Error Analysis of Sensor Geometric Factor for the Multi-node Cooperative Localization Accuracy

Transcription

1 INERNAIONAL JOURNAL OF CIRCUIS, SYSEMS AND SIGNAL PROCESSING Volume, 7 Error Analss of Sensor Geometrc Factor for the Mult-node Cooperatve Localzaton Accurac Yao Fan Abstract he localzaton accurac s ver mportant for the mult-node cooperatve localzaton sstem. In the localzaton sstem, the geometrcal factor has a great nfluence on the localzaton result. In order to solve ths problem, the error analss of the sensor geometr factor for mult-node cooperatve localzaton accurac s researched n ths paper. B usng the total dfferental method to analze the tme dfference equaton, the ntrnsc relatons of the localzaton error, the measurement error of sgnal arrval tme, the localzaton error of measurement platform and the sensors laout s determned, the localzaton accurac dstrbuton of dfferent sensors laout s analzed through the smulaton experments. Experments show that the method n ths paper provdes an mportant theoretcal bass for mprovng the mult-node cooperatve localzaton accurac. Kewords geometrc factor, mult-node cooperatve, total dfferental, localzaton accurac. I. INRODUCION HE mult-node cooperatve localzaton s the extenson of the sngle sensor node localzaton[], t can expand the scope of the montorng area and mprove the recognton capablt for the vbraton source at the long dstance. he technolog has been wdel used not onl n brdge montorng, envronmental montorng, regonal montorng, but also n mltar survellance, battlefeld detecton and so on []. he localzaton accurac manl s related to the measurement factor and the geometrc factor [3-4]. In recent ears, due to the contnuous mprovement of measurement technolog, the nfluence of the measurement factor to localzaton accurac was b constantl decreased [5], however the research of the sensor geometrc factor for the mult-node cooperatve localzaton accurac s less. It plas a vtal role for achevng the more accurate localzaton n mult-node cooperatve sstem how to choose the sensors laout. hs paper researches the nfluence of the sensor geometr factor on mult-node cooperatve localzaton accurac. B usng the total dfferental method to analze the tme hs work was supported b Natural Scence Foundaton of Chna (NO. 6768), bet Program for the Young eacher n Unverst (No. QCZ6-8), Foundaton of Xzang Mnzu Unverst (No. 6MYQP6). Yao Fan s wth the College of Informaton Engneerng, Xzang Mnzu Unverst, Xan Yang 78, Shaanx, Chna. (correspondng author; e-mal: fanao4983@63.com). dfference equaton, the ntrnsc relatons of the localzaton error, the measurement error of sgnal arrval tme, the poston error of measurng platform and the sensors laout s determned, the localzaton accurac dstrbuton of the dfferent sensor laout s determned through the smulaton experments. II. IME DIFFERENCE OF ARRIVAL PRINCIPLE Each node uses three (or more) known coordnates of vbraton sensor to samplng and collect the vbraton source n mult-node cooperatve localzaton sstem. he tme dfference value of the vbraton source and the dfferent sensors s obtaned accordng to the prncple of tme dela, and the vbraton source s on the two groups of hperbolc lne whch three sensors (or more) as the focus, the ntersecton of two groups of hperbolc lne s the actual locaton of the vbraton source accordng to the hperbola prncple n the mathematcs [6]. ake three sensors as an example and t s done the tme dfference locaton b usng the hperbolc theor, ts orentaton dagram as shown n fgure. S ( x, ) (, ) S x β β (, ) S x β ( x, ) Fg.. he sensor laout of tme dfference localzaton Fgure ncludes the prmar sensor S( x, ), the two secondar sensors S( x, ) ( x, ), β, β, β respectvel are the azmuthal angle of each sensor and the target, assumng the locaton of the measured target s x (, ), R, R, R respectvel are the dstance between the vbraton source and the sensor S ( x, ), S( x, ), S( x, ), the dstance expressons s: x ISSN:

