GPS/TDOA Hybrid Location Algorithm Based on Federal Kalman Filter

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1 GPS/TDOA Hybrid Locatio Alorithm Based o Federal Kalma Filter Cui-Xia Li, Wei-Mi Liu, Zi-Na Fu GPS/TDOA Hybrid Locatio Alorithm Based o Federal Kalma Filter Cui-Xia Li *1, Wei-Mi Liu *, Zi-Na Fu *3 *1, Correspodi author, School of Civil Eieeri ad Trasportatio, South Chia Uiversity of Techoloy, GuaZhou, PR Chia *3 Beiji GK Feida Traffic Eieeri CO, No8xi Tu Che Road, Haidia District, Beiji, PR Chia 183 licuier@siacom, miweiliu@16com, aa_c@163com doi: 14156/jcitvol5issue76 Abstract Geerally, accuracy of GPS(Global Positioi System) is hiher tha that of cellular etwork But i tall buildi urba area or idoor, relatively weak GPS sial makes the positioi ustable ad iaccurate TDOA(Time Differece of Arrival) is oe of the most widely used positioi methods i cellular mobile commuicatio systems, but its accuracy is ot hih eouh to meet the rowi demad I urba where there are more base statios to support bi commuicatio capacity tha i rural, the mobile termial positioi accuracy ad stability is relatively better I order to make up for these iheret deficiecy of GPS or TDOA separate positioi, this paper deduces a GPS/TDOA hybrid positioi alorithm based o federated kalma filter by ivi error model, state equatio ad measuremet equatio of each local kalma filters Simulatio results show that the alorithm effectively improves the data fusio reliability ad positioi accuracy 1 Itroductio Keywords: Federal Kalma Filter, Hybrid Positioi, GPS, TDOA I recet years, commuicatios techoloy, especially cellular mobile commuicatios techoloy is expadi rapidly, ad the umber of users is also icreasi rapidly This makes mobile termial positioi become a more uret demad It becomes a reat busiess ad market prospect to use the cellular etwork system to locate mobile termial The traditioal positioi methods are GPS i the sky ad cellular etwork positioi o the roud TDOA is oe of the most commoly used positioi methods based o the cellular etwork GPS method requires receivi at least four satellites sials to locate mobile termial I outdoor, the positioi ca be very accurate for eouh localizatio sials But i idoor, roads surrouded dese hih buildis, brides tuels or other areas where GPS sials are too weak to search or track, GPS positioi accuracy becomes istable or the positioi will fail TDOA positioi method is by meas of measuri the differece of trasmissio times from the mobile termial to base statios to calculate mobile termial locatio That required three or more cellular base statios whose system times must be sychroized for receivi mobile termial sials However, the commuicatios systems ad sial propaatio eviromet exert a reat ifluece o the positioi accuracy ad speed So the upper traditioal positioi techoloies all have obvious deficiecies i terms of accuracy, sesitivity, speed or usability, which reatly limit the developmet ad applicatio of positioi service To overcome those deficiecies, this paper costructs a hybrid positioi alorithm combii GPS ad TDOA to improve positioi accuracy ad availability Mathematical model of Federated Kalma Filter The core of GPS/TDOA hybrid positioi alorithm is data fusio mode How to fuse two kids of positioi data is the key of hih aviatio ad positioi accuracy Federated kalma filter is curretly the most widely used method of multi-sesor data fusio Federated kalma filter is actually a special kid of decetralized kalma filter Its particularity lies i cotaii a iformatio distributio process I this process, dyamic iformatio of master filter is 4

2 Joural of Coverece Iformatio Techoloy Volume 5, Number 7, September 1 allocated to each local filter I local filter, the observed iformatio collected throuh local sesor is combied with the feedback iformatio outputted from master filter for a local estimatio process The local estimatio result will be set to master filter for further iformatio fusio to make a lobal optimal estimatio [1] X, P, β 1 Z 1 X, P 1 1 Z X, P X, P, β Fiure 1 Federal Kalma Filter Model Fiure 1 shows the simplified structure of federal kalma filter Here GPS receiver is used as local sesor 1, ad TDOA observatios receiver as local sesor I fiure1, X 1, P1 ad X, P are estimatios of local filters X, P are lobal optimal estimatio outputted from master filter The data fusio alorithm is described as the followi: 1 1 = X ( k) P ( k)( P ( k) X ( k) P ( k) X ( k)) P ( k) = ( P ( k) + P ( k)) (1) β, β are the coefficiets of data allocatio betwee the local filters While the followi coditios 1 are met, X ( k) is the best lobal optimal estimatio result X ( k) = X ( k), β + β = 1 i 1 Q ( k) = βq ( k), P ( k) = β P ( k) i i i i () While federal kalma filter is used i GPS/TDOA hybrid positioi, both GPS sial ad TDOA observatio will be used i locatio solutio process simultaeously, so that the state of system ca be ameded cotiuously i the filteri process The output of hybrid positioi could provide more accurate iitial positio ad orietatio iformatio, thus whe oe positioi method is failure, aother ca maitai a hih positioi accuracy i a lo time [] 3 Alorithms Desi of Federal Kalma Filter of GPS/TDOA Hybrid Positioi System 31 Error Model, State Equatio ad Measuremet Equatio of Local Filter 1 GPS positioi errors iclude a variety of compoets, such as the distace error resulted from clock bias, speed error resulted from clock frequecy drift All of those errors show radom oise characteristics clearly It is eerally cosidered that the GPS positioi errors i the east ad orth directio are respectively equivalet to a first-order Markov process The error model is show as below [3] ~ ~ ε = τ ε + ω, ε = τ ε + ω (3) e ε e e εe ε ε 43

3 GPS/TDOA Hybrid Locatio Alorithm Based o Federal Kalma Filter Cui-Xia Li, Wei-Mi Liu, Zi-Na Fu I Eq (3), ε e ad ε respectively idicate total GPS positioi error i the east ad orth directio respectively, τ are correspodi to the reciprocal of time costat correlated with Markov processes, σ ε e ad σ ε τ ε e ad ε ω εe ad ε ω idicate Gaussia white oise whose mea is zero ad variace is It is eerally believed that the movemet model of mobile termial is compliat with Sier model Sier model is used i this case ad the system state variables ca be described as the followi X = [ l, v, a, l, v, a, ε, ε ] T (4) e e e e le, ve, aead l, v, a respectively idicate mobile termial s locatio, velocity, acceleratio i east ad orth directio ε e ad ε respectively idicate the total positioi errors caused by various error source alo east ad orth directio GPS positioi error model ca be derived as below: a = a () t + a (), t v = τ a () t + ω () t e e e e ae e ae a = a () t + a (), t v = τ a () t + ω () t e e a a (5) I Sier model, velocity of mobile termial is also cosidered as a first-order Gauss-Markov process The system state equatio ca be writte as: ~ X() t = AX() t + U + W() t (6) The locatio values outputted from GPS receiver, respectively amed as l e ad l i east ad orth, are cosidered as exteral observatios The relatioship of exteral observatios ad state variables is: le = le + ε e + τ1 l = l + ε + τ (7) I Eq (7), τ 1 ad τ are oise of positio observatio, l e ad l, outputted from GPS receiver, also are Gaussia white oise whose mea is zero ad variace is σ 1 ad σ So the measuremet equatio ca be writte as: le( k) le( k) + ε e( k) τ1( k) Zk ( ) = l ( k) = + l ( k) + ε ( k) τ ( k) (8) Eq (8) ca be abbreviated as: Ad: Z ( k) = H( k) X( k) + V( k) (9) 1 1 H( k) = 1 1 (1) 3 Error Model, State Equatio ad Measuremet Equatio of Local Filter Propaatio time from mobile termial to base statio i is deoted as the followi ti ti τ i τ ei = + + (11) 44

