Three-axis Attitude Control with Two Reaction Wheels and Magnetic Torquer Bars

Transcription

1 Thee-axis Attitude Contol with Two Reaction Wheels and Magnetic Toque Bas Byce A. Robets *, Jeffey W. Kuk, Thomas B. Ake, Tom S. Engla Johns Hopkins Univesity, Baltimoe, MD 2121, USA and Bian F. Class **, Daniel M. Rovne Obital Sciences Copoation, Dulles, VA 2166 USA The Fa Ultaviolet Spectoscopic Exploe satellite was launched in 1999 and began a thee-yea pime mission to collect high-esolution specta in the fa ultaviolet bandpass. Two and a half yeas afte launch, mechanical failue of two out of fou eaction wheels educed the satellite to two-axis contol, halting science obsevations. This failue pompted modification of the FUSE attitude contol system softwae to estoe thee-axis contol using a hybid of existing magnetic and eaction wheel actuatos. Pointing accuacy and stability ae once again at the sub-acsecond level, close to the pe-wheel failue pefomance. The ange of stable attitudes is limited, but gound-based modeling softwae now diects the planning pocess so that obsevations and maneuves stay within the limits of the actuatos. Despite these constaints, efficient science opeations ae ongoing, and ove the couse of a yea, the entie sky is available fo obsevation. Nomenclatue N = gavity gadient toque GM = eath gavitational constant R = satellite eath-centeed inetial (ECI) position vecto I = satellite inetia tenso µ = spacecaft dipole moment vecto B = geomagnetic field vecto N mag = magnetic toque vecto Ŝ = symmetic body axis  = anti-symmetic (magnetic) body axis k = unloading gain facto θ = angula attitude eo eo * Mission Planne, Depatment of Physics and Astonomy, bobets@pha.jhu.edu, Membe AIAA. Pincipal Reseach Scientist, Depatment of Physics and Astonomy Head of Instument Opeations, also with Compute Sciences Copoation. Independent Consultant to FUSE Poject. ** Senio Diecto, Attitude Contol Systems Depatment, Senio Membe AIAA. ACS Analyst, Attitude Contol Systems Depatment. 1 Ameican Institute of Aeonautics and Astonautics

2 I. Intoduction The Fa Ultaviolet Spectoscopic Exploe (FUSE) was launched into low-eath obit on 24 June 1999 and began a thee-yea pime mission to collect high-esolution astonomical specta in the fa ultaviolet bandpass 1. The science instument, a set of fou co-aligned telescopes and pime focus spectogaphs 2, is maneuveed by a spacecaft bus whose design evolved fom pevious Exploe Platfom spacecaft. The Johns Hopkins Univesity in Baltimoe, Mayland is esponsible fo satellite command and contol, and develops activity schedules on behalf of the intenational community of astonomes whose pee-eviewed pogams have been ganted obseving time by NASA. Commands ae uplinked by way of a gound station in Pueto Rico, and FUSE opeates autonomously as it obseves appoximately -6 astonomical tagets pe day. The science goals equie that the satellite be able to achieve absolute pointing accuacy of ~1", maintain pointing stability of ~.5", and autonomously slew between science tagets in a faction of one 1 minute long obit. To accomplish this, the attitude contol system (ACS) employs ing lase gyos and an optical fine eo senso fo attitude knowledge, and eaction wheel assemblies (RWAs) to geneate contol toque. Thee othogonal RWAs contol the yaw, pitch and oll body axes, and a fouth edundant skew wheel povides equal toque components along all thee body axes. Each RWA can povide up to 55 m Nm of toque until the 21 Nms momentum stoage limit is eached. Thee othogonal magnetic toque bas (MTBs) with a maximum dipole moment of ~14 Am 2 act against the geomagnetic field to unload momentum fom the RWAs and maintain a zeo wheel momentum bias. Finally, ti-axial magnetometes (TAMs) measue the geomagnetic field in the body fame, and ae used both fo coase two-axis attitude estimation as well as an input to the momentum unloading outine. FUSE is in a cicula obit with an inclination of 25º and altitude of 765km. Aeodynamic dag toque at this altitude fo a satellite of FUSE s size is negligible, so the only significant extenal distubance is gavity gadient toque, given by N = GM ˆ R IR ˆ (1) R whee R is the satellite s position vecto elative to the cente of the eath, R ˆ is the nomalized unit diection vecto vesion of R in the spacecaft body fame, and I is the spacecaft inetia tenso. The off-axis poducts in FUSE s inetia tenso ae 1% of the diagonal tems, so Eq. 1 can be appoximated by, GM R N y x x z z y ( I I ) zz yy ( ) xx zz I yy I xx ( I I ) whee x, y, and z ae the scala components of ˆ R. Inspection of Eq. (2) shows that gavity gadient toques ae geneally minimized when the spacecaft Z axis (the instument boesight) is pointed pependicula to the obital plane, at which point z is. The magnitude of the gavity gadient toque inceases as the boesight moves away fom the obit plane nomal vecto, eaching a maximum of ~5 m Nm; the components of this toque on each of the body axes depends on the attitude. Thee ae fou instantaneous gavity gadient toque minima when the boesight is pointed in the plane of the obit. The geomagnetic field can be accuately epesented in the eath centeed inetial (ECI) fame as a 1 th ode expansion in spheical hamonics, fo example the 1995 evision of the Intenational Geomagnetic Refeence Field (IGRF1995). Typical field stengths in FUSE s obital envionment vay between and T, and the field diection can also vay consideably as a function of obital phase. The toque geneated as the MTBs act against the geomagnetic field, (2) N mag = µ B, is highly dependent on the alignment of B in the body fame, but given the 14 Am 2 dipole moment of thee toque bas, is typically in the ange of 2-6 m Nm appoximately 1/1 th of the maximum RWA toque. 2 Ameican Institute of Aeonautics and Astonautics

3 II. Reaction Wheel Failues On 25 Novembe 21, the yaw RWA suffeed damatically inceased dag and ceased spinning, but science opeations continued with the edundant skew RWA contolling yaw. On 1 Decembe 21, the pitch RWA also suffeed a simila failue, leaving the spacecaft with only two axes of contol. The ACS was econfigued to use the emaining RWAs to stabilize the spacecaft in pitch and oll, but science opeations wee impossible, due to an uncontolled tumble in yaw. Effots to e-stat both the yaw and pitch RWAs have esulted in no detectable motion. Afte the yaw wheel failue in Novembe 21, while still opeating in thee-wheel mode, peliminay investigations began into the feasibility of using the MTBs to geneate contol toque. Geomagnetic toque has been used in conjunction with spin-stabilization fo quite some time, but this appoach is clealy incompatible with the existing design and thee-axis contol equiements of FUSE. It has also been suested as a method of contol fo satellites whose design povides inheent gavity gadient stabilization 4, but science demands equie that FUSE obseve tagets all ove the sky and hold attitudes that do not minimize gavity gadient toque. At any ate, puely magnetic pointing contol has only been applied to missions whee the toleances fo attitude contol wee at the 1º level 5. Since toque can neve be geneated about the instantaneous geomagnetic field vecto, any mission that uses magnetic contol toques must have additional actuatos, o accept attitude distubances about a vecto that is moving elative to inetial space. In the case of FUSE, the actuatos ae the two emaining eaction wheels. Initial calculations showed that the MTBs could be commanded with sufficiently high bandwidth fo fine pointing contol within the science equiements, and that they could poduce enough toque to cancel extenal distubances, but only at some spacecaft oientations. Afte the second pemanent RWA failue, simultaneous effots began to upgade the ACS softwae to accomplish magnetic contol, and to develop gound-based models useful fo pedicting stable spacecaft oientations. This development, as well as othe unelated ACS modifications, has also been summaized in Ref. 6. III. Attitude Contol System Upgades On 24 Januay 22, upgaded ACS flight softwae was installed, and has been incementally optimized ove the two following yeas. The most notable changes ae descibed below. A. Coodinate system change A new othogonal coodinate system was adopted, with a symmetic axis S ˆ = ( Xˆ + Yˆ) 2 and an antisymmetic o magnetic axis A ˆ = ( Xˆ + Yˆ) 2. The Z-axis emains the same, aligned with the stillfunctioning oll eaction wheel. The new S-axis is the pojection of the skew RWA axis onto the XY plane; the A- axis, which must be totally contolled with magnetic toque, is othogonal to both S and Z. This new SAZ coodinate fame thus completely decouples wheel and magnetic contol toques fom each othe. B. Contol toque geneated by MTBs The wheel momentum unloading algoithm was modified to calculate sepaate MTB dipole moment commands [ ] T, the fo unloading and contol. Given a geomagnetic field vecto otated into the SAZ body fame, B S B A B z basis vecto e ˆ 2 = [ B z B s ] T [ B z B s ] T () is pependicula to the A-axis and nealy pependicula to the instantaneous B field. Toques exeted about this diection will act to unload excess wheel momentum without geneating toque about the A-axis. A second basis vecto, e ˆ 1 = e ˆ 2 B e ˆ 2 B (4) is othogonal to both B and the unloading basis vecto, so toques about this diection affect contol ove the A-axis. Ameican Institute of Aeonautics and Astonautics

4 Using these basis vectos, dipole moment commands fo pointing contol and unloading ae µ contol N A axis e ˆ 1 B ( ) and µ unload H e ˆ 2 B ( ) (5) whee N A axis is the A-axis component of contol toque equested by the ACS, and H is a scaling facto deived fom the diffeence between the emaining RWAs momentum and thei commanded momentum bias. C. Inceasing MTB bandwidth The ACS hadwae can only contol the toque bas on o off and uses pulse width modulation (PWM) to geneate a time-aveaged dipole moment vecto that appoximates that equested by the unloading and contol algoithm. The PWM cycle was educed fom 1 to seconds in ode to incease the bandwidth of the A-axis toque command, and thus educe pointing jitte. D. Pioitizing contol ove unloading Each of the thee MTBs has a ~14 Am 2 dipole moment, and the diect addition of the desied contol and unloading dipole commands can exceed this satuation value. In this case, the final dipole moment vecto has a diffeent diection than desied, and optimal A-axis toque will not be ealized. To pevent this, contol dipole is decoupled fom and pioitized ove the unloading dipole by adding a linea combination of unload-only and contol-only dipoles by use of Eq. (5), whee µ = +, with k 1, such that µ x,y,z 14 Am 2 (6) MTB µ contol kµ unload µ MTB is the final dipole moment command that is to be ealized by the MTBs. E. Minimization of high-fequency contol components In Eq. (6), a satuated µ contol guaantees that k=, in which case thee will be no momentum unloading. If this condition exists long enough in a non-consevative gavity gadient envionment, the RWA momenta will gow unacceptably high. The ACS was modified to use a low bandwidth contolle at times when fine pointing contol was not equied. This pevents oscillation between positive and negative MTB-satuating dipole moment commands, which effectively blocks wheel momentum unloading. Eo (acsec) Figue 1. Measued A-axis attitude eo duing a one obit span of fine-pointing contol. Gay lines indicate ±.75 RMS jitte (1σ). Within minutes of the activation of magnetic toque contol, thee-axis contol was egained. Figue 1 shows typical measued attitude eo on the A- axis. The RMS pointing jitte of ~.75 (1σ), as indicated by the dashed gay lines in Fig. 1, is still too small to measuably affect the science data. IV. Pedictive Techniques The ACS eacts to A-axis distubances by geneating MTB commands as descibed above, but the available toque is limited and may not be sufficient to maintain pointing contol fo an abitay attitude at a given time. Gound-based pedictions must theefoe be used to constain science opeations so that obsevations ae scheduled at spacecaft oientations whee sufficient toque can be geneated to maintain fine pointing contol duing science exposues, and so that maneuves can follow thei planned tajectoy. The impotant aspects of these pedictive techniques ae descibed below. 4 Ameican Institute of Aeonautics and Astonautics

5 A. Static Pointing Even when the spacecaft holds a fixed attitude, the gavity gadient toque and geomagnetic field vectos change thei diection and magnitude duing the couse of one obit. No magnetic contol toque can be geneated when B is paallel to the A-axis; when the field stength dops, the maximum magnetic toque becomes popotionally smalle. Accounting fo all these effects mathematically, the ACS can maintain fine contol of the A- axis only when the elation N A = [ N + ( B) ] Â = µ (7) is satisfied. Checking that this elation will hold fo a given attitude ove a peiod of time, using an obit popagato and magnetic field model to pedict N and B, is now a fundamental planning constaint as science obsevations ae scheduled. In pactice, it is computationally feasible to check fo the satisfaction of Eq. (7) in 6 second time steps fo all tagets available fo planning. Equation 7 can be adapted to show the absolute magin of excess toque authoity available at any instant. Given a hypothetical toque command that fully satuates all toque bas and esults in a spacecaft dipole moment In Fig. 2, the gay shaded egion gaphically depicts the maximum toque envelope duing an obit-long dwell at an attitude whee the A-axis gavity gadient distubance toques ae consistently less than the maximum A-axis toque authoity. At this attitude, Eq. 8 would yield positive values thoughout, and the ACS will be able to maintain acsecond level fine pointing contol. Figue shows a one-obit simulation ove the same timespan, but at a diffeent spacecaft attitude whee the distubance toques ae at times geate than the maximum toque authoity. Even though the ECI geomagnetic field vectos ae identical in the simulations shown in Fig. 2 and Fig., the toque authoity envelope in µ, then, N max MTB µ (8) ( B) Â Nˆ Â magin = max Toque (m Nm) Toque (m Nm) Distubance toque Toque authoity envelope Figue 2. A-axis toque authoity envelope at a stable oientation Figue. A-axis toque authoity envelope at an unstable oientation. Fig. is moe esticted, because this paticula spacecaft attitude has esulted in a less favoable magnetic field vecto alignment with the MTBs. Additionally, the component of gavity gadient distubance on the A-axis is lage at the oientation in Fig. than in Fig. 2. The satellite s yaw and pitch ae detemined by the celestial coodinates of the taget being obseved, but in most obsevation scenaios, oll about the instument boesight can be adjusted though a ange of 5, while staying within powe and themal constaints. This fee paamete can often be used to impove the elative alignment of gavity gadient and B-field vectos and theeby incease the duation of positive toque authoity. 5 Ameican Institute of Aeonautics and Astonautics

6 Magin (m Nm) Attitude eo (acmin) Authoity loss The most consevative appoach would be to schedule obsevations only at times and attitudes when they ae pedicted to have positive toque authoity thoughout the duation of the visit. Since FUSE is in low-eath obit, the eath often blocks the telescope boesight duing a faction of each obit, at which time science exposues ae halted. Duing these peiods, modest (<5 ) attitude eos have no impact on scheduling efficiency o satellite health and safety, and can be allowed as long as they have been cancelled out by the time the science taget becomes visible again. Small (<.5º) attitude eos ae even acceptable duing peiods of taget visibility, as long as the eos can be pedicted duing planning, and science exposues can be halted and esumed at the appopiate times. Although losses of toque authoity in visibility do educe science efficiency, they often yield a damatic incease in scheduling flexibility. Once the stat and end times of the loss of toque authoity, t and t 1, have been identified by epeated examination of Eq. (7), the attitude angula eo that esults can be found by integation of the negative toque magin, α ω θ Attitude ecovey Pe-planned science downtime Figue 4. A pedicted loss of toque authoity. eo eo eo = I = = t1 t t1 t 1 ω (( µ max B) Â Nˆ Â ) α eo eo ( t) dt ( t) dt whee I -1 is the invese of the spacecaft inetia tenso. Although the time equied to coect a given attitude eo could also be found analytically, in pactice an empiical model elating θ eo fom Eq. (9) to ecovey time has poven suitably accuate. Figue 4 shows an example of a small (24 acmin) angula eo esulting fom a shot loss of toque authoity duing a peiod of taget visibility. The uppe panel of the plot shows pedicted toque authoity, as defined by Eq. 8, which is negative fo appoximately seconds. The shaded gay box in the lowe panel indicates the pedicted duation (epesented by the box s width) and angula eo magnitude (epesented by the box s height) esulting fom the loss of toque authoity. The solid line in the lowe panel shows actual measued angula eo. Since science exposues wee halted just befoe the stat of the loss of toque authoity, and wee esumed appoximately 85 seconds late, this pedicted loss of toque authoity had no effect on science collection. B. Maneuves The ACS must still cancel gavity gadient toques duing maneuves, but additional toque is needed to acceleate and deceleate the spacecaft body and cancel gyoscopic toques. Consequently, it is not necessaily tue that a slew will have sufficient toque authoity, even if the maneuve is between two stable spacecaft oientations. (9) 6 Ameican Institute of Aeonautics and Astonautics

7 6 Contol toque 6 Ramp up Toque (m Nm) Toque authoity envelope -6 Ramp down 1 2 a) 1 2 b) Figue 5. Toque authoity envelopes duing slews. a) stable maneuve, b) unstable maneuve. Figue 5a shows the A-axis contol toque of a slew that has been timed to phase the amp-up s negative A-axis acceleation toque with a positive local maximum in A-axis gavity gadient, and has a simila beneficial phasing duing the ampdown stage of the slew. At all times, the MTBs ae able to geneate enough A-axis toque fo the slew to follow the ideal tajectoy. Figue 5b shows an impopely phased slew in which the total A-axis toque equest exceeds toque authoity, both duing acceleation and deceleation. Duing the amp up, the A-axis position will lag the optimal path, and duing amp-down, the A-axis will oveshoot. FUSE Mission Planning tools simulate each slew as it is scheduled, and choose an optimized stat time that guaantees the maximum toque magin thoughout. In pactice, even 18º slews ae usually completed with little o no A-axis pointing eos duing the couse of the maneuve. V. Sky Coveage Equation 7 can be applied to a single spacecaft oientation ove a ange of times in ode to identify windows of stability in which an obsevation may be scheduled. The same equation can also be applied to ange of sky positions, each of which implies a specific spacecaft oientation. The Aitoff pojection all-sky maps in Fig. 6 show in black the locus of instument pointings that will have a net positive toque magin on the A-axis; in white egions, thee is insufficient toque authoity to keep the spacecaft stable. Figue 6a shows stable egions of the sky at one instant, and 6b shows stable egions at anothe instant, 4 minutes late. Clealy, egions of stability change damatically as the gavity gadient and geomagnetic field vectos change. Figue 6c shows in black only egions of the sky that will have continuous positive toque authoity thoughout an entie 1 minute-long obit. a) b) c) Figue 6. Aitoff pojection of sky egions with positive toque authoity (black) and insufficient toque authoity (white). a) At an instant in time, b) at anothe instant in time 4 minutes late, c) positive toque authoity continuously ove one 1 minute-long obit. The constaint of maintaining positive toque authoity thoughout an obsevation must be taken in combination with othe mission planning constaints such as luna, sola and boesight-to-am diection avoidance. The union of all these constaints leaves only small egions of sky availability on any given day, but ove time these egions cove diffeent pats of the celestial sphee, thanks to egession of the obit plane and motion of the eath about the sun. The total sky availability, shown in Fig. 7, illustates that, while some egions have less time available fo scheduling than othes, tagets at any point in the sky can be scheduled at some point duing a calenda yea. 7 Ameican Institute of Aeonautics and Astonautics

8 Availability (days) 6 <2 <1 <5 9h 6h h 21h 18h 15h <21 <5 - = -6 Figue 7. Cumulative sky availability ove one yea. VI. Conclusion Modifications to the FUSE attitude contol system and mission planning softwae have allowed the continuation of an efficient and poductive campaign of science obsevations afte the failue of two eaction wheels. While the new contol scheme allows less flexibility in scheduling than was enjoyed pio to the wheel failues, the entie sky is available fo science obsevations, and fine pointing jitte is nealy as low as befoe. Since ecoveing thee-axis contol in Januay 22, FUSE has conducted ove 25 science obsevations. Acknowledgments The authos acknowledge the suppot of the FUSE Opeations and Science teams, and thank the staff at Obital Sciences Copoation fo assistance in the development effots descibed in this pape. The FUSE poject at Johns Hopkins Univesity is funded by NASA contact NAS Refeences 1. Moos, H. W., et. al., Oveview of the Fa Ultaviolet Spectoscopic Exploe Mission, Astophysical Jounal, Vol. 58, 2, L1-L6. 2. Sahnow, D. J., et. al., "On-Obit Pefomance of the Fa Ultaviolet Spectoscopic Exploe Satellite", Astophysical Jounal, Vol. 58, 2, L7-L11.. Wetz, J. (ed) Spacecaft Attitude Detemination and Contol, Kluwe Academic Publishes, Nowell, MA, 2, pp Aduini, C., and Baiocco, P., Active Magnetic Damping Attitude Contol fo Gavity Gadient Stabilized Spacecaft, Jounal of Guidance, Contol and Dynamics, Vol. 2, No. 1, 1997, pp Wi niewski, R., Linea Time-Vaying Appoach to Satellite Attitude Contol Using Only Electomagnetic Actuation, Jounal of Guidance, Contol and Dynamics, Vol. 2, No. 4, 2, pp Kuk, J.W., Class, B.F., Rovne, D. and Westphal, J., Ake, T.B., Moos, H.W. and Robets, B., & Fishe, L. FUSE In-obit Attitude Contol with Two Reaction Wheels and No Gyoscopes. Poc SPIE, 4854, 22, 2. 8 Ameican Institute of Aeonautics and Astonautics