TDOA Based Trajectory Estimation using Recursive Least Square Algorithm

Transcription

1 Advanced Scence and echnology Letters Vol.9 (CA 3), pp DOA Based rajectory Estmaton usng Recursve Least Square Algorthm Kyunghyun Lee, Jungeun Oh, Mnwoo Lm, Kwanho You Sungyunwan Unversty, Suwon, , Korea {naman, ojorea, ddwtlr, Abstract. Recently, tme dfference of arrval (DOA) based locaton estmate method has been extensvely used n trajectory estmaton. In a geolocaton problem, the accurate poston estmaton of an emtter s sgnfcant snce DOA data contan measurement error. hs paper presents DOA based trajectory estmaton method usng recursve least square (RLS) algorthm. hs algorthm can derve an estmated poston of emtter precsely and reduce a calculaton speed. o confrm the effcency of the proposed algorthm, some smulaton results usng the RLS algorthm are provded. Keywords: rajectory estmaton, tme dfference of arrval, recursve least square, measurement error compensaton. Introducton he emtter trajectory estmate method s used n many types of modern locaton based servces such as navgaton, socal networng servce and augmented realty platform. Recently, locaton estmaton technque usng DOA method has been used broadly as t uses tme dfference of sgnal arrval from emtter to fxed recevers. he strength of DOA based localzaton method s that t s unnecessary to synchronze between recevers. Due to ths characterstc, DOA has been used extensvely n real tme locatng systems. [-] In DOA based trajectory determnaton technque, the estmaton of emtter s accurate locaton s also a sgnfcant problem. o estmate the emtter s accurate trajectory, Chan [3] proposed a tme of arrval (OA) and DOA based localzaton algorthm through an approxmate maxmum lelhood method. Chan showed the performance of the proposed algorthm compared wth Cramer-Rao lower bound (CRLB). In ths paper, we propose a DOA based trajectory estmaton of emtter usng the RLS algorthm to compensate for the measurement errors from DOA data. he RLS algorthm s an terated technque of least square (LS) algorthm. hs proposed algorthm provdes a rapd calculaton speed even when we get addtonal DOA data from each recever. On the bass of ths strength, we can treat much more DOA data for the estmaton of an emtter s locaton. As the teraton process s ncreased, we obtan more precse poston of emtter. ISSN: ASL Copyrght 3 SERSC

2 Advanced Scence and echnology Letters Vol.9 (CA 3) rajectory Estmaton by DOA In ths secton, we derve the DOA based trajectory estmaton equaton for applyng RLS algorthm. If there are fxed recevers of whch the locaton s supposed to be nown, we can get DOA data on the bass of specfc recever. [4] We express the unnown poston of an emtter whch we try to estmate as p = [ x, y] nown postons of recevers are represented as = [ x, y ] and the r, = {,,, M} n two dmensonal coordnates. We can obtan the DOA measurement values as d = p r d = ct = c( t t ), () where d means the dstance from emtter to -th recever and c s the propagaton speed of sgnal. t means the measured DOA between -th recever and the frst recever. [5] In a real case, the measured DOA data contans nose due to nonlne-ofsght (NLOS) problem. DOA data can be as followng t = col{ t + t, =,, M} = t + t, [ t] t = col{ t, =,, M}, E = () Gven M nosy DOA measurements, the unnown poston of an emtter can be obtaned by the followng equatons wth Ap = h (3) r d r, r h= b+ρd, A =, ρ =, q =, M d M M, r r rm r ( d) b= = ( q ρ ρ ) (4) rm ( dm) In equaton (3), the vector p s then the soluton of the system equatons. he measurement error n DOA data of real envronment causes that t has a dfference between the estmated locaton and the true locaton of an emtter. In order to solve ths problem, we propose the RLS algorthm n secton Copyrght 3 SERSC

3 Advanced Scence and echnology Letters Vol.9 (CA 3) 3 RLS Estmator for DOA localzaton he RLS algorthm s an terated method of LS algorthm. In ths secton, we derve the LS soluton of the DOA formulaton whch s obtaned n secton. When we get addtonal DOA data set n each recever, t s unnecessary to compute all over agan by usng the RLS technque. In order to apply RLS crteron, we need to express system equaton n a dfferent form. he LS soluton can be wrtten as A h p = A h + + (5) In equaton (6), A = [ A A A ] and h = [ h h h ] represent the values whch contan DOA data. A and + h are + ( + )-th DOA parameters. Usng ( + ) DOA data, the locaton of an emtter s obtaned through the LS algorthm as follows pˆ = ( A A ) A h (6) Wth the defnton of equaton s satsfed. ( ) S+ = A+ A for a smple notaton, the followng + S = S + A A (7) Usng the equaton (7), the equaton (6) s rewrtten as below pˆ = S [ Ah + A h ] = S [ S SAh + A h ] (8) Snce the term SAh n equaton (8) can be denoted as ˆ p, the equaton (8) can be rewrtten as follows pˆ = S [ S pˆ + A h ] p S A ( h A p ) (9) = ˆ ˆ where S represents + ( AA + A+ A + ) due to the equaton (7). herefore, the equaton (9) can be rewrtten as pˆ = pˆ + ( AA + A A ) A ( h A p ˆ ) () Copyrght 3 SERSC 55

4 Advanced Scence and echnology Letters Vol.9 (CA 3) Equaton () s the RLS soluton of a locaton estmaton problem. We can derve the locaton of an emtter more rapdly through equaton () when we receve addtonal DOA data sets. In equaton (), p ˆ denotes the estmated poston of emtter by usng DOA data. A s the parameter based on nformaton of A and h + data. hese values were calculated by the prevous -th teraton. + are addtonal measured data. We can rapdly estmate the locaton of an emtter wth prevous DOA data and addtonally receved nformaton. 4 Smulaton Results In ths secton, we demonstrate the performance of a proposed trajectory estmaton algorthm usng RLS method through a smulaton. In our smulaton, the locaton of an emtter s estmated by usng four recevers. he locatons of each recevers are (, ) m, (, ) m, (, ) m and (, ) m, respectvely. In a real case, each DOA sgnal contans measurement noses. We assume that the measurement noses follow the Gaussan dstrbuton wth a varance of.. We assume the sgnal propagaton speed c s for smplfyng the calculaton. Fgure descrbes that the estmated trajectory usng a proposed algorthm follows the true trajectory. he thn lne means the true trajectory of an emtter, the thc lne represents the estmated trajectory usng RLS method and the upward trangles express the recevers. 8 6 Y-poston [m] 4 - rue trajectory Estmated trajectory Recevers X-poston [m] Fg.. Estmated trajectory of emtter usng RLS method 56 Copyrght 3 SERSC

5 Advanced Scence and echnology Letters Vol.9 (CA 3) 5 Concluson hs paper ntroduces the DOA based trajectory estmaton algorthm through RLS method. hs proposed algorthm can perform whether the emtter moves or stops. Moreover, when we obtan addtonal DOA data from recevers, the proposed algorthm provdes rapd computng speed to estmate emtter s poston. We can treat more DOA data wth ths proposed algorthm. We confrm a hgh performance of proposed RLS algorthm based DOA localzaton method through some smulatons. As the teraton process s repeated over agan, we can obtan much more precse trajectory of an emtter. Acnowledgements. hs research was supported by Basc Scence Research pro-gram through the Natonal Research Foundaton of Korea (NRF) funded by the Mnstry of Educaton, Scence and echnology (3RAA678) References. Kay, S., Vanayalapat, N.: Improvement of DOA poston fxng usng the lelhood curvature. IEEE rans. Sgnal Process., Vol. 6, (3). ran, D.A., Nguyen,.: Localzaton n wreless sensor networs based on support vector machnes. IEEE rans. Parallel Dstrb. Syst., Vol. 9, (8) 3. Chan, Y.., Hang, H.Y.C., Chng, P.C.: Exact and approxmate maxmum lelhood localzaton algorthm. IEEE rans. Veh. echnol., Vol. 55, --6 (6) 4. Ho, K.C., Lu, X., Kovavsaruch, L.: Source localzaton usng DOA and FDOA measurements n the presence of recever locaton errors:analyss and soluton. IEEE rans. Sgnal Process., Vol. 55, (7) 5. Wu, S., L, J., Lu, S.: Improved localzaton algorthms based on reference selecton of lnear least squares n LOS and NLOS envronment. Wrel. Pers. Commun., Vol. 68, 87-- (3) Copyrght 3 SERSC 57