4 Joural of Coverece Iformatio Techoloy Volume 5, Number 7, September 1 I Eq (11), t i idicates sial LOS propaatio time, also is a liear fuctio of distace betwee mobile termial ad base statio τ i idicates system error ad it is a zero-mea Gaussia radom variable compliat with τ ~ N(, σ ), i which σ depeds o testi equipmet accuracy ad testi techiques τ ei idicates the additioal delay error caused by NLOS, it is a radom variable closely related to sial propaatio eviromet ad distace betwee mobile termial ad base statio By Eq (11), we ca deduce the error model of TDOA[4]: t = t + τ + τ i,1 i,1 i,1 ei,1 uei,1 = E( τei τel ) = uei uel σei, l = D( τei τel ) = σei + σel (1) i,1 t idicates the TDOA observed value i LOS eviromet t i,1 idicates system error ad its characteristic like τ i τ i,1 idicates the additioal delay error caused by NLOS, its mea is u ei,1 ad variace is σ ei, l TDOA errors cotai ot oly the Gaussia distributed radom variable, but also the expoetial distributed radom variables I order to facilitate statistics aalysis, ormal distributio method is be used to fit τ ei ad τ ej, chael eviromet errors which obeyi expoetial distributio The errors ca be cosidered as part of the state variables I order to elimiate the errors to improve the positioi accuracy, we use the secod-order Markov process to fit the TDOA positioi error model I TDOA positioi, movemet model of mobile termial is still compliat with Sier model System state variables ca be described as the followi X = [ l,,,,,,, ] T e ve ae l v a δθ δ s (13) Accordi to the physical meai of state variables, we ca deduce the dyamic equatio of termial mobility ad the state equatio of kalma filter as followi le ( k+ 1) 1 T le ( k) ve ( k 1) 1 ve( k) a + e = + l ( k+ 1) 1 T l ( k) v ( k+ 1) 1 v( k) a ~ Xt () = AXt () + Wt () (14) Mobile termial drivi directio ad distace iformatio are cosidered as exteral observatios The relatioship of exteral observatios ad state variables is: θ = T v + v + ω ek ( ) k ( ) 1 v s = arcta + k ( ) ω v ek ( ) (15) ω 1 ad ω respectively deote total error of motio directio ad distace of mobile termial, ad they are zero-mea Gaussia white oise ad their variaces are δθ, δ s Observatio equatio ca be described as the followi ad it is oliear [ ] Z( k) = h X( k) + V( k) (16) 45

5 GPS/TDOA Hybrid Locatio Alorithm Based o Federal Kalma Filter Cui-Xia Li, Wei-Mi Liu, Zi-Na Fu We must dispose liearizatio o it whe usi optimal estimatio method to estimate the system state variable 33 Data Fusio Alorithm Based o Federated Kalma Filter Accordi to patters of iformatio distributi ad feedback, federated kalma filter ca be classified ito four structural classes: o-reset, fusio-reset, zero-reset ad re-adjust Cosideri all the difficulties, such as system operatio efficiecy, fault tolerace ad reliability ad data fusio alorithm i master filter, No-reset structure is adopted i this desi [5] The master filter fuses the data outputted from local filters based o the iformatio distributio coefficiets, without iformatio feedback to effectively prevet the error back propaatio Summary of the fusio alorithm is: X = β X + β X k GPS GPS ( k ) TDOA TDOA( k ) (17) I Eq (17), β GPS ad β TDOA respectively idicate iformatio distributio coefficiets of GPS ad TDOA filter The key of data fusio alorithm is how to choose β GPS ad β TDOA Here, they are chose i accordace with the followi formula: β = P P, β = P P (18) 1 1 GPS ( k ) GPS ( k ) TDOA ( k ) TDOA( k ) I Eq (18), P = ( P + P ) ( P X + P X ) ( k) GPS ( k) TDOA ( k) GPS ( k) GPS ( k) TDOA ( k) TDOA ( k) β + β = 1 GPS TDOA (19) All of the above lead up to the followi equatio: GPS ( k ) TDOA( k ) GPS ( k ) GPS TDOA( k ) TDOA X ( k) ( P P ) ( P X ( k) P X ( k)) = + + () From the Eq (), we coclude the worse the estimatio quality of state i, the smaller the 1 correspodi iformatio matrix P i So i the data fusio process of GPS/TDOA hybrid positioi, if the accuracy of state estimatio of oe local filter is hiher, its correspodi availability weiht allocated by master filter is bier O the cotrary, the availability weiht should be smaller 4 Alorithm Simulatio We choose a test area covered by at least four cellular base statios ad a test mobile termial carryi a GPS receiver I the test process, supposi the iitial velocity is m/s, mobile termial is movi alo a particular east-orth path The movemet velocity may chaes i this process, that is to say it have a disturbed acceleratio which ca be cosidered as a zero-mea radom process ad be described with a first-order Markov process [6] The followi diarams respectively idicate the simulatio result of positioi errors of TDOA, GPS ad hybrid positioi based o federal kalma filter X 46

