Adaptive Methods of Kalman Filtering for Personal Positioning Systems

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1 Adaptiv Mthods of Kalman Filtring for Prsonal Positioning Systms E. Pulido Hrrra, H. Kaufmann, Institut for Softwar chnology and Intractiv Systms, Vinna Univrsity of chnology BIOGRAPHIES Edith Pulido Hrrra rcivd th lctrical nginr dgr from National Univrsity of Colombia and hr Ph.D. dgr in Computr Scinc from Univrsitat Jaum I (Spain). Sh currntly holds th position of Pos Doctoral Rsarchr at th Intractiv Mdia Systms Group, Institut of Softwar chnology and Intractiv Systms at Vinna Univrsity of chnology. Hr rsarch intrsts includ algorithms for fault dtction in prsonal positioning systms. Hanns Kaufmann finishd his PhD thsis in 4 on Gomtry Education with Augmntd Rality. Aftr his postdoc rsarch in th EU-IS projct Lab@Futur h got an assistant profssor position at th Intractiv Mdia Systms Group, Institut of Softwar chnology and Intractiv Systms at Vinna Univrsity of chnology in 5. H is now lading th VR/AR group at that institut at Vinna Univrsity of chnology. H publishd mor than 35 scintific paprs and was involvd in ovr rsarch projcts on computational gomtry, gomtry ducation, augmntd rality and psychological rsarch in AR/VR up to dat. ABSRAC Kalman filtring is vry fficint for data fusion, in which th dfinition of th procss and masurmnt noiss (i.. th matrics Q and R, rspctivly) gratly influncs th filtr prformanc. In rcnt yars svral studis rportd that adjustmnts of Q and R can b hlpful to rduc th rrors of th stimations. In this papr, various mthods for maing adjustmnts to th matrics Q and R ar introducd for th particular cas of Prsonal Positioning Systms (PPS). h aim is to obsrv th improvmnts achivd in an xtndd Kalman filtr whn adaptiv mthods ar applid, in othr words to obsrv thir influnc on th usr's path obtaind. hs adjustmnts ar considrd to b ndd bcaus nvironmntal conditions in such systms ar oftn not fixd. h mthods to b analyzd ar: () th wightd Kalman filtr; () scaling matrix Q (3) adjustmnts of Q and R basd on squnc-innovation; and (4) a combination of th mthod () and (3), i.. Q is stimatd by applying a scal factor and adjustmnts to R ar ralizd in accordanc with th mthod (3). Givn that th filtr may divrg, w us th χ tst to valuat validity of th stimations, which is basd on th analysis of th innovations. Individual componnts of th innovation vctor ar valuatd in ordr to corrct or liminat wrong information for data fusion. h PPS is basd on th Dad Rconing (DR) algorithm. h rrors of th DR paramtrs ar stimatd with an Extndd Kalman Filtr (), which combins th masurmnts of a GPS and an inrtial masurmnt unit (IMU). h rsults show that ach mthod allows us to obtain consistnt Kalman filtring and thy hlp to obtain bttr usr s trajctoris, but additional tchniqus and/or tchnologis should b usd. INRODUCION In a grat numbr of applications, prsonal positioning systms (PPS) ar vry usful,.g., in rscu wor. In such applications, accurat information of th position is rquird from ithr th victim or th rscu worr (.g. lifguard, firfightr, tc). In PPS, suddn or unxpctd changs of th bhavior of humans ar factors that can incras th complxity of th modling for xampl, whn a prson turns quicly. Hnc, conducting studis and dvloping tchniqus ar ssntial in ordr to hav bttr position stimations. Outdoor positioning information is usually providd by snsor systms, such as th global positioning systm (GPS) or inrtial navigation systms (INS). his information is oftn mrgd by Kalman filtring, sinc its fficincy for data fusion has bn dmonstratd. Nvrthlss, th filtr prformanc can b affctd by changs in th nvironmnt or in th systm dynamics [8]. Svral adaptiv mthods hav bn dvlopd in ordr to ovrcom this problm. hos mthods ar usually dfind to adjust th filtr paramtrs to th tim varying conditions; spcifically th procss covarianc matrix (Q) and th masurmnt nois covarianc matrix (R). his is du to th fact that prior nowldg of th tru valus of R and Q is ndd. Howvr, in practic w usually hav vry littl or no nowldg at all of th valus of R and/or Q. In this wor, w focus on four mthods to adapt th Kalman filtr in ordr to obtain bttr information of th prson s position. h first mthod is namd wightd h Institut of Navigation, Portland, OR, Sptmbr -4, 584

