New Approach to Achieving Stand Alone GPS Attitude Determination using Dual Short Baselines for Small-Satellite

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1 SSC99-XI- New Approach to Achevng Stand Alone GPS Atttude Determnaton usng Dual Short Baselnes for Small-Satellte S. Purvgrapong, M.S. Hodgart,Y. Hashda, M.J. Unwn Surrey Space Centre, UK Abstract Ths paper proposes a new approach to GPS (Global Postonng System) atttude determnaton for small satellte applcaton n LEO (low Earth orbt). Pror knowledge of atttude and nteger resoluton s not requred. The methodology of the new approach ncludes nteger ambguty search, ntal estmaton of atttude and lne bas, atttude ntalsaton, path dfference estmaton and fne atttude determnaton. The observable s the carrer phase dfference measurement between two GPS antennas. A dual short baselne (typcal baselne length up to 3 cm) s assumed n ths research. The key pont to ntalsng atttude s to estmated the atttude of ndvdual baselne vectors wth respect to the reference frame. Elmnaton of nteger ambguty s a smple task. Two set of vectors are requred to determne an ntal atttude. Once atttude s ntalsed, an estmaton algorthm based on the extended Kalman flter starts to determne the atttude. The nteger ambgutes and cycle slps can be resolved properly. The flter now s converged and, fne atttude s estmated. The robustness of the flterng estmator s tested wth smulated anomalous condtons. Nomenclature A b s λ η k β : transformaton atttude matrx : baselne vector : lne-of-sght unt vector to GPS satellte : L1 carrer wavelength : nteger ambguty : nteger ambguty ncludng effects of error : lne bas w : measurement nose ϕ : real phase dfference (- to + ) r : actual modulo path dfference (- / to + /) r ( m) : measured modulo path dfference r : actual path dfference r ( m) : path dfference ncludng error φ : roll angle θ ψ : ptch angle : yaw angle sup T : transpose of matrx sub B : body-fxed frame sub R : reference frame hat ^ : estmate Introducton Spacecraft atttude determnaton s generally based on the use of tradtonal atttude sensors such as Sun sensors, Earth sensors, star sensors, nertal sensors, and magnetometers. However, snce GPS has been successfully used for spacecraft navgaton, a new approach usng GPS atttude determnaton has been rapdly developng for space applcatons. In 1993, GPS atttude determnaton was demonstrated for the frst tme on the RADCAL satellte 1. Usng an onboard GPS recever and multple GPS antennas, GPS atttude sensng can be acheved through carrer phase dfference measurements between two antennas. The measured scalar phase dfference s used as an observable for the atttude algorthm for determnng spacecraft orentaton. However, there are several problems nvolved n the system mplementaton. Frstly, as a GPS recever can measure only a fracton of carrer phase cycle (- to + n radan or n equvalent - / to + / n range), the number of full 1

2 cycles are unknown. Moreover, f the fracton of phase measurement s very close to the edge, offset errors (e.g. lne bas and multpath effects) and nose may cause the unknown nteger number to slp to another number. Ths wll magnfy the dffculty of resolvng the nteger ambguty. Secondly, once the nteger ambgutes are resolved, the spacecraft atttude can be estmated. There are several optmal atttude algorthms that can be appled for spacecraft applcatons. However, the partcular choce of optmal algorthm depends on the msson requrements and practcal spacecraft detal such as computaton tme avalable, baselne confguraton and recever hardware desgn. For example, t has been reported that the three orthogonal baselne confguraton s the best confguraton. However, most satelltes can only accommodate a coplanar or near coplanar confguraton. Thrdly, once the atttude s determned, then the performance, relablty and accuracy of GPS atttude determnaton need to be mproved. Ths requrement depends on the effcency of the method used to mtgate the measurement error. Ths paper nvestgates the feasblty of the use of a new approach to estmate spacecraft atttude wthout pror atttude knowledge. The GPS atttude nformaton s measured from dual baselnes n the presence of measurement nose and lne bas. The ntegraton of three algorthms: a novel block ambguty search; followed by a standard algebrac algorthm and the extended Kalman flter. The block ambguty search conssts of a computaton of the atttude pontng of each baselne ndvdually, wthout regard to ther relatve atttude wthn the spacecraft, and ntally not even to