2 INERNAIONAL JOURNAL OF CIRCUIS, SYSEMS AND SIGNAL PROCESSING Volume, 7 R x x R x x R = x x + = + = + he tme dfference and can be obtaned b DOA measurement algorthm, s the tme dfference of S, s the tme dfference of S, equaton () can be converted nto equaton (). R R ( x x ) ( ) ( x x ) ( ) v R R = ( x x ) + ( ) ( x x ) + ( ) = v = + + = o solve the bnar quadratc equatons, the locaton of the target ( x, ) can be calculated. III. INFLUENCE ANALYSIS OF HE SENSOR GEOMERIC FACOR ON LOCALIZAION ACCURACY A. he sensor geometrc factor he nfluence of the geometrc factor on the target localzaton accurac s descrbed b usng the Geometrcal Dluton of Precson (GDOP)[7], t s an mportant ndex whch t s used to measure the sze of the postonng error. he geometrc factor ncludes these factors, for example, the dstance between the target and each sensor, the sensor laout and the measurement error. GDOP expresson s: x () () GDOP = σ + σ (3) Among formula (3), σ x and σ respectvel are the localzaton error whch the localzaton result on the x axs and axs. B. otal dfferental localzaton accurac algorthm In the passve tme dfference localzaton, the localzaton sstem needs more sensors to partcpate n the coordnaton localzaton, t s objectvel caused the dverst of error sources and the complext of sstem buldng [8], the geometrc localzaton accurac s obtaned through solvng the localzaton dfferental equatons, t can be analzed that the nfluence of the sensor laout and the measurement error on the localzaton result. Accordng to the three sensors localzaton prncple [9-] n the tme dfference localzaton, formula () can be changed nto the followng forms: = + + = + + ( x x ) ( ) ( x x ) ( ) ( x x ) ( ) ( x x ) ( ) (4) target s ( x, ), R, R and R respectvel are the dstance between the vbraton source and the sensor S ( x, ), S ( x, ), S ( x, ), and respectvel are the dstance dfferences between S, S. Solvng the partal dfferental for x, n formula (4), t can get the under formula: xx xx dx d = ( x x ) + ( ) ( x x ) + ( ) + d ( x x ) + ( ) ( x x ) + ( ) xx xx d = dx ( x x ) + ( ) ( x x ) + ( ) + d ( x x ) + ( ) ( x x ) + ( ) Among formula (5), the localzaton error components of the target are dx and d, and the measurement errors of tme dfference are d and d. he fgure shows that β s the azmuth angle of the prmar sensor and the target source, β s the azmuth angle of the secondar sensor and the target source. he azmuth angle formula (6) s concluded. cos β = sn β = x x ( x x ) + ( ) ( x x ) + ( ) In order to facltate to calculate for formula (5), formula (5) s represented b the product of matrx, and the azmuth angle formula (6) s brought nto formula (5), then t can be smplfed nto the formula (7). d cos β cos β sn β sn β dx = d cos β cos β sn β sn β d Set = [ dx d], dz = [ d d ], cos β cos β sn β sn β H = cos β cos β sn β sn β, the formula (7) can be represented as formula (8). hen dz (5) (6) (7) = H (8) H dz = (9) Formula () can be obtaned accordng to the nverse matrx calculatng method of the second order matrx: Among formula (4), the prmar sensor s S ( x, ), two secondar sensors are S ( x, ) ( x, ), the measured ISSN:

3 INERNAIONAL JOURNAL OF CIRCUIS, SYSEMS AND SIGNAL PROCESSING Volume, 7 H β + β β β β + β β β cos sn cos sn = H + β β + β β sn sn sn sn he H can be represented as: H β β β = 4sn sn sn () he formula (9) can be transformed nto formula (). + β + β cos cos dx d β β + β β sn sn sn sn = + β + β sn sn d d β β β β sn sn sn sn Set + β cos M = β β sn sn + β sn P = β β sn sn + β N = + β β sn sn, cos, sn, + β Q = β β sn sn () () Formula () can be abbrevated to: MN = dz (3) P Q Accordng to the covarance matrx of poston error, there s: σx σσ x P = = E σσ x σ (4) Put formula (3) to brng nto formula (4), t can be abbrevated to: P { E dzdz } MN MN = P Q P Q In formula (5), E dzdz = dag { σ, σ }, σ are the measurement error of dstance dfference, the (5) σ and covarance matrx P s obtaned through the matrx rule, t s as follows: P M σ + N σ MPσ + NQσ = MPσ + NQσ P σ Q σ + (6) In D plane, GDOP = trace[ P ], t s the root of the sum of squares for localzaton