A NOVEL SENSOR FOR ATTITUDE DETERMINATION USING GLOBAL POSITIONING SYSTEM SIGNALS

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1 AIAA A NOVEL SENSOR FOR ATTITUDE DETERMINATION USING GLOBAL POSITIONING SYSTEM SIGNALS John L. Crassds Senor Member AIAA Assstant Professor Department of Aerospace Engneerng Texas A&M Unversty College Staton, TX F. Lands Markley Fellow AIAA Aerospace Engneer GNC Center, Code 57 NASA-Goddard Space Flght Center Greenbelt, MD 77 Abstract. An entrely new sensor approach for atttude determnaton usng Global Postonng System (GPS) sgnals s proposed. The concept nvolves the use of multple GPS antenna elements arrayed on a sngle sensor head to provde maxmum GPS space vehcle avalablty. A number of sensor element confguratons are dscussed. In addton to the navgaton functon, the array s used to fnd whch GPS space vehcles are wthn the feld-of-vew of each antenna element. Atttude determnaton s performed by consderng the sghtlne vectors of the found GPS space vehcles together wth the fxed boresght vectors of the ndvdual antenna elements. Ths approach has clear advantages over the standard dfferental carrer-phase approach. Frst, errors nduced by multpath effects can be sgnfcantly reduced or elmnated altogether. Also, nteger ambguty resoluton s not requred, nor do lne bases need to be determned through costly and cumbersome self-surveys. Furthermore, the new sensor does not requre ndvdual antennas to be physcally separated to form nterferometrc baselnes to determne atttude. Fnally, development potental of the new sensor s lmted only by antenna and recever technology development unlke the physcal lmtatons of the current nterferometrc atttude determnaton scheme. Intal smulaton results ndcate that accuraces of about degree (3σ) are possble..copyrght c 998 by John L. Crassds. Publshed by the, Inc. wth permsson. Davd A. Qunn Aerospace Engneer GNC Center, Code 57 NASA-Goddard Space Flght Center Greenbelt, MD 77 Jon D. McCullough Aerospace Engneer GNC Center, Code 57 NASA-Goddard Space Flght Center Greenbelt, MD 77 Introducton The Global Postonng System (GPS) constellaton was orgnally developed to permt a wde varety of user vehcles an accurate means of determnng poston for autonomous navgaton. The constellaton ncludes 4 space vehcles (SVs) n sem-synchronous ( hour) orbts, provdng a mnmum of sx SVs n vew for ground-based navgaton. The underlyng prncple nvolves geometrc trangulaton wth the GPS SVs as known reference ponts to determne the user s poston to a hgh degree of accuracy. The GPS was orgnally ntended for ground-based and avaton applcatons, ganng much attenton n the commercal communty (e.g., automoble navgaton, arcraft landng, etc.). However, n recent years there has been a growng nterest n space-based applcatons. Snce the GPS SVs are n approxmately, km crcular orbts, the poston of any potental user below the constellaton may be easly determned. A mnmum of four SVs are requred so that n addton to the three-dmensonal poston of the user, the tme of the soluton can be determned and n turn employed to correct the user s clock. Snce ts orgnal ncepton, there have been many nnovatve mprovements to the accuracy of the GPS determned poston. These nclude usng local area as well as wde area dfferental GPS, carrer-phase dfferental GPS, and so-called pseudoltes (groundbased GPS transmtters). In partcular, carrer-phase dfferental GPS measures the phase of the GPS carrer relatve to the phase at a reference ste, whch dramatcally mproves the poston accuracy. Also, for spacecraft applcatons dynamcally aded GPS usng

2 orbt models wth GPS measurements n an extended Kalman flter can mprove poston accuracy. Early applcatons of ths concept to user spacecraft n Low Earth Orbts (LEOs) have demonstrated extremely useful results. 3 Recently, there have been nvestgatons of poston determnaton by user spacecraft from above the GPS constellaton. 4 Snce current GPS SVs transmt ther sgnals towards the Earth, ths concept poses a much more dffcult problem because the user spacecraft must rely on spllage sgnals receved from GPS SVs on the far sde of the Earth. Another aspect of space-based applcatons usng GPS that has ganed much recent attenton s atttude determnaton. One of the frst space-based applcatons was flown on the RADCAL (RADar CALbraton) spacecraft, 5 whch demonstrated a GPS atttude determnaton capablty usng post-processed measurements. To obtan maxmum GPS vsblty, and to reduce sgnal nterference due to multpath reflecton, GPS patch antennas were placed on the top surface of the spacecraft bus. Although the antenna baselnes were relatvely short for atttude determnaton (.67 meter separaton), atttude accuracy on the order of degrees per axs (3σ) was acheved. Another experment, Crsta-SPAS 6 provded the frst on-orbt demonstraton of real-tme atttude determnaton. The spacecraft contaned an accurate gyro reference, but the coordnate frame algnment was not measured relatve to the GPS atttude reference frame, whch means that dscrepances between the two reference frames mght account for slghtly dfferent measurements from the two systems. Over the course of the experment, the two sets of atttude solutons agreed to wthn degrees, whch was thought to be wthn the algnment tolerance of the two reference frames. The frst extended realtme GPS based atttude determnaton msson was flown on the REX-II spacecraft, 7 whch tested actual atttude control usng GPS atttude measurements. The dfferental carrer-phase measurement error has a standard devaton of about degrees, a small fracton of the standard wavelength. 8 However, many error sources can sgnfcantly contrbute to atttude naccuracy. These nclude: reflectons of the GPS carrer from the envronment surroundng the antennas (multpath), electrcal dsspaton nherent when passng carrer-phase sgnals over the lengths of the RF cables between antennas and recever (lne bas errors), antenna motons due to external dsturbances (e.g., thermal dstorton effects), constellaton avalablty, tropospherc refracton, and cross-talk errors. The most sgnfcant error source and the most dffcult to overcome s multpath. 9 In fact, multpath effects can be so pronounced as to be a major drver for the locaton of the GPS antennas on a vehcle. Despte lmted successes wth recent attempts at modelng multpath, ths error remans a lmtng factor n the performance of carrer-phase based GPS atttude determnaton. Ths s due to the complex physcal nature of the reflectng surfaces, whch depends mostly on antenna locatons. Lne bases can also adversely affect carrer-phase based atttude. These bases are typcally determned by performng extensve calbratons (self survey) of the flght system on the ground pror to launch. However, snce the space envronment can sgnfcantly alter the physcal propertes of the cable through large temperature gradents, a permanent soluton to ths problem remans elusve. Yet another error source for the carrer-phase based method nvolves shftng baselnes. In general, the attanable atttude accuracy mproves wth longer baselnes. If, however, satsfactorly separatng the GPS antennas requres mountng them on flexble structures (such as solar arrays, or deployable booms), then the atttude performance of the carrer-phase based method can be serously compromsed to the pont where the advantages of the longer baselne s compromsed. It s mportant to recognze that the aforementoned errors are prmarly a result of the physcal problems assocated wth usng carrer-phase based measurements for atttude determnaton. Before the actual GPS atttude determnaton can be performed, the correct number of nteger wavelengths between each par of antennas must be found. The resoluton of these nteger ambgutes has been extensvely nvestgated. Such nteger resoluton technques fall nto two general categores: nstantaneous and moton-based technques. Instantaneous technques provde mmedate nteger resoluton wthout vehcle moton; however, the unqueness of the soluton may be severely degraded wth sensor nose. Moton-based technques use a batch of data to determne the ntegers; however, they rely on suffcent vehcle moton to obtan system observablty. In ether case, t s essental that these ntegers are accurately resolved before atttude determnaton can occur. A new sensor approach for GPS atttude determnaton s proposed. Ths essentally nvolves usng an array of GPS antenna postoned to provde maxmum sky coverage. Ths array s used only to fnd whch GPS spacecraft are wthn the feld-of-vew (FOV) of each antenna. Atttude determnaton s performed by consderng the sghtlne vectors of the found GPS spacecraft together wth the boresght vector of the partcular antenna, unlke nterferometrc methods (see Refs. []-[4]). The boresght s used

3 snce the exact locaton of the GPS spacecraft n the body-frame of the antenna FOV s not known. The approach essentally s smlar to a star tracker, wth the GPS sghtlne vectors as the nertal reference vectors and the antenna boresght vectors as the body vectors. Multple antennas are used to ncrease atttude accuracy. The advantages of the new sensor approach nclude: ) dfferental carrer-phase measurements are not requred, ) atttude errors from multpath can be reduced or even elmnated, 3) nteger ambgutes do not need to be resolved, and 4) lne bases do not need to be determned. Therefore, the new sensor approach s easy to mplement and use for any applcaton. It wll be shown that the accuracy of the new sensor s better as the FOV decreases. Multple sensor confguratons are tested to nvestgate ths concept. Even though the accuracy n smulatons s currently not better than the standard carrer-phase approach, the new sensor s only lmted by technology. As technology advances, more GPS antenna can be used to further ncrease atttude accuracy. The organzaton of ths paper proceeds as follows. Frst, the new sensor concept s shown. A number of antenna confguratons are shown. Next, a revew of Wahba s problem s shown for atttude determnaton, as well as a method to determne the assocated weghts n the loss functon. Smulaton results are then presented, wth a dscusson of the procedures for actual hardware mplementaton. New GPS Sensor Concept In ths secton the concept of the Compound Eye GPS Atttude and Navgaton Sensor (CEGANS) s ntroduced. A number of sensor confguratons are shown for the new sensor. Next, an atttude determnaton algorthm s developed, whch s accomplshed by expandng upon current effcent methods. Fnally, smulaton results are presented. The Compound Eye GPS Atttude Sensor The basc concept underlyng the CEGANS s a relatvely smple one that uses GPS antennas n way smlar to methods employed by star trackers for many years. When consderng GPS for navgaton uses only, t s advantageous for a sngle antenna to cover as much of the vsble sky as possble, allowng sgnals from as many GPS SVs to be processed as are avalable to the user. In ths way, the best possble navgaton soluton can be ascertaned wth the mnmum amount of spacecraft hardware. The natural result of ths approach has been the development of patch antennas capable of trackng GPS SVs over a hemsphercal FOV. An deal soluton s to provde an atttude capablty wthout losng the navgaton functon, whle smultaneously avodng the constrants and requrements mposed by the nterferometrc method dscussed above. Ths s an approachable goal once a new and dfferent method of employng GPS patch antennas s consdered. Whle most antenna desgns tend to maxmze the avalable FOV to a gven antenna for navgaton and atttude determnaton purposes, a dfferent approach s ntroduced n ths paper whch uses a reduced FOV. Usng multple antennas dstrbuted over the surface of a hemsphere, and restrctng the FOV of each antenna to a predetermned cone can provde a workable soluton. In ths way, each antenna functons as a star tracker, whose stars are the GPS SVs themselves. Two such arrays of restrcted FOV antennas stll allow full sky coverage of the GPS constellaton thereby permttng navgaton solutons to be determned at any atttude. Snce the nomnal GPS navgaton soluton fxes the postons of the GPS SVs as well as the user vehcle n tme and space, the sghtlnes from the user to the GPS constellaton may also be determned. If each antenna can be polled to determne whch GPS SVs are vsble n each restrcted FOV at a gven tme, nformaton about where the known sghtlnes are relatve to the antenna array s also possble. Fnally, fxng the antenna geometry relatve to the vehcle body frame allows vehcle atttude nformaton to be determned from the orentaton of multple sghtlnes n the restrcted FOVs of the antenna array. Sensor Confguratons For the ntal feasblty study, a sx-element sensor array s employed. The computer model has each of the sx antenna elements