Impact of the Combined GPS + Galileo Satellite Geometry on Positioning Precision

Transcription

1 Imact of the Combned GPS + Galleo Satellte Geometry on Postonng Precson S. Daghay 1, M. Mons 1, C. Bruynnx, Y. olan and F. oosbeek 1 Abstract The mrovement n ostonng recson that can be exected from extendng the GPS constellaton wth Galleo satelltes wll be treated n ths aer. The nfluence of the ure geometrc asect,.e. the effect of havng more avalable satelltes, wll be assessed, whle the effect of the avalablty of new sgnals wll be gnored. From ths ont of vew, two scenaros have been studed: (1) the case of stand-alone ostonng, tycally based on code observables, and () the case of relatve ostonng, tycally based on double dfference carrer hase observables. The results show that the use of the addtonal Galleo constellaton mroves absolute ostonng wth about 40% n terms of formal errors when smulatng urban condtons. For relatve ostonng, the concet of elatve Dluton of Precson (DOP) allowed to demonstrate that usng GPS+Galleo only half the observaton tme s suffcent to get smlar recsons as wth GPS only. Introducton The future Euroean GNSS, Galleo, s desgned to be nteroerable wth GPS. Moreover the combnaton of Galleo and GPS wll rovde faster, more relable and more recse ostonng n comarson wth resent results obtaned usng GPS only. Postonng at locatons wth bad vsblty wll become feasble as twce as much satelltes wll be vsble. The enhanced geometry wll also mrove the recson of ostonng. Moreover, n the future, modernzed GPS and Galleo wll emt sgnals on 3 freuences whereas the current GPS system uses only carrer freuences. Ths extra freuency wll allow a better correcton of the atmosherc dsturbances, one of the most mortant error sources for hgh accuracy ostonng. The new Galleo system wll comrse a constellaton of 30 satelltes (7 oeratonal and 3 sares) dstrbuted over three crcular orbts at an orbtal alttude of nearly km above the sem-maor axs of the WGS84 reference ellsod. Galleo uses the same ostonng technues as those used by GPS. But to make fully use of the new GPS and Galleo sgnals and enhance ostonng, algorthms must be adusted or created to take the new sgnals nto account. Ths aer studes the effect of the geometry of the future GPS+Galleo constellaton on the recson of the estmated ostonng arameters and makes a comarson wth the current ostonng based on GPS. The comutatons are desgn studes only based on the geometry of the GPS and Galleo constellatons. The GPS orbts have been created usng the broadcast navgaton message, whle Galleo orbts were smulated for 7 oeratonal satelltes usng followng orbtal arameters: a sem-maor axs of km, an nclnaton angle of 56, the eccentrcty eual to 0, rght ascenson angles of -10, 0 and 10, a rate of rght ascenson of 0 a day, the argument of ergee eual to 0, a mean anomaly wth -160, -10, -80, -40, 0, 40, 80, 10, 160 as ossble values, and fnally a erod of 1 S. Daghay, Y. olan, Vre Unverstet Brussel, Plenlaan, B-1050 Brussels C. Bruynnx, M. Mons and F. oosbeek, oyal Observatory of Belgum, Av. Crculare 3, B-1180 Brussels 1

2 14 h 4 mn 4s. We also assume that the GPS and Galleo code observables have the same standard devaton as well as the GPS and Galleo hase observables. 1. Increase of vsblty twce as much satelltes In a frst ste, the ncrease of the number of vsble satelltes from any gven recever oston on earth s examned. Fgure 1 shows the daly mean of the number of vsble satelltes for any ont on the earth s ellsod. Usng an elevaton cut off angle of 0º, these mean values are comuted usng a grd of 10º 10º. For the frst lot, only the GPS system was consdered, whle the second lot reresents the results for the combned GPS + Galleo systems. The GPS system rovdes a worldwde mean of about 10 to 1 vsble satelltes over a day. For the combned system, an ncreased worldwde dstrbuton of vsble satelltes, more recsely 1 to 3, s seen. Fgure 1 : Worldwde dstrbuton of the daly mean of vsble satelltes usng a 0 cut off, GPS only combned GPS+GALILEO Fgure : usng a 30 cut off, Worldwde dstrbuton of the daly mean of vsble GPS satelltes Daly number of vsble satelltes n eykavk for all systems

3 Addng Galleo to GPS wll make esecally a dfference when we have a blocked horzon (e.g. n ctes). In ths case, a cut off of 30º s reresentatve. Fgure a emhaszes locatons where ontostonng s not ossble due to a lack of vsble satelltes at certan eochs usng GPS only. An examle of such a roblem locaton s shown n Fgure b, whch shows a lot of the number of satelltes vsble over a day n the EPN staton at eykavk, Iceland. We can see that there are eochs where only or 3 satelltes are vsble, makng t mossble to calculate a oston. Wth Galleo only, a mnmum of at least 4 satelltes s always vsble (even n our conservatve case where we dd not consder the 3 sare satelltes), whereas the combned system always rovdes more than enough satelltes.. Stand-alone ostonng based on code observables Stand-alone ostonng usng GPS code observables s utterly useful for navgaton alcatons wth a recson of several meters. At any tme, the code observable s obtaned by measurng the transmsson tme τ of the code emtted by satellte and receved by recever. Ths transmsson tme s the dfference between tme of arrval (measured on the recever clock) and tme of emsson (measured on the satellte clock) of the sgnal, and can also be characterzed as follows: τ = + T + I + MP c (1) wth the aroxmate geometrc dstance between satellte and recever, c the seed of lght, T and I the resectve trooshercal and onoshercal delays and fnally MP reresentng the multath errors and the delay of the electromagnetc sgnal whle assng through the recever hardware. The seudo-range measurement at tme t can be reresented by the followng observaton euaton: = t + c( δ + δ ) + I t + T t + MP t + ε () wth δ (t) en δ (t) the resectve recever and satellte clock errors and ε the measurement nose. Ths study consders an deal ostonng envronment, free of the tycal errors affectng ostonng. Ths means that we neglect the onosherc and troosherc errors I and T, as well as the multath errors. We assume the satellte clock error δ (t) to be known and eual to 0, and consder the smulated Galleo orbts and the orbts gven by the GPS navgaton message as correct. Ths smlfed seudo-range observaton euaton can not be solved mmedately for the oston coordnates, but has to be lnearzed frst. Havng an a ror recever oston ( o, Yo o ), the unknown recever oston (, Y ) can be wrtten as = 0 +, dem for Y and Z. Next, lnearzaton by a Taylor seres exanson wll be made around (, Y ), those arameters becomng our new unknowns. Conseuently the seudo-range observaton model becomes: t o t Y t Yo t Z t Z o t t o = Y Z cδ ε + (3) o (t) s the aroxmate geometrc dstance between satellte and recever calculated wth the a ror recever oston. Wrtng Euaton (3) n matrx notatons gves L = A + v, wth the vector (, Y, δ ) of the unknowns. Each element of the observatons vector L euals o and the desgn matrx A conssts of the coeffcents of the unknowns n each observaton euaton. Fnally, v s the vector of the resduals. 3

4 Gven the ostons of the vsble satelltes, the observaton euaton system can be solved usng the Least Suares Method. Mnmzng the weghted sum of the resduals ( v T Pv), results n the Least T 1 T 1 Suares Soluton = ( A PA) A PL. The weght matrx P s defned as = 0 Σ P L wth 0 the a 1 ror varance and Σ L the nverse of the covarance matrx of the observatons. For code-based ostonng, we wll consder a unt matrx as the covarance matrx of the observatons Σ L. The T 1 1 T 1 covarance matrx of the unknowns s Σ = ( A Σ L A) = ( A A), and ts dagonal elements rovde us nformaton on the formal errors of the estmated arameters. After transformng the matrx Σ nto ts toocentrc euvalent covarance matrx, usng the law of varance roagaton, we obtan n ne nu T ΣT = Σ = en e eu un ue u (4) wth the rotaton matrx. Ths covarance matrx deends only on the geometry of the vsble satelltes and allows to extract Dluton of Precson ( DOP ) values (Hust, 000): HDOP = n + e VDOP = u PDOP = n + e + u (5) referrng resectvely to the horzontal comonent, the vertcal comonent and the 3-dmensonal ostonng. Fgure 3 shows the worldwde HDOP values n smulated urban condtons wth a cut off angle of 30. Fgure 3 : Worldwde dstrbuton of the daly medan of HDOP value usng a 30 cut off GPS only combned GPS+GALILEO 4

5 To elmnate the nfluence of HDOP outlers, medan values are shown nstead of means. The results for the combned system show HDOP values below 1.5. For GPS only, most of the worldwde values are less than.5. That seems to be a good value, but for some locatons roblems occur when consderng the HDOP value over the whole day. As an examle, the daly evoluton of the HDOP value for the EPN staton Zelenchukskaya n ussa, s showed n Fgure 4b. Ths staton was chosen because n Fgure 3a as well as n Fgure 4a, t les resectvely n an area wth hgh nevertheless accetable medan HDOP values and n an area wth huge mean HDOP values. Fgure 4 : usng a 30 cut off Worldwde dstrbuton of the daly mean of the HDOP value wth GPS Daly dstrbuton of HDOP value n Zelenchukskaya for all systems In Zelenchukskaya, very huge daly mean HDOP values of and have been observed for both GPS only and Galleo only. These very bad values are caused by a lack of vsble satelltes at certan eochs, exlanng the outlers ( HDOP > 0) also vsble n Fgure 4b. esectvely 1 and 15 outlers for GPS only and Galleo only are vsble n Fgure 4b. On the other hand for the hybrd system, no outlers and not even HDOP values over 3 were observed, yeldng a very good daly mean HDOP value of Fgure 5 : usng a 30 cut off Worldwde dstrbuton of mrovement rato for daly medan of the HDOP value Worldwde dstrbuton of mrovement rato for daly medan of the VDOP value 5

6 Gong back to the results showed n Fgure 3, we conclude that the worldwde decrease of HDOP from less than.5 to less than 1.5, corresonds wth an mrovement of about 40% on the formal errors of the horzontal comonents of the staton oston when usng a combned GPS+Galleo constellaton comared to GPS only (see Fgure 5a). Fgure 5b shows the worldwde mrovement for the VDOP whch s smlar to the HDOP results. 3. elatve ostonng, based on double dfference carrer hase observables elatve ostonng dffers from absolute ostonng because the vector (, Y ) = (, Y Y Z ) between two recevers and s calculated nstead of one sngle recever oston. elatve ostonng reures the ntroducton of sngle and double dfferences ( SD & DD). These dfferences decrease or elmnate the nfluence of some of the error sources. The DD carrer hase observaton euaton between recevers and and satelltes and s: Φ = Φ Φ = t + λn + I + T + ε Φ (6) wth Φ the DD of the carrer hase observable, the DD of the aroxmate geometrc dstances between satelltes and and recevers and, λ the wavelength of the sgnal, whle N s the DD of the nteger ntal ambgutes. The frst order dfferental onosherc effect I s elmnated because we assume the use of an onoshere free combnaton. On the other hand the trooshercal delay T s consdered known. All other error sources, e.g. multath, are art of the measurement nose. Smlar to absolute ostonng, the carrer hase observaton Euaton (6) s lnearzed n order to solve for the unknown arameters. Once more, the recever oston wll be wrtten as (, Y ) = ( 0 +, Y0 + Y 0 + Z ) gven an a ror estmated oston o, Yo o ). Further, Taylor seres exanson wll be executed around (, Y ) and (, Y ) for resectve terms belongng to recevers and. Takng recever as the reference staton, ts coordnates are known and conseuently = Y = Z = 0. In ths test, we consder the ntal ambgutes N as beng fxed, yeldng to followng observaton model: Φ λn = a + ay Y + az Z + v The coeffcents a (t), a Y (t) and a Z (t) of the unknowns, are then gven as: (7) a = Y Y Y Y ay = Z Z Z Z a Z = (8) Every observaton can be wrtten as Euaton (7), yeldng a model reresented by the matrx euaton L = A + v. s the vector (, Y ) of the unknowns, A the desgn matrx contanng all 6

