DYNAMIC LOCATION ESTIMATION BY KALMAN FILTER

Transcription

1 Ubqutous Coputng and Councaton Journal ISSN DYNAMIC LOCATION ESTIMATION BY KALMAN FILTER Lubna Farh UBICC Publshers, Sr syed Engneerng Unversty, Karach, Pakstan ABSTRACT Ths paper descrbes an effectve ethod for dynac locaton estaton by Kalan Flter for range-based wreless network. The proble of locatng a oble ternal has receved sgnfcant attenton n the feld of wreless councatons. In ths paper, Kalan Flter wth TDOA technque descrbes the rangng easureent trackng approach. Kalan flter s used for soothng range data and tgatng the NLOS errors. The paper presents a sple recursve odel by usng te dfference of arrval based locaton easureent and ncorporatng state equalty constrants n the Kalan flter. Fro the process of Kalan flterng, the standard devaton of the observed range data can be calculated and then used n NLOS/LOS hypothess testng. The proposed recursve locatng algorth, copared wth a Kalan trackng algorth that estates the target track drectly fro the TDOA easureents, wll be coparatvely ore robust to easureent errors because t updates the technque that feeds the locaton correctons back to the Kalan Flter. It copensates for the easured geoetrcal locaton and decreases rando error nfluence to the locaton precson. Sulaton results show that the proposed locaton estaton algorth can prove the accuracysgnfcantly. Keywords: Kalan flter, rangng, TDOA,CWLS. 1 INTRODUCTION The te-based ethods for locatng oble statons n wreless systes usually nvolve the usage of locaton paraeters, such as te of arrval (TOA) and te dfference of arrval (TDOA). In general, the true range between a transtter and a recever n the wreless envronent can be correctly calculated only when the drect path of sgnal propagaton s present, whch ay not always be possble. Aong the error sources whch ay affect the accuracy of wreless locaton, non-lne-of-sght (NLOS) error s consdered the ajor one. Partcularly hgher accuracy s requred, the effects caused by NLOS error usually cannot be gnored [1]. In ths artcle, we concentrate on the rangng aspect of the dynac locaton estaton. In the TOA ethod, the dstance between the Moble Staton (MS) and Base Staton (BS) s deterned fro the easured one-way propagaton te of the sgnal travelng between the. For two-densonal (D) postonng, ths provdes a crcle centered at the BS on whch the MS les. By usng at least three BSs to resolve abgutes arsng fro ultple crossngs of the lnes of poston, the MS locaton estate s deterned by the ntersecton of the crcles. In the TDOA ethod, the dfferences n the arrval tes of the MS sgnal at ultple pars of BSs are easured. TDOA-based locaton systes are of ore nterest because of ts potental for hgh locaton estaton accuracy. We wll anly dscuss locaton algorth ncludng fundaental perforance n the presence of nose and ult path. Further ore we wll descrbe the cobnaton of optzaton strateges whch wll prove the overall accuracy [4]. Accurate locaton s one of the essental features that assst thrd generaton (3G) wreless systes n ganng a wde acceptance and trggerng a large nuber of nnovatve applcatons. Although the an drver of locaton servces s the requreent of locatng Eergency 911 callers wthn a specfed accuracy, oble poston nforaton wll also be useful n desgnng ntellgent transportaton systes. Global Postonng Syste (GPS) could be used to provde oble locatons. However, t would be expensve to be adopted n the oble phone network because addtonal hardware s requred n the MS. Alternatvely, utlzng the base statons n the exstng network for oble locaton s preferable and s ore cost effectve for the consuer. The basc prncple of ths software based soluton s to use two or ore BSs to ntercept the MS sgnal. Each TDOA easureent defnes a hyperbolc locus on whch the MS ust le and the poston estate s gven by the ntersecton of two or ore hyperbola. In general, the MS locaton s estated fro a set of nonlnear equatons constructed Volue 7 Nuber 5 Page

