin Ω k 2 = sin i cos Ω i cos Ω where e is the mean eccentricity i is the mean inclination (in radians) ω is the mean argument of perigee Ω is the mean right ascension of ascending node When a maneuver is performed, the effects on the orbit are determined by the magnitude and direction of the applied v, and the location (right ascension) in the orbit at which the burn occurs. If α represents the satellite right ascension at the time of the maneuver, and the radial, in-track (eastward) and cross-track (northward) components of the v vector are denoted ( R, I, C ), then the changes to the nonsingular elements are given by 3ω e d = 2 l = R I 2 I h1 = k 1 2 = I sinα R cosα + R cosα sinα h k 2 2 = C = C sinα cosα 3 of 3

4 where ω e is the earth s angular rotation rate and is the satellite speed in geochronous (geostationary) orbit. Ion Maneuver Planning Strategies When the satellite s inclination is close to zero, the luni-solar gravitational perturbation causes the inclination vector (k 2, h 2 ) to advance primarily in the h 2 direction; hence, to offset this effect, inclination maneuvers are optimally performed close to right ascensions of 90 or 270. Specifically, if the crosstrack component of v is negative (southward), the maneuver is ideally performed near 90 ; if the crosstrack component of v is positive (northward), the maneuver should be performed near 270. Because a negative cross-track component of v is provided by a thruster that is typically mounted on or near the north face of the satellite, such a component shall be denoted N. Similarly, a positive cross-track component of v provided by a thruster mounted on or near the south face shall be denoted S. In the strategies considered below, we shall assume that on any given firing day, one north thruster is fired near 90 and one south thruster is fired approximately 12 hours later near 270. A nadir (negative radial) component of v that is applied at a right ascension of 90 or 270 results in a change to the k 1 component of the eccentricity vector (k 1, h 1 ). Thus, the k 1 component of eccentricity can be controlled by simply adjusting the relative sizes of the north and south inclination maneuvers (while maintaining the sum of the burn magnitudes to provide the required inclination control). Assuming there is no tangential component of v in either burn, firing at 90 or 270 will not affect h 1. However, shifting the maneuver right ascensions away from 90 and 270 (i.e. towards 0 or 180 ) will produce a change in h 1 proportional to the sum of the nadir components of v. In shifting the maneuvers, the total v required for inclination control increases slightly because the burns are not performed at the optimal orbit locations. With the freedom to adjust the relative burn sizes and move the burn locations, three ion-thruster planning strategies become available: Inclination only (equal v s, optimal locations, drift and eccentricity control provided by chemical thrusters); Inclination + k 1 (unequal v s, optimal locations, drift and h 1 control provided by chemical thrusters); and Inclination + eccentricity (unequal v s, suboptimal locations, drift control provided by chemical thrusters). In addition to choosing which elements of the total orbit control to allocate to ion maneuvers, we can also define firing profiles that specify on which days ion thrusters will be fired, and on which days (if any) no firings will occur. A typical firing profile might be 6 days on and 1 day off in repeating 7-day cycles. 4 of 4

5 For a selected planning strategy and firing profile, the daily ion maneuver v s and right ascensions can be determined directly from the total required changes in (k 1, h 1 ) and (k 2, h 2 ) over the specified cycle. Ion Maneuver Planning Equations It is convenient to relate the nadir and cross-track components of v for each ion thruster firing by the cant angle between the +/- orbit-normal axis (in the spacecraft frame) and the thrust axis, where the reference axis is chosen such that the cant angle lies between 0 and 90 for each thruster. For crosstrack v components N and S of the north and south maneuvers, and north and south thruster cant angles C N and C S, the corresponding nadir v components are N tan(c N ) and S tan(c S ). It is also convenient to define the locations of the maneuvers (more specifically, the burn midpoints) in terms of angular offsets relative to the reference right ascensions 90 and 270. Thus the midpoint of the north maneuver occurs at a satellite right ascension of 90 + θ N, and the south maneuver midpoint occurs at θ S. Finally, after determining the total required changes in (k 1, h 1 ) and (k 2, h 2 ) for the specified planning cycle (accounting for both the effects of perturbations acting over the cycle and any initial orbit error to be corrected during the cycle), the required daily corrections to the four non-singular parameters are obtained by simply dividing the total cycle changes by the number of firing days in the cycle. Substituting the expressions for the v components and burn locations identified here into the equations defining the changes to the non-singular elements presented earlier (and summing over the two burns) yields the following system of equations that relate the cross-track v components and locations of the two burns with the required daily changes to the non-singular elements: N cos( θ N ) + S cos( θ S ) = A1 N sin( θ N ) + S sin( θ S ) = A2 N C1 cos( θ N ) + SC2 cos( θ S ) = N C1 sin( θ N ) + SC2 sin( θ S ) = A A 4 3 where C 1 = tan(c N ), C 2 = tan(c S ), and A1 = h 2, A2 + k2 =, A3 = + k1, A4 = + h1. These equations can be solved explicitly for four burn parameters N, S, θ N, and θ S in terms of trigonometric functions of the right-hand-side terms A i and the cant-angle tangents C 1 and C 2. 5 of 5