2 Kalman filtr, which was proposd by Andrson []. his consists of applying wightd xponntial valus. h scond mthod consists of applying a scal factor to Q [5]. h third mthod is an adaptiv stimation of th valus of R and Q basd on innovation squnc [8] [9]. h last mthod is a combination of th scond and third approachs: Q is stimatd by applying a scal factor and R is stimatd by using th innovation squnc as is givn in [9]. h mthods mntiond abov ar applid to an xtndd Kalman filtr (). h is usd to mrg th information in a prsonal positioning systm (PPS), which is mad up of a GPS rcivr and an Inrtial Masurmnt Unit (IMU). h information providd by ths snsors allows us to utiliz th dad rconing algorithm (DR), whos main paramtrs ar th strid lngth of th prson and th azimuth bias. By mans of an appropriat combination, th filtr stimats th rrors of th DR paramtrs [7] to finally corrct th stimations of th usr s position. o stablish th validity of th stimation w utiliz th χ tst. In this tst, filtr innovations ar valuatd vry instanc; whn thy fail th tst, on of th adaptiv mthods mntiond abov, is applid. POSIIONING Localization is basd on th dad rconing algorithm (DR), whos main paramtrs ar th strid lngth of th usr and th azimuth. h azimuth indicats th orintation of th usr s trajctory. h strid lngth is th distanc travld btwn two conscutiv foot stris of th sam foot [4]. An important aspct to calculat th valu of th strid lngth is to idntify whn a stp occurs. Svral mthods hav bn proposd to solv this problm; hr, w dtct th pas of th linar vrtical acclration [] whil th usr wals. In this cas th snsor is placd in th lowr bac of th prson. h strid lngth is obtaind through a nural ntwor. W us this tchniqu bcaus it is a good mchanism in cas th systm modl is not availabl or if modling th systm is vry difficult. Dtails of this procss can b found in []. Hnc, th main rror sourcs of DR ar th strid lngth and th azimuth bias. Bias rror can b du to th nvironmnt intrfrncs, body offst, among othrs [][4][7]. By applying a similar algorithm to that on givn in [7] w can stimat th strid lngth and th bias rrors. In [7] this is carrid out by fusing th information providd by a GPS rcivr with othr snsors through an xtndd Kalman filtr (). rquirs th stat modl and th masurmnt modl. For our PPS th stat modl is givn by th following quations: [ ] x = ast v north vn sl b () t t Φ= t β whr x is th stat vctor whos componnts ar: ast, ast vlocity ( v ), north, north vlocity ( vn ), strid lngth rror ( sl ), and bias rror ( b ); Φ is th transition matrix. h masurmnt modl is givn by: [ ] n () z = v n v s ψ (3) 5 [ ] h = ast v north vn h h ( ) h = v + vn t h6 = tan ( v ) + Bsnsor b vn 5 6 (4) h H = (5) x whr z is th masurmnt vctor whos componnts ar: ast ( ), ast vlocity v, north ( n ), north vlocity ( v n ), strid lngth ( s ) and azimuth (ψ ); H is th masurmnt matrix, which is th jacobian of h. h first 4 componnts of th vctor z ar providd by a GPS rcivr, s is obtaind by th nural ntwor and ψ is providd by th IMU. ADAPIVE MEHODS h following mthods ar valuatd and dscribd in th nxt sssions: i. Wightd Kalman filtr. ii. Scaling Q. iii. Estimation of R and Q basd on th innovation squnc. Wightd Kalman Filtr his mthod in combination with fuzzy logic has bn proposd for INS/GPS systms [3]. It was originally an approach by Andrson []. It addrsss th cas whr R and Q ar unnown and it assums thm to b xponntial wightd valus. By assuming R and Q in this way, in fact w ar actually wighting th filtr in gnral. h rsulting quations for th filtr, which hav bn adaptd from [] ar thn prsntd. h Institut of Navigation, Portland, OR, Sptmbr -4, 585