each other. Wth ths approach and by explotng the overdetermnaton of the atttude soluton usng four (or more) satelltes t s a relatvely smple matter to solve the so-called nteger ambguty problem wth respect to each baselne. As s well known the problem s to dentfy the ntally unknown nteger wavelengths that must be nserted nto each path dfference measurement: typcally n smulaton usng our new method only one or a few possble solutons emerge as lkely canddates. Then by lookng at a par wse combnaton of possble solutons the best and correct soluton s obtaned gven knowledge of the actual angle between baselnes. The correct number of nteger nsertons nto each path dfference measurement has now been establshed. Havng obtaned a correct soluton to the pontng of the baselnes (always n reference space) s then a smple matter to compute the actual atttude of the satellte, n the second phase, usng the standard TRIAD algorthm. Havng found what we now call an ntal atttude ths nformaton now feeds a standard extended Kalman flter to enable contnued atttude determnaton now explotng the dynamcs of the spacecraft. Ths thrd step s atttude refnement usng an extended Kalman flter. However, the measured path dfferences must contnue to be checked and modfed by an approprate nteger multples of wavelength durng flterng. Ths paper descrbes a technque to perform ths process; and other refnements to prevent dvergence; so that the overall process may be called extended Kalman flterng wth ntegrty check. Background In ths secton, the fundamentals of phase dfference measurement and the problems n GPS atttude determnaton are brefly ntroduced..1 Fundamentals of GPS Atttude Measurement The fundamental measurement n GPS atttude determnaton s the path dfference measurement between two antennas separated by the baselne length. For baselne and GPS satellte j, the actual ( j path dfference, r ), can be expressed as where r j ( ) ( j) r ϕ λ π. j j j = b s = r + η λ ( j ) ( ) ( ) ( ) R R (1) As the GPS measurement s always perturbed by measurement nose and bas, the realstc equaton can be expressed as j j ( ) ( j ) ( ) ( ) ( j) ( j ) ( j ) r = r + β + w + η λ r + k λ () ( m) ( m) where bas n equvalence wth path dfference s consdered to be wthn + /. ( j ) For Equaton (), only the r ( m ) term s gven by a GPS recever. The k ambguty cycles need to be resolved.

3 Note that the orgnal nteger resolutons may slp one cycle due to the error n measurement.. Problems n GPS Atttude Determnaton To acheve an atttude soluton from GPS measurements, several problems must be addressed. These nclude nteger ambguty resoluton, error sources, and the choce of the operatonal atttude algorthm. The problem of nteger ambguty has been ntroduced earler. Several methods based on ether ambguty search or moton technques have been proposed to resolve the nteger ambguty problem. The general concept of the ambguty search technque for atttude determnaton s to search for the true nteger ambguty set that mnmses the error between the estmate baselne length and the known length. The use of the double phase dfference approach n ambguty search not only removes the bas, but also reduces ambguty search space as well 3,4. The moton-based methods operate on a batch of measurements collected over a gven perod of tme, whch t s assumed that the nteger ambgutes stll reman constant over the gven perod 5,6,7. Error sources n GPS atttude sensng consst of multpath, lne bas, recever nose, phase centre varaton, and geometry of selected GPS satelltes. Multpath s usually consdered the domnant error source 8 n GPS atttude determnaton. A GTD (Geometrcal Theory Dffracton) method can be used to model and verfy the dfferental carrer phase error caused by multpath effects. 9 However, multpath error can be mtgated by usng sgnal to nose rato nformaton to correct multpath errors n carrer phase dfference measurements 1 or by usng a calbraton method 11. Lne bas s a phase offset between two antenna chans, caused smply by dfferent length cables or by dfferent RF front ends. Possble methods to remove lne bas nclude the use of double phase dfferences or a calbraton sgnal source. Antenna phase centre s one factor that affects the carrer phase measurement. Generally, the physcal centre of the antenna does not concde wth the 3 receved pont of GPS sgnals on the antenna. Furthermore, these receved sgnal ponts wll vary wth the elevaton angle, and also from one antenna to another antenna. 