5 th SASech 0, Khavaran Hgher-educaton Insttute, Mashhad, Iran. May -4. Satellte Atttude Determnaton Usng GPS Recever Based on Wahba Cost Functon Asghar Ebrahm, Malek Ashtar Unv. of ech., ehran, Iran Emal: ebrahm@mut.ac.r Mahmood Najaf, Iran Space Industres Group, ehran, Iran Emal: mahmoodnajaf@gmal.com Paper Reference Number: 497 Name of the Presenter: Mahmood Najaf Abstract After launchng and njectng a satellte nto the orbt, atttude determnaton and control s one of the most mportant tasks for the satellte to stay n ts orbt. hs s done by dfferent methods havng dfferent accuraces. Some examples for atttude determnaton and control technques are spn stabled, 3 axs stablaton, gravty gradent method, usng 3 gyros n 3 drectons, etc. GPS s usually used n satelltes to determne the satellte alttude and orbt parameters. It s used as a complementary for the atttude determnaton sensors. Atttude determnaton s crtcal for modern spacecrafts. An ordnary atttude determnaton system s composed of some rather cheap magnetometers, sun sensors, star trackers or earth horon gyros. GPS navgaton system can be valuable and effectve for these satellte sensors. It can also determne the satellte atttude solely. Here n ths paper a method s proposed for atttude determnaton usng GPS. hs Method s based on the carrer phase of a receved sgnal from two or more antennas mounted on the satellte s body. Phase dfference between antennas can be used to determne the atttude and drecton of the satellte. he rate of phase change shows the rate of atttude change. Based on the theores, ths method would have a hgh relablty and t can be used as an atttude determnaton technque. he number of GPS Antennas should be calculated to acheve the best accuracy. Its precson s acceptable and n some cases even hgher n respect of other methods. Key words: Atttude Determnaton, GPS, Satellte, Wahba cost functon. Introducton Atttude of a satellte means ts drecton n the space. Automatc atttude determnaton s necessary for modern spacecraft. Bascally moton of a satellte s descrbed by ts poston, velocty, atttude and relatve moton. Atttude assessment conssts of atttude determnaton, controllng and next moton predcton for satellte. Atttude determnaton contans atttude stablaton and controllng the atttude maneuver. he authors am s to present a method for atttude determnaton usng GPS wth a specal layout of antennas wth a good precson for satellte atttude determnaton..satellte atttude determnaton methods
5 th SASech 0, Khavaran Hgher-educaton Insttute, Mashhad, Iran. May -4. Earth, sun and stars are usually consdered as a source or reference for atttude determnaton. he precson depends on the hardware and process algorthm. An ordnary atttude determnaton system contans some relatvely cheap magnetometers, sun sensors, star trackers or Gyros (earth horon detector). Whle GPS navgaton system can be a valuable and effectve complementary system for the sensors. It can also be solely used as an atttude determnaton system [4]. For LEO satelltes a combnaton of two sensors (sun sensor and earth sensor) are used for atttude determnaton n 3 axes. Stars are used for very precse measurements (about some arc seconds). In table () there s a comparson between dfferent atttude determnaton methods. Method of Atttude Precson Process determnaton IMU (Inertal Measurement Unt) Earth sensor (especally nfrared sensor) (deg) 0. 3 Heavy and complcated Gyros are used to determne the angular velocty and 3 accelerometers are used to determne the lnear acceleraton [0]. 0. for LEO and 0.0 for GEO Comparng between earth atmosphere heat and deep space cold, shows the nfrared waves. So roll and ptch axes atttude can be determned [0,]. hs sensor has a varety of equpment.[,7] Sun sensor Better than 0. Measurng sun vson angle, satellte atttude s determned (usually for yaw axs) [0,9,7]. Star sensor 0.0 hs sensor senses lght ntensty and star se and tracks a specal star [0,]. hs sensor s heavy wth a hgh energy consumpton [9,7]. he map of the sky s nstalled n the memory of the sensor. Velocty sensor (Gyro) 00 deg./sec hs sensor measures the varatons of drecton as velocty or dsplacement. he man advantage of these sensors s satellte angular determnaton wthout need to space angle determnaton [9,7]. Magnetometer 0.5 hs sensor measures the satellte atttude by comparng local magnetc feld wth the values n t the memory (only for LEO). hey have less weght and less power consumpton wth a hgher Relablty [0]. GPS recever o be dscussed Usng phase dfference between receved sgnals by dfferent antennas. able (): Comparson between dfferent atttude determnaton methods 3- heorem of Atttude determnaton usng GPS Nowadays GPS s used n the spacecraft for many purposes. Orbt and atttude determnaton, locatng the spacecrafts n the space, launcher trajectory assessment and also tme synchronaton are some of the applcatons. Atttude determnaton usng GPS s done based on the carrer Phase of a sgnals receved from two or more antennas mounted on the satellte body. Phase dfference between antennas can be the determnant of atttude and orentaton of the spacecraft. he varaton rate of these phases gves the rate of Atttude varaton. Practcally two recevers s used on the satellte (or any other object havng GPS) to receve the sgnals from two antennas mounted on the satellte body. hey have a known dstance from each other. Indeed these sgnals are from a GPS satellte. he phase dfference s
