Implementaton of the Unscented Kalman Flter and a smple Augmentaton System for GNSS SDR recevers R. Capua, Soge A. Bottaro, Soge BIOGRAPHY Roberto Capua s responsble for GNSS R&D actvty at Soge S.p.A. He wored n several GNSS R&D and EC Framewor Projects for Galleo applcatons and serves as a delegate to the Galleo Servces consortum. Hs areas of actvty are: advanced hgh-accuracy GNSS and Augmentaton systems, GNSS SDR development and GNSS surveyng. Antono Bottaro s currently the Head of the Research and Development Department wthn Soge S.p.A. Hs man area of actvty concerns Advanced GIS and GNSS platforms Desgn and Development. ABSRAC he paper presents the results of the development of advanced PV solutons for the Soge s GNSS SDR. Such Real-me platform s based on an nternally desgned Front-End and s totally runnng on a General Purpose Noteboo. he EKF deals wth non-lnear systems through process and observatons functons lnearzaton. Non-lnearty errors can lead to poor Kalman flter performances or to dvergence. Lnearzaton does not allow correct state and measurements Covarance matrces propagaton n presence of sgnfcant dynamc condtons. Jacobans ntroduce severe computatonal burdens and cannot be appled n the presence of dscontnutes and sngulartes. In order to overcome such lmtatons, an Unscented Kalman Flter (UKF) algorthm has been developed and tested wthn the SDR. he UKF avods lnearzaton. Non-lnear transformaton s performed at each step on a set of determnstc sgma ponts extracted from the Covarance matrx, allowng predcton errors mnmzaton. Dependence on ntal condtons s reduced. Due to nherently quantzed measurements and computatonal load constrants, SDR can hghly beneft from the use of an UKF. In our mplementaton, a numerc effcent GPS L1 pseudoranges and Doppler measurements representaton has been developed. A fne tunng phase has been carred out n mxed statc and dynamc condtons. A test campagn has been carred out on a car equpped wth the GNSS SDR. rue trajectory was determned through RK. An accuracy of 5m and mproved robustness n case of abruptly changng routes s acheved. For SDR, a smple and nexpensve augmentaton technque for medum level accuracy s advsable. A Local Augmentaton based on Coordnates Correctons method has been mplemented. Correctons are calculated at SDR rover sde, usng Reference Staton measurements acqured through NRIP and RCM messages. An accuracy of m s reached. Such a soluton can pave the way for mplementng relable and low cost Augmentaton systems totally based on SDR recevers. INRODUCION Governmental applcatons are often requrng hgh performances (hgh accuracy, hgh relablty and hgh contnuty, securty), whle needng Open and customzable technologes. Such technologes should be also able to ncorporate legacy systems and propretary data formats, n order to facltate ther ntroducton wth mnmal mpact on well consoldated worflows and regulaton framewors. Soge, the IC Company of the Mnstry of Economy and Fnance of Italy, started n 8 a Project ntended to develop a GNSS SDR recever for hgh demandng applcatons.
Applcatons of nterest for the project are Land Admnstraton and Land Regstry/Cadastral Maps updatng through hgh precson GNSS surveyng, Border Controls and Dangerous Goods racng and racng for Customs operatons. Hgh accuracy applcatons requre the nstallaton and mantenance of extensve GNSS Reference Statons Networs at natonal or regonal level. Such Reference Statons are costly n terms of upgrade and mantenance. On the other hand, Freghts management applcatons for nsttutonal applcatons are requrng a hgh level of relablty, data securty and output customzablty. For such nd of applcatons, GNSS SDR s one of the most sutable cuttng-edge technologes, due to ts ntrnsc flexblty, scalablty and openness to the ntegraton of new sgnal waveforms. Soge GNSS SDR platform s able to perform Real me GPS, SBAS and GIOVE Acquston and racng. he archtectural components of the system are: - GNSS IF Front-End: nternally desgned and totally based on COS components and the Maxm MAX769 sampler, outputs IF samples on a USB - Acquston, racng and PV software modules: mplemented n C/C++, they are multthreaded and communcatng through CP/IP socets - Quad-Core Noteboo: runnng the GNSS SDR Software - GUIs: mplemented usng OpenSource wxwdgets and RKLIB tools Mult-Core programmng s mplemented usng the Cross- Platform OpenMP Parallel Programmng API, whle hgh computatonal tass le FF and multplcatons are performed usng SIMD (MMX and SSE) nstructons. In Fgure 1, the Soge GNSS platform Archtecture s represented. IF s set by default at 1.3 MHZ wth a samplng frequency of 4.9 MHz for GPS and a -bts (1-bt Sgn and 1-bt Magntude) quantzaton. he ntegraton of Galleo can be acheved, usng an IF of.46 MHz and a samplng frequency of 8.184 MHz. Acquston module performs a Parallel Code Phase Search. Sne and cosne Loo-Up ables and FF code replcas are generated off-lne and loaded nto the cache memory at start-up. Acquston s performed n two steps (Coarse and Fne), n order to provde a fner Doppler estmate. Warm start and Hot start are avalable. racng Loop s mplemented by default through a second order PLL and a frst order FLL. A thrd-order PLL can be actvated on request. he FLL s selectvely actvated at start-up or after a loss of loc and swtched off after a postve test on phase error varance. In order to mprove performances through data parallelsm, raw samples are pacet n two words of 3 Sgn and 3 Magntude bts. Code correlaton and Carrer mxng s therefore mplemented through XOR. In order to mnmze the number of needed correlators, the normalzed Dot product DLL dscrmnator s used ([R4]) for Code tracng. Prompt and Early-Mnus-Late Code Loo-Up tables are generated off-lne for 3 Code phase steps. Such tables are oversampled, due to the fact that the samplng frequency s hgher than the chp rate. Code Doppler s assumed to be zero n the Code-Loo-Up table. Sne and Cosne Intermedate Frequency Loo-Up tables for carrer wpe-off are generated wth π/8 phase steps and 175 Hz frequency steps. Carrer levels are represented by bts (Sgn and Magntude). Costas loop dscrmnator s mplemented through the classcal atan(q P /I P ), whle FLL uses the atan functon. he ntal PV was based on an Extended Kalman Flter (EKF). It s able to calculate navgaton soluton nto NMEA GGA message wth 5 ms update rate. Detals about the PV mplementaton wll be descrbed n the followng paragraphs. EXENED KALMAN FILER BASICS AND DRAWBACKS Let us gve a non-lnear dscrete system wth equatons: z( ) h f [ x( ), u( ), ( ), ] ~ N(, Q ) [ x( ), u( ), ] + w( ) ~ N(, R ) x( + 1) ν (1) Fgure 1 - Soge GNSS Platform System Archtecture
EKF predcton s performed usng aylor seres expanson: xˆ( + 1) f [ xˆ( ), u( ), ] () P( + 1 ) I P( ) I + Q( + 1) F F K+1 where IF s the Jacoban matrx of the state transton functon f. Kalman flter update equatons are: K 1 + 1 P( + 1 ) H ( HP( + 1 ) H + R ) xˆ ( 1 + 1) xˆ( + 1 ) + K 1( z Hx( )) + + P( + 1 + 1) P( + 1 1) K + 1HP( 1) where K +1 s the Gan matrx, P(+1+1) s the updated State Covarance matrx and: (3) Fgure EKF Covarance propagaton example For better estmatng true mean and Covarance matrx, as well as reducng computatonal load, the Unscented transformaton s used, as ntroduced n the followng paragraphs. h H x (4) UNSCENED KALMAN FILER BASICS AND ADVANAGES s the Jacoban of the measurement functon h. As can be seen, classcal Kalman Flter can be smply mplemented usng basc algebrac functons. Unfortunately, some drawbacs that lead to neffcency and possble flter dvergence are present: - Matrx nverson and Jacobans calculaton for Gan and Desgn Matrx calculaton mply hgh computatonal load - Predcton equatons provde very poor approxmaton of the true mean and Covarance matrx due to non-lneartes - Good qualty ntal condtons are needed for assurng flter convergence Matrx nverson computatonal load can be reduced assumng pseudorange and Doppler measurements decorrelated among satelltes, but recalculaton of Jacobans elements at each epoch s stll needed. EKF predcton can fal due to lnearzaton. hs can be seen for nstance n a crcular moton of a vehcle wth constant velocty (see Fgure and [R5]). As can be seen, the Covarance s lnearly projected n the ntal drecton of travel, leadng to loose the correct orentaton (the largest component of the uncertanty at +1 step s not n the current drecton of travel). Furthermore, due to the lnearzaton, EKF cannot be appled wth non dfferentable functons. he Unscented Kalman Flter tres to overcome the lmtatons of Extended Kalman Flter. It starts (e.g. [R4]) from the consderaton that t s easer to approxmate a Gaussan dstrbuton through a fxed number of parameters than a non lnear functon. In order to represent a gven dstrbuton wth mean ( x ) and State Covarance matrx, denoted as P xx for hghlghtng relevant terms, we can use the followng n+1 ponts named Sgma Ponts: χ x where x s the mean and columns of the matrces χ x + σ 1,.., n σ are taen from rows or ± ( n + ) P. he last multplcaton factor s due to the need of respectng the property: P xx n 1 ( n + ) 1 xx [ χ x][ χ x] s a scalng parameter to be chosen for scalng the error n the approxmaton of the fourth order moment. It can be demonstrated that choosng n+3, t s possble to mnmze the fourth order moment wth respect to a Gaussan dstrbuton ([R4]). Havng a non lnear transformaton yg(x), we can therefore wor n the χ doman and transform sngle sgma ponts, as showed n Fgure 3:
α ( n + ~ ) n M () S() ( n + )[ P() + Q()] [ M () M ()] χ ( ) xˆ() + σ (),..,n σ () th column of S() Cholesy decomposton s used for calculatng the square root matrx M. Fgure 3 Sgma ponts transformaton he predcted mean s computed as: n 1 1 y γ + + χ n 1 and the predcted Covarance s: P yy n 1 1 [ γ y][ γ y] + + [ γ y][ γ y n 1 ] Such representaton by ponts allows to correctly predct mean and Covarance matrx, as n the case of the prevous example (see Fgure 4).. Mean and State Covarance matrx nstantaton and predcton: χ ( ) xˆ( + 1 ) P( + 1 ) [ xˆ( ), xˆ( ) ± σ ( ) ] χ ( + 1 ) f ( χ ( ), u( ), ) where: w n + n n m w χ ( + 1 ) c α ( n + ~ ) n m w [ χ ( + 1 ) xˆ( + 1 )][ χ ( + 1 ) xˆ( + 1 )] c w + (1 α + β ) n + c m 1 w w 1,..,n ( n + ) 3. Measurements nstantaton and predcton: ζ ( + 1 ) h( χ ( + 1 ), u( ), t) zˆ( + 1 ) n o m w ζ ( + 1 ) Fgure 4 Unscented covarance propagaton example he UKF algorthm used for the mplementaton of the GNSS SDR PV s a scaled verson of the classcal UKF. he scalng s used for modulatng the spread of the sgma Ponts around the mean. In the followng, the Unscented Kalman Flter estmaton process s reported. 1. Intalzaton and parameters defnton: x ( ), Pxx () 4. Innovaton and Cross-Correlaton Covarance matrces calculaton: P ( + 1 ) vv P ( + 1 ) xz n n w w c [ ζ ( 1) zˆ( + 1 ) ][ ζ ( 1) zˆ( + 1 ) ] + R( + 1) [ χ ( 1) xˆ( + 1 ) ][ ζ ( 1) zˆ( + 1 ) ] + R( + 1) c where P xz s the cross-correlaton matrx. 5. Classcal Kalman Update: K( + 1) P xz ( + 1 ) P 1 vv ( + 1 ) xˆ( + 1 ) xˆ( ) + K ( + 1)( z( ) zˆ( + 1 )) P( + 1 + 1) P( + 1 ) K( + 1) P vv ( + 1 ) K ( + 1)
In the prevous equatons, s the scalng parameter, α determnes the spread of the sgma ponts around the mean and t s usually set to a small value (e.g. 1e-3), ~ s a secondary scalng factor and t s usually set to zero, whle β s a tertary scalng factor used for ncorporatng a pror nowledge about the dstrbuton of x and emphaszng the weght of zero-sgma ponts n the Covarance matrx calculaton (usually set to for Gaussan dstrbuton). As can be seen, UKF mplementaton does not requre lnearzaton (state transton functon and measurements functons are drectly appled to sgma ponts) and t can also wor n presence of dscontnutes. he predcton only conssts of lnear algebra operatons. All such advantages are fundamental for mnmzng computatonal load n an SDR mplementaton. Whle the classcal Kalman Flter mples the propagaton of n components for the state vector and n /+n/ components for the Covarance matrx, the UKF requres the propagaton of n+1 sgma ponts only. Furthermore, UKF s more nsenstve to ntal condtons wth respect to EKF. It has been demonstrated that UKF rapdly converge also n presence of an ntal poston error of several Klometers. DYNAMIC MODEL FOR KALMAN FILER IMPLEMENAION IN HE SDR PV he mplemented Kalman Flters ncludes poston, velocty, recever cloc bas and drft estmaton. State vector s defned as n the followng: x [ x y z x& y& z& b b& ] where x, y and z are the errors on Cartesan coordnates of the recever, b and b & are the recever code offset and drft estmaton errors. he classcal dscrete-tme dynamc model s: x I x + w w ~ N, I ( ) ( ) + 1, Q H x 1 v, v ~ N, R z + I I t I I I I t I Q c Q Q pv 3 Sb t + S f t / S f t / Q c S f t / S f t where t s the step tme, Φ, 1 s the transton matrx, Q and R and are the process and measurement nose covarance matrces and S b and S f are the spectral denstes related to the Allan varance parameters. H s the drectonal cosne vector from the recever to the satellte. he State evoluton s represented by the followng equatons: x+ 1 x + x& + wx y z x& z& b b& + 1 + 1 + 1 y& + 1 + 1 + 1 + 1 y z x& y& + y& + z& f + w x + w + + y z& + wz b + b& + ε b b& + ε Such dynamc model, where velocty varatons (e.g. acceleratons) are modeled as whte nose, s here used for mnmzng the computatonal load. It s sutable for medum dynamcs vehcles, as boat or cars. he measurement vector s: PR z h( x) D where PR and D are the vectors of relatve pseudorange and Doppler measurements for the observed satelltes. he relatve pseudorange, used as usual for SDR computaton ([R]), for satellte can be expressed n the followng way: off t s w w PR c( t + t t mn ) where c s the velocty of lght, t off s a bas of 68 ms (tang nto account that the travel tme between a GPS satellte and the Earth ranges s n the 67-86 ms range), z y
able to assure that all relatve pseudoranges calculated n ths way are postve, tt s the travel tme derved by the number of samples elapsed tll Subframe starts and represents the travel tme of the reference satellte. t s mn Classcal non lnear measurement model s used for pseudorange estmaton: PR + c( dt ( x x ) u d) + d + ( y + d y ) on trop u + ε + ( z z ) u + dρ + where dρ are the orbtal errors, dt s the satellte cloc error, d s the recever cloc error, d on and d trop are the onospherc and tropospherc delay for satellte and ε s the measurement nose. Usng UKF, relevant Jacobans for Gan and Covarance predcton, as n equatons (1) and (4), are no more calculated, whle the non lnear measurement functon s used wthout lnearzaton. ropospherc correctons are appled to pseudoranges usng a standard refracton model (Goad and Goodman model). Ionosphere propagaton error s mtgated through classcal Klobuchar method. For the Doppler measurement, the followng expresson s used: c f L1 D H ( v v ) + c b& + ε where D s the Doppler measurement for satellte, obtaned from the Carrer racng Loop, v s the satellte velocty, H s cosne drectonal vector from the recever to the satellte, v u s the user recever velocty and ε s the measurement error. Flter ntalzaton s performed through Bancroft algorthm. Sngle satellte measurements are assumed to be uncorrelated. Usng such assumpton, each measurement can be sngularly and sequentally processed, leadng to a very sgnfcant computatonal load reducton. In ths way, measurements vector sze s reduced to x1, whle low complexty x matrx nversons operatons only have to be performed. A fne tunng phase has been mplemented n order to fnd sutable values for scalng factors, Q and R covarance elements. Relevant results are descrbed n the followng paragraph. u UKF IMPLEMENAION AND UNING PHASE FOR GNSS SDR Startng from the same State and Measurement model used for EKF mplementaton, UKF has been developed for PV. A senstvty analyss for UKF fne tunng has been performed. Flters parameters choce (Q, R and scalng factors) vary dependng on operatonal condtons. At ths am, dfferent test cases have been performed n statc and dynamc condtons n order to determne sutable values for the flter desgn parameters. A fxed antenna, wth precse poston determned through a statc geodetc surveyng and relevant post-processng, was mounted on the roof of a buldng and used for statc envronment tests. Dynamc tests have been performed comparng poston, velocty and acceleraton behavors of a recever mounted on a testng car wth the reference ones obtaned from a geodetc recever operatng n RK mode fed by the same antenna. Q matrx represents the uncertanty n our nowledge about the process. It has a sgnfcant mpact at steady state, when convergence s acheved. Durng such phase, K s low and t depends on the ntal hgh Q elements manly. Hgher Q varance elements allow better followng acceleraton n dynamc condtons, but lead to hgher nose and ntal overshoot before achevng flter convergence. On the other hand, lower Q varance does not allow a good dynamc tracng, whle provdng a better dampng durng the ntal convergence. A trade-off between such condtons has to be found. A senstvty analyss revealed that Q varances between 1e-5 and 1e-4 are sutable values for statc condtons and.5-.1 for dynamc (e.g. n presence of acceleratons). In order to defne R matrx values, measurements varances have been derved from real data. Pseudorange error varances have been estmated, tang nto account quantzaton errors and typcal pseudorange measurements errors. Varances values n the range of 1 to 1 have been consdered sutable for a good flter convergence. Doppler measurements varances have been evaluated analyzng real measurements error dynamcs, as reported n Fgure 5 for one satellte channel. he range of Doppler measurements varances s therefore selected between 1 and 1. he State Covarance matrx characterzes the uncertanty about ntal condtons. ypcal ntal State Covarance Matrx varances have been used (1-1 for poston
state, 5 for velocty, and lterature values for cloc bas and drft). COORDINAES BASED LOCAL AUGMENAION For many applcatons (e.g. ADAS and lane eepng n the road sector), medum accuracy (m level) at low cost s suffcent. Affordable Augmentaton Systems based on low cost GNSS networs nfrastructures can mprove performances at local level and allow delvery of servces for Natonal and Local Publc Admnstratons. Fgure 5 Doppler measurements error ang nto account the dynamc model (acceleraton s assumed as whte nose), Q varances values ranges for the UKF are confrmed from the acceleraton analyss performed on real-tme test data (see Fgure 6). Here, the estmated acceleratons (n lght blue) are compared to the reference acceleratons derved from the geodetc recever worng n RK mode (n red) connected to the same antenna. For a GNSS SDR rover, mnmzaton of computatonal resources s the prmary constrant. Furthermore, phase measurement, for performng Carrer smoothng or RK solutons mplementatons, can be not drectly avalable n many mass maret GNSS recevers and n several GNSS SDR recevers. herefore, a smple augmentaton system based on drect Coordnates Correctons can be useful wthn ths framewor. Coordnates based correctons methods are nown snce the begnnng of GPS applcatons development (e.g. [R6]). Consderng a Reference Staton wth nown precse coordnates x, the estmated poston xˆ, affected by an error x, can be calculated through Least Mean Square as n the followng: xˆ x + x ( H x corr ( H H ) H ) 1 1 H ρ H ρ x Fgure 6 Acceleraton analyss As reported by the Unscented Kalman Flter theory, such flter s qute nsenstve to large ntal state errors. It has been demonstrated by real data, as shown n Fgure 7, provdng an ntal poston error of m. UKF has demonstrated to be able to converge also wth 1 Km ntal errors. he estmated error xcorr can be used as a Local correcton for a rover n the neghbors of the Reference Staton. Such correctons nclude mpacts of man error sources (e.g. onospherc delay error) n common between the rover and the Reference Staton, provded that same set of satelltes are used for poston computaton. he correctons can be appled to the rover poston as n the followng. x rov where xˆ rov x corr x rov + ( x rov x xrov s the true rover poston, xrov the rover poston computaton and Coordnate Correcton. corr ) s the error n xcorr s the appled Fgure 7 Intal state error mpact In order to respect the constrant of usng same satelltes for Reference Statons and Rover poston calculaton, as requred by classcal Coordnates Correctons algorthms, several correctons broadcastng methods can be mplemented.
For the sae of our nvestgatons, the followng two common Augmentaton archtectures have been taen nto account: - Archtecture 1: the rover transmts satelltes used for PV to the GNSS Reference Staton Networ Control Centre; t calculates relevant correctons and transmt them to the rover - Archtecture : the GNSS Reference Staton Networ Control Centre calculates the whole set of Coordnates Correctons for each possble satelltes combnaton and broadcast them to the rover Both the above solutons mply the avalablty of computatonal resource and the mplementaton of dedcated solutons at GNSS Reference Staton Networ Control Centre sde. Whle the Archtecture 1 mples the avalablty of a bdrectonal communcaton channel, the Archtecture broadcastng soluton requres sgnfcant computatonal resource (e.g. 386 combnatons solutons to be processed at each epoch for 1 satelltes n vsblty at Reference Staton sde and 6 to 1 satelltes of them vsble at rover sde). herefore, a thrd archtecture has been desgned and developed. Usng such archtecture, the only needed modfcatons are mplemented on the rover PV. Reference Staton poston estmaton and correctons calculaton s performed by the rover usng the same rover PV module. Only satelltes n common vsblty between the rover and the Reference Staton, wth relevant ephemers extracted by the rover PV, are used. In that way, there s not the need to calculate Reference Staton poston and relevant Correctons for all possble combnatons of n common satelltes. Once calculated, Coordnates Correctons are appled to the rover PV estmaton n order to produce a corrected poston n NMEA format. In order to estmate the spatal decorrelaton of correctons, a post-processng senstvty analyss has been performed. Precse rover poston, calculated through post-processng analyss, has been compared wth the corrected poston usng Coordnate Correctons calculated from several Reference Statons spread around the rover. It has been derved that correctons mantans ther valdty at least up to 1 Km of separaton, as reported n Fgure 9, where Grosseto Reference Staton s 9 Km far from the rover poston. Currently, RCM.3 and 3. standard messages do not nclude Coordnate Correctons transmsson. An adaptaton of the Coordnate Correctons method to the avalable messages s needed for mplementng such Augmentaton system. he used Archtecture s reported n Fgure 8. Fgure 8 Coordnate Correctons method Archtecture Reference Staton pseudorange measurements for all vsble satelltes are broadcasted usng NRIP protocol to the SDR rover usng RCM.3 #19 messages (or RCM 3. message 14). Pseudorange measurements are read and converted nto the data format used by PV through an Adapter module. Fgure 9 Dfferental Correcton post-processng analyss (ROMA and GROS Reference Statons) he above archtecture has been developed and tested usng raw measurements comng from exstng Reference Statons Networs as Local Augmentaton System. Pseudoranges calculated by the Reference Staton are dependent on embedded models used for errors detecton and mtgaton (e.g. onospherc correctons, tropospherc models or carrer smoothng). herefore pseudorange at Reference Staton and rover sde are not drectly
comparable ad can lead to slght msmatches n the correctons. A Local Augmentaton system usually mples hgh nstallaton and mantenance costs. A system totally based on SDR, both for rover and Reference Statons, could lead to sgnfcant costs savngs and smplfcatons for the mplementaton of medum accuracy servces and systems spread over a wde terrtory. Furthermore, such an mplementaton wll guarantee homogenous servce levels and raw measurements qualty, due to the use of same error models and PV algorthms at rover and Reference Staton sde. Future actvtes of our group wll wor around such soluton. Relevant results at steady state for a dynamc test are reported n Fgure 1, compared to classcal EKF mplementaton. After an ntal statc phase (frst tenth of s), the car starts movng and performs two curves wth relevant change of drectons and deceleraton/acceleraton. At the end, a new statc phase s mplemented. As can be seen, wth reference to the true trajectory, UKF s qute robust durng the whole trajectory and t s resstant to rapd change of drectons and n presence of acceleratons, as well as to satellte geometry change due to the shadowng of the surroundng buldngs, occurrng at the begnnng and at the end of the route. On the contrary, EKF does not react promptly to such dynamc condtons, leadng to a sgnfcant error ncrease. RESULS AND PERFORMANCES A testng car has been set-up n order to evaluate UKF performances. A sngle antenna s able to feed both a GNSS geodetc recever operatng n RK mode and the GNSS SDR through an antenna spltter developed for ths purpose (see Fgure 1). Fgure 1 UKF dynamc results Concernng the Coordnates Correctons method, a statc test has been carred on a pont wth nown coordnates. he RSS (Root Sum Square) planmetrc error and the 3D RSS error before and after correctons applcaton are reported n Fgure 14 and Fgure 14. Fgure 1 estng Car A reference trajectory has been followed (as shown n Fgure 11), performng change of drectons and acceleraton/deceleratons. Precse RK postons have been logged n the geodetc recever memory to be used as reference trajectory, whle SDR EKF and UKF outputs have been logged n a fle for post-processng. Under nomnal onospherc condtons, a fnal accuracy mprovement of at least 6% s envsaged usng such method. A 3D RMS error.3 m has been acheved applyng coordnates correctons, aganst about 9.1 m for absolute postonng. Fgure 11 Dynamc test trajectory Fgure 13 Error before and after correctons (D)
Applcatons le ADAS and lane eepng can benefcate from such method and wll be further nvestgated by the Authors n the followng years. ACKNOWLEDGMENS he authors acnowledge the essental wor performed by A. Caporale and L. Gattuso for ther contrbuton to the preparaton of materal for the present paper, as well as F. Frttella and C. D Amco for the desgn and development of the GNSS Front-End and relevant software modules. REFERENCES Fgure 14 Errors before and after correctons (3D) CONCLUSIONS A scaled Unscented Kalman flter has been desgned and mplemented wthn the SDR PV module of the Soge s GNSS SDR n order to overcome classcal lmtatons of an Extended Kalman Flter. UKF does not mae use of lnearzaton and performs Covarance Matrx propagaton by ponts (e.g. sgma ponts). It allows a better mean and Covarance matrx propagaton durng abruptly changng routes n dynamc stuatons. Relevant UKF parameters (Q varances, P varances, UKF scalng parameters) have been selected after a fne tunng phase. [R1] Fundamentals of Global Postonng System Recever, J. Bao-Yen su, Wley [R] A Software-Defned GPS and Galleo Recever K. Borre, D.M. Aos, N. Bertelsen, P. Rnder, S. H. Jensen [R3] Understandng GPS - Prncples and Applcatons, E. D. Kaplan, C. J. Hegarty, Artech House [R4] A New Extenson of the Kalman Flter to Nonlnear Systems, Smon J. Juler, Jeffrey K. Uhlmann, the Unversty of Oxford [R5] A General Method for Approxmatng Nonlnear ransformaton of Probablty Dstrbutons, S. Juler, J. K. Uhlmann [R6] Overvew of Dfferental GPS Methods, E. G. Blacwell, ION GPS 1985 he paper demonstrated through on-feld test that a sgnfcant mprovement n robustness and n poston and velocty error estmaton s acheved by UKF n dynamc stuatons wth respect to classcal EKF. Further mprovements are under nvestgaton and can be acheved through the development of automatc tunng methods for Q matrx elements. A smple Local Augmentaton system usng Coordnate Correctons has been developed for SDR based applcatons. he GNSS SDR rover recever pseudorange measurements are transmtted by a geodetc Reference Staton of a GNSS Networ usng standard RCM.3 messages and NRIP protocol. An mprovement n the order of 6% n poston estmaton s envsaged. Such results can pave the way for a possble mplementaton of low cost Local Augmentaton systems totally based on SDR recevers (both rover and Reference Staton sde).