AOA Cooperative Position Localization

Transcription

1 AOA Cooperatve Poston Localzaton Jun Xu, Maode Ma and Cho Loo Law Postonng and Wreless echnology Centre Nanyang echnologcal Unversty, Sngapore Abstract- In wreless sensor networs, t s very mportant to obtan an accurate poston of each sensor node. In ths paper, we consder the potental of Ultra Wdeband (UWB sgnals for postonng due to ts specal characterstcs. Accurate angle of arrval (AOA can be obtaned usng UWB antenna arrays. We propose a new cooperatve postonng method based on the AOA s. he method taes advantages of the par-wse AOA s nformaton among all the sensor nodes whle exstng postonng method only utlzes the AOA s from the. hen, the Cramer-Rao Lower Bound (CRLB for Cooperatve AOA localzaton s analyzed. hrough the smulaton experments, we show that our proposed method could acheve hgher postonng accuracy than exstng non-cooperatve postonng methods and ts accuracy ncreases wth the number of blnd nodes. Keywords- UWB, postonng, CRLB, AOA, DOA I. INRODUCION In recent years, there have been growng nterests n the applcatons of wreless sensor networs. In such applcatons, nodes are usually randomly placed over the area that s to be probed. Wthout node localzaton, the collected nformaton may suffer from unrelablty. UWB s a good canddate for poston localzaton [-3]. Wth the broad bandwdth, UWB can be used to provde extremely hgh data rate performance. Broad bandwdth also mples a hgh temporal resoluton. herefore, UWB can acheve hgh rangng accuracy hence good postonng precson. Many rangng technques based on UWB have been proposed. Lee [] suggested a UWB tme of arrval (OA rangng scheme whch mplements a search algorthm for the detecton of a drect path sgnal n the presence of dense multpath. Chung and Ha [] proposed a localzaton scheme based on the estmaton of tme of arrval of the frst multpath of UWB sgnals under the exstence of the lne-of-sght (LOS path n a multpath envronment. UWB rangng for non-lne-of-sght (NLOS propagatons was also nvestgated n [6]. AOA s one of the commonly used rangng technques. he AOA postonng technque s based on the measurement of angles between the target node and the reference nodes. It s can be acheved by applyng antenna arrays. he poston of the target node, n a two-dmensonal space, can be dentfed by measurng the angles of the straght lnes connectng the target node wth the reference nodes. here are ether teratve or non-teratve AOA localzaton methods proposed n lterature [7-]. he maxmum-lelhood (ML and the Stansfeld estmators have been analyzed n [7]. orrer [8] proposed a well-nown method the non-lnear equaton formed usng AOA nformaton have been lnearzed. It s an teratve method and requres the ntal poston guess. Nonteratve closed-form solutons for AOA localzaton have been presented n [9, ], and they elmnate the dsadvantage of non-convergence for teratve method. However, all of them only employ the AOA s from the and have gnored the nterconnectvty among the neghborng blnd nodes. hs paper ams to develop a new postonng algorthm whch taes advantage of hgh connectvty of the sensor networs as well as fne tme resoluton of UWB. he sgnfcant contrbuton of our proposed scheme s that t dentfes the locaton of a sensor node by AOA method employng all of the neghborng nodes to acheve hgher accuracy, whch maes our soluton outstandng from the exstng ones. It s assumed that all the nodes n the sensor networ have the ablty to determne the AOA s from ther neghborng nodes whch nclude both and blnd nodes. AOA can be obtaned by fndng the phase dfference of the arrval sgnal, and t can be measured by a smlar setup as that when measurng DOA. We transform the phase dfference nto AOA data and provde them to our proposed novel cooperatve AOA postonng algorthm. he advantages of our proposal are as followng.. he hgh accuracy of UWB n tme doman wll be fully utlzed.. he mnmum number of nodes requred to locate a blnd node n a -D postonng s only, whch s less than that of other postonng schemes. 3. he cooperatve postonng can be mplemented wthout synchronzaton. hs paper wll also perform an analyss on the theoretcal lower bound for the cooperatve AOA localzaton. o the best of our nowledge, ths s the frst pece of the research wor on the UWB cooperatve AOA localzaton usng all possble neghborng nodes to acheve hgher accuracy. And t s also the frst wor on the theoretcal lower bound analyss on the cooperatve AOA localzaton. he remander of the paper s organzed as follows. Secton II descrbes how to obtan the AOA nformaton from an exstng UWB DOA measurng system. Secton III proposes the cooperatve poston localzaton method based on AOA nformaton. Secton IV derves the theoretcal lower bound for AOA postonng. Secton V presents the smulaton results to show the advantages of the proposed scheme and Secton VI concludes the paper /8/$. 8 IEEE. hs full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE "GLOBECOM" 8 proceedngs.