5 th SASech 0, Khavaran Hgher-educaton Insttute, Mashhad, Iran. May -4. 3 calculated based on receved sgnals and ths s drectly related to satellte Atttude (more than two antennas s necessary for the system to work contnuously) [9]. he antenna s not always drected to the sky (sometmes drected to the earth), therefore a unque antenna can t be used to receve the sgnals n all drectons. wo omndrectonal antennas are nstalled on rght and left or up and down sdes of the satellte. Havng two antennas wll cause a Paraste regon because the sgnals are drectly mxed. Paraste one ranges ±5 degrees around the symmetry plane of the antennas. he GPS recever can t receve the sgnals n paraste one. hs causes a reduced precson and lack of relablty n atttude determnaton. hs one s rotatng as the satellte rotates around the earth. Paraste one causes the recever to recogne less GPS satelltes. Under these crcumstances, GPS recevers can be employed as master and slave. hey receve the sgnals from GPS antennas ndependently and demodulate them. Hence the sgnals from two antennas wll not be mxed and coverage area wll be whole the sky. If the Antennas are mounted n a proper poston, the GPS satelltes wll be always vsble durng the flght [6]. Precson of GPS s lmted by dstance of antennas, rgdty of antenna bases and phased nose multpath n recever. In the method Atttude determnaton usng GPS satelltes, carrer phase dfference measurement s used and ths causes the S/A (Selectve Avalablty) error to be small (f exstng) wth no reducton n precson. GPS precson s restrcted frstly by desgn parameters of the antenna structure and also recever electroncs lmtatons [5]. A precson better than 0. degrees s achevable by employng the best technologes. Necessary precson tolerance for satellte atttude determnaton s 5 degrees for smple satelltes and better than 30 6 degrees for spacecrafts lke Hubble telescope. A precson about 30 6 degrees s achevable by usng star sensors. In ths case GPS has a lower precson and can t replace the star sensor, but t s a good choce for most of msson requrements. Precson of ths atttude determnaton method s mentoned n dfferent references. Reference [8] notces the precson of about 0.4 up to 0. degrees for GPS. Reference [5] announces the precson of less than mnute usng GLONASS system. 3-- Descrbng Atttude determnaton usng GPS method by mathematc equatons hs method s to determne the atttude of a subject n the space n reference coordnaton [5]. wo GPS sgnal recevers (GPS antenna) are postoned n two ponts of a movng object (whch should be atttude determned) wth a dstance of d (ponts A and B n Fg.).
5 th SASech 0, Khavaran Hgher-educaton Insttute, Mashhad, Iran. May -4. 4 Fg. (): status of antennas and GPS satellte he poston of AB lne s known n relaton to the object (coordnaton system ntegrated to the object). By determnng stuaton of lne AB, atttude of object (Euler trple angles) n a reference coordnaton system can be determned. Status of lne AB s determned by ( cos, cos, cos ) and also the relaton to OXYZ geocentrc coordnaton system. x y Assume recevers n A and B measure ther dstance from NS (Navgaton Satellte) as D a, D b. Coordnaton of NS s (X, Y, Z). hen the phase dfference between A and B s: Da Db () When s wavelength of the sgnal sent from NS. s the angle between AB vector and SMvector whch connects center of lne AB to NS. he relaton between and s: () cos d he relaton between angle and cosnes vector of lne AB s as below: cos cos cos cos (3) x x y y Where x, y and are coeffcents calculated by equatons contanng coordnates of S and M by measurng D a, D b. Equaton (3) contans 3 unknowns ( cos x, cos y, cos ). Hence we need 3 equatons smlar to equatons () and () whch obtaned by A and B recevers (usng sgnals from two other GPS satelltes n dfferent postons). Now we have three ndependent equatons. Assume sgnals are receved from satelltes NS and NS 3. So we have these three equatons: Da Db x cos x y cos y cos