mounted to one face of a hemdodecahedron (see Fgure ). Ths confguraton has the advantage of allowng one reference element to be orented parallel to the sensor mountng plane, whle mantanng a unform separaton between all adjacent antennas. For the ntal study, the half-cone angle s degrees, effectvely dvdng half of the sky nto sx overlappng FOVs, entrely coverng half the sky whle avodng regons smultaneously observable by three elements. Agan, for the sake of smplcty n ths ntal study, the sensor s assumed to be mounted to a LEO spacecraft whch s drectly over the north pole of the Earth, and orented wth a zero degree offset n both azmuth and elevaton wth respect to the nertal frame (zero degree atttude error). The sensor has been presented wth a representatve scatterng of GPS SVs n nertal space. 3

4 Fg. Sphercally Symmetrc (Hem-Dodecahedron) Array of Restrcted FOV Patch Antennas Several smple computer models were run, wth successful atttude determnaton. Consderable mprovements can be made by dvdng the sky nto a greater number of smaller areas. Mantanng the full sky coverage permtted by the hem-dodecahedron desgn requres the ncluson of addtonal antenna elements. Sphercal symmetry makes the buckeyball a very attractve geometry. A buckeyball s a sold whch may be vewed as a combnaton of two regular solds, the dodecahedron and the cosohedron; upon the realzaton that a dodecahedron has faces and vertces, whle the cosohedron has faces and vertces. A three dmensonal fuson of the two solds renders a sold wth 3 faces, dentcal pentagons (half-cone angle of.7 degrees) regularly arranged as on a dodecahedron and dentcal hexagons (halfcone angle of 3.8 degrees) arranged as are the trangles of an cosohedron. For the next seres of feasblty studes, a 6-faced hem-buckeyball sensor has been employed (see Fgure ), agan wth all the half-cone angles set to avod regons smultaneously observable by three elements. The vehcle model was agan assumed to be a LEO spacecraft at the north pole wth the sensor algned wth the nertal reference frame (zero degree atttude error). Fg. A Hem-Buckeyball Another way to dvde the sky nto a greater number of smaller areas wthout addng more sensor elements s to enlarge reduced feld-of-vews (RFOVs) n the basc desgn to create areas of overlap, usng the nformaton provded by the resultng overlappng coverage (7% ncrease n the half-cone angles). The overlap regons and remanng regons now yeld effectve FOVs (EFOV) for the sky coverage (the RFOV and EFOV are the same when no overlap occurs). Two hem-buckeyballs orented n opposte drectons consdered as a sngle sensor can provde full 4π steradan coverage of the sky. Ths orentaton nvolves two hem-buckeyballs mounted to the user spacecraft (e.g., one to the zenth deck and the other to the nadr deck). Not only does ths confguraton provde the capablty of two ndvdual buckeyballs, but allows for the addtonal dvson of the full sky nto areas where the two halves overlap. Regons covered by only one element correspond to the faces of the buckeyball (3), regons covered by two correspond to the edges (9) and, not surprsngly, regons covered by three elements correspond to the vertces of the buckeyball (6). For the full buckeyball, ths dvdes the full sky up nto 8 unquely defned areas. A planar projecton of ths confguraton s shown n Fgure 3, where the element centers are labeled as S through S5 and the GPS SVs are labeled N through N3. The confguratons consdered n ths study are summarzed n Table (EFOV- corresponds to regons covered by one element, and lkewse for EFOV- and EFOV-3). 4

5 Table Geometry and Confguratons CEGANS Geometry Cov. EFOV- EFOV- EFOV-3 Total Areas A Full-Buckeyball 4π A Full-Buckeyball 4π 3 9 B Hem-Buckeyball π B Hem-Buckeyball π C Full-Dodecahedron 4π 3 6 C Full-Dodecahedron 4π 3 4 D Hem-Dodecahedron π 6 5 D Hem-Dodecahedron π S N3 N S4 S9 N3 S5 S S6 S7 S8 N6 N3 N S S N7 S3 Elevaton N S3 N S4 S N7 S3 N9 S5 S S9 N S S S3 N7 S8 N9 N5 S7 S9 S8 S7-45 N9-6 S6 N5 N6 S5 N3 N5 S4 N4 N4 S3 N8 N6 N4 S -75 S Azmuth Case Studes Fg. 3 -D Projecton of the Full-Buckeyball Sensor FOV wth Increased Half-Cone Angles To allow easy understandng of the envronment durng the sensor development phase, the frst set of refnement studes