1 The geometry of the GPS satelltes s another factor n achevng hgh qualty results. The geometry changes wth tme due to the relatve moton of the GPS satelltes. A measure for the geometry s generally descrbed as the Dluton of Precson (DOP) factor. In GPS atttude sensng, the uncertanty of the gven atttude solutons wll depend upon the geometry of selected GPS satelltes and the antenna baselne confguraton. To predct or evaluate the gven atttude solutons, the numercal mean called ADOP (Atttude Dluton of Precson) factor can be used to evaluate the atttude soluton. Ths paper concentrates on the nteger ambguty resoluton, lne bas and path dfference estmaton and operatonal atttude algorthms for dual baselne confguratons, accountng for measurement nose and lne bas. The known angle between baselne vectors wll be taken nto account n rejectng the wrong soluton from ambguty search technques..3 Approxmate Atttude Error A rough approxmaton to the rms atttude error,σ Θ, for effectve baselne length, L, can be estmated by 13 σ (3) σ L Θ r π λ whereσ r s the rms of path dfference error. RMS Atttude Error (degrees) L =.3 metre L = metre radan RMS Relatve Range Error (mm) Fgure 1: Approxmate atttude error of GPS-based atttude determnaton Typcally, the expected path dfference error s n the order of 5 mm approxmately 14. The expected pontng error of 1 metre baselne s then about.3 degrees as shown n Fgure 1. The atttude error wll ncrease to degree when the baselne length s reduced to 3 cm. 3 GPS Atttude Estmaton from Dual Baselne

4 3.1 Intal Step Estmaton - the block ambguty search Ths secton descrbes the baselne vector estmaton usng block ambguty search. The am s to determne the atttude of each baselne ndvdually and drectly n reference space. The measurements from each baselne and four selected GPS satelltes can be expressed as a matrx equvalent to Equaton (). where ( 1) s R ( ) s S = R ( 3) s R ( 4) s R xr b y = y R R β z R β Sy = r ( m) + λ k (4) ( 1) ( m) ( ) r = r ( m) m ( 3) r ( m) ( 4) ( ) r 4 4 ( m) ( 4 1) ( 4 1) r ( 1) k ( ) k k = ( 3) k k ( 4) ( 4 1 ) The nverse matrx of S matrx can be expressed as 1 S ( 4 4 ) (5) U v (6) where U s a ( 3 4) matrx and v s a ( 1 4) vector. Gven a tral set of ntegers n k, an estmated baselne vector $ b R can be obtaned by ( ( m) ) b$ = U r + λ k (7) R An ntal lne bas can be computed n parallel wth Equaton (9) from ( ( m) λ ) β$ = v r + k (8) For each tral choce of values n k a tral baselne vector s computed and also a tral lne bas. The most lkely correct soluton(s) may be determned by montorng two effects 1. a computed length of a tral baselne should be close to the actual baselne length.. a computed tral bas value should not be contradcted by whatever nformaton s known about that bas. 4 From detaled smulaton wth plausble values to typcal measurement errors we show that n most cases a unque and correct soluton s found to each baselne vector. For medum baselne lengths the range of search n k values typcally les between - and + and does not cause an excessve load for computaton n a systematc search, requrng only 65 trals. In some worst case solutons more than one contender emerges as a soluton to each baselne vector. To get round ths problem we explot the fact that the process so far has estmated each baselne vector ndependently. So we explot the known nformaton of the angle between the two baselnes by a par wse comparson. The correct par s dentfed wth a computed angle between them whch s closest to the true angle of the actual baselnes. Even f nformaton between computed tral bas and known bas s not used there s a hgh degree of success n the operaton and processng subsequent epochs wll ensure 1% acquston eventually. The computed bas s used n any case to update and montor the progress of the Kalman flter, once the correct soluton has been dentfed. Smulatons show that ths combnaton of procedures wthn the block ambguty search s always successful, leadng to a unque and correct soluton to the pontng of the baselnes. 3. Estmaton of atttude The block ambguty search has now dentfed the correct pontng (wth some error) of the two baselnes, expressed n reference space. The next step s to determne the actual atttude $ A of the satellte. A novel feature here s now to use the baselne vectors computed n reference space as observaton vectors. An atttude, to the actual satellte, can therefore be determned usng a standard TRIAD algorthm 15., usng the further knowledge of the pontng the drecton of these same vectors relatve to the body frame of the satellte. The top level dagram of the ntalsaton step s shown n Fgure.

5 phase dfferences from j satelltes select 4 measurements block ambguty search b$, b $ 1R R ( j) r ( m ) estmate β $, β $ 1( o) ( o) known lne-of-sght ( j) vectors s R select 4 vectors ( 1) ( 4) s, L, s R R ntalsng Kalman flter Equaton Models for Kalman Flter The Kalman flter presented here s mplemented to estmate spacecraft atttude and ts dynamcs for small rotaton angles. The equaton models are based on Euler angle representaton. State Vector The flter state vector conssts of three estmated Euler angles and ther rates. TRIAD b, b 1B B Fgure : Intal atttude estmaton It s mportant to be note that once the ntal atttude and ntal lne bas are estmated, the block ambguty search wll not be requred agan untl rentalsaton s requred. 3.3 Atttude Estmaton usng Kalman Flter Ths secton descrbes the atttude estmaton usng the extended Kalman flter. The flter s ntalsed by the atttude knowledge derved from the ntal step. For the frst epoch of flterng, the knowledge of atttude and lne bas from the ntal step wll be used to estmate path dfferences and feed nto the flter. An estmated atttude from the Kalman flter wll be used to estmate and update lne bas. The updated lne bas and estmated atttude also wll be used to estmate path dfferences n the next epoch. The process dagram of the flterng estmaton s shown n Fgure 3. phase dfferences from j satelltes ( j ) r ( m ) known lne-of-sght ( j) vectors s R path dfference estmatons $ ( j r ) $A Extended Kalman Flter $A p update mean lne bas $ β $ $ p+1) ( A( p+1) Fgure 3: Flterng estmaton process In ths secton, the equaton models of the extended Kalman flter are brefly ntroduced n the frst sub secton. The technque of path dfference estmaton s descrbed n the second sub secton. The reestmaton of lne bas s explaned n the second sub secton. β ( p ) System Equaton Snce the development of the flter follows the standard UoSat mcrosatellte platform 16 whch s a gravty gradent stablsed/nadr pontng spacecraft, the system model of the flter can be derved from the dynamcs model of Earth-pontng spacecraft for small rotaton angles 17. ( ) && φ + 4κ ω φ = 1 κ ω ψ& o o (9a) θ&& + 3κ ω θ = o (9b) ψ&& + ω φ& = o (9c) where ω o s the constant orbtal angular velocty of the spacecraft (assumng n crcular orbt), κ = ( 1 I z I t ), I z s a prncpal moment of nerta of z axs, and I t s a transverse moment of nerta. GPS Measurement Equaton The path dfference measurement from baselne and GPS satellte j can be expressed as where [ x y z ] b B B B B T ( j) T ( j ) r B R = b As (1) ( j) ( =, s R [ xr ] j ) ( s s j ) ( j ) = yr szr It s assumed that spacecraft s slowly rotatng (e.g.1 mnutes per revoluton) about the yaw axs for thermal control and solar power management purposes. Therefore, a small roll and ptch can be assumed. The GPS measurement equaton then can be gven by where ( j) r = ς1 cosψ + ς snψ + ς 3 (11) (j) (j) (j) (j) ( xr zr ) B ( yr zr ) (j) ( j ) (j) ( j ) B ( yr zr ) B( xr zr ) ( j ) ( j) ( j) B ( xr yr zr ) ς 1 = xb s s θ + y s + s φ ς = x s + s φ y s + s θ ς 3 = z s θ s φ + s T (1a) (1b) (1c) 5

6 The state transton matrx and process nose matrx of the developed flter are mplemented followng the standard Kalman flter formulaton Path Dfference Estmatons Phase dfference s provded by the GPS recever, whle the knowledge of atttude and lne bas s provded by the flterng estmaton. Durng operaton of the Kalman flter there s a contnual need to estmate and update the necessary nteger correctons on each phase dfference measurement. Fg 3 shows n block dagram form that ths s accomplshed n two ways. Usng the flter a predcted atttude s used to compute predcted path dfference measurements to