3 h matrics R and Q ar assumd to b: R R δ + ( ) = (6) Q Q δ + ( ) =, with δ (7) From [] th wightd rror covarianc matrix is dfind as: Aftr carrying out th corrsponding transformations, th rsulting quations for th prdiction and corrction phass ar: Prdiction: Corrction: (8) xˆ = Φ xˆ (9) P = δ Φ PΦ + Q () δ gain: δ R K = P H HPH + δ ( ) () xˆ = xˆ + K z Hxˆ () δ ( ) ( ) P= I KH P I KH + KRK (3) h aim is to giv mor wight to th last masurmnts, thrby avoiding th us of rronous valus from prvious stimations [] and facilitating th procss of obtaining a consistnt filtr. Scaling Q matrix his is an approach givn in [5], which is a variation of th algorithm proposd in [6]. It consists in simply applying a scal factor to th Q matrix. his solution is basd on th dfinition of th wighting of th associatd rror covarianc P (a prior); in fact what happns is that whn applying th scal factor to Q, P is also bing scald: P =Φ PΦ + λ Q (4) ϑ ϑ λ = (5) E whr is th filtr innovation, this is th diffrnc btwn th currnt masurmnt and th stat prdictd. If th scal factor λ is gratr than, th nw masurmnt will hav mor wight on th stimation. E is an mpirical valu basd on th xpctd sum of squars of th innovation [5]. Estimation of Q and R basd on Innovation Squnc Basd on th innovation squnc R is adaptd during th procss by mans of [8] [9]: ˆ ˆ R = Cυ HPH (6) h innovation covarianc Cˆ υ is obtaind in an stimation window of siz N [8]: Cˆ ϑ = ϑϑ (7) j j N j= N+ h matrix Q is adaptd at ach poch as follows: Qˆ FAUL DEECION ˆ = KC K (8) υ h validity of th stimation is valuatd by applying th χ tst [] in which th normalizd squar innovation (NIS) is valuatd. his is givn by []: ι ϑ S ϑ = (9) whr S is th covarianc associatd to th innovation, which is givn by: S = H PH + R () ι must hav a χ distribution with η dgrs of frdom, which ar givn by th dimnsions of th masurmnt vctor. In this wor vry componnt of th innovation vctor is valuatd, thus th dgr of frdom is on. EXPERIMENS AND RESULS h hardwar usd to carry out th tsts consistd of a GPS rcivr from Ublox, an inrtial masurmnt unit from Xsns and a laptop. h usr wor th GPS in a bag rcivr and th IMU was placd in his lowr bac; whil h was waling data wr collctd by th laptop. All data collctd wr procssd offlin in Matlab. A diagram of th data flow is illustratd in Fig.. h usr s trajctory was approximatly 5 m in lngth and consistd of straight lins and 9 dgrs turns. h tsts wr carrid out in strts in a dnsly populatd h Institut of Navigation, Portland, OR, Sptmbr -4, 586

ara in th city of Vinna. h rfrnc trajctory is illustratd in Fig.. GPS latitud, longitud, vlocity, tim IMU vrtical acc., azimuth Nural Nt. Strid Lngth Adaptiv DR Fig.

4 ara in th city of Vinna. h rfrnc trajctory is illustratd in Fig.. GPS latitud, longitud, vlocity, tim IMU vrtical acc., azimuth Nural Nt. Strid Lngth Adaptiv DR Fig. Simplifid bloc diagram of th systm SL & Bias rrors East & North mchanism [3]. On th othr hand, this fact allows us to valuat th adaptiv mthods undr ths conditions (i.. suddn chang in dirction of th trajctory.g. 9 dgrs turns). Fig. 3 shows a comparison of th NIS obtaind by th wightd (M. ) and th Q and R stimation basd on th innovation squnc (M. 3). It is obsrvd that in bias ι many valus fall outsid of th allowd limits and its valus wr spcially incrasd in th turns of th rout. h valus of bias ι wr considrably rducd whn any of th adaptiv mthods was applid, as can b sn in Fig 3 (dots in grn and yllow). Liwis, this provids validity to th stimations obtaind by th filtr. 7 6 M.. 5 M () 7 6 M.. 5 M Sampls () Fig. Rfrnc trajctory As th systm is mad up of low cost snsors, som aspcts wr obsrvd in th tsts. For instanc, th GPS did not provid accptabl information in aras with a lot of buildings. hrfor, a lot of simulation was rquird to calculat initial valus of R and Q in ordr to obtain an accptabl corrction of th trajctory. Nonthlss, this dos not guarant an optimal corrction, bcaus th modl usd has dpndnc on GPS data. In svral trials th first lin in th rout indicats vidnt rrors in GPS data, for instanc whil th usr wald in a straight lin th signal showd a curvd rout. his cratd a problm to dtrmin th bias rror. Liwis it causs an inappropriat bhavior of th bias rror in a statistical sns. o indicat this situation, w prsnt th rsults for th NIS associatd to th bias (dnotd by bias ι ) and th NIS associatd to th complt innovation vctor (dnotd by v ι ). h limit stablishd is th valu of 95% confidnc lvl of th χ distribution, which is It can b sn that th filtr had mor problms whn th usr mad a turn. his is xpctd bcaus w did not includ any mthod to corrct th information in ths situations and th filtr dos not hav any slf larning Fig. 3 Normalizd innovation squar for th bias rror () and innovation vctor (). M. rprsnts th NIS obtaind by th wightd ; M. 3 rprsnts th NIS obtaind by th adaptiv mthod basd on th innovation squnc [9] Fig. 4 illustrats th rrors ovr th total distanc travld in 6 simulations. h, th wightd and th scaling Q obtaind th minimal rrors. Howvr, th prformanc of this variabl is accptabl in all mthods, sinc in th worst cas th rror was.64% Fig. 4 Rsults of th rror (%) ovr th total distanc travld in 6 simulations for ach mthod. Wightd sq Innov. Basd unning R & Q h Institut of Navigation, Portland, OR, Sptmbr -4, 587

5 h trajctoris obtaind by all mthods ar prsntd in Fig. 5. Svral aspcts can b obsrvd in ths rsults. First, th trajctory stimatd, by corrcting th rrors of th DR paramtrs with th convntional, has larg rrors (blu lin). hat implis a corrction with any of th adaptiv mthods. According to th rsults obtaind in th χ tst th variabl which causs this fault was th bias rror. In othr words, th main problms appard in th orintation of th trajctory. Scond, as was prviously mntiond in th first lin of th rout th GPS signal could not provid propr information to corrct th trajctory. As a rsult it was not possibl to gt optimal stimations undr ths circumstancs. Nvrthlss, th mthods in which th stimation of R is basd on th innovation squnc providd a considrabl improvmnt in this part of th path (blac (M. 3) and grn (M. 4) lins). hird, w can also obsrv th situation whn th GPS can b a propr rfrnc, for xampl in th third lin th GPS prsntd roughly a straight lin of th path, which can b usd to stimat th bias rrors. Finally, in this tst th bst trajctory was obtaind by th mthod in which R and Q ar stimatd basd on innovation squncs (M. 3). Howvr, a bttr prformanc of th othr adaptiv mthods can also b achivd, but mayb dpnds on th GPS solution. Latittud (dgrs) Rfrnc GPS DR+ DR+M. DR+M Longitud (dgrs) Fig. 6 rajctoris obtaind by th GPS, DR in combination with: th, wightd (M.) and adaptd by scaling Q (M.). Fig. 7 includs th NIS of th bias rror and NIS of th innovation vctor obtaind in this tst. In this tst similar problms occurrd, compard to th formr tst. In placs whr th usr turnd th NIS valus wr highr. It can also b sn that th bias rror has svral points outsid th allowd limit. On th othr hand, onc th mthods wr applid most of th NIS passd th χ tst, ping th propr prformanc of th filtring and at th sam tim a bttr trajctory is obtaind. Latitud (dgrs) Rfrnc GPS DR+ DR+M. DR+M. DR+M.3 DR+M () () M. M. M. M Longitud (dgrs) Fig. 5 rajctoris obtaind by: GPS, DR in combination with, M. (wightd ), M. (scaling Q), M.3. (adaptiv stimation basd on innovation squnc) and M.4 (scaling Q and R is tund by using th tchniqu of M.3). Anothr tst was carrid out in ordr to obsrv th influnc of th initialization in all mthods. Fig. 6 shows th rsults obtaind by th wightd (M. ) and scaling Q (M. ); othr mthods wr not includd bcaus thy did not provid any improvmnt with rspct to th convntional. In this cas th trajctoris stimatd by M. and M. ar good spcially whn th GPS is a good rfrnc, for xampl in th middl of th scond lin or in th third lin Sampls Fig. 7 () NIS for th rror bias and () NIS for th innovation vctor obtaind by th, th wightd (M. ) and scaling matrix Q (M. ). SUMMARY AND CONCLUSIONS W mainly usd thr mthods that showd good prformanc in othr filds such as robotics or location vhicls. W adaptd thm for th particular cas of prsonal positioning systms. hs ar wightd Kalman filtr, scaling Q and stimation basd on innovation squncs. A fourth mthod that mas us of principls of scaling Q and stimation basd on innovation squncs was also includd in th analysis. h χ tst h Institut of Navigation, Portland, OR, Sptmbr -4, 588

6 was carrid out in ordr to dtrmin th validity of th stimations. o do this th normalizd innovation squar was valuatd at ach instant. In our tsts ths mthods wr dpndnt on th initialization, in othr words, it was ncssary to dfin initial valus for Q and R to achiv bttr stimations. h rsults indicatd diffrnt rsponss of th adaptiv mthods according to th initialization. hat is, in th first tst (Fig. 5) th mthod that providd bttr stimations was M. 4 (stimation of R and Q basd on th innovation squncs). In th scond tst with othr initialization th wightd and scaling Q providd bttr rsults. Each mthod prsnts som advantags, for xampl scaling is a vry simply mthod, or M. 4 is a rliabl mthod, which can provid good rsults as can b sn in Fig. 4 (blac lin). On th othr hand, thy also hav som disadvantags,.g. for scaling Q it could b ncssary to find a slf larning way to dtrmin th propr valu of E. h stimation basd on th innovation squnc is a mthod in which w could obtain bttr rsults if som additional tchniqus wr considrd. In othr words, in our tsts th GPS provids unaccptabl data at th bginning of th rout. In such a cas additional tchniqus basd on artificial intllignc can b usd, sinc w can ta advantag of th mpirical nowldg that w hav. For xampl, it is possibl to slct th information providd to th Kalman filtr by using fuzzy logic or nural ntwors. Analyzing vry variabl is also an intrsting aspct, bcaus only on variabl can b th sourc of th fault, causing rronous data fusion. his phnomna was prsnt in our tsts, th main sourc of th rrors in th trajctory was th bias rror. W can improv robustnss of th systm, if w can idntify th fault and corrct it. In addition this information can b usful for dcisionmaing systms. INS, Symposium of Position Location and Navigation PLANS, pp. 7-33, 4. [6] Hu C., Chn W., Chn Y. and Liu D., Adaptiv Kalman Filtring for Vhicl Navigation, Journal of Global Positioning Systms, vol. -, pp. 4-47, 3. [7] Jirawimut R., Ptasinsi P., Garaj V., Cclja F., and Balachandran W. A Mthod for Dad Rconing Paramtr Corrction in Pdstrian Navigation Systm. IEEE ransactions on Instrumntation and Masurmnt, vol. 5, pp. 9-5, 3. [8] Mhra, R. K., On th idntification of variancs and adaptiv Kalman filtring, IEEE rans. Automat. Contr., vol. AC-5,, pp , 97. [9] Mohamd A. H. and Schwarz K. P., Adaptiv Kalman Filtring for INS/GPS, Journal of Godsy, vol. 73-4, pp. 93-3, 999. [] Lvi R.W., and Judd., Dad Rconing Navigational Systm using Acclromtrs to Masur Foot Impact, US Patnt , 996. [] Lwis F. L., Optimal Estimation, Jhon Wily \& Sons, 986. [] Pulido Hrrra E., Improving Data Fusion in Usr Positioning Systms, Ph.D. Dissrtation, Univrsitat Jaum I, 9. [3] Sasiad JZ and Wang Q, Low cost automation using INS/GPS data fusion for accurat positioning, Robotica, vol. -3, pp. 55-6, 3. [4] Stirling R., Collin J., Fyf K., and Lachapll G., An Innovativ Sho-Mountd Pdstrian Navigation Systm, Procding of Europan Navigation Confrnc GNSS, pp. -5, 3. ACKNOWLEDGMENS his wor was fundd in part by th Europan Union (IS projct PROFIEX FP7-8855). BIBLIOGRAPHY [] Andrson B. D. O., Exponntial Data Wighting in th Kalman-Bucy Filtr, Information Scinc, vol. 5, pp.7-3, 973. [] Bar-Shalom Y., Li X.R., and Kirubarajan., Estimation with Application and racing and Navigation, John Wily & Sons,. [3] Brown R.G. and Hwang P.Y.C., Introduction to Random Signals and Applid Kalman Filtring, John Wily & Sons, 997. [4] Foxlin E., Pdstrian racing with Sho-Mountd Inrtial Snsors, IEEE Computr Graphics and Applications, vol. 5-6, pp , 5. [5] Hid C., Moor. and Smith M., Adaptiv Kalman filtring algorithms for intgrating GPS and low cost h Institut of Navigation, Portland, OR, Sptmbr -4, 589

EXST Regression Techniques Page 1

EXST Regression Techniques Page 1 EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy