THREE AXIS CON TROL OF THE HUB BLE SPACE TELE SCOPE USING TWO RE AC TION WHEELS AND MAG NETIC TORQUER BARS FOR SCI ENCE OB SER VA TIONS

Transcription

1 AAS HREE AXIS CON ROL OF HE HUB BLE SPACE ELE SCOPE USING WO RE AC ION WHEELS AND MAG NEIC ORQUER BARS FOR SCI ENCE OB SER A IONS Sn Hr-Diaz, * John Wirzbrger and Dan Smith he Hb ble Space ele scope (HS) is re nowned for its s perb point ing ac cracy of less than 10 milli-arcseconds ab so lte point ing er ror. o ac com plish this, the HS re lies on its com ple ment of for re ac tion wheel as sem blies (RWAs) for at ti tde con trol and for mag netic torqer bars (MBs) for momen tm man age ment. As with most sat el lites with re ac tion wheel con trol, the forth RWA pro ides for falt tol er ance to main tain three-axis point ing ca pabil ity shold a fail re oc cr and a wheel is lost from op er a tions. If an ad ditional fail re is en con tered, the abil ity to main tain three-axis point ing is jeopar dized. In or der to pre pare for this po ten tial sit a tion, HS Pointing Con trol Sb sys tem (PCS) eam de el oped a wo Re ac tion Wheel Sci ence (RS) control mode. his mode ti lizes two RWAs and for mag netic torqer bars to achiee three-axis sta bi li za tion and point ing ac c racy nec es sary for a con tin ed sci ence ob ser ing pro gram. his pa per pres ents the de sign of the RS mode and op er a tional con sid er ations nec es sary to pro tect the space craft while al lowing for a sb stan tial sci ence pro gram. INRODUCION he Hbble Space elescope (HS) was lanched in April 1990 and since that time has been renowned for its sperb pointing accracy of less than 10 milli-arcseconds absolte pointing error. o accomplish this, the HS relies on its complement of for reaction wheel assemblies (RWAs) for attitde control and for magnetic torqer bars (MBs) for momentm management. As with most satellites with reaction wheel control, the forth RWA proides for falt tolerance to maintain three-axis pointing capability shold a failre occr and a wheel is lost from operations. If an additional failre is encontered, the ability to maintain three-axis pointing is jeopardized. Oer the corse of the 18 year operational life of HS, there hae been two RWA anomalies on-orbit. De to the niqe ability for HS to spport sericing by the Space Shttle, these two RWAs were replaced in Febrary 1997 and March With the decommissioning of the Shttle planned prior to the HS end of life, the ability to serice the telescope will be greatly diminished. In order to prepare for this potential sitation, the HS Pointing Control Sbsystem (PCS) eam deeloped a wo Reaction Wheel Science (RS) control mode. his mode tilizes two * Emer gent Space ech nol ogies, Inc., Greenbelt, Mary land sn.hr-diaz@emergentspace.com. Honeywell echnology Soltions, Inc., Colmbia, Maryland jwirzbrger@hst.nasa.go. Lockheed Mar tin Mis sion Ser ices, Greenbelt, Mary land dsmith@hst.nasa.go. 335