be compared aganst the current measured dfferences. Unlke the standard Kalman flter ths nnovaton need to be adjusted when necessary by nteger shfts n each component of the observaton vector when a cycle shft has been detected. By backwards computaton also the proposed nteger correctons can be checked to see f a consstent bas value s generated, and f not a further correcton made. 3.4 Integrty Check Several crcumstances may cause the flterng estmaton to dverge. The developed Kalman flter presented n ths paper ncludes a protecton functon to prevent the dvergence or hgh error n estmated soluton. Ths functon s desgned to be actvated after the flter has been runnng for a whle. A smple dagram of the loop process of ths Kalman flter and ntegrty check s shown n Fgure 4. In normal operaton, the updated covarance matrx and updated state vector are used n the propagaton state for the next epoch estmaton, and ths nformaton wll be wrtten nto the buffer memory f the dfference of estmated atttude between two epochs s satsfed wthn the set boundary. If ths dfference s outsde the set boundary, the updated nformaton wll not be saved nto the buffer memory. Smultaneously, the prevous updated nformaton wll be used n the propagaton state. In ths protecton operaton, the propagated covarance matrx and propagated state vector wll be wrtten back to the buffer memory agan. The reason s that the anomalous operaton may last longer than one epoch. However, after a long perod of ths anomaly, the propagated nformaton may dverge from current spacecraft dynamcs. Therefore, a re-ntalsaton operaton wll be requred under ths condton. buffer memory for - covarance matrx - state vector wrte data read data wrte data N n tmes Y propagaton state - covarance matrx - state vector determnaton state dfferenced atttude between two epochs wthn boundary N reset Kalman flter ntal atttude estmatons usng block ambguty search and TRIAD agree Y update state - covarance matrx measurements Standard Kalman Flter flter ntalsaton Fgure 4: Loop process of Kalman flter wth dvergence protecton In such a re-ntalsaton operaton, the block ambguty search and TRIAD wll estmate agan an atttude for ntalsng Kalman flter. It may not necessary to re-estmate the lne bas agan. We can use the back up fgure whch was saved before resettng the flter, and re-estmate after the estmated atttude s gven by the flterng estmator. Wth experence ganed of the actual lne bas ths may also speed up the re-acquston durng the frst step of the block ambguty search. 4 Smulaton Results All smulaton results presented n ths paper are based on a gravty gradent satellte n a Sunsynchronous crcular orbt, 98 degrees nclnaton angle, and 8 km alttude. The nomnal smulaton parameters are gven n Table 1. The sx hours of smulated GPS measurements are used as the nput fle. The estmated atttude error (the estmated atttude from the extended Kalman flter compares to the reference atttude from smulaton) n roll, ptch and yaw are plotted n XY format n whch the X axs s the tme. 6

7 It s mportant to note that n ths paper, the measurement error s assumed as whte Gaussan wth 5 mm rms, whle the lne bas s assumed as constant. Table 1: Nomnal Smulaton Parameters ntal atttude roll ptch yaw degrees 1 1 moment of nerta x axs y axs z axs kg m 1 1 ntal angular velocty ω x ω y ω z deg/sec baselne coordnates b 1B b B [ x y z] metre [ 3. ] [ 3. ] lne bas β 1 β metre 9-8 Two anomalous operatons are smulated for testng the robustness of the flterng estmator. The frst anomaly (Case I) assumes that an error n the data memory may possbly caused by a sngle event upset and t ntroduces the sngle errors to the flter. The second anomaly (Case II) assumes that the smulated GPS measurement s perturbed by excessve nose for a short perod. The later crcumstance may be smlar to some cases under real operaton. For example a hgh gan antenna of an onboard RF transmtter s transmttng hgh power sgnals to communcate wth the ground staton. 