6 Joural of Coverece Iformatio Techoloy Volume 5, Number 7, September 1 Fiure Total Positioi Error of TDOA i East ad North Fiure 3 Total Positioi Error of GPS i East ad North Fiure 4 Total Positioi error of Hybrid Positioi i East ad North As ca be see from the fiures, federal kalma filter ca make the error lower tha TDOA ad GPS obviously The followi table 1 shows the comparative results of error before ad after filteri Table 1 Comparative results of error before ad after filteri stadard error stadard error before filteri after filteri TDOA GPS Hybrid Positioi i east i orth i east i orth i east i orth

7 GPS/TDOA Hybrid Locatio Alorithm Based o Federal Kalma Filter Cui-Xia Li, Wei-Mi Liu, Zi-Na Fu From the data of above table, we ca see that TDOA or GPS separate positioi error is hihly reduced based o kalma filter Also the hybrid positioi error is reduced to 1-1 meters, this is a striki result So, the simulatio results show the hybrid positioi alorithms is effective 5 Summary I this paper, a data fusio alorithm based o federal kalma filter is desied for GPS/TDOA hybrid positioi Theoretical aalysis ad experimetal result show that the federal kalma filter is a effective method to deal with data fusio i mobile termial positioi system I order to meet the dyamic characteristics of mobile termial movemet, optimizi the filter structure ad allocati coefficiet to esure performace of data fusio are still yet to be further studied [7] 6 Refereces [1] Seah, Chze E; Hwa, Iseok Performace aalysis of Kalma filter based hybrid estimatio alorithms IFAC Proceedis Volumes (IFAC-PapersOlie), v 17, 1, 8 [] Carlso, Neal A Federated filter for fault-tolerat iterated aviatio systems PLANS, Positio Locatio ad Naviatio Symposium, p , 1988 [3] Yamauchi, Seiji; Taaka, Toshiyuki GPS stadard positioi usi Kalma filter SICE- ICASE Iteratioal Joit Coferece, p , 6 [4] Dua, Kai-Yu; Zha, Li-Ju A TDOA/AOA locatio alorithm based o kalma filteri ale of arrival Diazi Yu Xixi Xuebao/Joural of Electroics ad Iformatio Techoloy, v 8, 9, p , September 6 [5] You, Keyou; Xie, Lihua; Su, Shuli; Xiao, Wedo Multiple-level quatized iovatio Kalma filter IFAC Proceedis Volumes (IFAC-PapersOlie), v 17, 1, 8 [6] Bartlett Steve; Rich Stato Aalysis ad simulatio of discrete Kalma Filter applied to a GPS receiver Naval Eieers Joural, v 18, 1, p 9-36, Ja [7] Liu, Guohai; Li, Qixue Applicatio of dyamic Kalma Filter i state estimatio of aviatio test Chiese Joural of Scietific Istrumet, v 3,, p 396-4, February 9 48

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