Journal of Global Postonng Systems (22) Vol. 1, No. 2: 85-95 GPS Atttude Determnaton Relablty Performance Improvement Usng Low Cost Recevers Chaochao Wang and Gérard Lachapelle Department of Geomatcs Engneerng, Unversty of Calgary Receved: 13 October 22 / Accepted: 4 December 22 Abstract. Ths paper descrbes dfferent methods to mprove relablty of atttude estmaton usng low cost GPS recevers. Prevous wor has shown that low cost recever atttude determnaton systems are more susceptble to measurement errors, such as multpath, phase center offsets, and cycle slps. In some cases, these error sources lead to severely erroneous atttude estmates and/or to a lower avalablty. The relablty control n the atttude determnaton becomes mperatve to users, as most atttude applcatons requre a hgh level of relablty. The three methods tested heren to mprove relablty are the use of a hgh data rate, fxed angular constrants, and a qualty control algorthm mplemented wth a Kalman flter. The use of hgh rate measurements mproves error detecton as well as ambguty fxng tme. Fxed angular constrants n a mult-antenna atttude system s effectve to reject ncorrect solutons durng the ambguty resoluton phase of the process. Utlzng a Kalman flter wth a hgh data rate, e.g. 1 Hz, not only ncreases relablty through an ncrease of nformaton, but also can mprove accuracy and avalablty. The smultaneous utlzaton of the above methods sgnfcantly mproves relablty, as demonstrated through a seres of hardware smulatons and feld tests. The low cost recever type selected s the CMC Allstar recever equpped wth a commercally avalable low cost antenna. Fnally, the use of statstcally relablty measures, namely nternal and external relablty measures, shows the nherent lmtatons of a low cost system and the need to ether use better antennas and/or external adng n the form of low cost sensors. Key words: GPS, Atttude Determnaton, Low Cost Recever 1 Introducton Mult-antenna GPS systems provde a hgh accuracy atttude soluton wthout error drft over tme [e.g., Lu 1994]. The performance of GPS atttude determnaton s a functon of recever frmware, satellte geometry, antenna carrer phase stablty, multpath rejecton ablty and nter-antenna dstances. Wth advances n GPS recever technology, low cost recevers equpped wth phase loc loops that output precsely tme-synchronzed carrer phase measurements are now avalable on the maret. The use of ths grade of GPS recever for atttude determnaton has proven feasble [e.g., Hoyle et al. 22]. However, t has been found that multpath, antenna phase center offsets and cycle slps are major error sources that mtgate the performance of low cost recever atttude solutons. In worst-case scenaros, these errors severely affect the ntegrated carrer phase measurements and lead to ncorrect atttude estmates n atttude determnaton. Therefore, the relablty of atttude estmaton becomes a major ssue. The objectve of ths paper s to nvestgate three methodologes to mprove the relablty performance of atttude determnaton usng low cost recevers. Three dfferent schemes, namely the use of hgh rate carrer phase measurements, fxed angular constrants and a Kalman flter wth a statstcal qualty control system, are used nteractvely to mprove relablty. These schemes are mplemented n a hgh performance, open archtecture atttude determnaton software, namely HEADRT+ TM, for testng [e.g. Hoyle et al 22]. The performances of dfferent methods are examned both n hardware smulaton mode and under feld statc and nematc condtons.
86 Journal of Global Postonng Systems 2 GPS Atttude Determnaton Because of the short nter-antenna dstance (generally less than 2 m), the spatally correlated orbtal and By defnton, atttude s the rotaton of a specfc frame atmospherc errors vrtually cancel out from the equaton. wth respect to a reference frame, whch s well defned n The errors sources remanng here are only multpath, space. In the case of a mult-antenna system, ths specfc antenna phase center offsets and carrer recever nose, frame s usually referred to as the antenna body frame, provded that the double dfference nteger ambgutes whle the local level frame s selected as the reference are correctly solved. frame. Once the antenna vector n the local level frame s precsely determned, the three Euler atttude angles n the rotaton matrx can be estmated usng Equaton 1. 3 Relablty Problems Usng Low Cost Recevers b ll x x b ll y = R2 ( r) R1 ( p) R3 ( h) y (1) b ll z z h, p, r denote headng, ptch and roll x, y, z are the coordnates of the antenna vector superscrpt b represents the body frame superscrpt ll stands for the local level frame The GPS recevers determne the nter-antenna vectors frstly n a Conventonal Terrestral frame, namely WGS- 84. The carrer phase measurements have to be used as observables n ths applcaton snce the atttude determnaton system requres hgh precson relatve postonng between the antennas. In the general case that ndependent (non-dedcated} recevers are used and each recever has a separate oscllator, the double dfferencng combnatons are formed so that not only the cloc errors but also the lne bases caused by the dfferent cable lengths can be removed. Wthout cloc and lne bas errors, the carrer phase double dfference observaton equaton s expressed as Φ = ρ + λ N + dρ d on + d trop : + ε mult + ε rx ε ant + (2) Prevous research has shown the advantages and lmtatons of usng low cost recevers such as the CMC Electronc Allstar, for atttude determnaton [e.g, Hoyle et al 22]. Wthout multpath and antenna mpact, ths recever type can acheve atttude estmaton performance comparable wth hgh qualty/hgh cost unts durng hardware smulaton testng. Ths s because, under hardware smulaton condtons, error sources can be separated. No multpath or antenna phase centre errors need to be ntroduced, thereby allowng a performance analyss of the recever nose and tracng loops. Under feld condtons, the low cost recever s more lely to suffer from carrer phase multpath and antenna phase nstabltes. In practce each of these two error sources range from a few mm to 1 cm (although hgher values are possble). In some severe cases, the two error sources, coupled wth cycle slps, sgnfcantly deterorate the carrer phase measurements and the wrong double dfference ambgutes could be produced from the ambguty resoluton. The ncorrect ambgutes eventually lead to the erroneous atttude estmates, whch mpar the relablty of the whole atttude determnaton system. In order to mprove the overall atttude performance, some measures should be taen to enhance the relablty of atttude determnaton usng low cost recevers. 4 Atttude Determnaton Algorthm represents the double dfference operator In the HEADRT+ TM software, the atttude determnaton Φ s the double dfference carrer phase measurement ρ s the double dfference range N s the double dfference nteger ambguty λ s the carrer wavelength (m) dρ s the orbtal error d on, d trop are the errors due to the onospherc delay and tropospherc delay s the double dfference multpath error ε mult ε rx s the double dfference carrer phase error due to the recever nose ε ant s the double dfference antenna phase center offset estmaton process s carred out n two phases. The frst phase determnes the correct double dfference carrer phase ambgutes for the antenna vector(s). After the coordnate transformaton from WGS-84 nto the local level frame, the atttude parameters are estmated from the vector components wth correspondng varancecovarance matrx n the second phase [e.g. Lu 1994]. The ambguty resoluton used n the software s based on the Least Squares Ambguty Search Technque (LSAST) [Hatch 1991]. Ths method has the advantages of a small number of canddate ambguty combnatons and hgh computatonal effcency. Gven that the vector lengths are small, ths technque s effectve for ths purpose. The ambguty search regon s defned as a sphere wth the
Wang and Lachapelle: GPS Atttude Determnaton Usng Low Cost Recevers 87 radus of the nter-antenna dstance(s). After formng all possble ambguty combnatons, dfferent dscrmnaton tests are conducted to solate the correct ambguty set usng the fxed antenna dstance(s) and some other statstcal propertes [e.g., Hoyle et al. 22]. Two statstcal tests are nvolved n the ambguty resoluton, namely the rato test and the Ch-square test. The underlyng assumpton of a rato test s that the resduals of the correct ambgutes should be sgnfcantly smaller than those of the ncorrect ones. Only f the rato of the two smallest resdual quadratc forms s greater than a preset threshold (normally 2.5 to 4), s the potental ambguty set wth the smallest quadratc form accepted as the correct ambguty set. For true ambgutes, t s assumed that the double dfference resduals are normally dstrbuted, and the sum of the quadratc forms follows the Ch-square dstrbuton, wth the degree of freedom beng the redundant measurement number. Therefore, a Ch-square test based on the resduals s conducted to verfy the double dfference ambgutes n the software. Once the nter-antenna vector ambgutes are fxed, the nter-antenna vector components are transformed from WGS-84 earth-fxed frame nto local level coordnates and the atttude parameters are computed from an mplct least squares estmaton. Currently, no dynamc constrants of the platforms are mplemented n the flterng process to permt an epoch-by-epoch assessment of atttude estmaton under any dynamcs, subject to the avalablty of unbased recever carrer phase). 5 Hgh Data Rate As the CMC Allstar recever can output raw tme synchronzed carrer phase measurements up to 1 Hz, t allows for hgh data rate processng n HEADRT+ TM, both for ambguty resoluton and atttude estmaton. The hgher data rate can beneft the ambguty resoluton process due to the hgh avalablty of phase measurements. Also, platform dynamcs can be predcted for short tme ntervals n many applcatons and outler estmaton n the antenna vector lengths can be easly detected and further rejected usng flterng of the hgh rate measurements. In ths secton, only the effect of the hgh data rate on ambguty resoluton wll be nvestgated. The mpact of the hgh data rate on Kalman flter estmaton wll be dscussed n the sequel. In order to evaluate the performance of ambguty resoluton, the tme to fxed ambgutes s utlzed. In ths test, two recevers were used, both for the hardware smulator and feld test. In the latter case, two AT575-14 low cost antennas were used. The hardware smulaton test was done usng a Sprent STR-476 smulator. As no errors were smulated, the only remanng error present was recever measurement nose. The feld test was conducted on the roof of Engneerng buldng at the Unversty of Calgary. The nter-antenna dstances were about 1 m n both tests. The data was collected at a 1-Hz rate. The double dfference ambgutes were ntentonally reset every 12 seconds durng the data processng to gather enough trals for a meanngful analyss. The Mnmum Tme To Ambguty Fx (MTTAF) was set to 1 epoch and the fxng rato was set to 3 n HEADRT+ TM. Percentage(%) 1 8 6 4 2 1Hz Data 1Hz Data <.1 <1 <5 <1 <2 >2 Ambguty Fxng Tme(Seconds) Fg. 1 Tme to Fx Ambguty n hardware smulaton 1 8 6 4 2 Cumulatve Percent(%) Fg. 1 shows the ambguty fxng tmes for the case of the hardware smulaton test. Wthout multpath and antenna phase center offset, the nteger ambgutes were successfully determned wthn a sngle epoch (1s or.1 s) durng each tral, demonstratng that the CMC recever measurement nose s not a sgnfcant factor that s affectng ambguty resoluton performance. Percentage(%) 1 8 6 4 2 1Hz Data 1Hz Data <1 <5 <1 <6 <12 >12 Ambguty Fxng Tme(Seconds) Fg. 2 Tme to Fx Ambguty n statc feld test 1 8 6 4 2 Cumulatve Percent(%) The correspondng statc feld test statstcs are shown n Fg. 2. Wth the exstence of multpath and antenna phase center errors, 19.6 % of the ambgutes were fxed n one second wth 1 Hz data. The nteger ambgutes were fxed n 5 seconds 84.9 % of the tme. Meanwhle, wth 1 Hz measurement rate, the correspondng values were 89.4 % and 93.4 % respectvely. The tme requred to fx the ambguty could be sgnfcantly reduced usng hgh data rate n ths case durng some trals. However, there are cases the fxng tme was larger than 6 second. Ths was related to the presence of tmecorrelated multpath and antenna phase center offset errors. The hgh data measurement s less effectve to these errors. Tab. 1 shows that the probablty of resolvng correct ambgutes for the feld test s 93% for 1 Hz data and
88 Journal of Global Postonng Systems 96% for 1 Hz data. Wth the hgher rate measurements, the ambguty resoluton relablty can thus be only slghtly mproved. Even though the ncorrect ambgutes were selected occasonally, they can be easly rejected n the atttude software ether by mprovng the MTTAF parameter n ambguty resoluton or by the relablty control n the atttude estmaton phase. Tab. 1 Performance of ambguty resoluton usng dfferent data rate measurements Correctness (%) 1 Hz data 1 Hz data Smulaton Test 1% 1% Statc Feld Test 96 93 6 Fxed Angular Constrant Scheme If one can assume that the antennas are mounted on a rgd platform, then ther relatve postons are fxed regardless of the platform moton. The full antenna frame geometry s nown a pror and approprate constrants can be used n the ambguty resoluton process to tae advantage of ths nowledge. Many geometrc constrants have been brought forward for the ambguty resoluton n mult-antenna GPS atttude determnaton system. [El- Mowafy 1994, Euler and Hll 1995] In ths research, the fxed angle between the antenna vectors, as well as the vector length, was employed to verfy the double dfference ambgutes. The mplementaton of the angular constrant s straghtforward. Frst, the fxed planar angles ( θ ) between antenna vector pars could ether be measured a pror or calculated usng the antenna coordnates n the body frame. Once the nteger ambgutes of the antenna vector pars have been determned, the angle between the pars can be drectly computed usng the antenna vector coordnates n the local level frame: r r LL LL E 1 bab b θ = cos r v LL LL b b AB Then, the estmated angle s compared wth the nown angle θ. If the ambgutes of two nter-antenna vectors are correctly solved, the two angles should be consstent wthn a certan tolerance. θ θ < δ θ E (4) The numercal value of the angular tolerance n (4) depends on the nter-antenna dstance and the qualty of phase measurements, whch are a functon of measurement nose, multpath and phase centre stablty. In the case of antenna vector lengths of 1-2 m and a moderate carrer phase measurement qualty, a 5-degree tolerance s approprate to detect the wrong ambgutes. If at least four antennas are used n the atttude determnaton system and only one vector ambguty s wrong, ths erroneous ambguty combnaton can be detected and dentfed by checng all the angles between the nter antenna vectors. A hardware smulaton test was conducted wth the 476 smulator to nvestgate the valdty of the angular constrant scheme. An antenna body frame was smulated usng nter-antenna dstances of 1 m. The angles between the antenna vectors were ntentonally set to 9 degrees n ths test. The ntal parameters used n the software were Fxng rato =3 MTTAF =1 epoch E δ LL LL LL LL LL LL 1 E AB E + N AB N + VAB V = cos (3) b b AB E θ s the estmated angle between the two antenna vectors r r LL LL b AB, b are the antenna vectors n local level frame b b AB, are the lengths of the antenna vectors E, N, V are three components of antenna vector n east, north, and vertcal drectons subscrpts A, B, C represent the prmary antenna and two secondary antennas Fg. 3 DOPs and SV number durng smulator test Fg. 3 shows the satellte s azmuth and elevaton DOPs durng the test. At GPS tme 216932 s, the loss of SV27 sgnal n one of the secondary recevers caused the falure of the Ch-square test and the re-ntalzaton of the double dfference ambguty for the correspondng nterantenna vector. Unfortunately, the wrong ambguty was determned due to the short MTTAF. When SV27 was reacqured by the recever, an ncorrect ambguty was frst
Wang and Lachapelle: GPS Atttude Determnaton Usng Low Cost Recevers 89 found, wth the true ambguty obtaned afterwards. The effect of ths error on the nter-antenna vector solutons durng ths perod s shown n Fg. 4. The nter-antenna length components are obvously ncorrect. However, the length tself was corrected solved and testng of the soluton wth that nown length faled to detect the ncorrect soluton n ths case. and rms agreements n headng, ptch and roll are 2.1,.3, -.2, 2.8, 3.4 and 3.3 arcmns, respectvely. Fg. 4 Effects of an ncorrect ambguty on an nter-antenna vector estmate Snce no qualty control procedure was performed n the least squares atttude estmaton, the erroneous nterantenna vector solutons nevtably led to the wrong atttude parameters. The error effects on the atttude component estmates are shown n Fg. 5. Fg. 5 Effects of an ncorrect ambguty on atttude component estmates After the angular constrant scheme was mplemented n the software, the wrong ambguty was easly detected and the erroneous vector soluton was successfully detected and excluded from the atttude estmaton. As shown n Fg. 6, the correct atttude components were estmated n the least squares soluton usng the other two nter-antenna vectors. The small shft n the atttude estmates s due to the excluson of SV27 and the resultng slght change of satellte geometry. The mean Fg. 6 Atttude results after mplementng angular constrants n statc smulaton test By employng the angle consstency chec n the ambguty resoluton, some ncorrect ambguty solutons can be effectvely rejected, whch sgnfcantly mproves the relablty of mult-antenna atttude determnaton. 7 Kalman Flter Estmaton Kalman flterng estmaton provdes a recursve method for the determnaton of atttude components through a predctng and updatng process. The general formulas n Kalman flterng can be wrtten as [Brown & Hwang 1992] z = H x + v (5) = x 1 x φ + w (6) z s the measurement vector at tme H s the desgn matrx x v φ w s state vector at tme s the measurement nose wth covarance R s the transton matrx s the process nose wth covarance Q In atttude determnaton usng vector components, the measurements are the antenna vector components n the local level frame. The desgn matrx s the partal dervatve of the rotaton matrx wth respect to the state vector n Equaton 1.
9 Journal of Global Postonng Systems R H = (7) x The state vector here ncludes the three Euler atttude parameters and ther angular rates: T ( ψ θ ϕ ψ& & θ & ϕ x = ) (8) The transton matrx expressed as follows 1 φ = Q = 1 1 σ dt 1 2 ψ& dt 1 σ 2 θ& φ dt 1 2 σ & ϕ and the process nose can be The numercal values of the angular rate varances n (1) represent the tghtness of the dynamc constrant of the Kalman flter. In vehcular atttude determnaton, the sgma of the angular rate n the Q matrx s emprcally selected as 2 degrees per epoch n 1 Hz samplng n the present case. Intutvely, one realzes that the effectveness of the flter n detectng ncorrect solutons, thus mprovng relablty, wll depend on our a pror nowledge of the vehcle dynamc and of the measurement rate. Usng ths model, the atttude parameters and ther angular rates can be correctly estmated n the Kalman flter as long as all the measurements are free of errors. As prevously mentoned, the measurements used n the Kalman flter are the nter-antenna vector solutons after ambguty resoluton. In the case that the wrong ambguty s determned, these quas-measurements are n error and the atttude estmates calculated from the Kalman flter may devate from the truth. In order to reject the ncorrect nter-antenna vector solutons from the Kalman flter and mprove the relablty of the atttude estmates, a qualty control system based on the flter nnovaton sequences s ntroduced heren. The nnovaton sequence s the dfference between the actual system output and the predcted output based on the predcted state (see Equaton 11) [Teunssen & Salzman 1988]. v ) f ( x ) = z (9) 2 χ α s the Ch-square probablty wth a sgnfcance (1) level of α. (11) Under normal condtons, the nnovaton sequence s a zero-mean Gaussan whte nose sequence wth nown varance. In the presence of erroneous measurements, such assumptons are no longer vald, and the nnovaton sequence devates from ts zero mean and whte nose propertes. Thus some statstcal tests can be conducted to detect and dentfy outlers or faults n the measurements. Frstly, an overall model test s conducted to detect the errors n the measurement vector. The test statstcs n ths global test are gven as T T 1 = v C v ~ χ 2 ( m,) (12) v a m s the number of observatons taen at tme, C s the covarance matrx of the nnovaton and v If the global test s rejected, the system error can be dentfed wth the one-dmensonal local slppage test. w = T 1 l C v v ~ Ν(,1) (13) T 1 l C l v l (,...,1,,...,) T = for =1,,m 1 +1 α The sgnfcant level n the local test s suggested to be.999, whch leads to a boundary value of 3.29. Thus the -th measurement s flagged for rejecton when w > 3.29 (14) When mplementng statstcal tests to dentfy outlers n the measurements, two types of errors may be made, as shown n Fg. 7. The frst type (Type I) s rejectng a good measurement. The probablty assocated wth ths type error s denoted by α. If a bad measurement s accepted by the test, a Type II error occurs. The probablty of a Type II error s expressed as. Fg. 7 Type I/II Errors Gven the probablty values of Type I and Type II errors, the Mnmum Detectable Blunder (MDB) can be calculated as the ablty to detect errors n the system as β
Wang and Lachapelle: GPS Atttude Determnaton Usng Low Cost Recevers 91 z = l δ T C l 1 v δ α β α (15) s a functon of and (see Fg. 8). In GPS nematc applcatons, and β are commonly selected to be.1 and.2 respectvely and 4.13. δ s then In the presence of strong multpath, the dentfcaton test (Equaton 15) may be too senstve and wll sometmes lead to a false alarm. In order to allevate ths problem, a further step was ntroduced by comparng the nnovatons wth the MDB. If the nnovaton s larger than the MDB, the measurement s dentfed erroneous, otherwse t s consdered a false alarm. The modfed Kalman-flter-based atttude determnaton software was tested wth the data collected wth the Sprent 476 hardware smulator usng four CMC recevers. A vehcle trajectory was smulated n ths test and the antenna confguraton s shown n Fgure 8. The maxmum atttude changes were about 2 degree/s n headng and several degrees per second n ptch. The true atttude durng the test s plotted n Fg. 9. Fg. 8 Smulated vehcle test antenna confguraton tght dynamc constrants usng a 1-Hz date rate, as over shootng effects occur. Wth a 1-Hz data rate, the performance of the flter s excellent, the atttude parameter estmates beng slghtly better than those of the least squares estmates. Fg. 1 Atttude estmate errors usng dfferent estmaton methods Tab. 2 RMS Kalman flter versus least-squares RMS Headng Ptch Roll 1 Hz LS 3.9 11.8 9.9 1 Hz KF 3.9 9.8 7.9 1 Hz KF1 31.9 11.8 9. In order to test the performance of cycle slp detecton usng the qualty control method mplemented by the Kalman flter, 8 cycle slps were ntroduced n the carrer phase measurements on dfferent recevers wth a magntude rangng from 1 to 8 cycles. Usng the tradtonal phase predcton detecton and nter-antenna length consstency chec, all the cycle slps but one were ether detected or recovered. The remanng cycle slp was removed when the Kalman flter was used, as shown n Fg. 12. Fg. 9 True atttude parameters durng hardware smulator test The results, summarzed n Fg. 1 and Table 2, show that the Kalman flter method dd not wor well wth Fg. 11 Cycle slp detecton usng Kalman flter
92 Journal of Global Postonng Systems 8 Feld Test And Result Analyss A nematc feld test was carred out usng two grades of GPS recever. The hgh grade system conssted of two NovAtel Beelne recevers and four NovAtel 51 antennas, whle the low grade system conssted of four CMC Electronc Allstar recevers and four AT575-7 antennas. The NovAtel Beelne recever s a hgh performance bantenna recever for 2-D atttude determnaton. The NovAtel 51 antenna has very good antenna phase center stablty. The AT575-7 actve antenna s a small sze low-cost (5 cm n dameter) OEM antenna often used wth CMC Allstar recevers. Two antenna frames were mounted wth smlar geometry on the roof of a mnvan, to create the moble platform n ths test, as shown n Fgure 12. The antenna confguraton used here was the same as n the above smulaton test (Fg. 9). The raw GPS measurements from two atttude systems were logged wth laptops n a 1-Hz rate. Fg. 13 DOPs and SV numbers durng vehcle test not calbrated. It s not realstc to do so for such low cost antennas that are lely to nclude unt-to-unt varatons. Fg. 12a Vehcle test Fg. 14a Resduals n nter-antenna vector solutons usng Beelne recevers Fg. 12b Antenna confguraton The azmuth and elevaton DOP and the number of satelltes traced are shown n Fg. 13. Durng the test, the number of satellte traced was mostly around sx to seven, except n some cases there was heavy folage near the road, and the satellte numbers dropped to fve or less. Fg. 14 show the resduals of double dfference pars at every epoch n the nter-antenna vector solutons. These resduals represent the overall effect of measurement errors, ncludng multpath and antenna phase center errors, assumng that the double dfference ambgutes are correctly solved. The average RMS values are gven n Tab. 3. The CMC unts have much larger double dfference resduals snce ther carrer phase measurements are more affected by multpath and antenna phase center errors than those of the Beelne unts. Note that the CMC antenna phase centre errors are Fg. 14b Resduals n nter-antenna vector solutons usng CMC recevers The three Euler atttude parameter estmates usng the Beelne unts are shown n Fg. 15. The blue dots are the
Wang and Lachapelle: GPS Atttude Determnaton Usng Low Cost Recevers 93 least squares atttude estmates and ther 3-sgma standard devaton envelopes, whle the red dots are the correspondng estmates from the Kalman flter wth the qualty control method turned on. Tab. 3 Resduals RMS (mm) for Beelne and CMC recevers Inter-antenna Beelne Rcvrs CMC Rcvrs Vector 1 5 17 2 5 12 3 5 15 however be easly dentfed by nspectng the 3-sgma standard devaton envelopes. Once a Kalman flterng wth the qualty control method and the angular constrants are mplemented, the wrong nter-antenna vector solutons are detected and excluded from the soluton. Ths elmnates erroneous atttude parameters from the output. The Kalman flter 3-sgma standard devaton envelopes are slghtly smaller than those from the least squares method due to the flter constrants. As can be seen n the fgures, the standard devaton mprovement s more sgnfcant n ptch and roll than n headng. Ths s because the ptch and roll dynamcs were lower than those of headng. Fg. 15 Atttude estmates usng the Beelne system Usng least squares estmaton, wrong atttude parameter estmates were output when heavy satellte blocages occurred. The reason for ths s that the least squares estmaton was severely affected by ncorrect vector solutons n such crcumstances. Once the base satellte s lost, the double dfference ambgutes have to be resolved at the next epoch. As s nown, ambguty resoluton performance s hghly correlated to the number of vsble satelltes and ther geometry. In a heavy sgnal blocage area wth strong multpath and phase center varatons, ambguty resoluton s more lely to result n an ncorrect soluton, whch leads to erroneous atttude parameter estmates. These ncorrect estmates can Fg. 16 Atttude estmates usng the CMC system The atttude results from the CMC unts are shown n Fg. 16. The overall atttude estmaton accuracy was slghtly lower than that obtaned wth the Beelne unts. Usng the Kalman flter augmented wth the qualty method, erroneous vector solutons, whch caused wrong atttude estmates n the least squares approach, were successfully dentfed and rejected from the atttude estmaton. As the phase measurements from CMC unts are more vulnerable to multpath, phase center errors and cycle
94 Journal of Global Postonng Systems slps, erroneous nter-antenna vector solutons were more frequently determned. When more ncorrect solutons were rejected by the Kalman flter, the avalablty of atttude estmates degraded due to the reducton of correct quas-observables compared wth the result from the least squares method. The lowered number of vector solutons nvolved n atttude estmaton, coupled wth the larger carrer phase errors, caused large varatons n the estmaton accuracy of the Kalman flter. The estmated atttude dfferences between the Beelne and CMC systems are shown n Fg. 17, and the correspondng statstcs are summarzed n Table 4. The estmated dfferences are mostly wthn 1.5 degrees n headng and 3 degrees n ptch and roll. The largest dfferences occur durng perods of poor satellte geometry. current low cost unts. In order to ncrease atttude component estmaton performance, hgher performance, but more expensve antennas could be used. The use of long nter-antenna dstances would also mprove accuracy. Adng wth external sensors s the other alternatve. Fg. 17 Atttude dfferences between the Beelne and CMC systems Tab. 4 Statstcs of atttude dfference between Beelne and CMC unts (Unts: degrees Dfference Headng Ptch Roll Mean.68 -.74 -.24 RMS.94 2.26 2.17 Max(abs) 4.56 8.64 9.44 Fgure 18 shows the external statstcal relablty of the two systems. External relablty s the mpact of the maxmum measurement errors that could occur and go undetected, on the atttude estmates, for two systems. Ths relablty measure s a functon of the qualty of carrer phase measurements and of the redundancy numbers n the Kalman flters. The external relablty of the Beelne system s farly consstent durng the test except durng tmes of poor geometry. The correspondng relablty of the CMC system s much poorer due to the hgher multpath and antenna phase center errors. It s mportant to note that the estmated atttude dfferences n Fgure 23 are wthn the relablty numbers of Fgure 24 and 25. Thus, one can conclude that the CMC unts have reached ther lmt n term of accuracy performance, f one assumes that the choce of antennas s lmted to 9 Conclusons Fg. 18 External relablty Multpath, cycle slps and antenna phase center nstablty are major error sources lmtng the relablty of standalone GPS-based atttude determnaton wth low cost recevers. Even f a requred level of accuracy can be acheved wth a gven multple recever confguraton, relablty becomes a major ssue. It has been demonstrated heren that the use of angular constrants and a Kalman flter wth a hgh data rate are effectve to sgnfcantly mprove relablty. However, the use of statstcal relablty analyss has also shown the lmtatons of the above technques. Another technque s currently beng assessed to mprove relablty and error detecton, namely the use of low cost rate gyros ntegrated wth the antenna assembly n varous confguratons. Gven a GPS data rate of 1 Hz, such low cost rate gyro should stll be useful for short term predcton between the GPS measurements, smoothng, error detecton and enhancement of avalablty. Early results ndcate these are possble enhancements ndeed.
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