2 II. AOA CONVERED FROM DOA In cooperatve localzaton, sensor nodes wor together n a par-wse manner to mae measurements and then form a map of the networ. In [], we measure the DOA ranges to determne the postons of the blnd nodes. However, t s not realstc to determne the peer-to-peer DOA ranges because t requres the synchronzaton among all the nodes n the sensor networ. It s not practcal to have all the nodes synchronzed n such sensor networs. We want to enjoy both of the advantages, whch are hgh tme precson of UWB and hgh nterconnectvty of the networs. he ey prncple of the system desgn s to avod the ntroducton of system components or structure that would degrade the fne tme resoluton of the UWB sgnal snce t s crtcal to precse of the poston localzaton. DOA measurements can be converted nto AOA nformaton, whch then elmnates the need for synchronzaton between the. Assumng that each node n the sensor networ s equpped wth a unform lnear array (ULA of antennas, and the dstance between two antennas s nown. he two receved sgnal can be consdered beng n parallel. Hence, the angle of arrvals can be derved from the tme dfference. We have desgned the UWB DOA measurement setup n []. A few technques can be deployed n order to extend ts applcablty to allow AOA estmaton as well. We consder a one-element antenna for transmsson and a lnear array of antennas wth W equdstant elements for recepton. It s assumed that the transmtter and recever communcate through an addtve whte Gaussan nose (AWGN channel. he sgnals and the nose at the recevng antennas can be modeled to be ndependent and dentcally dstrbuted (..d.. he transmtter and recevers are assumed to be suffcently dstant. ULA Recever Fg. AOA converted from DOA he transmtted me Hoppng Impulse Response (H-IR UWB sgnal s gven by [3] s( t d jbj / N w( t j f c j f c ( + j L c* L*sn d j represents the polarty and taes values ±, b s the nformaton symbol transmtted, w(t s a monocycle, f s the average pulse repetton tme, {c j } s a pseudo-random sequence s assgned to each user, c s chp nterval and N f represents the processng gan. For a sngle path, the receved sgnal s gven r ( t As( t τ ( W γ + n( t ( A s the path ampltude, s the arrval delay to the frst element of the ULA, s the tme delay between neghborng elements of the ULA, s(t s the transmtted sgnal and n(t s the nose. We can obtan the angle of arrval from Fg., and t s c* γ arccos( (3 L s the arrval angle, c s s the speed of lght, γ and L are the tme delay and dstance between neghborng antenna elements respectvely. Amplfer Amplfer ransmtter Fg. Setup for UWB AOA Measurement Pulse generator Aglent Osclloscope Functon generator III. AOA COOPERAIVE POSIIONING ALGORIHM Consder a networ wth N nodes, R nodes are reference nodes, and the remanng M nodes are blnd nodes (N M +R. he locatons of the n the networ are expressed as {( x, y,(,,,(, } x y x R y R and the locatons of the blnd nodes are {( xr, yr,( xr+, y,,(, } R+ xn y N blnd nodes It s assumed that the networ s fully connected so that cooperatve localzaton can be employed for the postonng. Dense nterconnectvty leads to redundancy n the range measurements. In dense obstacle envronments, the blnd nodes are less lely to be able to trangulate ther postons /8/$. 8 IEEE. hs full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE "GLOBECOM" 8 proceedngs.

