Internatonal Conference on Intellgent Systems Research an Mechatroncs Engneerng (ISRME 5) The Research an Realzaton of A Localzaton Algorthm n WSN Base on Multmensonal Scalng L Xang, Qanzh Hong, Luxao Hu College of Communcaton Engneerng, Jln Unversty, Changchun,Chna e-mal:lx4487@63.com,r.qzh@63.com, 6658585@qq.com Keywors: multmensonal scalng wreless sensor networs sensor localzaton TDOA Abstract. A new locatng algorthm NMDS-TDOA s propose, whch attempts to mprove the performance of noe localzaton n wreless sensor networs. As the rans of the noes base on TDOA shoul be monotonc wth ther rans base on true stance, we can combne TDOA an MDS-MAP. As a result, postonng error wll be reuce. The process of the computaton about turnng noes relatve coornates nto absolute coornates s ntrouce explctly. NMDS-TDOA realzes both less complexty an relatvely accurate postonng only base on few anchors, even n the stuaton of lacng nformaton on the stance between noes. Introucton In recent years, wreless sensor technology evelops raply. It s wely use n areas such as mltary, transportaton, envronment an nustral proucton, to fnsh the measurement of many physcal quanttes such as temeperature, humty, pressure an spee. For most of the measure quanttes, people not only nee to master the accurate measurement values, also nee to now the area or locaton of the measure quanttes. However, sensor noes of WSN are generally batch eployment, such as aeral seeng, an the structure an functon of each noe are exactly the same. So we can t separate each noe entfcaton. Therefore, to obtan the locaton of measure values, the sensor noes must be able to clear ther own locatons. The locaton nformaton of the sensor noes can also help to mprove the qualty of WSN routng effcency, coveragng qualty, realzes the networ loa balancng an snce topology confguraton,etc. How to get the noe locaton nformaton problem calle postonng problem of sensor noes. Now ths problem has become one [ ] of the hot ssues n the fel of WSN. MDS-MAP algorthm was propose by Shang, then [3 4] many more mprove algorthms were propose. In vew of the sensor noe postonng problem of WSN, ths paper proposes a new noe localzanton algorthm, an gves the specfc algorthm of how to turn noes relatve coornates nto absolute coornates. TDOA sgnal s use n ths propose metho to locate, so that the caculaton error an computatons brought by convertng sgnal strengh values nto stance values n prevous localzatong algorthms wll be reuce. NMDS-TDOA s prove a goo postonng result an certan robustness through the analyss of combnaton of the theory an practce. All algrthms ntrouce below are scusse n -mensonal space. Multmensonal Scalng(MDS) The multmensonal scalng(mds), a technque usually use for the analyss of exploratory ata analyss or nformaton vsualzaton, was frst use n ata analyss of psychometrcs. Now MDS s wely use n many fels as a general ata analyss technology. MDS uses versty between enttes to compose relatve coornate n mult-mensonal space. The more smlar two enttes, the shorter stance between the ponts whch the enttes correspon to n the space. The man avantage n usng MDS for locaton estmaton s that t can generates accurate locaton estmaton even base on few anchor noes. There are numerous varetes of MDS. We focus on classcal MDS an nonmetrc MDS. 5. The authors - Publshe by Atlants Press 8
Classcal Multmensonal Scalng In classcal multmensonal scalng, we assume p represents versty between entty an entty j. The versty matrx P s compose wth p whch s the euclcan stance between noes. The basc ea s to reconstute the coornates whch the noes coornate to n mult-mensonal space by usng matrx P, n orer to mnmze stress coeffcent. STRESS = ( f( p ) ) () If all parwse stances of sensors n P are collecte, we can use classcal multmensonal scalng algorthm to estmate the locatons of sensors: ) Compute the matrx of square stance P ) Compute the matrx J wth J = E n I, where E s entty matrx,an I s a full one square matrx () 3) Apply ouble centerng to ths matrx wth B= JP J.Double centre means every element n the matrx mnus the row mean value an the column mean value,then plus the mean value of the matrx, fnally multply by -/ 4) B s ecompose by sngular value ecomposton, so that we can get egenvalues λ, λ, λm whch compose agonal matrx Λ, an egenvectors e, e, em whch compose / matrx V. The coornate matrx of classcal scalng s X = V Λ. Nonmetrc Multmensonal Scalng The relaton between versty an stance s not requeste as strct as classcal multmensonal scalng n nonmetrc multmensonal scalng. It s only ase to meet the monotonc relaton, not quanttatvely presente. To entty, j, u, v, f there s p, then ( X) < uv ( X). It s a repettve teraton process to reconstute the coornates an stance base on versty between the enttes n nonmetrc MDS [-3] : ) Intalzaton phase. Calculate Euclean stance between enttes by ranom ntal coornate X. An each ntal coornate must be fferent from another ) A mean calle spartes s generate whch s expresse by. meet monotonc relaton, whch means to, j, u, v, f there s p < puv, there s an uv use PAV(pool-ajacent volators) algorthm to get : a) The enttes are sorte from small to large base on p b) Compare stance between enttes untl oes not meet monotonc relaton c) Raplace wth the average value of ) Deal wth the next p, then repeat b an c untl all 3) Change X to X an calculate Euclean stance are one an get all. p must. We coul Caculate an n the loop nteraton compose by the above 3 steps untl STRESS satsfes some certan request., j, j () STRESS = ( ) / 8
The NMDS-TDOA Algorthm NMDS-TDOA propose n ths paper meets the request of nonmetrc MDS,whch s the larger stance value between noes, the larger TDOA value we get. The etale process s as follows: Input: TDOA values between noes. Outpt: Absolute coornate of the noes n the networ. Step. Compose [s ] wth TDOA values Step. Get [ p ] by ealng [s ] wth Dstra algorthm, the shortest path algorthm Step3. Deal [ p ] wth nonmetrc MDS to get the relatve coornates: a) Gve noes ntal coornate ( x, y ) b) Caculate the Euclean stance c) Deal [ p ] an [ ] = ( x x ) + ( y y ) j j wth PAV algorthm to get [ ]: To envty, j, u, v, f p an >, = uv = ( + uv )/ uv uv If p an < uv, =, = uv ) =+, caculate the new coornates ( x, y ): α x = x + ( x ) j x n j M, j, (3) α y = y + ( y ) j y n j M, j, (4) n s the number of the noes, α s the teraton step, α = e) Upate the Euclean stance wth the new coornates -4 f) Caculate STRESS. If STRESS < ε, en the loop, else bac to c. ε = Step4. Turn relatve coornates nto absolute coornates base on the coornates of the anchors: The algnment usually nclues shft, rotaton, an reflecton of coornates. R = [ r ] n = ( R, R,, Rn ) enotes the relatve locatons of the set of n sensor noes. T = [t ] = ( T, T,, T ) enotes the true locatons of n sensor noes. In followng explanaton, n n we assume the noe,, 3 are anchors. A vector R may be shfte to R by R = R + X, where X = R R. It may be rotate counterclocwse through an angle α to R = QR,where cos( α) sn( α) Q = sn( α) cos( α). (5) It may also be reflecte across a lne cos( β / ) S = sn( β / ) (6) 3 to R = QR,where cos( β) sn( β) Q = sn( β) cos( β). (7) 83
Test results In our experments, we run NMDS-TDOA on varous topologes of networs n Matlab. The noes are place both ranomly an on a square gr. Locaton error coul be caculate by formula (8). n Xest Xreal = m+ error = ( n mr ) % (8) Ranom Placement In ths set of experments, 5 noes are place ranomly n a square.fgure shows an example usng a rao range of 3, whch leas to an average connectvty of.56. The green lnes represent the connectvty between neghbor noes. Fgure shows the fnal result of NMDS-TDOA whch s transforme base on 3 anchor noes, enote by re stars n the networ. The re lne represent the fference between the estmate poston an the true poston. The longer the lne, the larger the error.the average estmaton error n ths example s about 4.3%. Fg. True coornates of the noes Fg. Fnal poston estmaton by NMDS-TDOA Gr Placement In ths set of experments, we assume that the sensor noes are place as a square.we coul a ranom placement error to these noes. 49 noes are place on a 7 7 gr. Fgure 3, fgure 4 an fgure 5 show an example of no placement error. The rao range s R=3, whch leas to connectvty 7.6. Green lnes also represent the connectons between neghbors. Fgure 3 shows the true coornates of the noes. Fgure 4 shows the relatve coornates of NMDS-TDOA. Fgure 5 shows the fnal result of NMDS-TDOA. The average estmaton error n ths example s about.3%. Fg.3 True coornates of the noes n gr placement Fg.4 Relatve coornates of NMDS-TDOA 84
Fg.5 Absolute coornates of NMDS-TDOA Fgure 6 shows the average performance of NMDS-TDOA as a functon of connectvty. An [ 3] NMDS-TDOA s compare wth MDS-MAP whch only uses connectvty nformaton at the same tme. The rao ranges are,.5, 3, 3.5, 4, 4.5. From the fgure, we can see that the errors wth 3 anchors are qute goo when the connectvty level s. or greater. The result of gr placement n ths fgure s got wth 5% placement error. Concluson Fg.6 Average localzaton error as a functon of connectvty NMDS-TDOA propose n ths paper uses only few anchors to realze a relatvely accurate postonng. Ths algorthm taes full avantage of nonmetrc MDS, whch only nees sgnals eep monotonc wth stance between the noes. The caculaton error an computatons brought by convertng sgnal strengh values nto stance values n prevous localzatong algorthms wll be reuce. It s testfe by theory an practce that NMDS-TDOA has a smaller locaton error. References [] Y Shang, Wheeler Ruml, Yng Zhang, et al. Localzaton from Mere Connectvty n Sensor Networs [C]. In: Proc of the 4th ACM Int Symp on Moble A Hoc Networng & Computng. New Yor: ACM Press, 3. -. [] Y Shang, Wheeler Ruml, Yng Zhang. Localzaton from Connectvty n Sensor Networs [J]. IEEE Trans on Parallel an Dstrbute Systems, 4, 5(): 96-973. [3] Ma Zhen, Lu Yun, Shen Bo. Dstrbute Locaton Algorthm for Wreless Sensor Networs MDS-MAP(D) [J]. Journal of Communcatons, 8, 9(6):57-6. [4] Chen Sushen, Lu Jangang, Lou Xaochun. Localzaton Algorthm for Wreless Sensor Networs Base on MDS-MAP an Nonlnear Flterng [J]. Journal of Zhejang Unversty(Engneerng Scence),, 46(5): 866-87 [5] Zhang Lu, Fan We, Han Shuangxa, et al, Desgn an Realzaton of the Dstrbute Localzaton Algorthm Base on MDS-MAP for WSNs [J]. Computer & Dgtal Engneerng, 3, 4(6): 876-879. 85