error. So the geometrcal dluton of precson of the localzaton error s: σ σ GDOP = + β β β β 4sn sn 4sn sn (8) hrough the analss of the formula (8), the GDOP value of localzaton results s greater, the localzaton error s the greater and the accurac s lower. At the same tme, the nfluence of the dfferent target azmuth on the fnal measurement error also s ver bg,when β = β, β = β, β = β, the azmuth angle of each sensor and the target source are all same, the localzaton error s nfnt, t wll not be able to locate at ths tme. At the same tme as we can see that the measurement error of the dstance between each sensor and the dstance sze between the sensors all can affect the accurac of measurement IV. SIMULAION EXPERIMEN B the analss of the total dfferental postonng accurac algorthm, the factors, nclude the geometrc laout the target source locaton and the tme dfference measurement error, all nfluence the localzaton accurac of the postonng sstem. In order to determne the nner relatonshp of the geometrc accurac and the ste measurement error, the target azmuth angle and the sensor laout, the smulaton experment was done to calculate the GDOP value. o evaluate the performance of the algorthm, the smulaton experment was done b usng the total dfferental postonng accurac algorthm. A. Lnear sensor laout smulaton he smulaton analss of the geometrc accurac for lnear sensor arra was done to research the relatonshp of the localzaton accurac of sensor and the sensor laout. Select three sensors ( x, ), ( x, ), ( x, ), ts coordnates respectvel were (,), (, ), (,), the vbraton source coordnates was ( x,, ) the measurement error of each sensor ste s.5 m, the standard devaton of tme measurement error are equal, the all are. ms, the propagaton speed of wave on the surface s 3 m/s. Brought the parameters nto GDOP geometrc accurac formula (8) to do the smulaton, t can get the three-dmensonal mage of the source coordnate ( x, ) takng contnuous values n a fxed nterval, t as shown n fgure. GDOP = M σ + N σ + P σ + Q σ (7) Put M, N, P, Q to brng nto formula (7), t can be abbrevated to the formula (8). Fg.. Geometrc accurac dstrbuton ISSN:

4 INERNAIONAL JOURNAL OF CIRCUIS, SYSEMS AND SIGNAL PROCESSING Volume, 7 Fg.3. Precson error dstrbuton ransverse projecton was done for secton n the Z sem-axs, we made the judgment for the error value whch s obtaned b GDOP geometrc precson formula, the Z axs values s the threshold that the sstem set, t s unable to locate when GDOP value s greater than the threshold, the GDOP dstrbuton s shown n fgure 3. he fgure 3 shows that the localzaton accurac was the hghest for lnear sensor laout n the mdlne vertcal drecton, the accurac error was ver bg hghest on the extenson poston of the three sensors, so the lnear sensor arra was not able to adopted, t was unable to locate when the target was located on the sensor attachment and extenson poston, t should choose to the sensor laout whch the azmuth angle between target and sensor was larger. B. Four knds of sensors laout smulaton Common sensor ste laout ncludng Y laout, the damond laout,the ladder laout and the parallelogram laout, ths paper respectvel smulated for the four knds of tpcal sensor laout. he locaton of the four sensor laout coordnates as shown n table. he accurac error dstrbuton of the four sensor laout was as shown n fgure 4. Fgure 4(a), fgure 4(b), fgure 4(c) and fgure 4(d) respectvel were on behalf of the accurac error dstrbuton of the Y laout, the damond laout the ladder laout and the parallelogram laout. ab.. Four knds of sensor laout coordnate Laout Damond Ladder Parallelogram Y laout laout laout laout Coordnate (x, ) (,) (,) (,) (,) (x, ) (,5) (,) (,) (5,) (x, ) (-,5) (5,5) (,6) (3,5) (x 3, 3 ) (,-) (5,-5) (8,6) (8,5) (a) (b) (c) Fg.4. he accurac error dstrbuton of four sensor laout hrough the above ou can see, the accurac error dstrbuton of Y laout was more unform, t was the crcular dstrbuton of ( x, ) as center, the accurac ncreased when the dstance between the target source and the prmar sensor was bgger; the accurac was hgh n ± 5 area around axs for the damond laout and the localzaton error was bgger n other area; the postonng error was bgger n ± 5 area of the x axs and axs for the ladder laout and the accurac was hgh n other area; the accurac was hgh n ± 4 area of the target source and the man dagonal for the parallelogram laout and the accurac was low n ± 3 area of the target source and the other dagonal. In the mult-node cooperatve localzaton sstem, the localzaton accurac dstrbuton of Y laout was more unform and the accurac was hgh; the accurac dstrbuton of dfferent drect was nonunform when the sensor laout was the damond arra, the ladder arra or the parallelogram preach arra, but the postonng accurac was hgher than Y arra n a fxed drecton. herefore, t should be used Y arra when the target drecton was uncertant; t should be choose the sensor laout whch the postonng accurac s hghest when the target drecton s certant accordng to the target drecton and (d) ISSN:

5 INERNAIONAL JOURNAL OF CIRCUIS, SYSEMS AND SIGNAL PROCESSING Volume, 7 the regonal envronment V. CONCLUSION hs paper research the nfluence of the sensor geometr factor on mult-node cooperatve localzaton accurac. B usng the total dfferental method to analze the tme dfference equaton, these factors, nclude the sensor laout, the target source locaton, the azmuth angle of sensors and the target, can produce an effect on the localzaton accurac. hs paper respectvel smulate for the fve knds of tpcal sensor laout ncludng the lnear laout, the Y laout, the damond laout, the ladder laout and the parallelogram laout. Smulaton experments prove that: ) he lnear laout s unable to locate when the target was located on the sensor attachment and extenson poston, t should choose to the sensor laout whch the azmuth angle between target and sensor was larger. ) It should be used the Y arra when the target drecton s uncertant; t should be choose the sensor laout whch the postonng accurac s hghest when the target drecton s certant accordng to the target drecton and the regonal envronment. REFERENCES [] Ganesh Lavet, G. Sasbhushana Rao, D.eswara Chatana and M.N.V.S.S.Kumar, DOA Measurement Based GDOP Analss for Rado Source Localzaton, Proceda Computer Scence, vol. 85, no.3, pp , 6. [] Y. Wang and K.C. Ho, DOA source localzaton n the presence of snchronzaton clock bas and sensor poton erors, IEEE ransactons on Sgnal Processng, vol. 6, no. 8, pp , 3. [3] Yanl Chu, Luao He and Fan Yao, An mproved localzaton algorthm of talor seres expanson search targe, OPIK, vol. 7, no.9, pp , 6. [4] Y.. Chan and K. C. Ho, A smple and effcent estmator for hperbolc locaton, IEEE ransactons on Sgnal Processng, vol. 4, no. 8, pp , 3. [5] Feng Lje and Fan Yao, alor s seres expanson search vbrator source localzaton algorthm based on the gross error gra dscrmnant, Journal of Northwest Normal Unverst (Natural Scence). vol. 4, no., pp , 4 [6] Bn Xu, Guodong Sun, Ran Yu and Zheng Yang, Hgh-Accurac DOA-Based Localzaton wthout me Snchronzaton, IEEE ransactons on Parallel and Dstrbuted Sstems, vol. 4, no.8, pp , 3. [7] Chu Yanl, He Yuao and Fan Yao. Multple crcle lnkage s search target localzaton of the vbraton source algorthm, OPIK, 7, vol. 3, no.4, pp. 7 4, 7. [8] Shaddan SAA oltanzadeh H, Path plannng for two unmanned aeral vehcles n passve localzaton of rado sources, Aerospace Scence and echnolog, vol. 58, no.4, pp , 6. [9] J Gebbe, M Sderus and JS Allen III, A two-hdrophone range and bearng localzaton algorthm wth performance analss, Journal of the Acoustcal Socet of Amerca, vol.37, no.3, pp , 5. [] Ian Sharp and Ke gen Yu, GDOP Analss for Postonng Sstem Desgn, IEEE ransactons on Vehcular echnolog, vol., no., pp ,. Yao Fan was born on Nov. 4, 983. He receved the PhD degree n computer scence and technolog from Changan Unverst of Chna. Currentl, he s a researcher (assstant professor) at Xzang Mnzu Unverst, Chna. Her major research nterests nclude sgnal and nformaton processng. ISSN: