Sngle-Epoch Ambguty Resoluton for nematc GNSS Postonng hrstan Elng, Phlpp Zemetz, Hener Kuhlmann Insttute of Geodesy and Geonformaton, Unversty of Bonn, Germany Abstract Automatc machne control requres accurate and relable nformaton about the latest atttude and poston of the vehcle. In addton to nertal sensors and odometer Global Navgaton Satellte Systems (GNSS) are well establshed n the determnaton of these parameters. Besdes code observatons GNSS addtonally provde carrer-phase measurements, whch should be used to acheve hgh accuraces. ertanly, the ey to GNSS carrer-phase postonng s the ambguty resoluton. Ths s the process resolvng the unnown number of nteger cycles n the carrer phase data. Prncpally, dfferent approaches exst to resolve the ambgutes. Snce mult-epoch technques lead to a substantal loss of possble solutons, a sngle-epoch ambguty resoluton should be amed at. A common procedure that enables an ambguty resoluton for every sngle epoch s the Ambguty Functon Method (AFM). By means of a cost functon the AFM tests canddates correspondng to a generated search space, ncludng possble rover postons. However, by use of ths approach dsadvantages occur due to the computaton tme, ncreasng wth the sze of the search space, and the relablty, dependng on the decdedness of the complcated multpea-functon. Accordngly, the canddates n the search space have to be selected carefully. For ths purpose, poston approxmatons can be acheved by use of GNSS-veloctes, a vehcle moton model, dfferental code-solutons as well as Kalman flterng. Therefore, through the combnaton of these tools t was possble to develop a sngle-epoch ambguty resoluton algorthm that also shows good performances n urban areas wth a success rate of 96.59 %. Keywords GNSS, ambguty resoluton, Kalman flterng, GNSS veloctes 1 INTRODUTION In the feld of automated control as well as supported navgaton of machnes Global Navgaton Satellte Systems (GNSS) are of major mportance, snce they allow for the determnaton of absolute postons, atttudes and veloctes. On ths account most of the land- and constructon-machnes are nowadays already equpped wth at least one GNSS antenna. Besdes pseudoranges GNSS also provde more precse carrer phase measurements. Wthn dfferental postonng based on double dfferenced carrer phases, sub-centmetre-level precson GNSS postonng becomes possble. However, t s well nown that GNSS double dfferenced carrer phase measurements are ambguous by an unnown number of nteger cycles. To fully explot the hgh accuracy of the carrer phase observables, the ambgutes must be resolved to ther correct nteger value (Hoffmann-Wellenhof et al., 2008). Especally n urban areas obstacles le street canyons, brdges or vegetaton lead to frequent losses of loc, whch always necesstate a new ambguty resoluton. Therefore, for nematc applcatons, the duraton of the ambguty resoluton s of partcular mportance. In ths contrbuton we wll frst gve a short overvew of the exstng ambguty resoluton technques. Afterwards we wll present a sngle-epoch ambguty resoluton method for nematc postonng, whch s based on the combnaton of approaches le GNSS velocty determnaton and Kalman flterng wth an nstantaneous ambguty resoluton technque. By means of test runs, the procedure was tested successfully.