2 RWAs and for magnetic torqer bars to achiee three-axis stabilization and pointing accracy necessary for a contined science obsering program. With jst two operational reaction wheels, there exists one axis where no wheel control is aailable. Magnetic torqer bars can proide control abot the wheel-less axis, bt they mst also contine to dmp momentm from the wheels to preent wheel speed satration. Becase of the redced wheel momentm management capability of the bars, the wheel speeds can exceed those from normal operations and mst be tolerated by the ehicle and safing systems. Since the MBs can only generate a torqe in the plane perpendiclar to the Earth s magnetic field, no control torqe can be generated abot the wheel-less axis if the magnetic field aligns with the wheel-less axis. his means that certain maneers or attitdes may not be feasible and can lead to an ncontrolled ehicle state. herefore, carefl planning of science attitdes and maneers is necessary, and significant modification of the crrent planning and schedling tools is foreseen. Redced-wheel pointing capability had been considered in other missions as well, notably the Far Ultraiolet Spectroscopic Explorer (FUSE) 1,2,3. Unlike FUSE, HS can experience significant aerodynamic torqe which can be on the same order of magnitde as the graity gradient torqe depending on atmospheric density conditions. HS s pointing reqirement is also more stringent which places stricter constraints for the redced-wheel operations. his paper presents the design of the RS mode and the operation considerations necessary to protect the spacecraft while allowing for a sbstantial science program. his mode can operate with any pair of reaction wheels and assmes for magnetic torqer bars and at least three gyros. he paper begins with a brief backgrond followed by the algorithms and some simlation reslts. hen key isses in planning and schedling are discssed. BACKGROUND he location and direction of the for RWAs on HS along with the definition of the ehicle frame are shown in Figre 1. he boresight of the telescope is along the +1 axis. Figre 1 Reaction wheel assembly configration here are six possible pairs of wheels that may be in RS operation. he control athority, hence the performance, depends on the particlar pair of reaction wheels sed as well as the MB capability at arios leels of aerodynamic torqe. he maximm torqe from a single wheel is 0.82 N-m for wheel speeds below 3200 RPM. As the wheel speed increases aboe this ale, the torqe otpt begins to diminish. here is also a software limit on the wheel speed for safety. 336

3 he torqe otpt from the MBs depends on their orientation relatie to the Earth s magnetic field strength and direction. he maximm dipole moment from a single bar is 3500 amp-meter 2. he maximm torqe that can be prodced by all for bars is 0.34 N-m. Figre 2 shows the distribtion of maximm graity gradient and aerodynamic torqes in the body frame as well as the maximm MB torqe and acceleration capability. he wheel-less axis directions of the six possible pairs of operating wheels are also shown Graity Gradient Aerodynamic (=9.6e-12 kg/m 3 ) otal External Bar Acceleration Capability Bar orqe Capability Figre 2 External torqe and MB capability distribtion he MB acceleration capability distribtion shows that wheel pairs (1,2) and (3,4) hae the worst control athority abot the wheel-less axis as well as the worst external torqes. Performance for these two-wheel configrations is expected to be worse than the other for twowheel configrations. For adeqate control in all three axes, there needs to be enogh control athority abot the wheel-less axis by the MBs. his means that certain maneers or attitdes may not be feasible and can lead to an ncontrolled ehicle state. herefore, carefl planning of science attitdes and maneers is necessary. In addition to the maximm torqe realizable by the MBs, their dynamics are also important. Figre 3 shows the time response from a PSPICE model of PSPICE ME Response amp-m 2 to 1800 amp-m 2 (-450 ma to 450 ma) 600 Crrent (milliamps) deg C 24 deg C 71 deg C ime (seconds) Figre 3 Magnetic orqer Electronics ime Response 337

