GLONASS Double Difference Ambiguity Resolution in Real-Time
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1 GLONASS Double Dfference Ambguty Resoluton n Real-Tme do Rossbach IfEN Gesellschaft für Satelltennavgaton mbh (IfEN GmbH), Neubberg, Germany BIOGRAPHY do Rossbach holds a degree n aeronautcal engneerng from the Techncal nversty Aachen (Germany). From 1994 to 1998 he worked as a Research Assocate at the Insttute of Geodesy and Navgaton of the nversty of Federal Armed Forces Munch, dong research on GLONASS and GPS/GLONASS combnaton. Snce 1999 he s workng as a systems engneer wth IfEN GmbH. There he s nvolved n the development of GPS/GLONASS/EGNOS algorthms and software. Postonng wth GLONASS s also the topc of hs pendng Ph.D. thess. ABSTRACT ndoubtedly, the combned use of GPS and GLONASS brngs along a number of advantages as compared to the use of ether of the systems alone. However, the dfferent sgnal frequences of the GLONASS satelltes gve rse to problems n the determnaton of double dfference nteger ambgutes. Ths s the more concernng, snce double dfference carrer phase observatons are preferably used for geodetc applcatons. Therefore, ths topc s under nvestgaton by dfferent groups of scentsts all over the world. In bref, the problem wth GLONASS double dfferences s that ether the recever clock errors do not cancel, or the ambgutes are no longer of nteger nature. Proposed solutons to ths problem nclude confnng to sngle dfferences or scalng GLONASS carrer phase measurements to a common frequency. Ths paper ntroduces an mproved approach to tackle the problem of usng GLONASS and GPS/GLONASS combnatons for hgh-precson geodetc applcatons. It s sutable for real-tme operaton, especally also for Real- Tme Knematc (RTK) applcatons. The approach s descrbed n theory. Processng results are gven, and ts performance s evaluated. INTRODCTION GPS undoubtedly has become the means of choce for hgh precson navgaton and geodetc postonng worldwde. sng double dfference carrer phase measurements, decmeter accuracy can easly be reached. When the nteger ambgutes are fxed correctly, even centmeter accuracy s obtaned. There are, however, stuatons where a precse postonng soluton cannot always be guaranteed usng GPS. In areas wth lots of obstructons, such as urban canyons, mountanous areas or woodland, there may be cases when less than four GPS satelltes are n vew. Ths s not enough for a postonng soluton at all. And even f four satelltes are n vew, for fxng the nteger ambgutes t s desrable to have as many measurements as possble. Generally, at least sx satelltes are requred for ths purpose. To overcome ths potental vsblty problem, one possble soluton s the combnaton of GPS wth GLONASS. The Russan satellte navgaton system s very smlar to GPS, and the combned use of GPS and GLONASS therefore can be easly accomplshed. Bascally, the GLONASS satelltes can be treated as addtonal GPS satelltes. These addtonal satelltes may not only solve the problem of not havng enough satelltes n vew. Even n cases where enough GPS satelltes can be observed, the addtonal GLONASS satelltes may lead to a better satellte geometry and thus to a more precse postonng soluton. In a smlar fashon, the addton of GLONASS satelltes may mprove Recever-Autonomous Integrty Montorng (RAIM). These advantages are the reason why there s stll a strong nterest n the use of GLONASS and combned ION GPS 2000, September 2000, Salt Lake Cty, T 163
2 GPS/GLONASS for (precse) postonng, despte the recent demse of the GLONASS constellaton due to the ongong economc problems n Russa. Even the dscontnung of S/A, whch consttuted the man dsadvantage of GPS compared to GLONASS, dd not affect the nterest n GLONASS sgnfcantly. However, when usng double dfference carrer phase measurements for precse postonng, one dstnct feature of the GLONASS system gves rse to a dfferent knd of problem. Due to the dfferent sgnal frequences of the GLONASS satelltes, the recever clock errors do not cancel n the double dfference observaton equaton, when the equaton s set up n unts of cycles. Alternatvely, when expressed n unts of length, the carrer phase ambgutes lose ther nteger nature. Ths s a maor complcaton, snce double