6 With the cross-track v components evaluated, the required burn durations may be determined from the thrust of the ion thrusters and the spacecraft mass. Accounting for arc losses due to the fact that the v of each burn is applied over an arc of the orbit instead of being applied impulsively at the optimal right ascension, the required burn duration D for each maneuver is computed as where M is the satellite mass, 2 ω e D = arcsin ω e 2 M F cos ( C) is the cross-track component of v, F is the thrust produced by the ion thruster, C is the thruster cant angle, and ω e is the earth rotation rate. Nominal Drift Target As stated in the Introduction, each ion maneuver produces an eastward shift in mean longitude due to the nadir component of v. With ion maneuvers occurring as frequently as twice per day, there is a cumulative effect that must be offset to prevent the satellite from drifting eastward away from its desired longitudinal position. This is done by raising the nominal orbit altitude such that the westward drift due to the increased orbital period matches the eastward drift due to the daily ion thruster firings. The nadir (negative radial) component of v of each ion maneuver (- R ) produces an eastward shift in mean longitude (in radians) given by 2 l = On average, the daily cross-track component of ion v ( ) is the v required to offset the inclination R perturbation; that is, = i, where i is the average daily change in inclination (more correctly, the magnitude of the change in the (k 2, h 2 ) vector). Note that is computed as the average over all days of the cycle (not just the ion-thruster firing days) because the compensating effect of the increased altitude will be applied on every day of the cycle. The average daily radial component of ion v is computed from the thruster cant angle C: R = - tan C = - i tan C 6 of 6

7 Thus the nominal westward drift that is needed to offset the average daily longitude shift due to the ion maneuvers may be calculated as d N = - l = - 2 i tan C where d N (< 0) is in units of /day E if i is evaluated in units of degrees. As indicated above, d N does not depend on the number of ion-thruster firing days in the cycle; it is a function of only the average daily change in the inclination vector due to perturbations and the ion-thruster cant angle. Chemical Maneuver Planning In discussing the different ion maneuver planning strategies above, it was noted that chemical thrusters could be used to provide all required drift and eccentricity control, drift and h 1 control, or drift-only control. If chemical maneuvers are performed to control one or both components of the eccentricity vector, the classical method of planning and executing single-burn or double-burn drift and eccentricity maneuvers once every cycle (e.g. 7 or 14 days) may be used. However, if ion maneuvers are planned to fully control the eccentricity vector, it may be desirable to perform chemical maneuvers in conjunction with the ion maneuvers (i.e., daily). To provide drift-only control, two equal- v chemical maneuvers, separated by 12 hours, may be performed anywhere in the orbit without affecting the orbit eccentricity vector. In theory, if the spacecraft is at the correct stationkeeping longitude at the start of the cycle, and the startof-cycle drift is d N (the value required to offset the longitude shift due to the ion maneuvers), drift maneuvers should be necessary only to offset the effect of the triaxiality perturbation. In practice, there may be errors in both longitude and drift at the start of the cycle that must be corrected as well. Because these errors are independent, there will not be (in general) a single daily drift change that, when applied over each day of the cycle, corrects both the drift and longitude errors at the end of the cycle. One solution is to plan to correct the errors over two cycles; in this way, an intermediate drift target may be defined for the end of the first cycle that results in correcting the longitude error when the target drift d N is achieved at the end of the second cycle. Specifically, the intermediate drift target (d int ) is chosen such that the time integral of the resulting drift profile, relative to the nominal drift target d N, over the two cycles equals the longitude change required to achieve the target stationkeeping longitude at the end of the second cycle. As with ion maneuvers, one can define chemical-thruster firing profiles that specify on which days chemicals thrusters will be fired, and on which days (if any) no firings will occur. The time integral of drift, and hence the intermediate drift target, is dependent on the selected firing profile. By imposing the constraint that a single (constant) drift change is applied on each firing day of the first cycle, and a second (in general, different-valued) drift change is applied equally on all firings days of the second cycle, a unique intermediate drift target is easily computed. 7 of 7

