Sesors & rasducers Vol. 23 Special Issue July 203 pp. 94-98 Sesors & rasducers 203 by IFSA http://www.sesorsportal.com WSN Node Localizatio Regularizatio Algorithm Based o Quasi Optimal Criterio Parameter Selectio Wag Lei Cai Che School of Cotrol Sciece ad Egieerig Shadog Uiversity Jia Shadog 25006 Chia E-mail: sduleiwag@63.com Received: 5 April 203 /Accepted: 20 July 203 /Published: 30 July 203 Abstract: Node localizatio techology is oe of the basic research fields i Wireless Sesor Networks (WSN) applicatios. he coordiates of the ukow odes ca be determied by the Least Square Estimate () which is commoly employed i the WSN ode localizatio. Due to the ifluece of multi-path fadig the distace ca be obtaied from the Received Sigal Stregth Idicator (RSSI). But i the experimets ad applicatios it is foud that differet spatial positios of the achor odes ad the errors of the distace measuremet sometimes lead to large locatio errors which is called ill-posed problem. o solve this problem coditio umbers are selected to diagose the ill-posed degree. Whe the ill-posed degree is weak the method ca be utilized to localize. Whe the ill-posed degree is serious the ridge estimate method is proposed to weake the ill-posed problem ad quasi optimal criterio is proposed to choose the regularizatio parameter. est results idicate that the ridge estimate method dilutes the ifluece of ill-posed problem obviously ad the locatio errors ca be reduced to aroud 4 meters. Copyright 203 IFSA. Keywords: Wireless sesor etworks Ridge estimate Quasi optimal criterio Regularizatio parameter Ill-posed problem.. Itroductio Wireless Sesor Networks (WSN) have broad applicatio prospects i the idustrial military evirometal ad other fields [-3]. As a importat supportig techology WSN ode localizatio becomes oe of the research focuses [4-6]. he Rage-based localizatio methods are oe promisig localizatio way which has the beefit of good locatio accuracy. here are several methods to measure the distaces betwee two odes: Received Sigal Stregth Idicator (RSSI) ime of Arrival (OA) ad ime of Differece of Arrival (DOA). he RSSI method is widely used i which kowledge of the power of the trasmitted sigal the path loss model ad the power of the received sigal are used to determie the distace betwee the receiver ode ad the trasmitter ode [7-]. I the process of localizatio usig RSSI it is foud that the covetioal Least Square Estimate () localizatio algorithm may produce large locatio errors. he spatial locatio ad the error of the measured distace cause the ill-posed problem where the method is t available. his pheomeo is especially apparet i the threedimesioal localizatio. he ill-posed problem is correspodig to the well-posed problem. he wellposed problem meas: ) he existece of the solutio; 2) he uiqueess of the solutio; 3) he solutio depeds cotiuously o the data. he problem which does ot fulfill all the requests above is defied as a ill-posed problem [2-3]. he ill-posed problem is studied i the WSN ode localizatio ad the ridge estimate method is proposed to weake the ill-posed problem. he 94 Article umber P_SI_409
Sesors & rasducers Vol. 23 Special Issue July 203 pp. 94-98 quasi optimal criterio is proposed to choose the regularizatio parameter. est results idicate that the ridge estimate method based o quasi optimal criterio dilutes the ifluece of ill-posed problem obviously ad improves the locatio accuracy of the ill-posed problems. 2. Node Localizatio Model WSN ode localizatio is based o the spatial geometric relatioship betwee ukow odes ad achor odes ad the coordiate of the ukow odes ca be obtaied through the spatial locatig algorithm. I the ideal case the achor ode is set as the ceter of a ball ad the radius is the distace betwee achor ode ad ukow ode. he these balls ca itersect to oe poit which is the locatio of the ukow ode. However the distace errors betwee achor odes ad ukow ode make the itersectio become a regio or eve o itersectio. he the ode localizatio becomes multi-solutio or eve o solutio. Suppose the coordiates of the achor odes ad the ukow odes are x y z x2 y2 z2 x y z ad x y z respectively. he distace from the achor odes to the ukow ode are d d2 d3 d. he localizatio model is described by x x y y z z d x x y y z z d x x y y z z d By subtractig oe equatio from the rest we get where 2 A 2 () A B (2) x x 2y y 2z z x x 2y y 2z z x y z b d d z y x z y x B b d d z y x z y x he distaces from the achor odes to the ukow ode ca be determied by the RSSI i the localizatio process. Wireless chael geerally uses a logarithmic distace path-loss model [4]: d Pi d Pi d0 0 dbm i d0 i 0 log 2 (3) he distace from achor odes to the ukow ode ca be measured by Eq. 3 where d i deotes the distace from achor odes to the ukow ode d 0 deotes the referece distace is the path-loss expoet P i d deotes the received power at distace of d 0 [6]. dbm deotes the error because of the multi-path fadig which cause B i Eq. 2 differs from the real value. So the right side of the Eq. 2 is defied as B. With the defiitio B B Eq. 2 ca be described as A he the of ca be deoted as B () (5) NW AA AB where N A A W A B. We ca obtai: max Q Q A B (6) mi where Q deotes the orthogoal matrix 0 deote the eigevalues of the max mi matrix N. is the best liear ubiased estimatio whe the measured value obeys ormal distributio. But the result is o loger a best liear ubiased estimatio whe there is ay eigevalues of matrix N ear to zero where the ill-posed problem exists. he the coditio umber is proposed to diagose the ill-posed degree i the ode localizatio. he coditio umber of matrix N is cod N N N. If the matrix N is positive-defiite matrix it's easy to obtai cod N /. he coditio umber max mi measures the distributio extet of the eigevalue of matrix N which ca be employed to diagose the ill-posed degree. Statistical applicatio experiece 0 cod N the ill-posed problem shows: if 00 is weak; if 00 cod N 000 the ill-posed problem is i moderate or strog degree; if cod N the ill-posed problem is i serious 000 degree [2]. o solve this problem the ridge estimate method is proposed to modify the deviatio caused by the eigevalue. 3. Localizatio Algorithm Based o the Quasi Optimal Criterio here are several ways to solve the ill-posed problems such as Regularizatio method Iterative 95
Sesors & rasducers Vol. 23 Special Issue July 203 pp. 94-98 Regularizatio method rucated Cojugated Gradiet method Laczos method Pre-coditioed Iterative method etc. [5-6]. Because of simple ad low calculatio cost the regularizatio method is more suitable for WSN positioig tha the others. he ridge estimate regularizatio method is adopted to overcome the ill-posed problem i the WSN ode localizatio. Accordig to the regularizatio theory: 2 2 F mi{ A - B } (7) he the ridge estimate is: y A A I () d d Make calculated by 2 0 the ca be d x A A I x (2) d where is the regularizatio parameter that makes y miimum. ˆ ( ) A A I A B (8) where x is the estimatio of is the regularizatio parameter. Compared with the method the ridge estimate turs A A ito A A I. Whe the miimum eigevalue is ear to zero the degree of the miimum eigevalue of A A I will be improved. Whe there are illposed problem i the system the ridge estimatio ca overcome the istability of the. Differet causes differet estimatio result. he smaller is the smaller athropic factor is itroduced ad vice versa. So the key poit of ridge estimate is the selectio of. here are two types of strategies prior ad posteriori. As to the prior strategy it's difficult to verify the qualificatio for it depeds i practice. I the process to determie the regularizatio parameter the method of posteriori policy is more practical. It is based o certai priciples ad the error level of origial data. By appedig qualitative or quatitative iformatio to the solutio the error level of the regularizatio parameter matches the origial data. However the error level is ofte uable to be obtaied i practice. So some approaches to select regularizatio parameter caot be used such as Morozov Deviatios Geeralized variatio priciple etc. he the Quasi Optimal Criterio is proposed to choose the regularizatio parameter as followed where equatio M dˆ mi d B (9) is the solutio of Euler (0) A A I A B ake partial derivative of Eq. 0 with respect d y to make d the 4. Experimetal Results Some WSN odes i good quality are chose for the localizatio experimets. he radio frequecy chip is cc2520. A ope space is selected as experimetal site ad the achor odes are located i a 3-dimesioal space. 0 typical achor odes are chose ad marked which are show i able. he coordiate of ukow ode is 24.0. able. Coordiates of achor odes. Uit: meter No. x y z No. x y z 0 0 0.54 6 4 7 0.54 2 8 4.6 2 7 5 0.54 3 22 5 22 8 23 24 0.54 4 23 0.54 9 23 7.0 5 5 0 0.54 0 6 20.0 I order to elimiate the impact of radio wave trasmissio oise a large umber of experimetal data are employed to revise the wireless chael model. he revised chael model is i 63.79 20 log 2 P d d i (3) I the test we radomly choose four or five achor odes as a group to localize ode with ridge estimate method. o evaluate the locatio result the localizatio error is itroduced to compare the experimetal result with the WSN ode real positio. he error e is defied by the real positio (x y z) ad the localized positio (x y z) as follows ' ' ' 2 2 2 e xx y y z z (4) 2 groups of typical experimetal results are show i able 2. he results show that the ill-posed serious degree icrease as the coditio umber icreasig. he largest locatio error is 908.83 meters i No. 7 group 96
Sesors & rasducers Vol. 23 Special Issue July 203 pp. 94-98 localizatio which has o sese. he coditio umber of group ad 2 are both less tha 00 the ca be selected directly to localize. Usig the same umber of achor odes to localize with positioig results varies differetly. his is the feature of ill-posed problem i WSN ode localizatio. No. able 2. Localizatio result compariso betwee ad ridge estimate. Uit: meter Ridge estimate based o Achor ode group Coditio umber error 52 error Quasi Optimal Criterio error [3578] 2 2.55 --- --- --- 2 [3580] 3.46 --- --- --- 3 [579] 743 60.0 2.547 75 2.269 4 [570] 8975 35.87 2.90 42 2.836 5 [780] 3475 8.0 2.23 9 2.94 6 [789] 2660 95.2 3.646 33 3.545 7 [689] 274560 908.8 4.856 44 4.842 8 [5780] 2538 7.53 2.547 44 2.525 9 [5689] 286 63.7 5.508 46 4.254 0 [5680] 239 45.54.670 64.47 [5790] 4048 87.4.673 56.58 2 [7890] 56 9.57 3.555 23 3.6 he selectio method of ridge estimate parameter based o quasi optimal criterio is show i Fig.. is selected whe is miimum i differet groups. I order to verify the validity of the quasi optimal criterio varies from 0 to 300. he WSN localizatio error variatios are show i Fig. 2. he WSN ode localizatio error reaches miimum whe is aroud the ridge parameter. his shows that appropriate ridge estimate ca be obtaied through the quasi optimal criterio. As is show i Fig. 2 the error goes dow ad the goes up to a certai value whe varies from 0 to 300. Ad almost each error reaches miimum whe is aroud 50. I order to reduce the computatioal cost i WSN localizatio applicatios is set to a certai value. he the computatioal cost of the proposed locatio method is equal to that of method. Accordig to the test data the average value of regularizatio parameter is 52. It ca be foud i able 2 that whe is 52 the locatio accuracy is improved ad the computatioal cost is the same as. Results show that selectig the appropriate regularizatio parameters effectively overcome the ill-posed problem. he most obvious case is the third group the locatio error drop sigificatly from 59.9844 meters to 2.2689 meters. It is foud that improper spatial distributio of achor odes cause the ill-posed problem i the localizatio process. If the ill-posed degree is weak the ca be employed to localize. If the ill-posed degree is serious the ridge estimate is utilized. If the positioig accuracy requiremet is high the regularizatio parameter ca be selected by quasi optimal criterio. If the positioig accuracy requiremet is low the regularizatio ca be set to 52 so that the computatioal cost is same as method. y value Error 40 35 30 25 20 5 0 5 Selectio of the ridge parameter ɑ Group3 Group4 Group5 Group6 Group7 Group8 Group9 Group0 Group Group2 0 0 50 00 50 200 250 300 350 400 450 500 ɑ value 60 50 40 30 20 0 Fig.. he selectio of ridge parameter.. Error varyig with the ɑ chagig 0 0 50 00 50 200 250 300 ɑ value Fig. 2. Error varyig with the chagig. Group3 Group4 Group5 Group6 Group7 Group8 Group9 Group0 Group Group2 Due to the ill-posed problem foud i the WSN ode localizatio the locatio process is proposed as Fig. 3. 5. Coclusios It is foud that there are ill-posed problem i WSN ode localizatio. he traditioal localizatio result is ot always a best liear ubiased estimatio whe the ill-posed problem exists. o solve the ill-posed problem i WSN ode localizatio the coditio umber is proposed to diagose the ill-posed degree. If the ill-posed degree is weak the ca be employed to localize; if the ill-posed degree is serious the ridge estimate method is proposed to reduce the positioig error based o quasi optimal criterio. 97
Sesors & rasducers Vol. 23 Special Issue July 203 pp. 94-98 Fig. 3. Flow chart. A locatio process is proposed to reduce the computatioal cost. he regularizatio parameter ca be selected to a costat value of 52 so that the computatioal cost is as much as the method; if the positioig accuracy requiremet is high the regularizatio parameter ca be selected by quasi optimal criterio to obtai a better. est result shows that whe the regularizatio parameter is 52 the locatio error of ill-posed problem ca be reduced to aroud 4 meters which meet part of the WSN ode localizatio requiremets. Refereces []. M. Fraceschiis M. A. Spirito R. omasi G. Ossii M. Pidala Usig WSN echology for Idustrial Moitorig: A Real Case i Proceedigs of the Coferece o Sesor echologies ad Applicatios (SENSORCOMM' 2008) pp. 282-287. [2]. Liu Ya Research o rust Cotrol echologies i Iteret of higs Process Automatio Istructio Vol. 32 Issue 8 20 pp. 60-63. [3]. Jihog Sog he Applicatio of WSN echology i the Space Locatio System i Proceedigs of the Coferece o Iteratioal Cogress o Image ad Sigal 20 pp. 633-636. [4]. Dai G. L. Zhao C. C. Qiu Y. A localizatio scheme based o sphere for wireless sesor etwork i 3D Acta Electroica Siica Vol. 36. Issue 7 2008 pp. 297-303. [5]. Peg Yu Wag Da A review: wireless sesor etworks localizatio Joural of Electroic Measuremet ad Istrumet Vol. 25 Issue 5 20 pp. 389-399. [6]. Kuo-Feg Ssu Chia-Ho Ou Hewiji Christie Jiau Localizatio With Mobile Achor Poits i Wireless Sesor Networks Vehicular echology IEEE rasactios Vol. 54 Issue 3 2005 pp. 87-97. [7]. Su iaolig Li Weiqi Qiu Hogbig Simulatio ad Aalysis of two Rage-Free WSN Localizatio Algorithm Microcomputer Iformatio Vol. 27 Issue 0 20 pp. 05-07. [8]. Wag Lei Wag iaopeg Diagosis ad solutio of the ill-posed problem i WSNs ode localizatio Chiese Joural of Scietific Istrumet Vol. 33 Issue 3 202 pp. 32-38. [9]. Shi Zhag Lirog Re Ju iao Jidog a A High Precisio Localizatio Algorithm i Wireless Sesor Network i Proceedigs of Cotrol Automatio Robotics ad Visio (ICARCV '2006) 2006 pp. -6. [0]. Hui Suo Jiafu Wa Lia Huag ad Caifeg Zou Issues ad Challeges of Wireless Sesor Networks i Proceedigs of the Coferece o Computer Sciece ad Electroics Egieerig Localizatio i Emergig Applicatios 202 pp. 447-45. []. Fag Zhe Zhao Zha Guo Peg Zhag Yu-Guo Aalysis of Distace Measuremet Based o RSSI Chiese Jourals of Sesors ad Actuators Vol. 20 Issue 2007 pp. 2526-2530. [2]. ikhoov A. H. Solutios of ill-posed problems Geological Publishig House Beijig 974. [3]. Wag Yafei Computatioal Methods for Iverse Problems ad heir Applicatios Higher Educatio Press Beijig 2007. [4]. Raida Al Alawi RSSI Based Locatio Estimatio i Wireless Sesors Networks i Proceedigs of the Coferece o Networks (ICON 20) 20 pp. 8-22. [5]. Iria F. Goroditsky Bhaskar D. Rao Aalysis of error produced by trucated SVD ad ikhoov regularizatio methods i Proceedigs of the Coferece o Sigals Systems ad Computers 994 pp. 25-29. [6]. Wag L. Qi N. Q. Du.. et al. WSNs 3D localizatio algorithm based o ikhoov regularizatio method Chiese Joural of Scietific Istrumet Vol. 3 Issue 4 200 pp. 770-775. 203 Copyright Iteratioal Frequecy Sesor Associatio (IFSA). All rights reserved. (http://www.sesorsportal.com) 98