5 th SASech 0, Khavaran Hgher-educaton Insttute, Mashhad, Iran. May -4. 5 Da Db (4) x cos x y cos y cos Da3 Db3 x 3 cos x y 3 cos y 3 cos he object atttude s determned by solvng equaton (4) n terms of cos, cos and cos. Most of the tme the equaton wll be smplfed usng the followng equaton: (5) cos x cos y cos x y 3-- Dfferental carrer phase measurements for atttude determnaton he dfference between receved sgnals from two antennas (separated from each other by a base lne) s very mportant for atttude determnaton. Angle of wave front and wavelength s used to obtan the phase dfference. (Fg. ()). Accordng to Fgure () the phase dfference s: () be cos ( n) Where be s the baselne length, s the angle between baselnes and lne of sght of GPS satellte, n s the nteger number of wavelengths between two antennas, s the phase dfference and s the wavelength of GPS satellte sgnal [3]. Fg. () : angle of wave front and wavelength to obtan the phase dfference. here are two carrer frequences for GPS. One s n L band n 575.4 MHZ and the other s on L band n 7.6 MHZ. L s usually used n cvl applcatons. Assumng the known offset (nteger number n) and compensated effect s the measured phase dfference ( ) s: () b As
5 th SASech 0, Khavaran Hgher-educaton Insttute, Mashhad, Iran. May -4. 6 Where s a ero mean whte Gaussan process. he standard devaton assumed as [3]. 3 S R s the normal sght lne vector for the GPS satellte n the nertal coordnate system. 3 br s also the normal baselne vector whch shows the relatve poston vector from one recever to the other one. he atttude matrx A s obtaned from unt vertcal matrx so that: (3) det A, A A I 33 3-3- Obtanng Wahba Cost Functon for atttude determnaton he most common methods for atttude determnaton are the ones n whch Wahba problem s solved (ths problem ntally suggested by Wahba). Suppose m baselnes and n vsble GPS satelltes. For th baselne and jth satellte: (9) b As When: (0) So: () V N ~ (0, ) ~ ( N b AS j, ) Consequently: () ( ) exp Assumng ndependence of measurements errors: P (,..., ) P ( )... P ( ) P ( ) m,..., mn mn mn mn j m jn ( b AS j ) exp j jn b AS j (3) Now assumng the observed parameters as, the problem s to fnd the matrx A so that the term (,..., ),..., mn P be maxmed wth the Maxmum lkelhood crtera. mn Maxmng equaton (3) s equal to maxme ts natural logarthm. So: m j n m j n b AS j (4) LnP ( ) L n j j s consdered constant, hence the problem s mnmng the second term, presented below: m jn (5) J A b AS j j As we see n the equaton (5), s the same weght coeffcent n Wahba problem. 3-4- How to solve Wahba problem for atttude determnaton he problem s to fnd a vertcal matrx mnmng the equaton (6):
5 th SASech 0, Khavaran Hgher-educaton Insttute, Mashhad, Iran. May -4. 7 m (6) J A w b As Where b s the th unt vector of EFOV, b and s are known vectors. But the weght coeffcents w should be determned. A s ndeed the matrx of the drecton cosnes whch should be determned. If the measurements are equally mportant, then w coeffcents wll be equal. Assumng w smplfes the problem. A smple answer for atttude determnaton matrx n equaton (7) usng S-V-D (sngular Value decomposton) method s: (7) n F w b s U V he optmum answer for matrx A s: (8) Aopt U V U, V are obtaned from the followng equatons: (9) U U dag (,,det U ) (0) V V dag (,,det V ) Assumng s error vector of the nsde angle, the error covarance s obtaned as: m () P E b s FA opt When b, s are standard devatons of measurement error process and sght axs n sequence. Poston of the GPS satelltes are known exactly, therefore we can assume and so. b s b 3-5- Atttude determnaton Consderatons Common atttude determnaton systems use nterferometrc methods whch have some physcal lmtatons such as, Multpath Error, Lne Bas Error, Antenna movements because of envronmental turbulences (.e. thermal dstorton), Combnaton of accessble satelltes, troposphere scatterng and Crosswalk Errors. Multpath error s the man error source and also the most complcated to solve. Despte progresses n modelng and reducton, ths error s stll the man source of error n evaluaton of sgnal phases. Lne bas errors whch are due to Power downfall n RF cable, can be elmnated by calbraton on the ground. he other source of error s the baselne slp. Generally longer baselnes wll cause more precson for atttude determnaton. But t s necessary to nstall the antennas on the flexble surfaces (such as solar arrays) to remove them easly. In these condtons, atttude determnaton s related to the pont, antenna s mounted. In the old methods and tradtonal applcatons, t was desred to use antenna wth the most rado foresght. But n new methods multple antennas are used so that every antenna has a porton of the space n ts rado foresght. herefore usng multple antennas causes the coverage of whole the space. By determnng the satellte n foresght of every antenna and knowng Foresght vector of the antenna, matrx A can be obtaned so that the GPS cost functon (Wahba problem) become mnmed.