dfference carrer phase measurements are preferred for precse postonng. Ths topc therefore s under nvestgaton by dfferent groups of scentsts all over the world. Ths paper descrbes a unque approach to deal wth ths problem and gves some frst experences. Ths approach bases on the algorthm as ntroduced n [6]. GLONASS DOBLE DIFFERENCE CARRIER PHASES The observaton equaton for a carrer phase observaton at a user to a satellte n unts of length can be wrtten as: λ ϕ = ρ + λ N + cδ t cδt cδt + cδt ε wth λ Sgnal carrer frequency of satellte, ϕ Phase measurement from user to satellte, ρ Geometrcal range between user and satellte, N Integer ambguty between user and satellte, c t Speed of lght, δ Clock offset of satellte, δ t Clock offset of user recever, δ t Tropospherc delay between user and sat., δ Ionospherc advance between user and sat., t ε Nose of measurement from user to satellte. + (1) The sngle dfference carrer phase observaton between user and reference staton R to satellte then reads: λ ϕ = ρ + λ N + c δt wth the abbrevaton = R + c δt c δt + ε Formng double dfference carrer phase observatons between user and reference staton R to satelltes and, the observaton equaton becomes: λ ϕ ρ + c δt λ ϕ + λ N wth the abbrevaton = λ N c δ = t + ε Gven two satelltes wth dentcal sgnal frequences and thus wavelengths λ = λ = λ, equaton (3) smplfes to λ ϕ = ρ + λ N + c δt c δt + ε Tropo, Ths s the common observaton equaton for double dfference GPS carrer phase measurements. But as soon as at least one GLONASS satellte s partcpatng n the double dfference observaton, the wavelengths are no longer dentcal, and equaton (3) can no longer be smplfed n ths way wthout losng the nteger character of the ambguty terms. Therefore, sngle dfference terms reman n the double dfference observaton equaton (3). In the past, a number of dfferent ways have been proposed to treat ths problem. Proposed solutons are to confne to floatng ambgutes [3] or to confne sngle dfference observatons and obtan the clock offsets from pseudorange measurements [7]. However, the pseudorange measurements are less precse than the carrer phase measurements. Another proposal was to scale all GLONASS carrer phase observatons to a common (center) frequency [4]. The proposed teratve ambguty resoluton [2] s not sutable for real-tme applcatons such as e.g. RTK. In [6] a method was descrbed to form carrer phase double dfferences for dfferent sgnal frequences wthout losng the nteger nature of the ambgutes. The core of ths method s the ntroducton of an auxlary sgnal wth a wavelength, of whch the wavelengths of the partcpatng satelltes are nteger multples. Ths method does not (2) (3) (4) 164
3 requre much addtonal effort n processng the phase measurements. It s thus sutable for real-tme applcatons and was therefore chosen as the bass for the GLONASS ambguty resoluton. It s brefly recaptulated n the followng. An expresson of the double dfference observaton equaton wth nteger ambgutes but wthout sngle dfference terms n t can be found by ntroducng an auxlary wavelength λ, of whch both λ and λ are nteger multples: λ λ λ = = = k k c k f, wth k, k I (5) Insertng ths defnton nto equaton (3) yelds a modfed double dfference observaton equaton: λ k, k ρ ϕ + c δt = + λ wth the abbrevaton k, k N c δ ab, = a b t + ε The modfed double dfference ambgutes k k, N are stll of nteger type. They refer to the auxlary sgnal wth the wavelength λ. The coeffcents k, k must be chosen dependng on the sgnal frequences nvolved. For GLONASS satelltes they depend on the frequency letters of the satelltes nvolved. The L 1 and L 2 carrer frequences of GLONASS satellte are determned by the relatons ( + n ) f 1 f L1 = 2848 L f = n f ( ) L2 L2 wth f L1 = MHz and f L2 = MHz and n beng the frequency letter of satellte. In case of two GLONASS satelltes, accordng to the defnton (5), λ thus must be λ = c k n f = c k n f ( ) ( ) Ths condton s fulflled for the par of coeffcents k = n, k = n. Therefore, the GLO- NASS/GLONASS double dfferences become: λ n,2848+ n ρ + c δt + λ ϕ = n,2848+ n c δt N + ε (6) (7) (8) wth the auxlary wavelength λ = c 1 f n n ( )( ) These equatons hold for both L 1 and L 2. Therefore, the respectve ndces were omtted. For the double dfference between a GPS and a GLONASS satellte the coeffcents must be chosen as k = 75 ( n ), k = for L 1 and k = 35 ( n ), k = for L 2. However, these auxlary wavelengths are very small. For GLONASS/GLONASS double dfferences on L 1, ths wavelength s about 65 µm, for GPS/GLONASS double dfferences t s even n the order of 0.9 µm. For L 2, the respectve wavelengths are about 84 µm and 2.4 µm. AMBIGITY RESOLTION The treatment of GLONASS double dfference carrer phase measurements as descrbed n the prevous chapter was mplemented n a GPS/GLONASS postonng software package. For the ambguty search tself, the Euler- Landau method [1] was chosen. Ths s a fast way of searchng ambgutes, sutable for real-tme applcatons. Due to the small wavelength of the GPS/GLONASS double dfferences, the ambgutes were deemed to be very dffcult to fx correctly. Besdes ths, n GPS/GLONASS double dfferences effects of dfferent recever hardware bases for satelltes of the two systems may be found [5]. Therefore, only GLONASS/GLONASS double dfferences are formed. In combned GPS/GLONASS scenaros, ths means that two reference satelltes are used, one for GPS and one for GLONASS. The auxlary wavelength for GLONASS/GLONASS double dfferences s much larger and mght therefore be easer to fx. Stll, t s far off from the wavelength of an orgnal satellte sgnal. The dea was that wth such small wavelengths, ambguty fxng to ntegers wll not be necessary, but fxng to thousands of cycles mght be suffcent. The rato of a typcal GLONASS wavelength to a typcal auxlary wavelength s approxmately 3000 for both L 1 and L 2. Tests were made to verfy the ambguty resoluton. Two Ashtech GG24 sngle-frequency GPS/GLONASS recevers had been set up at two known locatons, approxmately 450 m apart. One of them was stuated on a concrete observaton pllar. Ths was treated as the reference staton. The other was regarded as the user whose poston was to be determned. Data had been recorded at both statons wth a data rate of 1 Hz. These data were (9) 165
4 processed off-lne, but under real-tme condtons. The man reason for ths was to be able to test the effects of dfferent parameter values. Ths s best done wth dentcal data sets for the dfferent processng runs, whch s not possble n actual real-tme. Data analyss was done twofold: Separate postonng solutons were computed for GPS only and for the GPS/GLONASS combnaton. Due to the low number of GLONASS satelltes, no GLONASS only soluton was possble. These two solutons (GPS only and ) were compared. As was noted before, the dea was to fx the GLONASS ambgutes (whch refer to the auxlary sgnal wth the small wavelength) only to thousands of cycles. To test the effect of ths quantzaton on the postonng soluton, several processng runs were done wth dfferent values for the grd spacng, namely 500, 1000, 1500, 2000, 2500 and 3000, whch s approxmately the rato of physcal satellte wavelength to the auxlary wavelength. The grd orgn was also vared to nclude zero n one case or the rounded float soluton n another case. For a grd spacng of 1000 e.g., ths means that n the frst case ambgutes 0, ±1000, ±2000, and so on were possble canddates for ambguty resoluton. In the second case, the rounded float soluton k k, N was a possble canddate, k, k along wth N ± 1000, k, k N ± 2000, and so on. Of course, for the GPS double dfferences the grd spacng was 1 and the grd contaned nteger numbers n all cases. 0 Float soluton Rounded float soluton Search canddate Grd spacng Fgure 1: Organzaton of search space for GLONASS nteger ambgutes. Results of one of these tests are presented n the followng. These results are from a half-hour data set, contanng measurements from sx GPS and three GLONASS satelltes. Specal mportance was attached to fxng the ambgutes correctly. Thus, ambgutes were only fxed, when there was a sgnfcant dfference between the resduals of the best and second-best combnatons of nteger ambgutes. TEST RESLTS The varaton of grd spacng and orgn for the GLONASS ambguty search dd not have any effect on the GPS ambgutes, of course. The mpact on the GLONASS ambgutes was twofold. Wth the grd orgn at zero, no successful ambguty resoluton could be acheved. Obvously n ths case the examned grd ponts were too far off from the rounded ambguty (where the mnmum resdual s located) to yeld sgnfcantly dfferent mnmum resduals. Ambguty resoluton could only be acheved when the rounded float soluton was ncluded n the grd. In ths case, however, the grd spacng dd have no effect on the tme when the ambgutes were fnally fxed. Also, the found nteger ambguty was the rounded float soluton, no matter what the grd spacng was. On the other hand t was notced that the GPS ambgutes were fxed to the rounded float soluton, too. Thus, at the pont of fxng the float soluton was already such precse that a smple roundng would have suffced, rather than a costly ambguty search. Fgures 2-4 show the results of the postonng soluton for lattude, longtude and heght, respectvely. Fgure 5 shows the estmated ambguty values. To obtan a more compact dagram, the value, to whch the ambguty fnally was fxed, s subtracted from each ambguty value. A successful ambguty fxng was acheved only very late nto the data set. Ths s most probably due to mult-path effects. At least the ste of the reference staton s known to be apt to mult-path. But t can be notced that ambguty resoluton for the combned soluton was acheved earler than for GPS alone. Ths s most clearly vsble n fgure 3. Ths effect of earler ambguty resoluton n a combned soluton s consstent wth the fndngs of [3]. However, they confned to floatng GLONASS ambgutes and fxed only the GPS ambgutes. Furthermore, as soon as the ambgutes are fxed the poston s very close (wthn 2 mm for lattude/longtude and 6 mm for the heght component) of the known user poston. Durng the approxmately 15 mnutes untl the end of the data set, no sgnfcant drft n the poston can be observed. Ths s a strong hnt for a correct ambguty resoluton. A closer look at the fxed ambgutes reveals dentcal values for the GPS ambgutes n the combned soluton and the GPS only soluton. The GPS observatons were also processed wth an ndependent software package. Here agan, the ambgutes were fxed to the same nteger values. So t s safe to assume that the fxng really was done correctly. 166
5 Lattude [ ] GPS only GPS Tme [s] Fgure 2: Lattudnal component of poston soluton Longtude [ ] GPS only GPS Tme [s] Fgure 3: Longtudnal component of poston soluton. 167
6 Ellpsodal Heght [m] GPS only GPS Tme [s] Fgure 4: Heght component of poston soluton GPS Ambgutes [-] GLONASS Ambgutes [-] GPS 4 GPS 7 GPS 15 GPS 18 GPS 25 GPS 29 GLO 3 GLO 13 GLO GPS Tme [s] Fgure 5: Dfference of estmated double dfference ambguty values to fnally fxed values, as obtaned from the combned GPS/GLONASS soluton
7 Lattude [ ] GPS Tme [s] Fgure 6: Lattudnal component of poston soluton, GLONASS ambgutes fxed ncorrectly Longtude [ ] GPS Tme [s] Fgure 7: Longtudnal component of poston soluton, GLONASS ambgutes fxed ncorrectly. 169
8 Ellpsodal Heght [m] GPS Tme [s] Fgure 8: Heght component of poston soluton, GLONASS ambgutes fxed ncorrectly. ndoubtedly a correct fxng would have been possble already much earler than t actually happened n the tests. As can be seen from fgure 5, the floatng ambgutes very soon are wthn ±1 cycle for GPS respectvely ±2000 auxlary cycles for the GLONASS double dfference ambgutes. To evaluate the effect of an ncorrect GLONASS ambguty fx, all GLONASS double dfference ambgutes were delberately offset by 500 (auxlary) cycles as soon as they were successfully fxed to nteger. Ths equals approxmately 1/6 of a physcal wavelength. Ths value was chosen because t also corresponds to the maxmum possble dstance between the floatng ambguty and the nearest canddate for fxng n the case of fxng to thousands of cycles. The results of the postonng soluton under these crcumstances are shown n fgures 6-8. The computed postonng solutons now show a notceable offset from the known poston. Ths offset s approxmately 5 mm n lattude and 9 mm n longtude. Wth ntally 15 mm t s most vsble n the heght component. The heght component also shows a notceable drft away from the true poston. However, wth approxmately 6 mm over the 15-mnute perod untl the end of the data set, ths drft s not sgnfcant enough to clearly ndcate an ncorrect ambguty fx. The same drft can be found n the postonng