4 the magnetic torqer electronics at three different temperatres. he general response is that of a second-order system. o characterize the torqe capability abot the wheel-less axis, the following parameter is defined: AWX w x B ( eh / M M ) w x GG AERO ( I) IaMR (1) w ( h where a w B x MR eh / RWA h RWA x eh / RWA I ehicle inertia RWA magnetic field ector in ehicle frame maneer acceleration command nit ector in the wheel - less axis in ehicle frame eh / M transformation matrix from MB frame to ehicle frame M bar dipole moments in MB frame GG graity gradient torqe AERO aerodynamic torqe ehicle anglar rate in ehicle frame transformation matrix from wheel to ehicle frame nominal wheel momentm in wheel frame A positie AWX assres controllability abot the wheel-less axis. For planning and schedling prposes, some of the ariables are assmed nominal ales. For maximm torqe from the bars abot the wheel-less axis, the bar moments in the aboe eqation are set to sign( w B (2) M max x ) where [ ] X represents the cross-prodct matrix of its argment. For maximm wheel gyroscopic torqe abot the wheel-less axis for a gien maneer axis, wheel momentm is set to h h sign w (3) RWA RWA _ nom X M ) ( x X eh / RWA Dring non-maneer periods, the acceleration command is zero and the two gyroscopic terms containing ehicle anglar elocity are negligible. Dring maneers, the acceleration command and the ehicle anglar rate are the expected ales compted by the maneer command generator. he expected maneer attitde profile is also sed in the comptation of the magnetic field in the ehicle frame as well as the graity gradient torqe and the aerodynamic torqe. ALGORIHMS One of the main considerations in the design of the RS mode for HS was to minimize changes to the existing flight software (FSW). herefore, the existing proportional-integralderiatie control architectre for the normal 3- and 4-wheel modes was left intact as mch as possible. A major deiation is the comptation of the feedback control in a wheel-orthogonalframe (WOF) to maximize controllability in the wheel plane. Feedback control is compted in 338

5 WOF with gains in the wheel-plane generally higher than the gains in the wheel-less axis since the wheels hae higher actator capability than the bars. Figre 4 illstrates WOF defined by a pair of any two wheels A and B. Wheel Plane Normal (Wheel-less axis) ŵ X ŵ N rb Wheel B w Wheel A A r A Figre 4 Wheel-Orthogonal-Frame (WOF) he transformation from the wheel orthogonal frame to the ehicle frame is gien by / WOF w A w N w X r A r B r A r B r A r A r A r B r A r B where the parameters with the ^ symbol indicate nit ectors in the ehicle frame and the other ectors are defined in Figre 4. RS Control Wheel Plane Becase the MBs are needed for both ehicle control and wheel momentm management, the primary problem to sole is how to distribte these two potentially competing tasks. hree possible approaches were explored and are described below. Method 1. he first method takes the feedback control that was transformed to the ehicle frame and sms it with the feedforward control torqe compensation comprised of momentm management torqe and other known torqes. his total torqe is sed for compting the wheel command as well as for compting the deficit torqe, which is the reqired control torqe not achieable by the wheels. he total magnetic dipole command for the torqer bars is compted from a combination of the deficit torqe and momentm management torqe. he dipole moments of the torqer bars are compted from a modified cross-prodct law described in a later section. o inclde in the feedforward control torqe, the momentm management torqe is processed throgh the cross-prodct law, and the actal momentm management torqe that can be prodced from the bars is compted as follows: B MM MM (5) 2 B MM B where MM is the dipole moments of the bars also gien in the ehicle frame. he negatie of this momentm management torqe is fed forward along with other known feedforward torqes so that the total desired control torqe is gien by (4) _ ACUAL MM (6) 339

6 C FB FB FF MM _ ACUAL FF _ OHERS (7) he control torqe that is realizable by the wheels is gien by / lim / (8) C eh RWA LIMRW RWA eh where eh / RWA is transformation from the wheel frame to the ehicle frame, and RWA/ eh is the transformation from the ehicle frame to the wheel frame. hese transformations are shown below: where C eh / RWA ra rb (9) 1 RWA / eh eh / RWA eh / RWA eh / RWA r A r B (10) nit ector nit ector of wheel A in ehicle frame of wheel B in ehicle frame Note that the total control torqe transformed to the wheel frame is limited by LIMRW before transforming back to the ehicle frame. he control torqe that is not realizable by the wheels is the deficit torqe gien by DEFICI eh / WOF lim WOF / eh C C (11) RSLMM which is to be compensated by the bars as mch as possible. he limit RSLMM, applied in WOF, controls how mch deficit torqe is compensated. he total torqe to be prodced by the bars is then a combination of the deficit control torqe and the momentm management torqe. he corresponding bar moment command is then gien by: lim M MAX lim M / eh M / eh MAX t DEFICI B B MM DEFICI 2 where M / eh is the (4x3) transformation from the ehicle frame to the MB frame, Max is the maximm moment limit, and is a scaling factor applied to achiee the fll desired control torqe (within the bar limit) and is gien by DEFICI MM (12) 2 2 B DEFICI (13) 2 B he scaling of the bar torqe is illstrated in Figre 5. With scaling, the projection of the magnetic torqe along the deficit torqe direction prodces the desired magnitde proided that 340