5 th SASech 0, Khavaran Hgher-educaton Insttute, Mashhad, Iran. May -4. 8 4. Expermental applcatons of atttude determnaton usng GPS In atttude determnaton usng GPS, t s expected to have a precson less than degree n each drecton, but t s not stll relable because of error n magnetc feld model. Anyway the basc dea on measurng the phase dfference between GPS recever sgnals to determne the atttude n three axes has been absolutely successful as t s tested on some satelltes. One of frst applcatons of ths method was RADCAL satellte. In ths satellte GPS recevers were used for atttude determnaton. hs measurement was post processed. In order to have the maxmum foresght for GPS recever and reducng the nternal sgnals (they cause multpath repercusson) 3 patch GPS antenna nstalled on 3 sdes of the satellte. Although baselne of the antenna s shorten, but the precson of atttude determnaton was about degrees for each axs. he dstance between antennas was 0.67 meters. he other experment was on Crsta-SPAS satellte. It augmented the frst real-tme atttude determnaton. In ths experment satellte conssts of a precse gyroscopc orgn, but justfcaton of orbt coordnaton toward reference coordnaton was not measured. hs means small dfferences between coordnaton systems may ntrude n the calculatons as small dfferences. hen both systems were measured. Durng the experment the algnment tolerance between two coordnaton systems was about degrees. he other expermental case used GPS recever for atttude determnaton, was UoSA-. hs satellte was the frst small satellte used 3-axs atttude determnaton, wth a small budget. It also had the 3-axs stablaton. UoSA- was launched to orbt n Aprl 000 (alttude 650 km, nclnaton of 64.5 deg). hs satellte was equpped wth many sensors for atttude determnaton. A multchannel GPS recever was also used for orbt determnaton and precse tme synchronaton. hs recever was able to determne the atttude usng an array of antennas (consst of 5 patch antennas). MICROLAB, GANE, OAS-Flyer and ORBCOMM are some of the satelltes used GPS recevers for atttude determnaton [3]. 4-- Comparson between atttude determnaton methods Precson of GPS based Atttude determnaton s better than 0. degrees. Consderng able () and comparng the methods, we can say atttude determnaton usng GPS s an acceptable method n precson and even better n some cases. In ths method, atttude can be assessed n all 3 axes (Roll, ptch and Yaw) because phase dfference of GPS sgnals s used. hs s an advantage over other methods whch can only gve the atttude n one or two axes. hs method s also more economc because t uses the same GPS for navgaton and tme synchronaton of the satellte. Indeed a GPS recever n a satellte can be multpurpose and elmnate the need to other nstruments and so reducng the cost. he other advantages are hgh relablty and ease of use compared to other methods. 5. Conclusons Based on theores descrbed, a GPS can be used to determne the atttude of flyng objects and specally satelltes. It can be ndependent or supplementary for other methods. Atttude determnaton usng GPS s due to phase dfference n receved sgnals by antennas. At least recevers and 4 sgnal recevng antenna s necessary. he accessble precson would be about 0.0 degree. References [] P. Forteseue and J. Stark, 995, Spacecraft systems Desgn and Engneerng, nd Edton Chapter 0.
5 th SASech 0, Khavaran Hgher-educaton Insttute, Mashhad, Iran. May -4. 9 [] john L. Grassds, Davd A. Qunn, F. Lands Markley, john D. McCullough, A Novel Sensor For Atttude Determnaton Usng Global Postonng System Sgnals, 998, Publshed by the Amercan Insttute of Aeronautcs and Astronautcs, Inc. [3] john L. Grassds, E. Glenn Lghtsey and F. F. Lands Markley, Effcent and Optmal Atttude Determnaton Usng Global Postonng System Sgnal Operatons. [4] Nat l Academy Press, 995, he Global Postonng System, Chapter. [5] Atttude determnaton system usng GLONASS system (n Russan). By: V. Kharsova, A. E. Perova and V. A. Boldna, 999-560. [6] We Qnm, Lbo,Zhangyng, Proceedngs of the 5 th Asa-Pacfc Conference on Multlateral Cooperaton n Space echnology and Applcaton - 999, An Omndrectonal Master-Slave GPS Recever. [7] M. j. Sd, 997, Spacecraft Dynamcs Control, Chapter 0 Appendx B, Pages: 39-345-35 [8] Atttude determnaton usng GPS, L. Vallon, B. affet. L. Gulcch, A. Pasett, 4 th Internatonal ESA Conference on Spacecraft CGS, Oct, 999, Noord wk. [9]Ed.By:J.R.Wert,999, Spacecraft Atttude Determnaton and Control, Part-II,Chapter6 [0] Ed. By: Wley j.l. Space Msson Analyss and Desgn, Chapter.