2 Ubqutous Coputng and Councaton Journal ISSN fro TDOA easureent. In [5], a ethod of locaton and velocty estaton wth sgnal strength and TOA easureents was suggested. By usng a lnear recursve odel of oblty and by soothng the locaton va Kalan flter, an accurate estated track could be acheved. A Kalan trackng ethod based on TDOA easureents for UMTS oble locaton has been deonstrated n [6]. Usng a dfferent way of convertng nonlnear equatons to lnear equatons wthout ntroducng constrans, have developed a sple LS AOA-based locaton algorth [7]. However, all these ethods gnore the constrans on the oveent of oble users. The goal of ths work s vehcle poston optzaton by usng NLOS error dentfcaton and tgaton by usng based Kalan flter, as shown n Fg.1. easureent based on t n the presence of dsturbance, denoted by r TOA, s odeled as r TOA, = d ntoa, = ( x x ) ( y y) ntoa, (3) = 1,,, M n TOA, s the range error n r TOA,. Eq. (3) can also be expressed n a vector for as rtoa = f TOA ( x) ntoa, (4) (5) and (6) Fgure 1: Dynac Locaton Archtecture. The rest of the paper s organzed as follows. Secton descrbes the easureent odels. Secton 3 presents locaton estaton odel and Secton 4 descrbe locaton easureent odel NLOS dentfcaton, the Kalan flter technques for soothng, trackng and NLOS tgaton technques are presented n Secton 5. Secton 6 deonstrates the sulaton results. Conclusons are suarzed n secton 7. MEASUREMENT MODELS Ths secton descrbes the TOA and TDOA T easureents. Let x = [ x, y] be the MS locaton to be deterned. The coordnates of the th BS s T x = [ x, y], = 1,,, M, the superscrpt T denotes the transpose operaton and M s the total nuber of the recevng BSs. The dstance between the MS and the BS, denoted by d, s gven by d = ( x x ) ( y y ) (1) = 1,... M The TOA s the one-way propagaton te taken for the sgnal to travel fro the MS to a BS. In the absence of dsturbance, the TOA easured at the BS, denoted by t, s d t = = 1,,, M () c c s the speed of lght. The range.1 TDOA easureent algorth (7) The TDOA algorths are based on easurng the dfference n the te of recepton of sgnals at dfferent BSs, wthout requrng a synchronzaton of all the partcpatng BSs and MT. Actually, the uncertanty between the reference te of the BSs and that of the MT can be reoved by eans of a dfferental calculaton. Therefore, only the BSs nvolved n the locaton estaton process ust be tghtly synchronzed. For a couple of BSs, let say and j, the TDOA,, s gven by, and are the absolute tes taken by the burst to reach the BSs and j, respectvely. Assung that the MT s n LOS wth both the BSs, and j, t follows that the MT ust le on a hyperbola. A second hyperbola on whch the MT ust le can be obtaned usng a further TDOA easureent nvolvng a thrd BS. The user poston can therefore be dentfed as the ntercepton pont of the two hyperbolae, as shown n Fg.. The soluton of the relevant equaton syste can be found wth an teratve ethod and least squares nzaton. Volue 7 Nuber 5 Page

3 Ubqutous Coputng and Councaton Journal ISSN Fgure : TDOA ethod. Assgnng the frst BS as the reference, t can be easly deduced that the range easureents based on the TDOAs are of the for: (8) s the range error n. Notce that f the TDOA easureents are drectly obtaned fro the TOA data, then In vector for, Eq. (8) becoes 3 LOCATION ESTIMATION (9) (10) (11) (1) In general, the MS locaton s not deterned geoetrcally but s estated fro a set of nonlnear equatons constructed fro the TOA, RSS, TDOA, or AOA easureents, wth the knowledge of the BS geoetry. Generally optzaton approaches for locaton are Nonlnear Least Square (NLS), Maxu Lkelhood (ML), Gaussan ML (GML), Genetc Algorth Method (GAM ) and Constraned Least Square ethod (CWLS) [3]. Bascally, there are two approaches for solvng the nonlnear equatons. The NLS approach s to solve the drectly n a nonlnear least squares or constraned weghted least squares fraework. Although optu estaton perforance can be attaned, t requres suffcently precse ntal estates for global convergence because the correspondng cost functons are ultodal. The CWLS approach s to reorganze the nonlnear equatons nto a set of lnear equatons. In ths