soluton usng the unbased GLONASS double dfference ambgutes. CONCLSIONS A method for fxng GLONASS/GLONASS double dfference carrer phase ambgutes has been ntroduced. It s based on an auxlary sgnal wth a wavelength, of whch the orgnal wavelengths of the satelltes nvolved are nteger multples. Contrary to prevously ntroduced algorthms, whch could only be used n post-processng, ths method does not requre much addtonal treatment of the GLONASS double dfference carrer phase measurements. It s therefore also sutable for real-tme operaton, even though t was not operated ths way for the tests shown. Thus, t may be used n RTK applcatons as well. Ths method has been appled to the GLONASS observatons n a combned GPS/GLONASS scenaro. It was shown that n the combned scenaro a successful ambguty resoluton was possble earler than n a GPS only scenaro. The fxed nteger ambgutes for both GPS and GLONASS double dfferences must be assumed to be correct. Wth the nteger ambgutes resolved, accuracy below one centmeter could be acheved. These results are com- 170
9 parable to postonng solutons from GPS only wth fxed nteger ambgutes. Even wth a delberate offset of 500 cycles (of the auxlary sgnal) on all fxed GLONASS double dfference nteger ambgutes, a poston accuracy of less than 3 cm was acheved. Instead of extendng the costly ambguty search to GLONASS, t mght be suffcent to round the transformed GLONASS double dfference carrer phase ambgutes to the nearest nteger. But ths stll has to be nvestgated further. Anyway, the fxed value must be wthn less than 500 cycles of the auxlary wavelength to obtan centmeter accuracy of the poston soluton. Ths may suggest that smple roundng may be possble n case the floatng ambguty s sgnfcantly less than 500 cycles away from the true value. But then crtera must be developed to determne, whether the floatng ambguty s suffcently close to the true value or not. For the results presented n ths paper, a correct resoluton of the nteger ambgutes was preferred over a fast resoluton. Ths s why ambgutes have been fxed rather late. Even though t s not shown here, an earler fx s possble wthout obtanng any dfferent results. Despte the current depleton of the GLONASS constellaton, the nterest n GLONASS to be used n postonng s stll strong. Ths ncludes hgh precson postonng usng carrer phase measurements. Wth a successful resoluton of GLONASS double dfference nteger ambgutes n real-tme, t s possble to open access to GLONASS also for RTK and other applcatons, where hgh precson n real-tme s requred. [3] Landau, Herbert; Vollath, lrch: Carrer Phase Ambguty Resoluton usng GPS and GLONASS Sgnals, Proceedngs of ION Satellte Dvson GPS- 96 Internatonal Techncal Meetng, Kansas Cty, Mssour, September 17-20, 1996, pp [4] Leck, Alfred; L, Jnye; Beser, Jacques; Mader, Gerald: Processng GLONASS Carrer Phase Observatons - Theory and Frst Experence, Proceedngs of ION Satellte Dvson GPS-95 Internatonal Techncal Meetng, Palm Sprngs, Calforna, September 12-15, 1995, pp [5] Rossbach, do: Postonng and Navgaton sng the Russan Satellte System GLONASS, Ph.D. thess, nversty FAF Munch, Faculty of Cvl and Surveyng Engneerng, Neubberg, Germany (to be publshed) [6] Rossbach, do; Hen, Günter W.: Treatment of Integer Ambgutes n DGPS/DGLONASS Double Dfference Carrer Phase Solutons, Proceedngs of ION Satellte Dvson GPS-96 Internatonal Techncal Meetng, Kansas Cty, Mssour, September 17-20, 1996, pp [7] Walsh, Davd; Daly, Peter: GPS and GLONASS Carrer Phase Ambguty Resoluton, Proceedngs of ION Satellte Dvson GPS-96 Internatonal Techncal Meetng, Kansas Cty, Mssour, September 17-20, 1996, pp ACKNOWLEDGMENTS Some of the work presented n ths paper has been conducted for MAN Technologe AG. REFERENCES [1] Euler, H.-J.; Landau, H.: Fast Ambguty Resoluton On-The-Fly for Real-Tme Applcatons, 6 th Internatonal Geodetc Symposum on Satellte Postonng, Columbus, Oho, March 1992 [2] Habrch, Henz; Beutler, Gerhard; Gurtner, Werner; Rothacher, Markus: Double Dfference Ambguty Resoluton for GLONASS/GPS Carrer Phase, Proceedngs of ION Satellte Dvson GPS-99 Internatonal Techncal Meetng, Nashvlle, Tennessee, September 14-17, 1999, pp
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