Source Localization by TDOA with Random Sensor Position Errors - Part II: Mobile sensors

Transcription

1 Source Localzaton by TDOA wth Random Sensor Poston Errors - Part II: Moble sensors Xaome Qu,, Lhua Xe EXOUISITUS, Center for E-Cty, School of Electrcal and Electronc Engneerng, Nanyang Technologcal Unversty, , Sngapore Emal: {xmqu,elhxe}@ntuedusg College of Computer Scence and Technology, Southwest Unversty for Natonaltes, Chengdu, Schuan, 64, Chna Emal: mathsgrl@63com Abstract For the purpose of source localzaton, we have proposed a constraned weghted least squares (CWLS) source localzaton method n our companon paper, whch uses statc sensors by accountng for random uncertantes n sensor postons Ths paper s devoted to developng two recursve algorthms to deal wth the source localzaton problem by usng tme dfference of arrval (TDOA) measurements receved by moble sensors More specfcally, the frst one uses the current TDOA measurements to estmate the unnown source poston and then treats the estmate as a measurement to update the source localzaton For the second approach, we estmate an auxlary varable wth the current TDOA measurements and then rearrange the nonlnear TDOA equatons nto a set of lnear measurement equatons to update the source localzaton An llustratve example s gven to demonstrate that the second algorthm outperforms the frst one Index Terms Source localzaton, tme dfference of arrval, weghted least-squares, recursve localzaton algorthm I INTRODUCTION In our companon paper [], we have proposed a constraned weghted least squares (CWLS) source localzaton method whch uses tme dfference of arrval (TDOA) measurements receved by a networ of passve statc sensors Although the proposed CWLS source localzaton method s able to reach the optmal accuracy, e the Cramer-Rao lower bound (CRLB), the performance of the localzaton stll could be poor as the CRLB depends on the specfc locaton geometry formed by sensors and the source When the locaton geometry s not desrable, such as the postons of sensors are close to each other or the source s far away, the CRLB wll be large Modern localzaton system often uses passve sensors whch could be born by arplanes or unmanned aeral vehcles (UAVs) The postons of these moble platforms wll change dynamcally and the nformaton of ther postons can be avalable wth GPS [] So source localzaton based on moble sensor networ may be traced over tme from multple TDOA measurements The problem of tracng a statonary or movng source usng multple TDOA collected over tme s not a trval tas due to the nonlnear nature of the TDOA measurements Oello and Musc [3] tacled ths problem based on only two UAVs They used the TDOA to defne a hyperbolod on whch Ths wor was supported by Defense Scence and Technology Agency, Republc of Sngapore the emtter must be located the hyperbolc measurement error regon s approxmated by a sum of weghted Gaussan dstrbutons They appled unscented Kalman flters (UKFs) ntated wth Gaussan mxture measurement (GMM) components Ths method has been extended to nclude frequency dfference of arrval (FDOA) n addton to TDOA n [4] Fletcher et al [5] have nvestgated the recursve localzaton problem usng a smple extended Kalman flter (EKF) based on two UAVs A comparson of these recursve algorthms for a par of UAVs s gven n [6] All of the above mentoned methods are based on dealng wth the nonlnear TDOA measurement equatons drectly In our companon paper [], we dealt wth the nonlnearty of TDOA by rearrangng the nonlnear TDOA equatons nto a set of lnear equatons However, ths rearrangement needs to ntroduce an auxlary varable whch depends on the source poston Actually, ths reorganzaton dea was ntally proposed n [7] for source localzaton wth statc sensors, and mproved n [8] [] whch mae use of the relatonshp between the unnown source poston and the auxlary varable to acheve better localzaton performance Inspred by ths reorganzaton dea to remove the nonlnearty n the TDOA measurement equatons, the present paper s devoted to developng two lnear recursve source localzaton algorthms for a statonary target by usng TDOA measurements receved from a moble sensor networ The frst recursve algorthm s referred to as recursve localzaton algorthm and the second as mproved recursve localzaton algorthm In the recursve localzaton algorthm, we rearrange the nonlnear TDOA measurement equatons nto a set of lnear equatons wth the TDOA measurements at the current samplng step and use the weghted least-squares (WLS) method to estmate the unnown source poston and ts estmaton error covarance Then we treat the estmate as a measurement to update the source localzaton estmaton obtaned at the last samplng step In the mproved recursve localzaton algorthm, we use the weghted least-squares (WLS) method to estmate the auxlary varable at the current samplng step frstly and then substtute t nto the set of rearranged TDOA equatons whch gves rse to a new set of lnear measurement equatons wth respect to the unnown source poston only We then use Kalman flter to update the 54

2 source localzaton estmate of the last samplng step The rest of the paper s organzed as follows Secton II formulates the recursve source localzaton problem wth moble sensors and ntroduces the symbols and notatons used Secton III presents the proposed two recursve source localzaton algorthms Secton IV contans the smulaton results to demonstrate the performance of the two algorthms Secton V s the concluson Throughout ths paper, the transpose and nverse of matrx X R m n are denoted by X and X respectvely, and X ( ) means that X R n n s symmetrc and postve semdefnte (postve defnte) The symbols I and represent the dentty matrx and zero matrx wth approprate dmenson, means the Eucldean norm, and E( ) means the mathematcal expectaton II PROBLEM FORMULATION The source localzaton scenaro s 3-D a pont source at unnown poston u = [x, y, z] radates a sgnal to a networ of n moble passve sensors At each samplng step, the TDOAs of the receved sgnals wth respect to the sgnal at a reference sensor are transmtted to the central processor Wthout loss of generalty, let the frst sensor be the reference The true poston of the th sensor at samplng tme s denoted as s = [x, y, z ], =,, n The TDOA measurement model between sensor par and at step s gven by t s the true TDOA, and t = t + t, () t = t t () The TDOA nose vector t = [ t,, t n] s zeromean Gaussan nose wth covarance Q t After multplyng by the propagaton speed c, we have the range dfference of arrval (RDOA) measurement r = r + c t, (3) r = ct, r = ct The true RDOA measurement r s r = r r, (4) r ( =,, n) s the dstance between the source to the true poston of sensor, e, r = u s (5) In the sequel of ths paper, TDOA and RDOA wll be used nterchangeably The collecton of all RDOA measurements at samplng tme s denoted by an (n ) vector as r = [r, r 3,, r n] = r + r, the RDOA error vector r = c t s zero-mean Gaussan nose wth covarance c Q t The postons of the sensors change dynamcally at each step to collect multple TDOA measurements over tme However, at samplng step, the true sensor postons s are not avalable but only nosy versons of them are nown, whch are denoted by s = [x, y, z ], s = s + s, (6) wth s beng the random uncertanty n s The dstance between the source and the avalable poston of th sensor s denoted as r, e, r = u s (7) We collect the avalable sensor postons as a vector s = s + s, s = [s, s,, s n ] and the correspondng uncertanty vector s = [ s, s,, s n ] s zero-mean Gaussan wth covarance matrx Q s As n our companon paper [], we also adopt the assumpton that the sensor poston uncertanty s s ndependent of the TDOA nose t n ths paper for ease of llustraton In addton, we gnore synchronzaton, quantzaton and delays of data n ths wor The objectve of ths paper s to answer the second queston proposed n companon paper [], e, for a moble sensor networ, gven the nosy TDOA measurements r together wth the nosy sensor postons s n each samplng tme step =,,, how to fuse these nformaton effcently to