A Study of Hybrid Modernized GPS and Galileo Positioning in Japan

Transcription

1 1 A Study of Hybrid Moderized GPS ad Galileo Positioig i Japa Fali WU*, Nobuaki KUBO** ad Akio YASUDA** Abstract he moderizatio of the GPS ad the advet of the Europea Galileo will ehace the capability of quickly ad correctly resolvig the iteger cycle carrier phase ambiguities i precise differetial positioig. I this paper the performaces of hybrid moderized GPS ad Galileo positioig are aalyzed for differet scearios by usig the LAMBDA method i Japa. he scearios iclude moderized GPS system, future Galileo system ad combied GPS ad Galileo system over differet baselies. he results show that extedig a preset dual frequecy GPS system to hybrid moderized GPS ad Galileo system i future will sigificatly improve the capability of Real-ime Kiematic (RK) positioig. O the short baselie very high ambiguity success rate ca be easily obtaied, eve with a sigle epoch of data. O the medium baselie the ambiguity success rate could icrease by the icreased umber of satellites ad the improved satellite geometry. O the log baselie, for which differetial atmospheric delays have to be dealt with, reliable istataeous ambiguity resolutio is feasible oly if a striget coditio is met o either the code measuremet accuracy or o the accuracy of atmospheric correctio iformatio. Keywords: Precise Positioig, Moderized GPS, Galileo 1. Itroductio Udoubtedly, the GPS moderizatio program as well as the setup of the aticipated Europea couterpart Galileo will prove to be highly beeficial for Real-ime Kiematic (RK) positioig. GPS Block IIR-M ad IIF will both trasmit the uecrypted L civil sigal o the secod carrier frequecy makig the trackig of this sigal much easier ad reliable. he moderizatio of the GPS ad the advet of Galileo will together lead cooperatively to a truly multi-frequecy civil Global Navigatio Satellite System (GNSS). his paper focuses o the performace of itegral moderized GPS ad Galileo positioig i Japa. I the paper, the sigal structure ad parameters of moderized GPS ad Galileo are briefly reviewed. RK positioig usig itegral moderized GPS ad Galileo are itroduced. he ambiguity success rates are calculated with a geometry-based model for differet scearios usig the LAMBDA method (1)(4). he scearios chose for aalyses are itroduced, ad the system s performace o ambiguity resolutio will be preseted over time ad for locatios i Japa. he performace will be quatified for differetial positioig over short, medium ad log distaces. Sice oly istataeous ambiguity resolutio is cosidered, which is based o a sigle epoch of data, o distictio betwee a movig ad a statioary receiver is ecessary.. GPS Moderizatio ad Galileo he moderizatio of GPS has bee proceedig for the past several years based o discussios, recommedatios, ad plas that have occurred over most of * Studet Member: okyo Uiversity of Mercatile Marie (-1-6, Etchujima, Koto-ku, okyo ) ** Member: okyo Uiversity of Mercatile Marie (-1-6, Etchujima, Koto-ku, okyo )

2 the last decade. Also, coordiatio, plaig ad activities i Europea Uio to the implemetatio of the Galileo avigatio satellite system have cautiously but steadily advace cosiderably durig this same period..1 GPS Moderizatio.1.1 Satellite Costellatio he origial Block II ad IIA operatioal GPS satellites have all bee lauched, so o modificatios or chages to these satellites are possible. Seve of the GPS Block IIR repleishmet satellites have bee lauched ad six are actively ow supportig the GPS operatioal costellatio. wety-oe of these satellites were acquired by Air Force from Lockheed Marti. Part of the curret moderizatio pla is to modify the last twelve of these completed satellites to provide the capabilities. he civil L5 capabilities will ot be icorporated ito the modified IIR-M s because of a satellite prime power limitatio (6)..1. Sigal Structure he curret costellatio of GPS Block II ad