paper, the CWLS approach s adopted and we wll focus on a unfed developent of accurate poston algorths for TDOA easureent. For TDOA-based poston systes, t s well known that for nose condtons, the correspondng nonlnear equatons can be reorganzed nto a set of lnear equatons by ntroducng an nteredate varable, whch s a functon of the source poston. Ths technque, s coonly called sphercal nterpolaton (SI) [8]. However, the SI estator solves the lnear equatons va standard least squares (LS) wthout usng the known relaton between the nteredate varable and the poston coordnate. To prove the locaton accuracy of the SI approach, Chan and Ho have proposed [4] to use a two-stage CWLS to solve the source locaton, whle [3] ncorporates the relaton explcty by nzng a constraned LS functon based on the technque of Lagrange ultplers. Accordng to [3], these two odfed algorths are referred to as the quadratc correcton least squares (QCLS) and lnear correcton least squares (LCLS), respectvely. The QCLS estator and LCLS yelds an unbased soluton wth sall standard devaton that s close to oderate nose level. But when nose s practcally strong, ts bas s consderable and ts varance could no longer approach the stablty crtera. We have proved the perforance of the QCLS estator and LCLS estator by ntroducng a weghtng atrx n the optzaton, whch can be regarded as a hybrd verson of the QCLS and LCLS algorths. We observe n sulaton that the perforance of the CWLS estator s superor to that of the LCLS and QCLS by reducton n the MSREs, for the whole range of nose power. Therefore, consderng the constrants, we can elnate spurous poston estates and prove the accuracy of estaton. In ths work, we developed a unfed approach for oble locaton by utlzng TDOA easureent and provng the rangng accuracy wth the use of CWLS [7]. The CWLS locaton estaton approach for the TDOA easureent s: wthout dsturbance, Eq. (8) becoes Volue 7 Nuber 5 Page

4 Ubqutous Coputng and Councaton Journal ISSN It can be rewrtten as (13). Optu estate of the locaton s coputed by nzng the followng CWLS cost functon: Squarng both sdes of Eq. (13) and ntroducng an nteredate varable, whch has the for We obtan the followng set of lnear equaton Wrtng Eq. (15) n atrx for gves (14) (15) (1) s the covarance atrx of TDOA's easure values, whch s a functon of the estate of, denoted by [6]. The dea of CWLS estator s to cobne the key prncples n the CWLS and LCLS ethods, the MS locaton estate s deterned by nzng Eq. (1) subject to Eq. (0). For suffcently sall easureent errors, the nverse of the optu covarance atrx for the CWLS algorth s found usng the best lnear unbased estator [] as () (16) (17) (3) (18) And, denotes eleent-by-eleents ultplcaton. Snce contans the unknown, we express and the varable, conssts of the MS locaton as well as [8]. In the presence of easureent errors, the SI technque deternes the MS locaton by sply solvng Eq. (16) va standard LS and the locaton estate s found fro and approxate by. Slar to [6], the CWLS proble s solved by usng the technque of Lagrange ultplers and the Lagrangan to be nzed s (4) (19) s the lagrange ultpler to be deterned. The estate of s obtaned by dfferentatng Eq. (18) wth respect to and then equatng the results to zero [3] as s an optzaton varable vector [6]. An proveent to the SI estator s the lnear correcton least squares (LCLS) ethod, whch solves the LS cost functon n Eq. (19) subject to the constrant of or equvalently, (0) The soluton s: (5) s real value. Put the n Eq. (5) and obtan the sub-estates of. Then choose the soluton fro those sub-estates whch akes the cost functon nu. Volue 7 Nuber 5 Page 131