4.1 Atttude Acquston from block ambguty search Smulaton follows a smpler approach n the block ambguty search that the general procedure outlned n secton 3.1 : the approach conssts smply n selectng the one soluton for each baselne whch s closest n length to the known length. Bas nformaton s only used as far as predcton of ts sgn. Ths procedure ncludes the test of known angle between baselnes aganst angle between tral baselnes. Ths smplfed approach sometmes rejects correct solutons on the frst epoch. The remedy s smply to repeat the acquston on the next epoch block of data untl successful acquston. Ths smplfed approach sometmes accepts ncorrect solutons on the frst epoch. The remedy s smply to 7 repeat the acquston on the next epoch of data and test for consstency of atttude s delvered. In a typcal heurstc test the angle between baselne s 9 degree and tolerance between baselne angles was set at + 15 degrees. Then t was seen that 75% of solutons from the block ambguty search were correct from the very frst epoch of data used The method was of course always successful n eventually acqurng the atttude n the few followng epochs for those 5 % of trals that were ntally unsuccessful. Fgures and tabulaton show the success n drect atttude determnaton on a successful acqure epoch. The estmated error n roll s shown n Fgure 5. For the estmated error n ptch and yaw, the rms fgure shown n Table s used to represent the characterstcs nstead of XY plot. The estmated fgure of ntal lne bas s shown n Fgure 6. Estmated Roll Error (degrees) Fgure 5: Estmated roll error of ntal atttude Table : RMS Error of Intal Atttude angle roll ptch yaw error (1 ) degree From the rms error as shown n the table, t can be seen that the accuracy of ntal atttude knowledge s adequate to ntalse Kalman flter. Estmated Lne Bas (cm) baselne #1 baselne # Fgure 6: Intal lne bas from block ambguty search

8 From these graphs, the error of ntal lne bas s wthn + cm. Ths confrms that the knowledge of the ntal lne bas can be used n path dfference estmaton for the frst epoch of flterng estmaton. 4. Estmaton under Kalman Flterng Ths secton shows the smulated results of flterng estmaton under normal condton. The estmated atttude error s shown n Fgure 7, and the estmated lne bas s shown n Fgure 8. Estmated Roll Error (degrees) Estmated Ptch Error (degrees) Estmated Yaw Error (degrees) Fgure 7: Estmated atttude error (normal condton) Table 3: RMS Error of Flterng Estmaton angle roll ptch yaw error (1 ) degree Estmated Lne Bas (cm) baselne #1 baselne # Fgure 8: Estmated lne bas (normal condton) From these graphs, t can be seen that the flterng estmaton performs very well. The error of estmated lne bas s wthn + 1cm. 4.3 Anomalous Condtons wthout Integrty Check Ths secton shows the smulated results of the flterng wthout ntegrty check. Two examples of anomaly as descrbed prevously are smulated Case I : Sngle Error In ths example, t assumes that a sngle errors occur at 1 th mnute and 4 th mnute of the operaton. For the frst event, t s assumed that each estmated Euler angle from the flterng estmaton s perturbed by an error of degrees (~34.5 degrees of pontng error). For the second event, an error of 3 degrees for each angle s used for the same purpose. The RMS of atttude error n roll, ptch and yaw wll be shown n Table 4. The estmated atttude errors n roll s shown n Fgure 9. The estmated lne bas s shown n Fgure 1. Table 4: RMS Error of Case I wthout Integrty Check angle roll ptch yaw error (1 ) 16 5 degree Estmate Roll Error (degrees)