IIA satellites provides C/A code o L1 oly, whereas the ecrypted P-code is modulated o both carriers, L1 ad L. he moderized GPS will trasmit two ew additioal coded civil sigals, the civil code o the L frequecy ad the ew sigal at L5. GPS Block IIR are lauched sice 1997 to replace the older Block II (ad IIA) vehicles. A moderized versio (Block IIR-M) is plaed for 003 with the C/A code ad a L civil sigal beig implemeted o L1 ad L, respectively. Furthermore, the military M-code is expected to be modulated o both carriers. Albeit GPS Block IIR-M will ot offer a third frequecy, trackig the L carrier will become sigificatly easier ad the sigal-to-oise ratio o the secod frequecy will improve. his fact is also importat for kiematic applicatios sice ioospheric disturbaces may cause loss-of-lock especially o L much easier for curret GPS tha for GPS IIR-M due to the accessible civil code o L (6).. Galileo..1 Satellite Costellatio he space segmet of Galileo is iteded to cosist of a total 30 Mea Earth Orbitig (MEO) satellites able 1 Parameters for Galileo satellite Semi-major axis a 9994 km Icliatio i 56 o Eccetricity e 0.0 Right ascesio Ω Rate of right ascesio Ω! -10 o,0 o,10 o 0.0 o /day Argumet of perigee ω 0.0 o Mea aomaly (1 st orbit pla) M -160 o, 10 o,..., 10 o, 160 o able Galileo carrier frequecies Carrier Ceter Frequecy (MHz) E5a(L5) E5b [ ] E E-L1-E cofigured as Walker costellatio, i.e. distributed over three orbital plaes. he altitude is 3616 km, ad the icliatio is 56 o. able 1 gives the almaac parameters for Galileo costellatio (3)(9)... Sigal Structure Ceter frequecies of Galileo are preseted i able. Galileo will provide te avigatio sigals i Right Had Circular Polarizatio i the frequecy rag MHz (E5a ad E5b), MHz (E6) ad MHz (E-L1-E1), which are part of the Radio Navigatio Satellite Service allocatio. All the Galileo satellites will share the same omial frequecy, makig use of Code Divisio Multiple Access (CDMA) compatible with the GPS approach. Six sigals, icludig three data-less chaels, so-called pilot toes, are accessible to all Galileo Users o the E5a, E5b ad E-L1-E1 carrier frequecies for Ope Services ad Safety-of-Life Service. wo sigals o E6 with ecrypted ragig codes, icludig oe data-less chael are accessible oly to some dedicated users that gai access through a give Commercial Service provider. Fially, two sigals with ecrypted ragig codes ad data are accessible to authorized users of the Public Regulated Service (3). 3. Hybrid Moderized GPS ad Galileo

3 3 Positioig he measured rages, by pseudorage ad carrier phase respectively, are related to the ukow parameters via the followig geeric measuremet equatios (8) : f = ρ... (1) s s s, i L1 s s s P r, i r + dr, i d + Ir + r + e r, i fi f Φ...() s s s, i L1 s s s s r, i = ρ r + δr, i δ Ir + r + λi N r, i + εr, i fi where Φ ad P are the carrier phase ad pseudorage, respectively; ρ is the geometric rage from satellite s to receiver r; i is the L-bad frequecy sigals, i =L1, L, L5, E1, E5a, E5b ad E6; I ad are the ioospheric ad tropospheric delay, respectively; d ad δ are the satellite ad receiver clock error, respectively; λ ad N are the wavelegth ad ambiguity of sigal i carrier phase; ε ad e represet the effect of receiver oise o the carrier phase ad the pseudorage, respectively. 3.1 Sigle-Poit Positioig he liearized pseudorage measuremet is give by the followig: ρ = G x + e.. (3) where ρ is the vector of predicted mius actual psudorage measuremet. he vector x has four compoets, the first three are the positio offset of the user from the liearizatio poit; the fourth is the offset of the user time bias. he m 4 geometry matrix G oly depeds upo the lie-of-sight effect. e is the residual error vector after the kow biases have bee removed. he Dilutio of Precisio (DOP) is a measure for the geometrical stregth of the observatio model. Differet types of DOP values are distiguished. Here oly geometric DOP, GDOP is cosidered. he DOP values deped o the cofactor matrix, ( ) 1 he GDOP is defied as: Q = G G. GDOP = trace( Q) = Q + Q + Q + Q.