7 coeffcents of the unknowns, L the vector contanng all observatons and fnally v s the vector of resduals. Usng the observaton model from Euaton (7), we can comute the assocated covarance matrx of the T 1 1 unknowns Σ = ( A Σ L A) and convert t to ts toocentrc euvalent Σ T, smlar to what was done for absolute ostonng. The covarance matrx of the observables Σ L s not a unt matrx, but the mathematcal correlatons between the double dfference measurements are now taken nto account. Agan, the covarance matrx of the unknowns rovdes nformaton on the recson of the soluton. The DOP (elatve DOP ) s smlar to the PDOP value for the case of absolute ostonng, but wll be calculated n a dfferent way wth the formula (Goad, 1988): DOP = trace ( Σ ) DD (9) In addton, contrary to the comutaton of the PDOP, the observatons wll now be accumulated over sessons varyng between ½ and 4 hours, usng a 60 seconds measurement nterval. Gong back to formula (9), DD s the uncertanty of a DD measurement. Ths defnton mles that the DOP wll not deend on the a ror varance of the carrer hase measurements. For the matrx Σ as well as for the value DD, a factor can be set aart, whereas those factors aearng n denomnator and nomnator of Euaton (9) can be removed. Conseuently, we wll not have to make assumtons about ths value. The unts of DOP are meters/cycle. Theoretcally, the uncertanty of a DD measurement multled by DOP wll therefore yeld a relatve oston error. Followng the same rncle as for HDOP and VDOP, we consdered north, east and u comonents for DOP. We have chosen two sets of baselnes between EPN statons, long dstances of about km (Fgure 6a) and short dstances of about km (Fgure 6b), each set of observatons dvded n three knds of ars: statons wth eual lattude, statons wth eual longtude and randomly chosen ars. The observatons accumulated over sessons of 1 hours show the same mrovement for north, east as well as for u comonents. Ths mrovement s about 30% for both long and short baselnes n comarson wth revous results obtaned for GPS only. 7

8 North comonent North comonent East comonent East comonent U comonent Fgure 6 : DOP mrovement rato For baselnes of about km For baselnes of about km U comonent 8

9 Another nterestng result s that the same DOP values can be obtaned for the hybrd system n only half of the observaton tme needed when usng GPS only. Table 1 llustrates ths result for the examle EPN-statons Brussels and La Palma. tdsnterval DOP_gs DOP_gg 1/ u u / u u u u u u Table 1 : DOP values for GPS only and for the hybrd system for varyng nterval lengths Concluson Ths aer comared the formal errors of absolute and relatve ostonng erformed usng a GPS only satellte constellaton and usng a combned GPS+Galleo constellaton. As exected, usng GPS+Galleo more vsble satelltes are avalable worldwde and the combned system reduces or even elmnates the resence of roblem locatons worldwde. Usng the DOP values, we demonstrated that the contrbuton of the satellte geometry to the total ostonng error budget s reduced wth about 40% only by addng the Galleo constellaton to the GPS constellatons. For relatve ostonng, we showed that the combned system reaches smlar formal errors as stand-alone GPS, but usng only half of the observaton tme n comarson wth GPS only. As mentoned before, these results are only vald n an deal envronment because the effect of other error sources on the total error budget has been neglected. eferences C.C. Goad, Investgaton Of An Alternate Method Of Processng Global Postonng Survey Data Collected In Knematc Mode, n GPS-Technues Aled to Geodesy and Surveyng, Arl G.J. Hust, Global Postonng System, een nledng,