2 AMBIGUITY RESOLUTION BAKGROUND In the last decades nteger ambguty resoluton was the focus of many researchers, snce the ambguty resoluton s the ey to precse GNSS postonng. Therefore, many dfferent technques exst to determne the unnown nteger cycles of the observed double dfferences. Generally, these approaches can be classfed nto three categores: ambguty resoluton n the measurement doman, search technque n the ambguty doman and search technque n the coordnate doman (Km and Langley, 2000). The ambguty resoluton n the measurement doman s prncpally based on code observatons. Snce code observatons are more nexact than carrer phase measurements, these approaches are ordnarly not very sutable. Only the processng of nterfrequency lnear combnatons enables a more or less relable ambguty resoluton. However, ths requres the observaton of at least 2 frequences. The second class of ambguty resoluton technques contans approaches searchng n the ambguty doman. Generally, they are based on the so called nteger least squares (ILS) theory (Teunssen, 1993). The ILS-approaches consst of three steps. By means of a float soluton the cycles are estmated va a least squares adjustment. Usng the resultng float ambgutes and the varance-covarance matrx the odd number of ambgutes can afterwards be fxed wthn a search process n the nteger ambguty estmaton step. As soon as the ambgutes are set to nteger values a fxed soluton follows to determne the precse baselne parameters. The most famous and relable ILS ambguty resoluton technque s the LAMBDA method (e.g. Teunssen, 1995). Further well nown approaches are the FASF (hen and Lachapelle, 1995), the FARA (Fre and Beutler, 1990), the OMEGA (Km and Langley, 1999) and the LSAST (Hatch, 1990) method. Except the LSAST method, all of these search technques are mult-epoch approaches. Ths s because the float soluton step necesstates the usage of observatons from more than one epoch, snce the number of parameters defntely exceeds the number of avalable double dfferences n one sngle epoch. The thrd class of ambguty resoluton technques contans approaches searchng n the coordnate doman. The most famous of these methods and smultaneously one of the earlest ambguty resoluton search technques n general s the Ambguty Functon Method (AFM) (ounselman and Gourevtch, 1981; Remond, 1984; Mader, 1990). On the bass of approprate crtera, canddates of a predefned search space have to be tested, only regardng the fractonal part of the observed carrer phases of one sngle epoch. In ths paper, ths method s the bass for further nvestgatons. 3 INSTANTANEOUS AMBIGUITY RESOLUTION In case of nematc applcatons the rapdty of the ambguty resoluton s of partcular mportance. Therefore, fxng the nteger number of cycles of the observed double dfferences wthn one sngle epoch should be amed at. Moreover, not only long term sgnal nterruptons appear very often durng nematc applcatons. Even cycle slps and nterruptons to the sgnal between epochs occur, whch result n new sets of ntegers (orbett and ross, 1995). To avod addtonal processng durng cycle slp detecton a sngle epoch ambguty resoluton s nevtable. Snce the AFM s not n need of float ambgutes or a varance-covarance matrx t s well-suted for nstantaneous ambguty resoluton and therefore also resstant to cycle slps. For these reasons, we now le to ntroduce the basc prncple of the AFM. As mentoned above, the ambguty search usng the AFM taes place n the coordnate doman. Maxmzng the Ambguty Resoluton Functon (ARF) enables the assessment of canddates of a predefned search space, contanng possble rover postons. The ARF, for the sngle frequency case, can be wrtten as (Lachapelle et al., 1992): ARF N 1 j j ( X, Y, Z ) = cos 2 ( φ [ E1 E2] φ [ E1 X, Y, Z ]) m= 1 π (1) obs whereas φ j obs s the observed and φ j calc a calculated double dfference of the satelltes and j. Snce E1 s the nown poston of the master antenna, the ARF s only dependent on the coordnates of the canddates, whch are located n the search space around the true rover poston E2 (see fgure 1). In case the poston of the canddate (X, Y, Z ) s smlar to E2, the dfference between the observed and the calculated double dfference corresponds to the unnown ambgutes n order that the result of the cost functon s equal to 1. onsderng the carrer phase measurements of N satelltes on w calc