have been executed assumng the CEGANS to be affxed to a statc user spacecraft wth a zero degree atttude error under a statc GPS constellaton. Ths allowed for realstc yet comprehensble results to be obtaned, whle provdng a consstent comparatve bass of results. Development of ncreasngly complex CEGANS types followed. Once the desred level of sensor complexty has been successfully modeled, refnements n the envronmental model could be addressed. Up to ths pont, all atttude solutons are obtaned assumng the CEGANS to be affxed to a statc LEO (7 km alttude) user spacecraft wth a zero degree atttude error under a statc GPS constellaton. Movng to the next level of complexty nvolves settng the user spacecraft n moton about the Earth. For ths step, the user spacecraft s assumed to be an Earth pontng vehcle, wth no atttude errors, mantanng the CEGANS n a zenth pontng orentaton as the spacecraft moved under a statc constellaton. A polar orbt s used to provde the wdest varety of geometres wth respect to the GPS constellaton. In each case, sghtlnes from the user spacecraft to each SV n the GPS constellaton are determned, wth those behnd the Earth (from the user spacecraft s perspectve) elmnated from subsequent consderaton. The GPS SV sghtlnes are then compared to the boresght vectors (and cone angles) of each sensor element to establsh whch GPS SVs fall wthn the 5

6 RFOV of each sensor element. In ths way, a truth model can be developed whle collectng the sghtlne data to be made avalable for atttude determnaton. Ths sghtlne data takes the form of a bnary vsblty matrx (: SV vsble, : SV not vsble) wth GPS SVs along one axs and CEGANS element along the other. Snce the nomnal GPS navgaton functon permts nertal poston determnaton of all the GPS SVs as well as the user spacecraft, vector dfferences allow determnaton of the vector sghtlnes from the user spacecraft body to the GPS SVs n the nertal frame. Comparng the GPS SV sghtlnes to the known geometry of the varous CEGANS elements n the user spacecraft body frame through the vsblty matrx allows determnaton of a unque atttude whch permts the correct GPS SVs to be seen by the correct CEGANS elements at a partcular tme. Atttude Determnaton Once the body boresght vectors and spacecraft sghtlne vectors are gven, then the atttude can be determned. Ths s accomplshed by mnmzng the followng loss functon (frst posed by Wahba 5 ) n J A w b As () where b now denotes th unt vector to the center of the EFOV, s denotes the normalzed th sghtlne vector, and w s a weghtng factor. The optmal choce of weghts wll be dscussed below. The error ntroduced usng the new sensor confguraton s mostly due to the ncorrect knowledge of the actual lne-ofsght to the GPS spacecraft n the body frame, snce all vsble GPS spacecraft n an antenna FOV are assumed to have a body vector n the center of the EFOV. It s possble to have overlappng crcles so that that all EFOVs have approxmately the same area. If the areas are equal for each correspondng boresght, then Equaton () can be smplfed by settng w. Once the weghts have been chosen, then the soluton for the atttude can be found usng standard technques that mnmze Wahba s problem. A smple soluton for the atttude matrx n Equaton () s gven by performng a sngular-valuedecomposton of the followng matrx 6 n F w bs T T U V () The optmal soluton for the atttude matrx s gven by the followng 6 where Aopt U V T (3) U U dag,,det U V V dag V (4a),,det (4b) The covarance of the estmaton error angle vector n the body frame s gven by E T n b s I F Aopt T P (5) where corresponds to a small error angle, and b and s are the standard devatons of the body and sghtlne measurement error processes, respectvely. Snce the GPS spacecraft postons are well known, t s reasonable to assume that b s (for the remander of the paper b ). Snce the z-axs of the sensor coordnate system s outward along the boresght, then the reconstructed unt vector n the body frame s gven as a functon of the coelevaton and azmuth sn cos b sn sn cos The true (error-free) unt vector s gven by true b If the error dstrbuton s axally symmetrc about b true (6) (7) (whch s a reasonable assumpton for the GPS sensor), then the varance of the body measurement process for a unform dstrbuton over a crcle of radus can be determned by E sn whch leads to cos dcos cos 6 cos cos (8) (9) 6

7 Note that f s small, then the standard devaton can be accurately approxmated by. Determnng the optmal weghts n Equaton () s not straghtforward. An ntutve approach uses w. Ignorng overlap regons, the error for each antenna encompasses a small crcle on a curved surface of the unt sphere. The area of a small crcle of angular radus s gven by 7 cos () Now consder the case where the FOV of two antennas overlap. The overlap area between two small crcles of angular rad and, separated by a center-to-center dstance s gven by 7 coscos cosacos cos - sn sn coscos cos acos cos - snsn coscos acos cos - snsn wth () The overlap regon can also be used to defne another boresght vector. Suppose that two areas overlap, and each area has each center boresght vector gven by b and b. Then, the boresght vector of the overlap regon s smply gven by b3 b b b b () Ths allows another measurement set to be made avalable smply by overlappng the FOV of two antennas. Also, the non-overlappng part of antenna FOV area decreases smply by. Choosng weghts for the overlappng case becomes extremely dffcult, snce the error dstrbuton s no longer unform n general. Snce ths paper focuses on the applcaton of the sensor and not on a purely theoretcal analyss, a number of smplfcatons have been made. Frst, for the non-overlappng case, Equaton (9) can be approxmated by a sold angle gven as the projected surface area dvded by the total surface area of the sphere, so that cos w 4 (3) cos Ths s a good approxmaton even for large values of (see Fgure 4). Next, t s assumed that the same approxmaton holds true for the overlappng case; so that the weght for the overlappng regon s gven by w 4, and the weght for the non-overlappng regon s gven by w 4. Therefore, as the area of the small crcle decreases, more weght s placed on that measurement n the atttude determnaton, whch ntutvely makes sense. The case for trple overlaps becomes ncreasngly complex; however, for ths study ths case yelds areas that are approxmately equal so that Equaton (3) s a good approxmaton for the EFOVs. Inverse Weght Actual Approxmate Angular Radus (Deg) Fg. 4 Actual and Approxmate Inverse Weghtng The performance of the atttude determnaton algorthm may be enhanced. Ths s accomplshed by assurng that vectors formed by mappng the sghtlne vectors nto the body frame (usng the determned atttude) are wthn the correspondng antenna FOV centered at the assumed body-frame boresght vector. The procedure s as follows: ) Determne any overlap regons and correspondng boresght vectors. ) Determne the optmal weghts usng area formulas. 3) Determne the avalable GPS spacecraft n each area and form sghtlne vectors. 4) Determne the atttude ( A ) by mnmzng Equaton (). 5) Map the sghtlne vectors nto the body frame,.e., b As. 6) Determne the angle between the mapped vector and actual boresght acosb b. 7

8 7) Determne f each mapped vector b s outsde of ts correspondng FOV. If a mapped vector s not wthn ts correspondng FOV, then the weght assocated wth the correspondng boresght vector and sghtlne vector should be decreased by some factor (e.g., ). The procedure s contnued untl all mapped vectors are wthn ther correspondng FOVs. Ths ensures that the physcal nature of the determned atttude s correct. Smulaton Results In ths secton, smulaton results are presented for a number of sensor confguratons. The frst test case nvolves a smulated non-movng spacecraft at the zenth poston usng the hem-dodecahedron sensor (D n Table ). There are nne avalable GPS sghtlnes wth one overlappng SV n the S and S3 FOVs. Wth the weghtng scheme developed n the prevous secton, t was determned that the found atttude provded mapped sghtlne vectors wthn ther respected FOVs. Therefore, the atttude s a consstent wth the sensor confguraton. Atttude accuracy and 3σ bounds usng Equaton (5) are shown n Table. Clearly the smple sensor approach provdes a feasble method for atttude determnaton. The 3σ bounds are large due to the assumpton of a unform error dstrbuton, whch results n an absolute worst case scenaro (.e., when all actual body measurements are at the sensor edge of vew). Table Results for Case