5 Ubqutous Coputng and Councaton Journal ISSN LOCATION MEASUREMENT MODEL AND NLOS IDENTIFICATION The range easureent between a oble staton and the th base staton correspondng to TOA data can be odeled as: r ( t ) = L ( t) n ( t) NLOS( t ) (6) r ( t ) s the easured range at the saplng te t, L ( t) s the true range, n ( t ) s the easureent nose and can be odeled as a zeroean addtve Gaussan rando varable wth varance ; NLOS ( t ) s the NLOS error coponent n the receved sgnal. There s no NLOS error f the lneof-sght exsts, n whch case, the easureent error n ( t ) becoes the only source of range easureent error. To tgate the NLOS errors, the exstence of non-zero NLOS coponent needs to be dentfed. To dentfy the change of channel stuaton between NLOS and LOS, the standard devaton of the estated range data r ) can be calculate as ˆ ˆ = 1 N k = 1 ( r ( t ( t ) Lˆ ( t )) (7) L ( t ) s the estated range data of the th BS at te nstant t soothed by the unbased Kalan flter. The standard devaton estated n Eq. (1) can then be used n the sple hypothess testng to deterne the LOS/NLOS BSs H o : ˆ < LOS condton (8) H ˆ 1 : NLOS condton (9) s a scalng factor, and > 1 experentally for the UWB channels. 5 KALMAN FILTER s chosen necessary for optalty actually exst, and yet the flter apparently works well for any applcatons n spte of ths stuaton. Flterng s desrable n any applcatons. For exaple, rado councaton sgnals are corrupted wth nose. A good flterng algorth can reove nose fro electroagnetc sgnals whle retanng the useful nforaton. Kalan flter s a tool that can estate the varables of a wde range of processes and estates the states of a lnear syste. It s the one that nzes the varance of the estaton error. Kalan flterng provdes a tool for obtanng that relable estate. It estates a process by usng a for of feedback control: the flter estates the process state at soe te and then obtans feedback n the for of (nosy) easureents. As such, the equatons for the Kalan flter fall nto two groups: te update equatons and easureent update equatons. The te update equatons are responsble for projectng forward (n te) the current state and error covarance estates to obtan the a pror estates for the next te step. The easureent update equatons are responsble for the feedback for ncorporatng a new easureent nto the a pror estate to obtan an proved a posteror estate. The te update equatons can also be thought of as predctor equatons, whle the easureent update equatons can be thought of as corrector equatons. Indeed the fnal estaton algorth resebles that of a predctor-corrector algorth. A Kalan flter s recursve so that new easureents can be processed as they arrve. It s essentally a set of atheatcal equatons that pleent a predctor-corrector type estator that s optal n the sense that t nzes the estated error covarance. A Kalan flter can be used n estatng the state vector of a oble target fro the observed range data and therefore estatng the true range value. The state vector can be represented n [1] X 1) = A( k ) X ( k ) B( k ) W( ), (30) ( k k X (k ) = [ L ( k) L ( k)] s the state vector of the oble related to the th sensor. The W (k) s the drvng nose vector wth covarance atrx Q = I, and w The Kalan flter s a set of atheatcal equatons that pleent a predctor-corrector type estator that s optal n the sense that t nzes the estated error covarance. The Kalan flter has been the subject of extensve research and applcaton, partcularly n the area of autonoous or asssted navgaton. Ths s lkely due n large part to advances n dgtal coputng that ade the use of the flter practcal, but also to the relatve splcty and robust nature of the flter tself. Rarely do the condtons The easureent process s s the easured data and (31) (3) (33) Volue 7 Nuber 5 Page

6 Ubqutous Coputng and Councaton Journal ISSN and s the easureent nose wth covarance. The operaton of the Kalan flter can be suarzed as follows: Kalan flter te update equatons: Kalan flter easureent update equatons (34) (35) (36) (37) (38) s the Kalan gan vector and s the covarance atrx of. The odfed based Kalan flter s proposed to process the range easureent accordng to the feedback dentfcaton result fro the prevous processed data. Before coputng kalan gan, the easureent nose covarance or the range predcton covarance s adjusted as follows: Fg.3 llustrates the applcaton of Kalan flter to tgate the NLOS error, the sold curve and dshed curve denote the true dstance and the estated dstance respectvely. Fg.4 