7 the bars are not satrated. If the magnetic field ector and the deficit torqe are aligned, then is set to 1. DEFICI orqe from nominal cross-prodct law orqe from cross-prodct law with scaling Figre 5 Scaling of the bar torqe command for control Any extraneos torqe in the wheel plane prodced by the bars is compensated by the wheels. his torqe is the desired magnetic torqe mins the actal: RWA _ M _ COMP DES ( eh / M M ) B (14) where the desired magnetic torqe is gien by (15) DES DEFICI MM _ ACUAL he total command to the wheels is pdated with the compensation torqe: (16) C C RWA _ M _ COMP he total torqe to be prodced by the wheels is gien by the pdated command lim (17) RWA RWA _ Max RWA / eh Method 2. In the second method, the desired torqe is decomposed into a non-orthogonal UN frame defined by the nit ector n along the intersection of the plane of magnetic control athority and the plane of the wheel control athority which is gien by w X B n, (18) w B X the nit ector in the plane of wheel control athority gien by w X n, (19) w n X and the nit ector in the plane of magnetic control athority gien by B n. (20) B n Figre 6 illstrates the UN coordinate frame. C 341

8 342 Figre 6 UN Coordinate Frame he control torqe that is originally compted in the wheel-orthogonal-frame is conerted to the ehicle frame and then conerted to this non-orthogonal UN frame as follows: n n n 2 2 (21) he commands to the wheels and the bars are then allocated as follows. n n eh RWA RWA Max RWA lim / _ (22) MM eh M M B B MAX 2 / lim (23) he control torqe component abot the n direction (intersection of the wheel plane and the magnetic control athority plane) is allocated flly to the wheels since the wheels hae a higher torqe capability. here is a singlarity when the magnetic field ector is aligned with the wheel-less axis. In this case, torqe abot the wheel-less axis is zero and we only command the control torqe in the wheel-plane. Method 3. he third method inoles a least-sqares soltion to optimally combine the wheel torqe and the bar torqe to prodce the desired control torqe. Since momentm management torqe mst be proided by the bars in order to redce the system momentm, it is not combined with the control torqe when sing this concrrent method.

9 he system of eqations to sole for is: C B eh / RWA B X eh / M RWA M _ C (24) where is the nknown control ariable. he aboe eqation represents an nder-determined system since there are 6 nknowns and 3 eqations, so there is no niqe soltion. One possible soltion is the minimm norm soltion which can be soled by forming an objectie fnction agmented with the aboe eqality constraint sing a Lagrange mltiplier: min J W B (25) where W is a sqare matrix of weights on the nknowns. We take the partial of J with respect to and, set each partial to zero, and sole for. After some matrix algebra, we get C 1 BW B C 1 1 W B (26) which is essentially a psedo-inerse of the eqality constraint eqation. he wheel torqe command for control and the bar moment command for control are then gien by RWA lim (1: 2) RWA _ Max (27) (4 : 6) M _ C he bar command is limited after the momentm management component is added: lim (28) M MAX t M _ C M / eh he matrix BW 1 B is not inertible when the magnetic field is aligned with the wheel-less axis since the rank of the matrix wold then be only two. In this case, we can only command the torqe in the wheel plane. he psedo-inerse soltion becomes no longer optimal when the control ariables begin to satrate, and control limits wold hae to be incorporated. For example, the problem can be posed as a qadratic programming problem with linear and ineqality constraints and soled sing established nonlinear programming methods. Comparison of Methods. he three methods of torqe allocation presented aboe were simlated in Matlab. In a case where there is no actator satration, all methods performed identically. In a case where there is a period of little or no torqe athority abot the wheel-less axis, methods 2 and 3 performed worse than method 1. he torqe athority for the second case and the attitde errors for all three methods are shown in Figre 7(a) and (b), respectiely. MM 343

10 0.2 orqe Athority Abot Wheel-Less Axis (Wx) 4.5 x 104 re Error Magnitde Nm re Error, arc-seconds Methods 2 and 3 Method t (sec) (a) t (sec) (b) Figre 7 (a) AWX, (b) Attitde Error he lighter cre in Figre 7(b) corresponds to methods 2 and 3 whose attitde control error is abot 10 times worse than that of method 1. he reason is becase the wheels satrate in methods 2 and 3 and ths lose controllability. Figre 8(a) shows the wheel torqe history for Method 1, and Figre 8(b) shows the wheel torqe history for Methods 2 and Wheel orqe RWA 1 RWA 2 RWA 3 RWA Wheel orqe RWA 1 RWA 2 RWA 3 RWA 4 N-m 0 N-m t (sec) t (sec) (a) (b) Figre 8 Wheel orqe for (a) Method 1 (b) Methods 2 and 3 For Method 2 or 3 to be iable, actator limits will hae to be addressed. his will be the sbject of a ftre paper. Momentm Management For minimal FSW change, the nominal momentm management (MM) proportionalderiatie control law is maintained except that the MM command is limited to be only along the direction that wold not corrpt control. If the control torqe can be flly achieed, i.e., there is sfficient torqe athority abot the wheel-less axis, then sch restriction is not necessary. Howeer, to aoid redcing this torqe athority with nnecessary corrption from the momentm management torqe, only the n -component of the nominal momentm management torqe,, is applied: MM 344

11 MM n n (29) When the wheel-less axis is aligned with the magnetic field ector within a tolerance, the intersection line is not defined. In this case, most of the realizable momentm management torqe is already in the wheel plane and only the component of momentm management torqe that is in the wheel plane is commanded. Maneer Planning For maneer planning in RS, the same 3- and 4-wheel command generator algorithm is sed which is based on an eigen-axis maneer from an initial attitde to a final attitde. he command generator determines the ehicle rate profile for the control system to follow that satisfies the maximm rate, maximm acceleration, and maximm jerk parameters. For 3- and 4- wheel operations, these parameters are fixed for all maneers. For 2-wheel operation, on the other hand, these parameters hae to be specified per maneer becase the amont of control athority abot the wheel-less axis is dependent on the wheel configration, the ehicle attitde and orbit, atmospheric density, as well as the magnetic field strength and alignment. he comptation of these parameters, which will be performed on the grond by planning and schedling, is discssed in the next section. OPERAIONAL CONSIDERAIONS With jst two reaction wheels, there exists one axis where no wheel control is aailable. Magnetic torqe bars can proide control abot the wheel-less axis, bt they mst also contine to dmp momentm from the wheels. Becase of the redced wheel momentm management capability of the bars, the wheel speeds can get high. o allow for operation in this mode, the wheel speed safing limit is increased bt to a leel that wold still allow enogh margin for safe mode. For arios operational modes of RS, the minimm torqe athority has to be specified AWX AWX Min (30) Becase of the dependency on the aerodynamic drag torqe, the minimm torqe athority aries with the expected leel of the atmospheric density. Attitde Hold and Science For attitde hold and science modes, AWX mst be sfficient to preent nallowable attitde errors and maintain RWA momentm adeqately to allow for ehicle maneers and preent safemodes. It is obios that AWX mst be greater than 0 N-m at all times dring science interals. Once AWX becomes negatie, the external torqes become larger than the control torqes and the ehicle wold incr attitde errors and loss of lock on the target. At 0 N-m, there is no additional torqe aailable for the feedback control loop as all the control torqe aailable is dedicated to controlling expected external torqes. Some attitde error is acceptable dring attitde hold portions, as long as the schedling system protects for the largest error possible. Howeer, once AWX becomes negatie, momentm management of the RWAs is no longer garanteed and wheel speeds increase beyond acceptable limits. wo major performance criteria are attitde errors and wheel speeds. Figre 9 shows the maximm attitde errors relatie to minimm AWX from a Monte Carlo simlation of 1000 random initial conditions for wheel configration (1,2) with each case simlated for 3.5 orbits and MM 345

12 with an aero density of 3.5e-12 kg/m 3. Figre 10 shows the wheel speeds relatie to the minimm AWX for all 1000 cases. (a) (b) Figre 9 Maximm attitde error (a) all 1000 cases shown, (b) Close-p 346

13 Figre 10 Maximm Wheel Speeds Based on the simlations, setting AWX Min to N-m wold maintain attitde errors below the science acqisition search radii and maintain RWA speeds below the proposed safing speed limit. Once HS transitions to RS science mode, the absolte pointing error is maintained below 20 milli-arcseconds for the worst case wheel geometry with the inclsion of attitde information from the Fine Gidance Sensors. his increase in pointing error oer the nominal science pointing error with three RWAs of less than 10 milli-arcseconds is primarily de to the lagged response of the MB as compared to the RWAs. Figre 11 shows the probability of haing a certain torqe margin for the entire dration of a 3.5 orbit science obseration at a specific time for wheel configration (1,2) and (1,3) for aero density ale of 3.5e-12 kg/m 3. As expected, the efficiency is lower for the (1,2) configration becase the torqe athority abot the wheel-less axis is less. Positie torqe margin occrs approximately 40 percent of the time for wheel configration (1,2) and 60 percent of the time for wheel configration (1,3). When performing the calclation of AWX oer attitde hold or science interals, the selection of atmospheric density is critical. he density chosen mst be the maximm density expected to be seen on-orbit dring the interal or higher, not a mean ale. his ensres torqe athority and bilds in a slight pad for times when the actal density is lower than the schedling density. If a mean density ale is sed, AWX Min mst be reealated. 347

14 Wheel Configration (1,3) Wheel Configration (1,2) Figre 11 Science Planning Efficiency One of HS s defining performance characteristics is its jitter leel below 7 milli-arcseconds allowing for clear scientific images. Since HS is relatiely insensitie to jitter abot the boresight axis, this metric is defined as the Root-Smmed Sqared of the 60-second standard deiation of the pointing error in 2 and 3. With the increase in actator rise time incrred by sing the MB as opposed to a RWA, the fine control of HS is degraded. he amont of degradation of jitter is dependent on the remaining two RWAs. he RWA 1-2 and the RWA 3-4 configrations place the wheel-less axis in the 2/3 plane. his reslts in the maximm jitter transferred to the boresight as the total affect of the increased jitter de to the MBs is captred in the jitter calclation. Anticipated jitter leels for the worst-case wheel configrations are on the order of 12 milli-arcseconds, while jitter for the other configrations meets the 7 milli-arcseconds goal. he impact on the science mission de to the increase jitter wold be minimal shold HS fall into a worst-case wheel configration. Similarly, the ability for HS to track a moing target, sch as a planet or comet, cold be affected. his moing target track makes se of feedforward acceleration to keep the target in the field of iew. hrogh carefl gain selection, no significant degradation was obsered in moing target tracking. hrogh carefl design of the attitde hold and science modes, there is minimal performance degradation in RS mode. Howeer, a transition to RS is not withot cost. he biggest impact to these modes is the ability to schedle attitdes that hae sfficient torqe margin in the wheelless axis to control attitde errors and RWA speeds. Maneers For a gien maneer, the command generator parameters max, maximm elocity; Amax, maximm acceleration; and start, maneer start time, that satisfy the minimm AWX 348

15 reqirement will hae to be determined. A combination of these command generator parameters that optimizes an objectie fnction, e.g., minimm end-of-maneer time, can be fond: min start slew (31), max,amax start sbject to AWX AWX Min (32) he parameter start is to allow delays in the maneer for a possible improement in the magnetic field and orbit, and slew is the maneer dration compted in the command generator. he lower and the pper limits of the sole-for command generator parameters in the search space will hae to be specified for the arios 2-wheel configrations. In addition, the minimm torqe-athority abot the wheel-less axis AWX Min has to be specified. Becase aerodynamic torqe can be a significant torqe on the HS, all of these parameters will hae to be specified for different leels of atmospheric density. A method for soling the optimization problem is a global parametric search based on discrete ales of the sole-for parameters where each combination of the sole-for parameters is simlated oer the maneer dration. Cases that iolate the AWX constraint are rejected, and the combination that minimizes the objectie fnction is selected. Recall that gyroscopic torqe in the comptation of AWX in Eqn. (1) is negligible except dring maneers. Since the momentm of the wheels at the start of a maneer cannot be predicted, AWX is compted sing a nominal wheel momentm ale as shown in Eqn. (3). Figre 12 shows plots of AWX from a sample simlation. he top plot shows AWX withot the wheel gyroscopic term sing nominal ales as well as the actal ales. he bottom plot shows AWX with the gyroscopic term sing a nominal momentm in each wheel of 250 N-m-s as well as the actal wheel momentm. Figre 12 shows the attitde error corresponding to this case. 0.2 Nominal and Actal abot Wx Nm 0 M -abs(gg+aero-wcxiwc) M -abs(gg+aero-wactxiwact) t (sec) Nominal and Actal AWX abot Wx 0.2 Nm 0 M - wcxhnom -abs(gg+aero-wcxiwc) M - wactxhact -abs(gg+aero-wactxiwact) t (sec) Figre 12 Nominal AWX sing h RWA_nom =250 N-m-s and the actal AWX for a sample case 349

16 6 otal attitde error 5 4 deg ime, seconds Figre 13 Attitde error corresponding to the case shown in Figre 12 he gyroscopic torqe from the npredictable momentm of the wheels can hae a significant effect on the control athority abot the wheel-less axis. For this particlar case shown, the actal wheel momentm reaches a maximm ale of abot 350 N-m-s per wheel. o assre that cases sch as this are not planned, the optimization problem gien by Eqns. (31) and (32) is soled sing a larger ale of the nominal wheel momentm or a larger ale of AWX Min. It is possible that a feasible soltion does not exist for a gien desired maneer. If no sitable maneer exists, the science timeline wold hae to be modified accordingly. CONCLUSION A preliminary design of the two-reaction wheel science (RS) mode for the Hbble Space elescope was presented. hree methods of combining wheel torqe and magnetic bar torqe for RS control were deried and compared. he first method was chosen for the preliminary RS design becase it has the least impact to the existing flight software code. Withot accommodation of actator limits, the latter two methods perform worse than the first when there is a period of negatie torqe athority abot the wheel-less axis. Consideration of actator limits in the latter two methods is the sbject of a ftre paper. Operational considerations in terms of the torqe-athority abot the wheel-less axis were also presented. Simlation reslts of both attitde hold and maneers show that acceptable performance can be had when there is sfficient torqe athority. ACKNOWLEDGMEN he athors wold like to thank Landis Markley and Peiman Maghami of NASA Goddard Space Flight Center for their peer reiew and sggestions of the RS design. REFERENCES 1. Roberts, Bryce A., et al, hree-axis Attitde Control with wo Reaction Wheels and Magnetic orqer Bars, AIAA Gidance, Naigation, and Control Conference and Exhibit, Paper 5245, Proidence, Rhode Island, Agst Krk, Jeffrey W., et al, FUSE In-Orbit Attitde Control with wo Reaction Wheels and No Gyroscopes, SPIE, ol. 4854, Paper 72, Sahnow, Daid J., Operations with the new FUSE obseratory: three-axis control with one reaction wheel, SPIE, ol. 6266, Paper 2,