(4) 3. Real-ime Kiematic Positioig 3..1 Sigle-Baselie Solutio he liearized double differece observatio equatios are collected i the followig liear system of equatios (1) : y = Bb + Aa + e..(5) where y is the vector of observed mius computed double differece carrier phase measuremets, m y R ; b is the vector that cotais the icremets of the p baselie coordiates; a is the vector of double differece ambiguities; B is the m p desig matrix for the baselie coordiates; A is the m desig matrix for the ambiguity terms; e is the vector of umodelled effects ad measuremet oise. he least-squares priciple will be used to compute estimates for the baselie coordiates ad the iteger double differece ambiguities (7) : mi ba, p y Bb Aa 1 withb R ad a Z Qy...(6) he parameter estimatio is carried out i three steps: the float solutio, the iteger ambiguity estimatio ad the fixed solutio. I first step, the float solutio, the equatio (6) is p carried out with b R, a R. he real valued estimates ad variace-covariace matrix will be obtaied (1) : a Qa Q ab b, Qba Q (7) b he secod step, the iteger ambiguity estimatio, cosists of: 1 Qa mi a a with a Z...(8) a his miimized yields the iteger least-squares estimates for the vector of ambiguities: a". he computatio of the iteger estimate will be used LAMBDA method. I the third step, the fixed solutio, the fial solutio will be obtaied with the ambiguities fixed to their iteger least-squares estimates a" (1), b" = b Q Q a a"...(9) ba a 1 ( ) he least-squares estimates b " ad a" are the solutio to the costraied miimizatio equatio (6). 3.. LAMBDA Method he LAMBDA method (1) essetially cosist of a efficiet implemetatio of iteger least squares esti-

4 4 matio, where part of the efficiecy is caused by performig a decorrelatig ambiguity trasformatio. he ambiguity search space ca be defied by (7), a { a Z ( a a) Q ( ) } 1 a a a χ Ω =...(10) with χ a to be chose positive costat. he boudary of this search space is ellipsoidal. It is cetered at â, its shapes is govered by the variace-covariace matrix Q ad its size is determied â by χ. he search space is usually extremely elogated, due to the high correlatios betwee the ambiguities. Sice this extreme elogatio hiders the computatioal efficiecy of the search, the search space is first trasformed to a more spherical shape (7), z { z Z ( z z) Q ( ) } 1 z z z χ Ω =...(11) usig the admissible ambiguity trasformatio z = Z a, Qz = Z Qa Z. Sice the bootstrappig estimator is so easy to compute ad at the same time gives a good approximatio to the iteger least-squares estimator oce properly decorrelated, the bootstrapped solutio is a excellet cadidate for settig the size of the ambiguity search space. Followig the decorrelatio step z = Z a, the LAMBDA method therefore uses the bootstrapped solutio z" for settig the size of the ambiguity search space as χ ( z z ) Q ( z z ) = " "...(1) 1 B z B Usig the triagular decompositio of Q, the ẑ left-had side of the quadratic iequality i equatio (11) is the writte as a sum of squares (1) : i= 1 ( z z ) ii i χ σ ii...(13) Oce the search has completed, oe ca either output the trasformed iteger least-squares solutio z" or, by usig the iverse trasform a= " Z z", output the iteger least-squares solutio of the origial ambiguities. More iformatio about the LAMBDA method ca be foud i (1) Ambiguity Success Rate he ambiguity success rate is the probability, or chace, that the iteger ambiguities are correctly estimated. It ca be writte i equatio form as (4) Pa (" a) p( xdx ).. (14) = = Sa a where p a ( x) is the probability desity fuctio of the float ambiguities; S a is the pull-i regio, or area aroud the correct iteger for which ay float solutio gets pulled towards the correct fixed solutio. he ambiguity success rate depeds o three cotributig factors, the fuctioal model, the stochastic model, ad the chose method of iteger estimatio. Chages i ay oe of these will affect the ambiguity success rate. I geeral, the itegral i equatio (14) is difficult to evaluate, however i case of the bootstrappig estimator, the probability of correct iteger estimatio could be give explicitly as (7) : 1 P( a" B = a) = Φ (15) i= 1 σ ii where is the umber of ambiguities ad x 1 1 Φ ( x) = exp υ dυ π...(16) he coditioal stadard deviatios σ ii ca be obtaied directly as the square-roots of the etries of the diagoal matrix D i the triagular decompositio of the variace-covariace matrix Q a = LDL he ambiguity success rate ca be evaluated oce the fuctioal model ad stochastic models are kow. Similar to the usage of dilutio of precisio measures, it ca be computed without havig the actual measuremets available, that is, before actual field operatio. By meas of the success rate, the user is give a rigorous way of assessig how ofte oe ca expect ambiguity resolutio to be successful. It has bee prove that the iteger least-squares estimator with the LAMBDA method has the largest success rate of all admissible iteger estimators (10).. 4. Results ad Aalysis I this sectio the expected performace of hybrid moderized GPS ad Galileo positioig are aalyzed for various scearios i Japa. he scearios iclude

5 5 GPS aloe with dual ad triple frequecy, Galileo aloe with dual ad triple frequecy ad evetually itegrated GPS ad Galileo. First the model parameters ad basic assumptios are briefly reviewed. 4. Spatial Variatios Before cosiderig temporal variatios performace of hybrid moderized GPS ad Galileo, the spatial variatios performaces are aalyzed. Fig. 1, ad 3 show the GPS, Galileo ad itegrated GPS-Galileo visible satellite umber with geographic locatio, respectively. he visible satellite 4.1 Set Up I all computatios it is assumed that the double differece ambiguities remai costat durig the complete time spa. o compute the positios of the GPS satellites ad to simulate the positios of the Galileo satellites, a YUMA almaac was used, i the same way as was doe for the itegrated GPS-Galileo computatios (). For 4 GPS satellites ad 7 Galileo satellites costellatios were used, cotiuously tracked at March 6, 003, 1:00 for spatial variatios ad from March 6, 003, 00:00 to March 7, 003, 4:00, with a samplig iterval of 10 secods, for temporal variatios. he receiver-satellite geometry was simulated over Japa for spatial variatios ad at okyo ( 35 39'59'' N, '3'' E ) for temporal variatios. he visible satellites were masked by 15 o cut-off elevatio. he stadard deviatios of all phase ad code observatio were set at m ad 0.30 m, respectively. hree differet baselie legths, short baselie, medium baselie ad log baselie, are cosidered. he short baselie is typically oly of a few kilometers legth, the medium baselie some tes of kilometers ad the log baselie ca be hudreds of kilometers. O the short baselie differetial atmospheric delays are assumed to be completely abset (zero). hese delays are to be accouted for o the medium ad log baselie. A tropospheric zeith delay ad ioospheric slat delays are icluded as ukow parameters, but the ucertaity i these parameters values has bee restricted. Variatios i the delays are tolerated to a reasoable small extet o a medium baselie ( σ = 0. 01m ad σ I = 0. 0m ), ad to a much larger extet o a log baselie ( σ = 0. 10m ad σ I = 0. 40m ) (5). Sigle epoch solutios are cosidered, i.e. istataeous positioig. Fig. 1 Visible satellite umber spatial variatios (GPS) Fig. Visible satellite umber spatial variatios (Galileo) Fig. 3 Visible satellite umber spatial variatios (itegrated GPS-Galileo) umber of GPS varies betwee 5 ad 9. Galileo gives similar results with values betwee 6 ad 10. If it was possible to use the itegrated GPS ad Galileo, the visible satellite umber above 13 at ay locatio could