shows the range easureent whch s optzed by applyng CWLS optzaton technque to prove the estaton. It can be observed that the estaton by applyng CWLS technque can great prove the estaton accuracy. The Sulaton result shows that the bg value of velocty can reduce the "transent te" of the Kalan flter especally n the case of based soothng. In ths paper the sulaton s conducted n the followng two cases: Fg.5 shows the locaton estaton error wth constraned and unconstraned flters represented by the dashed and the sold curve respectvely. It can be seen that the constraned flter results n uch ore accurate estates than the unconstraned flter. The unconstraned flter results n average poston errors of about 5.8, whle the constraned flter results n poston errors of about 3.3. Case 1: f (39) (40) Case : f 6 SIMULATION RESULTS (41) (4) Fgure 3: NLOS tgaton usng based KF. Sulaton s perfored for poston estaton by usng MATLAB. It s assued that 00 data saples are easured wth the saplng perod 3s. The vehcles have a velocty of 4/s. The range data are created by the true dstance fro each vehcle poston n the trajectory to the known three BSs. The easureent nose s assued to be AWGN and NLOS nose s added to the true calculated range to get the easured range data. The easureent nose s assued to be Gaussan dstrbuted wth zero ean and a standard devaton. The ntal value for the state vector s. Volue 7 Nuber 5 Page

7 Ubqutous Coputng and Councaton Journal ISSN effectveness of ths ethod. If the state constrants are nonlnear they can be lnearzed. The proposed ethod gves a real-te constraned least squares estaton algorth for estatng the locaton usng TDOA easureents. The NLOS errors can be tgated effectvely for achevng hgher accuracy n range estaton and wreless locaton. 8 REFERENCES [1] B. L. Le, K. Ahed, and H. Tsuj, "Moble locaton estator wth NLOS tgaton usng kalan flterng," In Proc.IEEE Wreless Councaton and Networkng, vol. 3, pp. 16 0, 003. Fgure 4: NLOS tgaton wth CWLS optzaton. [] H. C. So and S. P. Hu, "Constraned locaton algorth usng TDOA easureents," IEICE Transactons on Fundaentals of Electroncs, Councatons and Coputer Scences, vol. E86-A, no. 1, pp , 003. [3] G. W. Elko, Y. Huang, J. Benesty, and R. M. Mersereat, "Realte passve source localzaton: a practcal lnear-correcton least-squares approach," IEEE Transactons on Speech and Audo Processng, vol. 9, no. 8, pp , 001. [4] Y. T. Chan and K. C. Ho, "A sple and effcent estator for hyperbolc locaton," IEEE Transactons on Sgnal Processng, vol. 35, no. 8, pp , [5] M. Hellebrandt and R. Mathar, "Locaton trackng of obles n cellular rado networks," IEEE Transactons on Vehcle Technology, vol. 48, pp , Fgure 5: Poston estaton error wth the constraned and unconstraned flters. 7 CONCLUSION The technques and algorth n ths paper has been developed to be used for dynac locaton estaton. GPS often use to localze objects; however, the accuracy of the GPS recever s locaton estate s not relable when the satellte sgnal s copletely lost or dstorted by trees, hgh buldngs, or tunnels. The absence of satellte sgnals has been studed n uch of the research and a Kalan Flter has been eployed n order to ntegrate the GPS easureents wth other easureents and to overcoe the outage of the satellte sgnals. However, the dstorton of the satellte sgnal stll causes sleadng nforaton for the locaton. In ths paper, we present a range estaton algorth wth NLOS error tgaton usng based Kalan flter. Sulaton results deonstrate the [6] M. Najar and J. Vdal, "Kalan trackng based on tdoa for uts oble," IEEE Internatonal Syp. Personal, ndoor and Moble Rado Councaton, vol. 1, pp , 001. [7] A. Pages-Zaora, J. Vdal, and D. R. Brooks, "Closed-for soluton for postonng based on angle of arrval easureents," 13th IEEE Internatonal Syposu on Personal, Indoor and Moble Rado Councatons, vol. 4, pp , 00. [8] J. O. Sth and J. S. Abel, "Closed-for leastsquares source locaton estaton fro rangedfference easureents," IEEE Transactons on Acoustcs, Speech and Sgnal Processng, vol. 35, no. 1, pp , Volue 7 Nuber 5 Page