3 precsely due to the rado blocage from fxed. In such stuaton, cooperaton among the blnd nodes can sgnfcantly mprove the localzaton capablty []. A sngle AOA may be measured at each recever of the locaton system. In a networ wth enough, the absolute map of the networ can be determned just usng the AOA measurements. In the absence of nose and nterference, the drecton lnes from two or more wll ntersect to determne a unque locaton. In the presence of nose, more than two drecton lnes wll not ntersect at a sngle pont, as llustrated for a two-dmensonal confguraton n Fg. 3. Consequently, processng s requred to determne the optmal poston estmate. he measured AOA from node at node can be expressed as follows α + n, ( y y α arctan(, s the actual AOA from node x x at node. n, s the nose (Fg.. Gatherng angle measurements from dfferent blnd nodes and, we have the followng equaton + n (,, α, α, α, N, N,, N, α, α, N M, M, M, N α M, αm, αm, N and n s matrx showng the nose of the AOA s. It should be noted that matrx ncludes the AOA s from the reference nodes as well as the AOA s from the other blnd nodes. We can form a cost functon (Eq.( usng the measured and calculated AOA s. f M N ( p ( (6 α, Fg. 3 he blnd node s located n the grey area formed by the AOA nformaton from three. Fg. he actual AOA calculated from the actual coordnates he locatons of the unnown nodes thus can be estmated by mnmzng the cost functon [3]. Wth ntal guess of the coordnates of blnd nodes p, both cost functon f (p and partal dervatve f (p can be calculated, and therefore conjugate gradent method can be used to do the mnmzaton. Conjugate gradent method s an teratve method. he computatonal complexty s O ( N. We measure the performance of the postonng method wth root mean square error (RMSE, whch s obtaned by comparng estmated postons wth actual postons. Lower RMSE means better performance. RMSE ε s expressed as Eq.(6. M RMSE ( pest ( pactual ε RMSE (7 M p est and p actual are estmated and actual coordnates, respectvely. IV. CRAMER-RAO LOWER BOUND Cramer-Rao Lower Bound (CRLB gves a theoretcal lower bound of the varance of the poston estmates and t s defned as the nverse of the Fsher nformaton matrx (FIM []. C s the covarance matrx of the poston estmator of pˆ the unnown node C ˆ [(ˆ ( ˆ p E p p p p ] (8 p [ p, p ] [ x, y], E[ ] s the expected value. I(p represents the FIM and can be found usng ln( f ( r p ln( f ( r p I( p E[( ( ] (9 p p r [ r,, r3,,..., r M, ], s a vector of the DOA ranges, f ( r p s the pdf functon of the ranges based on the unnown node poston p [ x, y]. he postonng error bound satsfes σ blnd tr[ I ( p] (, /8/$. 8 IEEE. hs full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE "GLOBECOM" 8 proceedngs.

4 tr [ ] s the trace of a square matrx, defned to be the sum of the dagonal elements. It s assumed that the AOA s are dstorted by Gaussan nose for smplcty. Hence the jont lelhood functon s ( α p exp( ( πσ σ σ, s the varance of the AOA from observed at. he jont pdf s f ( p N N R N R R p N ( p * p R blnd nodes ( ang the logarthm of Eq. (, we wll have two components representng the nformaton from and nformaton among the blnd nodes. y(m y(m x(m (a x(m (b Fg. Comparson of the CRLB for (a Cooperatve and (b noncooperatve localzaton ln f ( p N R R N N p + p R R (3 We can express the FIM as I I I( x, y I I ( Assumng all the AOA s are corrupted by Gaussan nose wth varanceσ, the components of FIM can be derved usng Eq. (9 as follows R N N ( y y ( y y ( y y ( I ( + + σ d, R d, R d, ( anchornodes N ( y y d, blnd nodes N ( y y ( + σ R d, Smlarly we can obtan the other three components N N ( x x ( x x ( I ( + (6 σ d, R d, ( I ( I N N ( x x( y y ( x x( y y (7 ( + σ d, R d, We have derved the CRLB for both OA and DOA for range-ndependent case n [], and t has been found that postonng errors are ndependent of the dmenson of the networ, and only related to the geometry and number of. Contour plot n Fg. (a s an example for cooperatve AOA localzaton, showng the theoretcal postonng accuracy. he AOA measurements are all corrupted wth a Gaussan nose term ofσ.873rad. Fg. (a s an extreme case that all the blnd nodes are n cooperaton. It can be observed from Fg. that cooperatve AOA postonng gves much smaller postonng error compared wth normal AOA postonng (Fg. (b. However, observng Eq. (-7 and Fg., we can fnd that the errors of AOA based postonng are related wth the actual networ sze even though the errors of AOA are ndependent. he postonng error bound of any blnd node s also dependent on the locatons of the other blnd nodes and the number of the neghborng blnd nodes. V. SIMULAION RESULS In ths secton, we examne the performance of AOA cooperatve poston localzaton. We smulated the above postonng algorthm usng MatLab. It s assumed that the rado range of the reference nodes covers the whole area. he ntal postons of the blnd nodes are guessed as long as they are wthn the mxm rectangular area. he scenaro selected for numercal smulatons conssts n R reference nodes unformly dstrbuted on a crcle of radus m centered at the orgn of coordnates. M blnd nodes are placed randomly n the area. he AOA measurements are all corrupted wth a Gaussan nose term of σ.873rad /8/$. 8 IEEE. hs full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE "GLOBECOM" 8 proceedngs.

5 Fg. 6 shows Comparson of the RMSE of the proposed method wth the theoretcal postonng error bound. In ths fgure, we can see the performance of the proposed algorthm when t s ntalzed at the optmum locatons (.e. the actual locaton of the blnd nodes. Here the proposed method s not compared wth the other non-cooperatve AOA postonng methods. We can demonstrate the superorty of our method by comparng t wth the CRLB for non-cooperatve case snce CRLB gves the lowest bound of the postonng errors. It should be noted here that ts accuracy almost ncreases exponentally wth the number of blnd nodes. Normally the postonng accuracy s seen only related to the number of. However we utlzed the nterconnectvty between the neghborng blnd nodes, and hence mprove the performance of the postonng algorthm. Fg. 7 shows an example of the node localzaton n the nterested area. blnd nodes are placed randomly n a mxm area. It s observed that most of the nodes can be located accurately n ths case. he smulaton results demonstrate the better performance acheved by the newly proposed algorthm. RMSE(m Cooperatve AOA postonng method Lower bound for Cooperatve Postonng Lower bound for non-cooperatve Postonng 3 3 number of queryng nodes Fg. 6 Comparson of the RMSE of the proposed method wth the theoretcal postonng error bound. Y Poston (m - actual locaton estmated locaton VI. CONCLUSION It has been wdely nown that UWB can mprove the tmebased postonng. AOA can be obtaned by measurng the DOA between the neghborng elements of a ULA. In ths paper, cooperatve postonng concept has been ntroduced n the postonng process. he our proposed postonng method employs the par-wse AOA nformaton to mprove the AOA based postonng performance, whch has been proved by smulaton results as well as the theoretcal analyss. REFERENCES [] R.J., Fontana, Recent system applcatons of short-pulse ultra-wdeband (UWB technology, IEEE ransactons on Mcrowave heory and echnques, Volume, Issue 9, Part, Sept. Page(s:87 - [] M.Z. Wn, R.A., Scholtz, Impulse rado: how t wors, IEEE Communcatons Letters, Volume, Issue, Feb. 998, Page(s:36-38 [3] S. Gezc, Zh an, G.B. Gannas, H. Kobayash, A. F. Molsch, H.V. Poor and Z. Sanoglu, Localzaton va ultra-wdeband rados: a loo at postonng aspects for future sensor networs, IEEE Sgnal Processng Magazne, Volume, Issue, Page(s:7 8, July [] Joon-Yong Lee, R.A. Scholtz, Rangng n a dense multpath envronment usng an UWB rado ln, IEEE Journal on Selected Areas n Communcatons, Volume, Issue 9, Dec., Page(s: [] W. C. Chung and D. S. Ha, An Accurate Ultra Wdeband (UWB rangng for precson asset locaton, n Proc. IEEE Conf. Ultra WdebandSystems and echnologes, Nov. 3 [6] B.Dens, J.Kegnart and N.Danele, Impact of NLOS propagaton upon rangng precson n UWB systems, IEEE Conference on Ultra Wdeband Systems and echnologes, Page(s: ,6-9 Nov. 3 [7] M. Gavsh, A.J. Wess, Performance analyss of bearng-only target locaton algorthms, IEEE ransactons on Aerospace and Electronc Systems, Volume 8, Issue 3, pp.87 88, 99 [8] D.J.orrer, Statstcal heory of Passve Locaton Systems, IEEE ransactons on Aerospace and Electronc Systems, Publcaton,Volume:AES-, Issue, pp.83-98, March 98 [9] A. Pages-Zamora, J. Vdal, D.H. Broos, Closed-form soluton for postonng based on angle of arrval measurements, he 3th IEEE Internatonal Symposum on Personal, Indoor and Moble Rado Communcatons, vol, pp. - 6, [] A Urruela,. A.Pages-Zamora, J.Rba, Dvde-and-Conquer Based Closed-form Poston Estmaton for AOA and DOA Measurements, IEEE Internatonal Conference on Acoustcs, Speech and Sgnal Processng, Volume, pp.iv IV., 6 [] J. Xu, M. Ma and C. L. Law, Poston estmaton usng ultra-wdeband tme dfference of arrval measurements, IE Scence Measurement & echnology, IEEE 6 Internatonal Conference on UWB, page(s: 3-8, Jan. 8 [] D.B.Jourdan, D.Dardar, M.Z Wn,. Poston Error Bound for UWB Localzaton n Dense Cluttered Envronments, IEEE Internatonal Conference on Communcatons, Volume 8, pp.37 37, June 6 [3] W. H. Press, B. P. Flannery, S. A. euolsy and W.. Vetterlng, Numercal recpes n C : the art of scentfc computng, Cambrdge Unversty Press [] Steven M. Kay, Fundamentals of statstcal sgnal processng: Estmaton heory, Prentce Hall, Englewood Clff, New Jersey, USA [] J. Xu, M. Ma, C. L. Law, heoretcal Lower Bound for UWB DOA Postonng, IEEE Global elecommuncatons Conference, 7 - X Poston (m Fg. 7 blnd nodes are placed n the nterested area /8/$. 8 IEEE. hs full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE "GLOBECOM" 8 proceedngs.