9 Estmated Lne Bas (cm) Fgure 9: Estmated roll error of Case I wthout ntegrty check Fgure 1: Estmated lne bas of Case I wthout ntegrty check From these graphs, t can be seen that the flter can survve from the frst event, however t takes tme to re-converge agan. But n the second event upset, the estmaton completely dverges Case II : Contnuous Error In ths example, two perods of 5 mnutes perturbaton are smulated. The frst perod s snce 1 th mnute to 15 th mnute, and the second perod s snce 4 th mnute to 45 th mnute. Table 5: RMS Error of Case II wthout Integrty Check angle roll ptch yaw error (1 ) degree Estmated Lne Bas (cm) baselne #1 baselne # Fgure 1: Estmated lne bas of Case II wthout ntegrty check From these graphs, t can be seen that n the presence of excessve nose for short perods, the flter dverges after the frst perod of perturbaton and contnues to dverge. 4.4 Anomalous Condtons wth Integrty Check Ths secton shows the mproved performance of the flterng estmaton under dvergence protecton. The example n secton and secton 4.3. are redemonstrated Case I : Sngle Errors Wth the ntegrty check n place t can be seen that the flter recovers not only from the frst but also the second sngle upset event. As the large error caused by a sngle event errors can be detected by the ntegrty check, the flterng estmaton can be expected to contnue normally. The estmated atttude error s shown n Fgure 13. The rms of estmated atttude error s shown n Table 6. Estmated Roll Error (degrees) Fgure 11: Estmated roll error of Case II wthout ntegrty check Estmated Roll Error (degrees) Case I wth Integrty Check

10 Estmated Ptch Error (degrees) Case I wth Integrty Check Estmated Roll Error (degrees) re-ntalsaton Case II wth Integrty Check re-ntalsaton - - Estmated Yaw Error (degrees) Case I wth Integrty Check Fgure 13: Estmated atttude error of Case I wth ntegrty check Table 6: RMS Error of Case I wth Integrty Check angle roll ptch yaw error (1 ) degree From these graphs, t can be seen that the atttude error n roll and ptch s the same as the results under normal condtons. The reason s that the rate of change n roll and ptch between two epochs s not so hgh, and the updated covarance matrx of prevous epoch can cope wth ths rate. However, there s a short perod of fluctuaton n yaw as shown n Fgure 13. The reason s that the yaw rate s qute hgh, and the use of prevous nformaton may not cope wth the current yaw rate perfectly Case II : Contnuous Errors In ths case, the anomalous operaton wll contnue to occur wth the set count n the ntegrty check regularly beng exceeded. So regular ntalsaton s called and for and suppled by the overall algorthm Estmated Ptch Error (degrees) Estmated Yaw Error (degrees) re-ntalsaton re-ntalsaton Case II wth Integrty Check re-ntalsaton re-ntalsaton Case II wth Integrty Check Fgure 14: Estmated atttude error of Case II wth ntegrty check As we expected, several perods of anomalous operatons are detected by ntegrty check. Once the number of the detecton exceeds the lmt of set count (e.g. 1 epochs), the protecton functon resets the Kalman flter, and requrng the ntal step to estmate the atttude to re-ntalsng Kalman flter. The smulated results n ths sub secton and the prevous sub secton ndcate that the developed Kalman flter wth ntegrty check provdes a robust soluton, and the flterng tself ncludng the ntal step estmaton performs very robustly under anomalous condtons Conclusons In ths paper, the prncple and problems of atttude determnaton usng GPS sgnals were descrbed. A new approach based on the ntegraton of three