6 6 be obtaied. Fig. 7 emporal variatios of visible satellite umber Fig. 4 GDOP spatial variatios (GPS) Fig. 8 emporal variatios of GDOP Fig. 5 GDOP spatial variatios (Galileo) Fig. 4, 5 ad 6 show the spatial variatios of GDOP for the GPS, Galileo ad itegrated GPS-Galileo, respectively. he GDOP of GPS varies betwee 1.74 ad Galileo gives similar results with values betwee 1.81 ad 5.. If it was possible to use the itegrated GPS ad Galileo, GDOP could be obtaied betwee 1.38 ad.65. Fig. 6 GDOP spatial variatios (itegrated GPS-Galileo) 4.3 emporal Variatios Visible Satellite Number ad GDOP Fig. 7 presets the variatio of visible satellite umber for the GPS, Galileo ad itegrated GPS-Galileo, over a full two days period. he satellite umber of GPS varies betwee 4 ad 10. Galileo also gives similar values betwee 6 ad 10. he values of itegrated GPS-Galileo are betwee 11 ad 17. Fig. 8 shows the GDOP of GPS, Galileo ad itegrated GPS-Galileo as a fuctio of time. he GDOP of GPS varies betwee 1.67 ad Galileo gives values betwee 1.74 ad he values of itegrated GPS-Galileo are betwee 1.30 ad.94. he GDOP mea values of GPS, Galileo ad itegrated GPS-Galileo are 3.00, 3.03 ad 1.90, respectively. Comparig Fig. 8 with Fig. 7, the user would immedi-

7 7 ately otice the jump i GDOP of GPS, whe the umber of GPS satellites drops to four Ambiguity Success Rate able 3 ad Fig. 9 summarize the average ambiguity success rates of all scearios he GPS Case O the short baselie, the ambiguity success rates are 99.43% ad 99.97% for dual ad triple frequecy, respectively. O the medium, the ambiguity success rates are improved from 90.93% to 95.78% by usig triple frequecy. From dual frequecy to triple frequecy yields a improvemet from 13.6% to 5.01% for log baselie. able 3 Ambiguity success rates of various scearios (%) Baselie Legth Short Medium Log Dual GPS riple Galileo Dual riple Itegrated Dual GPS-Galileo riple he Galileo Case O the short baselie, the ambiguity success rates are 98.06% ad 99.89% for dual ad triple frequecy, respectively. O the medium, the ambiguity success rates are improved from 89.95% to 95.78% by usig triple frequecy. From dual frequecy to triple frequecy gais oly a improvemet from 13.4% to 1.73% for log baselie he Itegrated GPS ad Galileo Case O the short baselie, the ambiguity success rates are % ad % for dual ad triple frequecy, respectively. O the medium, the ambiguity success rates are improved from 99.83% to 99.86% by usig triple frequecy. O the log baselie, ambiguity resolutio is defiitely ot feasible usig just a sigle epoch of data. From dual frequecy to triple frequecy yields oly a improvemet from 6.4% to 7.6% for log baselie. able 3 ad Fig. 9 demostrate that the ambiguity success rates decrease with icreasig baselie legth. O a short baselie, very high ambiguity success rate levels ca be easily obtaied. O the medium baselie, ambiguity success rate could be icreased by trackig more satellites ad improvig satellite geometry. O the log baselie, it is difficult to resolve the ambiguities o the basis of just oe epoch of data, either with dual frequecies GPS, or with triple frequecies itegrated GPS-Galileo. he ambiguity success rates of itegrated GPS ad Galileo are ot higher tha for the two stadaloe systems. If, for example, GPS o its Fig. 9 Ambiguity success rates of various scearios ow provides high success rates, additio of the Galileo observatios may eve result i slightly lower success rates. 5. Coclusio I this article it has bee show that a itegrated use of the ew sigals i space of the moderized GPS ad the upcomig Galileo system will drastically improve the capability of RK positioig i Japa. O a short baselie, very high ambiguity success rate levels ca be obtaied, eve usig a sigle epoch of data. O a medium baselie ambiguity success rate could be icreased by the icreased umber of satellites ad the improved satellite geometry. O log baselie, the istataeous success rate is still to low for ay practical applicability. herefore for a future RK system atmospheric, especially ioospheric, modelig is of utmost importace. Ackowledgemets Discussio with Mr. Peter Jooste ad Ms. Sadra Verhage, Delft Uiversity of echology, were very useful for uderstadig the LAMBDA algorithm. Refereces (1) de Joge, P. ad iberius, C. (1996): he