frequences the outcome of the ARF would deally be (N-1) w, tang nto account that due to multpath and recever nose ths maxmum wll never be reached. Fgure 1: Depcton of a possble search space for the AFM. However, there are two drawbacs of the AFM. Frst, the computatonal effcency s hghly dependent on the sze of the search space, whch s defned by the accuracy of the approxmate poston. Therefore, the computaton tme can potentally be very long. And second there may be several maxma ponts that the AFM must dscrmnate between to fnd the optmal soluton (Han and Rzos, 1995). onsequently, the basc AFM approach has to be mproved to mae t sutable for practce. 4 EFFIIENY IMPROVEMENT OF THE AFM The ey to relable and fast ambguty fxng by means of the AFM s a reducton of the sze of the search space. In so dong, the computaton tme decreases heavly. Furthermore, false canddates can be excluded wth the result that the decson-mang of the AFM wll be smplfed. In the creaton of the search space two cases can be decded. On the one hand the frst tme ntalzaton or re-ntalzaton after a GNSS gap and on the other hand the transfer between two epochs as long as GNSS s avalable. 4.1 Frst-tme ntalzaton or re-ntalzaton of the ambgutes In the begnnng of an applcaton as well as n consequence of a gap of the GNSS sgnals, a frst-tme ntalzaton or a re-ntalzaton of the ambgutes s necessary. The startng pont for ths search process s an approxmate rover poston to generate a search volume. Wthout the use of addtonal sensors, there are only few opportuntes for determnng these prelmnary coordnates. In most cases code observatons are preferably used to cope wth ths tas. By means of dfferenced code sgnals accuraces n the range of a few decmetres to metres are achevable. Lnear combnatons such as the wde lane also enable the determnaton of an approxmate poston (Abdn, 1994). However, ths requres the observaton of at least two frequences, whch s not always gven, e.g. low-cost recevers. Therefore, we use dfferental code-observatons for the determnaton of an approxmate rover poston n case of frst-tme or re-ntalzaton of the ambgutes. Even f the ambgutes could be fxed correctly n the frst epoch after a loss of loc, ths ntalzaton process wll be delayed for more than one epoch to avod an ncorrect frst-tme or re-ntalzaton. Durng ths tme the prelmnary coordnates are fltered n an Extended Kalman Flter (EKF) to mprove the relablty of the ambguty resoluton (see chapt.5). In order to consder the balance between the computatonal effort of the search process and the sze of the search space, whch has to be large enough to contan the true rover poston, the confguraton of the canddates should be carefully selected. Snce the canddates vary on the bass of dfferent ambgutes, the possble rover postons n the search space are also dependent on dfferent sets of nteger ambgutes. Therefore, the approxmate rover poston s used to determne the ambgutes of the observed double dfferences, whch can be rounded to nteger cycles. Afterwards these ambgutes need to be vared for dfferent values to determne possble rover postons. In order to lmt the number of canddates n the search space, not all double dfference ambgutes, but only three should be used for generatng the search space. The selecton of ths three prmary observatons occurs n consderaton of the poston dluton of precson (PDOP), the elevaton angles as well as resduals of the code soluton. In case the range of the ambgutes s set from mnus fve to plus fve values the
search volume conssts of 1331 ([(2*5)+1] 3 ) sets of ambgutes, whch lead to possble postons, ncludng the true rover poston (orbett and ross, 1995). Dependng on the PDOP, the edges of the cubc search space reach lengths up to 10 m. Therefore, the search space s frst defned n the ambguty doman before t s used to generate a physcal space, defned n the coordnate doman. Fnally, the ambguty resoluton of all double dfferences occurs by testng the canddates by means of the AFM. Snce the ARF s a mult pea functon the results are not nevtably unambguous (see chapt.4.3). In case the maxmum of the ARF cannot be clearly dstngushed from sde-lobes, further nvestgatons are necessary to fnd the correct set of ambgutes. rtera are for example the varances of a least squares adjustment n the determnaton of the baselne parameters. 4.2 Poston update usng GNSS-veloctes Once the ambgutes were fxed and at least four GNSS satelltes are vsble n two successve epochs, the determnaton of an approxmate rover poston can occur by use of ntegrated GNSS-veloctes n combnaton wth a Kalman flter. Ths s because the GNSS-veloctes are also based on precse carrer phase measurements, whereas they are not n need of an ambguty resoluton. Ths becomes obvous regardng the observatons used for the velocty determnaton, consstng of the frst order dfference approxmaton of the carrer-phase observatons (Serrano et al., 2004): j j j φ φ t φ (2) t where φ s the fractonal carrer-phase of the satellte j, s the observaton epoch and t the samplng rate. The velocty determnaton occurs by use of a least squares adjustment based on the followng objectve functon: j ( j ) j φ = h j + B v V + ε (3) Where V s a vector contanng the unnown recever veloctes (V x,v y and V z ). v represents the satellte velocty vector, whch can be computed by use of the ephemers. h stands for the drectonal cosne between the recever and the satellte: h j T j ( S X ) = (4) j S X whereas S s the poston vector of satellte j and X the recever poston vector. Furthermore, B are the recever cloc drft and ε the recever nose. By means of equatons (2)-(4) the recever velocty can be determned n every observaton epoch. Of course the resultng veloctes are not the true ones for the actual epoch, snce the frst order dfference approxmaton n equaton (2) enables the estmaton of the mean velocty between the epochs and - t, but n case of samplng rates hgher than 1 Hz, they are stll accurate enough to delver a sutable approxmate poston for the AFM. Fgure 2: Dstances between ntegrated GNSS velocty postons and fnal postons. To underlne ths, the dstances between the approxmate postons, calculated by ntegraton of the veloctes, and the fnal postons durng a nematc experment are presented n fgure 2. Except for a few outlers, the total devatons are mostly less than 1 cm. The mean of the dstances between the approxmate and the fnal postons s about 3 mm. The clearly vsble outlers are based on poor GNSS measurement condtons. However, these devatons are less than 6 cm whereas the
wavelengths of GNSS sgnals are n the order of 20 cm. Accordng to ths, the fltered approxmate postons are stll well suted to reduce the sze of the search space. Therefore, as long as GNSS s avalable the GNSS veloctes are used to defne the search volume. 4.3 Reducng the number of maxma n the AFM One drawbac of sngle-epoch ambguty resoluton approaches s the susceptblty n case of poor GNSS measurement condtons. Bases le multpath, resdual atmospherc effects and satellte orbt errors lead to devatons n the observed carrer phases (Km and Langley, 2000). Therefore, t cannot be excluded that false ambgutes wll be selected from the set of canddates n the search space, snce the mult pea ARF does not allow for an unambguous decson n such epochs. To mprove the performance of sngle-epoch ambguty resoluton technques approaches employng lnear flters for the resduals or tme averages for the objectve functon showed good performances n earler studes (e.g. Borge and Forssell, 1994; Martn-Nera et al., 1995). Therefore, the decson-mang of the AFM should also be mprovable. In our sngle-epoch ambguty resoluton approach we use a Kalman flter to predct the resduals of the observed double dfferences n every epoch. Snce the resduals cannot be descrbed by any moton behavour a random-wal-process s appled as moton model n ths procedure. Accordngly, the dscrete system equaton follows: xr ( + 1) 1 = x ( + 1) 0 R t xr ( ) t + w 1 x ( ) 1 R ( ) x ( + 1) = T( ) x( ) + S( ) w( ) (6) whereas x() s the state vector, contanng the fltered resdual x R and the dervatve of the resdual x R. T() s the transton matrx, S() the system nose couplng and w() the system nose. By use of ths Kalman flter the resduals of every observed double-dfference can be predcted from epoch -1 to epoch. Therefore, the performance of the ambguty resoluton can be mproved by reducng the devatons of the observatons n the AFM usng the predcted resduals. (5) Fgure 3: omparson of results of the ARF usng and dsregardng the predcton of the resduals. In fgure 3 a comparson of the outcome of the AFM dsregardng and usng the resduals predcton durng a nematc test s presented. Accordng to ths, the maxma of the ARF ncrease by use of the Kalman flter, wth the result that they obvously come closer to the nomnal value of (N-1) w. However, ths does not mply a smplfcaton n the dscrmnaton of false and correct solutons n the search space, snce a smultaneously ncrease of the sde-lobes s also possble. In order to demonstrate the actual mpact of the flter process, the outcome of the ARF for every canddate of the search space durng one epoch s presented for two cases, usng and dsregardng the fltered resduals, n fgure 4. Therefore, not only the sze of the maxma but also the dfference to sde-lobes ncreases by use of the predcton step. Summarzng ths secton, there are two reasons why the sngle-epoch ambguty resoluton has become better. On the one hand, the maxmum of the outcome ncreases. And furthermore, the correct soluton s now n greater contrast to ncorrect sets of ambgutes. Therefore, there are no more nvestgatons necessary to come to a decson, whch canddate of the search space leads to the best and correct ambgutes.
Fgure 4: Results of the ARF dsregardng (left) and usng (rght) the resduals predcton for one epoch. 5 EXTENDED KALMAN FILTER As mentoned above, an EKF s used to mprove the relablty of the determned rover postons n ths system. Generally, an EKF s a recursve algorthm that enables the combnaton of nosy measurements wth a pror nown moton behavour of a vehcle, to estmate an optmal state vector as tme progress on the bass of external observatons, (e.g. Grewel et al., 2007). Dependng on the correctness of the used moton model the EKF s well suted to reduce whte nose and to detect outlers. Snce vehcles mostly move on streets, whch are compled of the basc elements straght, arc and clothode, we assume a unform crcle movement as system dynamcs model (Aussems, 1999; Echhorn, 2005): G xˆ East G G yˆ + RL North G zˆ Up x+ 1 = ˆ α + ˆ α vˆ ˆ α hˆ East North t vˆ = t vˆ = ( cos( ˆ α + ˆ α ) cos( ˆ α )) ( sn( ˆ α + ˆ α ) sn( ˆ α )) Up ˆ α ˆ α = hˆ whereas x,y,z are the cartesan coordnates from the epoch, α s the headng, Δα the headng change, v s the velocty n drvng drecton and Δh s the alttude change. R G L represents the rotaton matrx to transform the coordnate changes ΔEast, ΔNorth and ΔUp from the local level frame to the global geocentrc coordnate frame (Hoffmann-Wellenhof et al., 2008): sn λ snϕ cosλ cosϕ cosλ G R = cosλ snϕ sn λ cosϕ sn λ. (8) L 0 cosϕ snϕ Thereby λ stands for the longtude and ϕ for the lattude. Accordng to ths, the transton matrx T conssts of the dervatves of equaton (7) wth respect to the states x = x G y G z G α v Δα Δh T. The system nose couplng S contans the dervatves of equaton (7) wth respect to v, Δα and Δh, ntegrated over the samplng rate Δt, wth the result that the dscrete system equaton (6) allows for acceleratons, updates of the headng changes as well as updates of the alttude changes. The measurement equaton (9) establshes the connecton between the state vector x and the observatons, gven by the desgn matrx H and a whte nose ε. The observaton vector l= x GPS,y GPS,z GPS,v GPS T conssts of the GPS rover poston and the rover velocty. Ths rover velocty s the norm of the East and North component of the transformed GPS rover velocty vector V, whch can be estmated by means of eq. (3). D s a unt vector D = 1 0 0. T I GPS GPS GPS GPS 3x3 03x4 G G G T x + y + z + v + = x + 1 y + 1 z + 1 α + 1 v + 1 α + 1 h + 1 + ε (9) 1 1 1 1 01x4 D Outlers, whch are attrbutable to multpath or ncorrect fxed ambgutes, can be detected by an nnovaton test. Besdes the reducton of whte nose as well as the detecton of outlers, the system dynamcs model of the EKF also allows for brdgng GNSS gaps. In fgure 5 two examples for ths (7)
type of applcaton are shown. In the left chart, the GNSS gap only lasts 25 epochs, wth the result that the predcton s worng very well. Therefore, n case of a short term GNSS outage, whch can for example be caused by a tree, t s well suted to detect outlers, whch are not uncommon durng the frst epochs after a sgnal nterrupton. However, n the rght chart, a long term sgnal nterrupton s presented (146 epochs). In ths case, the predcton does not completely agree wth the true moton behavour, n order that the EKF has to be restarted, once the ambgutes are fxed. By addng further sensors, le gyroscopes or odometer, the brdgng of long term GNSS gaps can also stll be mproved, n order that the devatons to the true poston would only ncrease very slowly. Fgure 5: Predcton of the states durng GNSS gaps by use of the system dynamcs model. The entre sngle epoch ambguty resoluton approach for the determnaton of GNSS postons durng nematc applcatons s presented n a flow chart n fgure 6. Accordngly, the ambguty resoluton ether occurs by use of a Kalman fltered code poston or a poston update based on GNSS veloctes. In case no GNSS observatons are avalable, the states are predcted by the system dynamcs model presented n equaton (7). 6 RESULTS The developed sngle epoch ambguty resoluton approach for nematc GNSS postonng was tested by means of dfferent experments. In order to dentfy the correctness of the ambguty resoluton, the results were compared to commercal software. The outcome of ths commercal software of a test run, carred out on freeways and cty avenues n and near Bonn, s presented n fgure 6. Especally n urban areas ths post processng led to less pleasant results, snce most of the postons are based on mprecse code observatons. The reason s that the commercal software could not fx the ambgutes qucly after each of the numerous sgnal nterruptons. Therefore, ths example emphasses the necessty of a sngle-epoch ambguty resoluton n case of nematc applcatons, snce a mult-epoch approach leads to long-term GNSS gaps after sgnal nterruptons. Fgure 6: Flow chart of the developed approach (left) and test run processed by com. software (rght)
The same dataset was also analyzed by means of the presented sngle-epoch approach. The samplng rate durng ths test run, n whch speeds up to 140 m/h were reached, was 10 Hz. In table 1 the success rates of the ambguty resoluton for all of the 20294 epochs are shown. As expected, the relablty of the ambguty resoluton depends on the number of vsble satelltes. Ths s manly because of the dstnctness between correct and false sets of ambgutes n the AFM. The less observatons are avalable the ncreased s the nfluence of multpath effects of ndvdual sgnals. However, stll 89.46% of the epochs, n tmes only four satelltes were vsble, led to correct ambguty resolutons. In case more than sx satelltes were vsble, the success rate ncreases 98 % whereas n 99.65% of all the epochs, where 9 satelltes were vsble, the ambgutes could be fxed correctly. Table 1: Success rates of the ambguty resoluton by use of the sngle-epoch approach. vsble satelltes epochs ncorrect resolutons success rate 4 2240 236 89.46 % 5 2320 183 91.98 % 6 3153 136 95.69 % 7 4492 85 98.11 % 8 4961 42 99.15 % 9 3128 11 99.65 % sum 20294 693 96.59 % Besdes the success rates the tme to fx the ambgutes s also of nterest, snce the objectve of the sngle-epoch approach s to fnd the correct set of ambgutes as fast as possble after a sgnal nterrupton. In table 2 the number of epochs needed to fnd the true ambguty resoluton s presented for dfferent re-ntalzatons durng the nematc experment. For comparson, the number of epochs elapsed untl re-ntalzaton s also shown for the outcome of the commercal software. In case of the sngle-epoch approach t should be notced that not all of these epochs led to ncorrect ambgutes, but rather the ambguty resolutons were nconstant durng these lsted epochs. Table 2: omparson of epochs needed to re-ntalze the ambgutes after GNSS gaps. epochs elapsed untl re-ntalzaton re-ntalzaton sngle-epoch approach commercal software 1 2 52 2 3 324 3 27 85 4 2 37 5 3 51 6 3 58 7 2 58 8 37 20 9 12 50 10 10 403 11 20 47 12 14 85 13 7 38 14 5 206 15 2 57 16 10 73 17 2 278 Accordng to ths, the sngle epoch approach s mostly a lot faster than the commercal software. In many cases, t was possble to fx the ambgutes n the second epoch beyond a sgnal nterrupton. Especally n case of the second, the tenth, the fourteenth and seventeenth re-ntalzaton the dfference between both approaches s partcularly evdent, snce the commercal software requres above two hundred epochs more to mae the user carrer phase postons avalable agan. Instead, the number of epochs elapsed untl re-ntalzaton s not of nterest, f the moble object s movng very
slowly or the samplng rate s very hgh. Therefore, to llustrate the advantage of the developed ambguty resoluton procedure, the postons of the seventeenth re-ntalzaton process are presented n the left chart of fgure 7. Whlst the sngle-epoch approach could fx the ambgutes n the second epoch, the commercal software only provded code postons for 277 epochs untl the ambgutes could be fxed. Durng ths tme, the vehcle covered a dstance of almost 400 metres. Fgure 7: omparson of results of the sngle-epoch approach to the outcome of a com. software. However, t s also conspcuous that there are often few outlers n the frst epochs after a sgnal nterrupton vsble, whch are based on poor GNSS condtons due to the proxmty to the obstacle, whch prevously produced the loss of loc. Furthermore, the recever generally needs dfferent lengths of tme to allocate the current carrer-phases. Hence, there are mostly only few observatons avalable n the frst epochs after a loss of loc. ertanly, regardng the Kalman flter as well as the varances of the postons, dscontnutes become apparent. Ths s also true for the seventh rentalzaton process, whch s shown n the rght chart of fgure 7. In the frst epoch after the gap only three poor double dfferences were avalable wth the result of an ncorrect ambguty resoluton. Nevertheless, already one epoch later these ambgutes were fxed correctly, whereas the commercal software requred 58 epochs (ca. 32 metres) to provde carrer-phase postons. oncludng, by combnng the approaches to lmt the search space wth the predcton of the resduals, a fast and relable procedure to fx the ambgutes could be mplemented. Furthermore, the procedure wors ndependently of the applcaton feld (low speed n urban areas or hgh speed on freeways). In most cases only less than 5 epochs are requred to fnd the correct ambguty resoluton, whch leads to an ambguty resoluton success rate of 96.59%. 7 ONLUSIONS In ths contrbuton we presented an ambguty resoluton approach for nematc GNSS postonng. Especally n urban areas obstacles lead to frequent losses of loc, whch always necesstate a rentalzaton of the double-dfference ambgutes. Therefore, we developed a sngle epoch ambguty resoluton to provde carrer-phase postons as fast as possble. In our approach we used the AFM for the determnaton of the nteger number of unnown cycles. To overcome defcences caused by the computatonal effcency of ths procedure we used a combnaton of an ambguty resoluton n the ambguty and the coordnate doman for frst-tme and re-ntalzatons as well as GNSS veloctes as long as no nterrupton occurred, to generate a well-suted search volume. Furthermore, by predcton of the resduals, the unambguousness of the ARF could be mproved. onsderng experments durng dfferent applcatons the approach was tested successfully. Despte frequently poor GNSS condtons an average ambguty resoluton success rate of 96.59% was reached. Wth few exceptons the nteger cycles were re-ntalzed durng the frst 10 epochs after every sgnal nterrupton, n order that the approach leads to a relable and fast ambguty resoluton.