11 algorthms: block ambguty search; TRIAD; and extended Kalman flter was presented. A technque to resolve nteger ambgutes and cycle slps caused by nose and lne bas was developed for path dfference estmatons. Dvergence protecton was mplemented as an ntegrty check on the fltered soluton. The new approach requres only short dual baselnes whch can be accommodated on small satelltes. The block ambguty search requres only four selected GPS satelltes for ntal lne bas and baselne vector estmaton. The technque of resolvng nteger ambgutes ncluded cycle slps allows total atttude error up to + 18 degrees for 3 cm baselne. The flterng estmaton was tested n the presence of measurement nose and bas, and under smulated anomalous condtons. Smulated results ndcate that the flterng estmator provdes a robust soluton n presence of sngle errors, and re-ntalses tself after contnuous errors. References 1 Lghtsey, G.E., Cohen, C.E., Feess, W.A. and Parknson, B.W., Analyss of Spacecraft Atttude Measurement Usng Onboard GPS, Advances n the Astronautcal Scences, Vol. 86, Sutton, E., Optmal Search Space Identfcaton for Instantaneous Integer Cycle Ambguty Resoluton, Proceedngs of ION GPS-97, Kansas Cty, MO, September, Qunn, P.G, Instantaneous GPS Atttude Determnaton, Proceedngs of ION GPS-93, Salt Lake Cty, Utah, September, Martn-Nera, M., Tledo, M., and Pelaez, A., The Null Space Method for GPS Integer Ambguty Resoluton, Proceedngs of DSNS 95, Bergen, Norway, Aprl, Cohen, C.E. and Parknson, B.W., Integer Ambguty Resoluton of the GPS Carrer for Spacecraft Atttude Determnaton, Advances n the Astronautcal Scences, AAS Gudance and Control Conference, Keystone, CO, February, Conway, A., Montgomery, P., Rock, S., Cannon, R., and Parknson, B., A New Moton-Based Algorthm for GPS Atttude Integer Resoluton, Journal of the Insttute of Navgaton, Vol. 43, No., Crassds, J.L., Markley, L.F., and Lghtsey, G.E., Global Postonng System Integer Ambguty Resoluton Wthout Atttude Knowledge, Journal of Gudance, Control, and Dynamcs, Vol., No., March-Aprl, Lghtsey, G.E., Spacecraft Atttude Control Usng GPS Carrer Phase, Global Postonng System: Theory and Applcaton, Parknson, B.W. and Splker, J.J. (ed), Vol. II, Gomez, S.F., Panneton, R.J., Saunders, P.E., Hwu, S.U. and Lu, B.P., GPS Multpath Modelng and Vertcaton Usng Geometrcal Theory of Dffracton, Proceedngs of ION GPS-95, Plam Sprngs, CA, September, Axelrad, P., Comp, C.J., and Macdoran, P.F., SNR-Based Multpath Error Correcton for GPS Dfferental Phase, IEEE Transacton on Aerospace and Electronc Systems, Vol. 3, No., Aprl Cohen, C.E. and Parknson, B.W., Mtgatng Multpath Error n GPS Based Atttude Determnaton, Advances n the Astronautcal Scences, AAS Gudance and Control Conference, Keystone, CO, Schupler, B.R., Allshouse, R.L., and Clark, T.A., Sgnal Characterstcs of GPS User Antennas, Journal of the Insttute of Navgaton, Vol. 41, No. 3, Cohen, C.E., Atttude Determnaton, Global Postonng System: Theory and Applcaton, Parknson, B.W. and Splker, J.J. (ed), Vol. II, Cohen, C.E., bd, Shuster, M.D. and Oh, S.D, Three-Axs Atttude Determnaton from Vector Observatons, Journal of Gudance and Control, Vol.4, No.1, Hodgart, M.S., Hashda, Y. and Steyn, H., Enhanced Low-cost Atttude Control of Mcrosatelltes, Proceedngs of Spacecraft Gudance, Navgaton and Control System, ESTEC, Noordwjk, September, Hashda, Y., RDDC4A-ADCS for Future UoSat Standard Platform, nternal document, Surrey Satellte Technology Lmted, January

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