estmate the source locaton u? III RECURSIVE LOCALIZATION ALGORITHMS FOR MOBILE SENSOR NETWORK In ths secton, we shall extend the CWLS source localzaton method for statc sensor networs proposed n [] to moble sensor networs A Revew of CWLS source localzaton The CWLS source localzaton has two stages For ease of notaton, we smply drop the tme step n r, s and ther correspondng components n ths subsecton Frst Stage: Reorganze the nonlnear TDOA equatons nto a set of lnear equatons Ths process s acheved by squarng both sdes of the RDOA equatons r + r = r and ntroducng an auxlary varable r that depends on the source poston The RDOA measurement equatons can be smplfed as: r + R = (s s ) T u r r, (8) R x = + y + z, and =,, n Because only the nosy values of r and s are avalable, we express them as r = r c t and s = s s, =,, n Usng the Taylor-seres expanson to expand r around the nosy sensor poston s up to lnear error term, we have r r + g T u,s s 55

3 r = u s and g u,s = u s u s By defnng the unnown vector as u = [x, y, z, r ], the set of equatons (8) can be rewrtten as h = ɛ = h Gu (9) r + R rn n + R (s s ) T r G = (s n s ) T r n R = x + y + z From the defnton of the equaton error vector ɛ n (9), t can be wrtten n terms of t and s as ɛ = cb t + c t t + D s + s s () cb t + D s () the second order error terms have been gnored and represents the element by element multplcaton The matrx B s gven by r r3 B = () rn and D s shown at the bottom of the next page n (3) Second Stage: Incorporate the relatonshp between u and r as a second order equalty constrant n the weghted LS estmaton strategy, whch can be formulated as the followng optmzaton problem: mn (h Gu ) W(h Gu ) st (u s ) C(u s ) = (4) C = dag([,,, ]), s = [x, y, z, ] and W s the weghtng matrx defned as W = E[ɛɛ ] (5) Although the CWLS source localzaton problem (4) s actually a nonconvex quadratcally constraned quadratc programmng We have nvestgated the hdden convexty of the optmzaton problem (4) n our companon paper [] and the global optmal source locaton estmate u can be effcently obtaned Meanwhle, as a byproduct, an estmaton of error covarance P of the CWLS localzaton s gven by: P = V (V G WGV) V, (6) ([ ]) V = dag u () x, u () y, u (3) z, ˆr = u () x u V = () y u (3) z ˆr ˆr ˆr (u () x ) + (u () y ) + (u (3) z ) B The recursve localzaton algorthm Assume at the every samplng tme step =,,, we can get all the nosy TDOA measurements r together wth the nosy sensor postons s of the sensor networ At the frst samplng tme step =, we can use the CWLS source localzaton method to get an ntal estmate of the unnown source poston and ts estmaton error covarance, whch s denoted as û and P However, because the followng TDOA measurements r, ( =, 3, ) are nonlnear wth respect to u, t s mpossble to use them drectly to update source localzaton In ths recursve source localzaton algorthm, we frstly use the current TDOA measurements r and nosy sensor postons s to estmate the unnown source poston u, and then treat t as a lnear measurement on u, so the optmal recursve WLS algorthm s applcable The followng Algorthm summarzes the recursve localzaton algorthm Algorthm : Recursve localzaton algorthm ) Let =, and calculate the ntal localzaton û and the ntal localzaton error covarance P ) Set = +, and calculate the current localzaton denoted as û and ts error covarance denoted as P 3) Treat the current locaton estmate û as a measurement y of the unnown source poston u at tme step : y = u + u, the covarance of measurement nose u s P 4) Update the source localzaton and ts error covarance as follows: Go bac to step ) û = û + K(y û ) K = P (P + P ) P = (I K)P In step ) of Algorthm, we can use the CWLS source localzaton method to get the current localzaton û and ts error covarance P However, ths needs to solve a convex optmzaton problem n each samplng tme step, whch has a large computatonal load Instead of ths, we can also neglect the constrant and drectly use WLS method to get the current localzaton, whch has a closed-form soluton [] and can be calculated much faster Our extensve smulaton results ndcate that the localzaton accuracy s relatvely nsenstve 56

4 to gnore the sde nformaton between source poston and the auxlary varable n the recursve localzaton algorthm C An mproved recursve localzaton algorthm In the mproved recursve localzaton algorthm, the ntal step = s the same as n Algorthm At samplng tme step, nstead of treatng the current localzaton based on only the current TDOA measurements as a new measurement of u, we use the current TDOA measurements to estmate the auxlary varable r usng WLS method frst, and then substtute the estmaton nto the TDOA equaton to rearrange t as a lnear measurement of the unnown source poston u whch can be used to update the source localzaton More specfcally, we rearrange of the nonlnear RDOA measurement equatons nto lnear equatons (8) at samplng tme step wth respect to u and r as r + R = (s s ) T u r r (7) R = x + y + z, ( =,, n) In the presence of TDOA measurement nose n r, the sensor poston errors s and the frst-order approxmaton of r, the equaton (7) could be vewed as a lnear measurement equaton on unnown parameter u = [x, y, z, r] as n (9) of CWLS source localzaton method: r + R = (s s ) T u rr + ɛ, (8) R = x + y + z and ɛ s the measurement error We can apply the WLS estmaton method to get an estmate of the auxlary varable r denoted as ˆr, the estmaton error s denoted as r = ˆr r and the estmaton error covarance s Qr Now we substtute ˆr nto (8) and rearrange the lner equaton only wth respect to u as: r + R + r ˆr = (s s ) T u + ɛ (9) The correspondng lnear vector equaton s h = r r n ɛ = h G u () + R + r ˆr n + R + r n ˆr (s s ) T G = (s n s ) T ɛ = [ ɛ,, ɛ n ] The measurement nose vector ɛ can be wrtten n terms of t and s as ɛ = cb t + c t t + D s + s s + E r + c t r cb t + D s + E r, r E r3 = () rn B and D are smlar wth B and D n () and (3) Generally, r s correlated wth t and s, here we neglect ths correlaton and the correspondng covarance of measurement nose ɛ can be approxmated as Q e = c B Q t B D Q sd + Qr E E () The resulted measurement equaton () s a lnear model only wth respect to the unnown parameter u, so the Kalman flterng can be appled to recursvely update the source localzaton The followng Algorthm summarzes the mproved recursve localzaton algorthm Algorthm : Improved recursve localzaton algorthm ) Let =, and calculate the ntal localzaton û and the ntal localzaton error covarance P ) Set = +, and use the WLS method to calculate the estmate of current auxlary varable r as ˆr and ts error covarance denoted as Qr wth current TDOA measurements 3) Rearrange the nonlnear TDOA measurement equatons nto a set of lnear equatons only wth respect to the unnown source poston u as n (9)-() 4) Update the source localzaton and ts error covarance wth Kalman flterng as follows: û = û + K(h G û ) K = P G T (G P G T + Q e) Go bac to step ) P = (I KG )P Remar : It s noted that the measurement equaton () s an approxmaton of the actual TDOA measurement equaton rg u,s (u s ) (u s ) r3g u,s (u s ) (u s 3 ) T D = rng u,s (u s ) (u s n ) (3) 57

5 Meanwhle, n comparson wth the standard Kalman flterng, Q e n () s not the true covarance of measurement nose ɛ, so generally the update n step 4) of the mproved recursve localzaton algorthm s not the optmal update wth all avalable TDOA and sensor poston nformaton However, ths s an effcent sub-optmal recursve algorthm to overcome the nonlnearty of TDOA measurement equaton and smulaton results ndcate that the performance degradaton due to the approxmaton of Q e s nsgnfcant Remar : Both the two localzaton algorthms utlze the reorganzaton dea to rearrange the TDOA equatons to deal wth the nonlnearty of TDOA at each samplng tme The man dfference s that n the mproved recursve localzaton algorthm, the auxlary varable s estmated n advance before usng the TDOA equatons to update the source localzaton, but n the recursve localzaton algorthm, the auxlary varable s regarded as a completely unnown varable n the update of source localzaton Intutvely, the mproved recursve localzaton algorthm should have better performance snce the measurement equaton (9) maes use of more nformaton on the auxlary varable n each update of source localzaton, ncludng the estmate of the auxlary varable as well as ts estmaton error covarance Ths s valdated n the followng smulatons IV NUMERICAL SIMULATIONS Ths secton contans smulaton results of the proposed two recursve source localzaton algorthms The smulaton scenaro contans n = 8 moble sensors, and ther nomnal postons at samplng tme step = are gven n Table I The sensor poston noses at dfferent coordnates and for dfferent recevers are assumed to be ndependent Gaussan noses wth varance σ s, e Q s = σ s I, I s the 3n 3n dentty matrx We consder that the emtter source s far away from the sensor networ and s located at [, 5, ] m The eght sensors move towards the target wth a step sze of [4,, ]m for each step The TDOA measurements are obtaned by addng Gaussan nose wth covarance matrx Q t to the true values, Q t = σ t T and T s the (n ) (n ) matrx wth n the dagonal elements and 5 otherwse The TDOA nose and sensor poston nose are ndependent TABLE I NOMINAL POSITIONS (IN METERS) OF SENSORS sensor no x y z We mplement the proposed recursve localzaton algorthm and mproved recursve algorthm followng the steps as descrbed n the Algorthm and Algorthm The localzaton accuracy s evaluated by the average range error (ARE) and the standard devaton (SD) of localzaton error whch are defned as L ARE(u) = û l u /L, SD(u) = L û l u /L, l= û l denotes the unnown source poston estmate at ensemble l and L = s the number of ensemble runs Fg shows the localzaton accuracy of the proposed two recursve localzaton algorthms for samplng tme steps when σ t = mcro-second and σ s = / 3 m For comparson purpose, the performance of the recursve localzaton n Algorthm s shown n dashed lne, and that of the mproved recursve localzaton n Algorthm s shown n sold lne It s evdent from the fgure that both the two algorthms can sgnfcantly mprove the locaton accuracy of the ntal localzaton Meanwhle, both the ARE and SD of the mproved recursve algorthm s smaller than that of the recursve algorthm Fg s the correspondng result for 3 samplng tme steps when σ t = mcro-second and σ s = / 3 m As expected, the localzaton accuracy s generally worse n the case of a hgher nose level However, the proposed recursve localzaton algorthms can sgnfcantly mprove the accuracy, especally n the frst a few steps Also, the performance of the mproved recursve localzaton algorthm s better than that of the recursve algorthm V CONCLUSION In ths wor, we have provded two recursve source localzaton algorthms by usng TDOA measurements receved from moble sensor networ Compared wth centralzed source localzaton wth statc sensors, the recursve localzaton algorthms can sgnfcantly mprove the locaton accuracy by updatng the localzaton at each samplng step Smulaton results showed that the mproved recursve localzaton algorthm gves better localzaton performance VI ACKNOWLEDGEMENT The authors are grateful to the comments and suggestons from Dr Ng Gee Wah, Mr Ng Hang Sun, Mr Zhang Xng Hu and Mss Tan SHong Sharon REFERENCES l= [] X Qu and L Xe, Source localzaton by tdoa wth random sensor poston errors-part : Immoble sensors, submtted to 4th Internatonal Conference on Informaton Fuson, [] A Fnn, K Brown, and T Lndsay, Mnature uav s & future electronc warfare, Proceedng of the land warfare conference, pp 93 6, [3] N Oello and D Musc, Emtter geolocaton wth two uavs, Proceedng of Informaton Decson and Control, 7 IDC 7, pp 54 59, 7 [4] D Musc, R Kaune, and W Koch, Moble emtter geolocaton and tracng usng tdoa and fdoa measurements, IEEE Transactons on Sgnal Processng, vol 58, no 3, pp , [5] F Fletcher, B Rstc, and D Musc, Recursve estmaton of emtter locaton usng tdoa measurements from two uavs, th Internatonal Conference on Informaton Fuson, pp 8, 7 58

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