REFERENES ABIDIN, H.Z.: On-the-Fly Ambguty Resoluton, GPS World, Aprl 1994, pp. 40-49, 1994 AUSSEMS, T.: Postonsschätzung von Landfahrzeugen mttels Kalman-Flterung aus Satellten- und Koppelnavgatonsbeobachtungen, Veröffentlchung des Geod. Insttutes der Rhensch-Westfälschen Technschen Hochschule Aachen, Nr. 55, 1999 BORGE, T.K., FORSELL, B.: A new real-tme ambguty resoluton strategy based on polynomal dentfcaton, Proc. of KIS 94, Banff, anada, pp. 233-240, 1994 HEN, D., LAHAPELLE, G.: A comparson of the FASF and least-squares search algorthms for on-the-fly ambguty resoluton, Navgaton: Journal of the Insttute of Navgaton, Vol. 42, No.2, pp. 371-390, 1995 OUNSELMAN,.., GOUREVITH, S.A.: Mnature nterferometer termnals for earth surveyng: ambguty and multpath wth Global Postonng System, IEEE Transactons on Geoscence and Remote Sensng, Vol. GE-19, No. 4, pp. 244-252, 1981 ORBETT, S.J., ROSS, P.A.: GPS sngle epoch ambguty resoluton, Survey Revew, Vol. 33, No. 257, 1995 EIHHORN, A.: En Betrag zur parametrschen Identfaton von dynamschen Struturmodellen mt Methoden der adaptven Kalman Flterung, DGK, Rehe, Heft 585, München, 2005 FREI, E., BEUTLER, G.: Rapd statc postonng based on the fast ambguty resoluton approach FARA : theory and frst results, Manuscrpta Geodaetca, Vol. 15, No. 4, pp. 325-356, 1990 GREWAL, M.S., WEILL, L.R., ANDREWS, A.P.: Global Postonng System, Inertal Navgaton and Integraton, Second Edton, John Wley and Sons, 2007 HAN, S., RIZOS,.: Improvng the effcency of the ambguty functon algorthm, Journal of Geodesy (1996) 70, pp 330-341, 1995 HATH, R.: Instantaneous ambguty resoluton, Proc. of KIS 90, Banff, anada, pp. 299-308, 1990 HOFMANN-WELLENHOF, B., LIHTENEGGER, H., WASLE, E.: GNSS Global Navgaton Satellte System, SprngerWenNewYor, Graz, 2008 KIM, D., LANGLEY, R.B.: A search space optmzaton technque for mprovng ambguty resoluton and computatonal effcency, In: Earth, Planets and Space, Vol. 52, No.5, pp. 807-812, 1999 KIM, D., LANGLEY, R.B.: GPS Ambguty Resoluton and Valdaton: Methodologes, Trends and Issues, Proc. of the 7 th GNSS Worshop Intern. Symposum on GPS/GNSS, Seoul, Korea, 2000 LAHAPELLE, G., ANNON, M.E., ERIKSON,., FALKENBERG, W.: Hgh Precson /A ode Technology for Rapd Statc DGPS Surveys, Proc. of the 6 th Intern. Geodetc Symposum on Satellte Postonng, olumbus, Oho, USA, pp 165-173, 1992 MADER, G.L.: Ambguty functon technques for GPS phase ntalzaton and nematc solutons, Proc. of 2 nd Intern. Symposum on precse postonng wth the global postonng system, Ottawa, anada, pp. 1233-1247, 1990 MARTIN-NEIRA, M., TOLEDO, M., PELAEZ, Z.: The null space method for GPS nteger ambguty resoluton, Proc. of DSNS 95, Bergen, Norway, 1995 REMONDI, B.W.: Usng the Global Postonng System (GPS) phase observable for relatve geodesy: modellng, processng and results, PhD thess, enter for Space Research, Unversty of Texas, Austn, 1984 SERRANO, L., KIM, D., LANGLEY, R.B., ITANI, K., UENO, M.,: A GPS Velocty Sensor: How Accurate an It Be? A Frst Loo, Proc. of ION NTM 2004, San Dego, pp. 875-885, 2004
TEUNISSEN, P.J.G.: Least-squares estmaton of the Integer GPS ambgutes, Invted lecture, Secton IV Theory and Methodology, IAG General meetng, Bejng, hna, 1993 TEUNISSEN, P.J.G., DE JONG, P.J., TIBERIUS,..J.M.: The LAMBDA-method for fast GPS surveyng, Proc. of Intern. Symposum GPS technology applcatons, Bucharest, pp. 203-210, 1995 ontact: hrstan Elng, Insttute of Geodesy and Geonformaton, Unversty of Bonn Nussallee 17, D-53115 Bonn, Germany phone: +49 (228) 73 3565 fax: +49 (228) 73 2988 Emal: elng@gg.un-bonn.de