8 An example of a drift profile over two 7-day cycles is depicted below. In this example, drift maneuvers are performed on days 1, 2, 5 and 6 of each cycle. On each firing day of the first cycle, two drift maneuvers are performed to provide a combined drift change of d 1 ; on each firing day of the second cycle, the drift is adjusted by d 2. For this simplified example, the effect of the triaxiality perturbation has not been included. In implementing the planning algorithm, the calculation of d int accounts for the longitude changes over the two cycles due to the triaxiality acceleration, and after determining the drift target for each firing day, the drift changes d 1 and d 2 are adjusted slightly to account for the changes in drift between firing days due to the triaxiality effect. d int t 3 d N t 2 t 1 d i t 0 d 0 t 4 t 0 t 1 T d 1 Sample 2-Cycle Drift Profile t 2 t 3 t 4 d 2T 2 Software Implementation The combined ion and chemical maneuver planning strategies and algorithms described above have been successfully implemented in Telesat Canada s OnOrbit Flight Dynamics System. The OnOrbit FDS is a powerful and easy-to-use suite of tools that can determine, predict, and control the orbits and spin vectors of geostationary satellites. With a rich heritage of almost 30 years, Telesat s FDS has been designed by experienced orbital analysts to both maximize the propellant life of satellites and efficiently accommodate the actual day-to-day operational tasks of analysts responsible for maintaining the orbits of geostationary satellites. arious versions of the FDS have been used to control more than 40 satellites built by three different spacecraft manufacturers. As one of the key applications in the OnOrbit FDS suite, the Hybrid Stationkeeping Maneuver Planning tool is used to plan the stationkeeping maneuvers for a satellite equipped with both chemical and ion thrusters. Combined daily ion and chemical maneuvers may be planned for stationkeeping cycles of up to 28 days at once, using the ion-maneuver planning strategies presented in this paper. 8 of 8

9 Along with the choice of stationkeeping strategies, the user may choose independent firing profiles for the ion and chemical maneuvers which define the days in the stationkeeping cycle that maneuvers occur. Application output consists of FDS event file records representing the planned maneuvers for the stationkeeping cycle, maneuver message files or maneuver tables that can be electronically transmitted to the real-time ground control system, and plots of predicted orbit dynamics over the cycle. Images of the key Graphical User Interface dialogs are shown below, with some sample plot output of predicted orbital parameters for a 7-day stationkeeping cycle. Hybrid Stationkeeping Maneuver Planning Main Window and Planning Input Dialogs Hybrid Stationkeeping Maneuver Planning Sample Plot Output 9 of 9

10 The OnOrbit FDS is the latest in a series of FDS Products developed by Telesat Canada for distribution to satellite operators around the world, seeing its inaugural release to L-3 Storm Control Systems, Inc. in December As a new planning function available only in the OnOrbit FDS, the Hybrid Stationkeeping Maneuver Planning tool has not yet been used operationally. However, the OnOrbit FDS is an integral component of the ipstar Ground Control System that is being developed by L-3 Storm for Shin Satellite, and the Hybrid Stationkeeping Maneuver Planning application will be used operationally to plan the orbital maneuvers for the ipstar-1 satellite beginning later this year. Summary A technique for simultaneously planning daily ion and chemical maneuvers to control the orbital effects of perturbations on geostationary satellites equipped with both ion-electric and chemical propulsion systems has been developed and implemented in Telesat Canada s OnOrbit FDS Flight Dynamics System. The implementation allows for three possible ion-thruster planning strategies: Inclination control only (chemical drift and eccentricity control) Inclination and k 1 control (chemical drift and h 1 control) Inclination and eccentricity control (chemical drift control) The software supports independent, user-specified firing profiles for ion and chemical maneuvers for planning cycles of up to 28 days. References [1] Bernard M. Anzel, Controlling a Stationary Orbit Using Electric Propulsion, DGLR/AIAA/JSASS 20 th International Electric Propulsion Conference (Garmisch-Partenkirchen, Germany, October 3-6, 1988). [2] Ahmed Kamel, Walter Gelon, Keith Reckdahl, A Practical Stationkeeping Method for Modular Geochronous Satellites with a Xenon Propulsion System, AAS/AIAA Astrodynamics Specialists Conference (Girdwood, Alaska, August 16-19, 1999). [3] B. Sauer, M. Chow, Geochronous Satellite Collocation at Space Systems/Loral, AAS/AIAA Astrodynamics Specialists Conference (Quebec City, Quebec, Canada, July 30 - August 2, 2001). 10 of 10