Roll Ptch Yaw Atttude Errors σ Bounds The second test case nvolves the same spacecraft at the zenth poston usng the hem-buckeyball (encompassng both B and B n Table ). For ths case, there are a total number of avalable GPS sghtlnes, wth three overlappng spacecraft. In order to quantfy the concept of usng overlappng FOVs, two dfferent solutons were determned. The frst assumes that no overlappng occurs (B). Results for the atttude accuracy and 3σ bounds are shown n Table 3. Clearly, decreasng the sensor FOV ncreases atttude accuracy (as expected). The next soluton uses the overlappng regons (B), wth an effectve boresght centered n each overlappng regon. Results for the atttude accuracy and 3σ bounds are also shown n Table 3. Clearly atttude knowledge mproves for yaw, but more mportantly the 3σ bounds are dramatcally reduced. Ths shows that sgnfcant mprovements are possble by consderng the overlappng regons, wth areas much less than the non-overlappng regons. Table 3 Results for Case Roll Ptch Yaw Atttude Errors (B) σ Bounds (B) Atttude Errors (B) σ Bounds (B) A dynamc test run has also been performed for a smulated Earth pontng spacecraft at one revoluton per orbt (RPO). The sensor confguraton s gven by the full-buckeyball (A n Table ) wth ncreased halfcone angles, as seen n Fgure 3. Increasng the halfcone angles results n approxmately equal areas for the overlappng regons. The sensor measurements are sampled at. degree ncrements. A plot of the number of avalable GPS sghtlnes s shown n Fgure 5. In general, the more avalable SV s the more accurate the atttude (the separaton angle affects atttude accuracy as well). A plot of the atttude errors wth 3σ bounds s shown n Fgure 6. Clearly, the theoretcal weghtng choce n Equaton (3) provdes accurate atttude error bounds. Also, the atttude errors are greatest when there are the fewest avalable number of SV s. For ths sensor confguraton case atttude accuracy wthn 5 degrees s possble. In order to further mprove the accuracy a smple atttude flter has been mplemented. Ths s a smple frst-order Kalman flter that combnes a propagated model wth the determned atttudes. Snce gyros are not used for ths case, the angular velocty s assumed to be perfect (.e., gven by the one revoluton-per-orbt moton). Ths assumpton s not exact, snce external dsturbances and control errors are present n the actual system. These general nvolve dynamc couplng n the roll/yaw axs for Earth pontng spacecraft, whch are modeled by addng a bas to the ptch rate and sne wave to the roll and yaw axes wth a 9 degree phase dfference (see Ref. [8] for detals). The smple flter s gven by q q q~ q exp t q k k (4a) (4b) k k k 8

9 where t s the samplng nterval n seconds, q ~ s the k determned atttude at tme t k, q s the estmated k atttude at tme t k, s the vehcle s angular velocty, and s a scalar gan. Ths gan can be determned by mnmzng the atttude errors from the smulated runs. A value that s too small adds too much model correcton, and tends to neglect measurements. A value that s too large adds too much measurement nose, and tends to neglect model correctons. A value of. was determned to be optmal. Also, a frst-order approxmaton n the atttude-error covarance for the smple flter yelds the followng propagaton expresson 8 T Pk Pk Pk (5) where s a state transton matrx, and P denotes the atttude-error covarance of the smple flter. Snce s assumed constant and s nearly the dentty matrx, the dagonal elements of Equaton (5) approach the followng steady-state condton P P (6) Note that Equaton (6) s only vald for optmal values of (see Ref. [8] for more detals). A plot of the atttude errors and 3σ bounds usng the smple flter s shown n Fgure 7. Clearly, the atttude accuracy can be mproved by nearly a factor of four. Ths smulaton case clearly ndcates that atttude determnaton usng the smple sensor scheme s vable. Number of Avalable GPS Satelltes Orbt Fg. 5 Number of Avalable GPS Spacecraft Roll (Deg) Ptch (Deg) Yaw (Deg) Roll (Deg) Ptch (Deg) Yaw (Deg) Orbt Fg. 6 Atttude Errors and 3σ bounds Orbt Fg. 7 Atttude Errors and 3σ bounds usng a Smple Flter Conclusons The concept behnd the CEGANS sensor was presented. Theoretcal results were obtaned whch are suffcent to demonstrate the feasblty of the CEGANS sensor concept as a vable means of provdng an autonomous on-board atttude determnaton capablty usng GPS. The tradtonal nterferometrc method requres long baselnes (on the order of a meter or more) to be effectve, thereby lmtng the sze of the vehcle upon to whch t can be employed, and can be senstve to multpath nterference. The smple CEGANS concept has the potental to overcome these dffcultes. As technology evolves, GPS recevers and antennas become more capable, allowng further refnement of ths method. Ths s n stark contrast to the potental growth nherent n dfferental carrer phased based methods whch are approachng the lmts mposed by physcal constrants. 9

10 References [] Elrod, B.D., and Van Derendonck, A.J., Pseudoltes, Global Postonng System: Theory and Applcatons, Volume, edted by B.W. Parknson and J.J. Splker, Progress n Astronautcs and Aeronautcs, Vol. 64, AIAA, Washngton, DC, 996, Chapter. [] Parknson, B.W., Dfferental GPS, Global Postonng System: Theory and Applcatons, Volume, edted by B.W. Parknson and J.J. Splker, Progress n Astronautcs and Aeronautcs, Vol. 64, AIAA, Washngton, DC, 996, Chapter. [3] Cretaux, J.F., Investgaton of Orbt Determnaton usng the GPS Constellaton, Proceedngs of the AAS/GSFC Internatonal Symposum on Space Flght Dynamcs, Volume (Greenbelt, MD), NASA- Goddard, Greenbelt, MD, Aprl 993, AAS #93-7. [4] Hanes et. al., A Novel Use of GPS for Determnng the Orbt of a Geosynchronous Satellte: The TDRS/GPS Trackng Demonstraton Proceedngs of the 994 ION-GPS (Salt Lake Cty, UT), The Insttute of Navgaton, Alexandra, VA, Sept. 994, pp. 9-. [5] Axelrad, P., and Ward, L.M., Spacecraft Atttude Estmaton Usng the Global Postonng System: Methodology and Results for RADCAL, AIAA Journal of Gudance, Control, and Dynamcs, Vol. 9, No. 6, Nov.-Dec. 996, pp. -9. [6] Brock et. al., GPS Atttude Determnaton and Navgaton Flght Experment, Proceedngs of the 995 ION-GPS (Palm Sprngs, CA), The Insttute of Navgaton, Alexandra, VA, Sept [7] Lghtsey et. al., Flght Results of GPS-Based Atttude Control on the REX-II Spacecraft, Proceedngs of the 996 ION-GPS (Kansas Cty, MO), The Insttute of Navgaton, Alexandra, VA, Sept. 996, pp [8] Cohen, C.E., Atttude Determnaton Usng GPS, Ph.D. Dssertaton, Stanford Unversty, Dec. 99. [9] Cohen, C.E., Atttude Determnaton, Global Postonng System: Theory and Applcatons, Volume, edted by B.W. Parknson and J.J. Splker, Progress n Astronautcs and Aeronautcs, Vol. 64, AIAA, Washngton, DC, 996, Chapter 9. [] Braasch, M.S., Multpath Effects, Global Postonng System: Theory and Applcatons, Volume, edted by B.W. Parknson and J.J. Splker, Progress n Astronautcs and Aeronautcs, Vol. 64, AIAA, Washngton, DC, 996, Chapter 4. [] Cohen, C.E., and Parknson, B.W., Integer Ambguty Resoluton of the GPS Carrer for Spacecraft Atttude Determnaton, Advances n the Astronautcal Scences, Vol. 78, AAS #9-5, pp [] Crassds, J.L., and Markley, F.L., New Algorthm for Atttude Determnaton Usng Global Postonng System Sgnals, AIAA Journal of Gudance, Control, and Dynamcs, Vol., No. 5, Sept.-Oct. 997, pp [3] Bar-Itzhack, I.Y., Montgomery, P.Y., and Garrck, J.C., Algorthms for Atttude Determnaton Usng GPS, Proceedngs of the AIAA Gudance, Navgaton, and Control Conference, (New Orleans, LA), AIAA, Reston, VA, Aug. 997, AIAA #97-366, pp [4] Crassds, J.L., Lghtsey, E.G., and Markley, F.L., Effcent and Optmal Atttude Determnaton Usng Recursve Global Postonng System Sgnal Operatons, Proceedngs of the AIAA Gudance, Navgaton, and Control Conference, (Boston, MA), AIAA, Reston, VA, Aug. 998, AIAA # [5] Wahba, G., A Least Squares Estmate of Spacecraft Atttude, Problem 65-, SIAM Revew, Vol. 7, No. 3, July 965, p. 49. [6] Markley, F.L., Atttude Determnaton Usng Vector Observatons and the Sngular Value Decomposton, The Journal of the Astronautcal Scences, Vol. 36, No. 3, July-Sept. 988, pp [7] Wertz, J.R., Sphercal Geometry, Spacecraft Atttude Determnaton and Control, edted by J.R. Wertz, D. Redel Publshng Co., Dordrecht, The Netherlands, 978, pp [8] Crassds, J.L., Markley, F.L., Kyle, A.M., and Kull, K., Atttude Determnaton Improvements for GOES, Proceedngs of the Flght Mechancs/Estmaton Theory Symposum (Greenbelt, MD), NASA-Goddard, Greenbelt, MD, Aprl 993, pp