8 8 LAMBDA method for iteger ambiguity estimatio: implemetatio aspects. LGR-Series 1, Delft Geodetic Computig Ceter, Delft Uiversity of echology, he Netherlads, August () Eissfeller, B., iberius, C., Pay,., Biberger, R., Schueler,., ad Heirichs, G. (001): Real-ime Kiematic i the Light of GPS Moderizatio ad Galileo. I Proceedig of the 14 th Iteratioal echical Meetig of the Satellite Divisio of the Istitute of Navigatio (ION GPS-001), pp , Salt Lake City, U. (3) Hei, G. ad Pay,. (00): he Europea Satellite Navigatio System Galileo. I ext for GPS Symposium 00, pages 3 34, okyo, Japa. (4) Jooste, P. ad iberius, C. (000): Fixig the Ambiguities - Are You Sure hey re Right? GPS World, 11(5): (5) Odijk, D. (000): Weightig Ioospheric Correctios to Improve Fast GPS Positioig Over Medium Distaces. I Proceedig of the 13 th Iteratioal echical Meetig of the Satellite Divisio of the Istitute of Navigatio (ION GPS-000), pp , Salt Lake City, U. (6) Shaw, M., urer, D. A., ad Sadhoo, K. (00): Moderizatio of the Global Positioig System. I ext for GPS Symposium 00, pp. 3 1, okyo, Japa. (7)euisse, P., Jooste, P., ad iberius, C. (00): A Comparisio of CAR, CIR ad LAMBDA GNSS Ambiguity Resolutio. I Proceedig of the 15 th Iteratioal echical Meetig of the Satellite Divisio of the Istitute of Navigatio (ION GPS-00), pp , Portlad, Orego, 00. (8) iberius, C., Pay,., Eissfeller, B., de Jog, K., Jooste, P., ad Verhage, S. (00): Itegral GPS-Galileo Ambiguity Resolutio. I ENC -GNSS 00 PROCEEDINGS, Copehage, Demark. (9) Verhage, S. (00a): Studyig the Performace of Global Navigatio Satellite Systems - A New Software ool. GPS World, 13(6): (10)Verhage, S. (00b): Performace Aalysis of GPS, Galileo ad Itegrated GPS-Galileo. I Proceedig of the 15 th Iteratioal echical Meetig of the Satellite Divisio of the Istitute of Navigatio (ION GPS-00), pp , Port- Portlad, Orego, 00. Questios ad Aswers Kazuaki Hoshioo (Electroic Navigatio Research Istitute, Idepedet admiistrative Istitutio): hree differet baselie, short baselie, medium baselie ad log baselie, are cosidered. How log are the short baselie, medium baselie ad log baselie? Fali Wu: he short baselie is typically oly of a few kilometers legth, the medium baselie some tes of kilometers ad the log baselie ca be hudreds of kilometers. Kazuaki Hoshioo (Electroic Navigatio Research Istitute, Idepedet admiistrative Istitutio): O the log baselie, why are the ambiguity success rates of itegrated GPS ad Galileo ot higher tha for the two stadaloe systems? Fali Wu: Because the ambiguity success rate depeds o three cotributig factors, the fuctioal model, the stochastic model, ad the chose method of iteger estimatio. Chages i ay oe of these will affect the ambiguity success rate. here are more ukow ambiguities for itegrated GPS ad Galileo tha GPS or Galileo stadaloe system. he effects of the larger umber of ukows may offset the improvemet due to better geometry. Nobuaki Kubo (okyo Uiversity of Mercatile Marie): I computatio, the stadard deviatios of the all phase ad code observatio were set at ad 0.30m, respectively. How do you deal with the multipath i your research? Fali Wu: Multipath is the error caused whe the sigal arrives at the receiver via more tha oe path, ormally caused by reflectios ear the receiver. As a result, it is highly depedet upo the coditios surroudig the receiver atea, the type of atea that is used, ad the iteral trackig loop algorithms of the receiver. Because there is o model that ca be used for the geeral case, the multipath has t bee take ito accout i this study.