Open Access Research on Ranging Optimization and Twice-Weighted Centroid Calculation in Wireless Sensor Network Localization Algorithm

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1 Send Odes fo Repints to 154 The Open Automation and Contol Systems Jounal, 013, 5, Open Access Reseach on Ranging Optimization and Twice-Weighted Centoid Calculation in Wieless Senso Netwok Localization Algoithm Leina Gao * and Jianshe Peng School of Industial Manufactuing, Chengdu Univesity, Chengdu, Sichuan , China Abstact: In pactical applications of anging technology commonly used in wieless senso positioning algoithm, anging between nodes will inevitably poduce eos due to envionmental factos and measuement hadwae effects. The wieless senso netwok localization algoithm DGC-TWCL poposed in this pape takes advantage of the Cayley-Menge deteminant used to povide distance geomety constaints(dgc in two-dimensional eal space to optimize and coect the anging eo, combined with the twice-weighted centoid localization(twcl calculation, the final position of the unknown node can be obtained. The expeiment esults show that the algoithm can get the positioning esults with highe accuacy and bette obustness. Keywods: Localization, wieless senso netwok, inaccuate anging, Cayley-Menge deteminant, twice weighted centoid localization computation. 1. INTRODUCTION As an intenational hot eseach field attacting a geat deal of attention nowadays, wieless senso netwok (WSN is a new patten of infomation acquisition and pocessing. The netwoked computing devices with low pice and low powe povide a novel design mode to the connectivity between the physical wold and human society in ubiquitous ea. The node localization is one of the most impotant issues afte many WSN system layouts have been complete. The monitoing data fo many applications with unknown location have no meaning. In usual case, WSN positioning algoithm obtains the unknown node position estimation though measuement o connection between the unknown node and ancho nodes with pioi position infomation. Accoding to the cuent status, WSN positioning technology can be divided into anging-based and non-anging-based in view of whethe the metics between two nodes is necessay duing localization pocess. The fome positioning mode has elatively highe accuacy and cost, while the latte based on the connectivity of netwok has less accuacy and hadwae equiement. In accodance with hops between the unknown node and ancho nodes in connection, the localization algoithm is divided into single-hop and multi-hops. Fom the viewpoint of pocessing manne to the measuement data among nodes, WSN positioning can be divided into the centalized and distibuted method. The centalized algoithm uses the cental pocesso to get all anging infomation esulting in the entie netwok location mapping and the distibuted algoithm achieves self-positioning by node acquisition of local infomation to the neighbo nodes. The typical *Addess coespondence to this autho at the School of Industial Manufactuing, Chengdu Univesity, Chengdu, Sichuan , China; Tel: ; Fax: ; leinna@163.com location algoithms and systems include centoid method [1], least squaes estimation method [], the MIT Ciket [3], the convex pogamming location algoithm of the Univesity of Califonia, Bekeley [4], DV-Hop [5] and DV-Distance [6] distibuted localization algoithm of the Rutges Univesity. The common anging technology of WSN localization algoithm includes RSS, AoA [7], ToA and TDoA[8]. In pactical application envionments, anging has the eos inevitably because of the impact of multi-path, eflection, diffaction, scatteing and othe envionment o measuement hadwae factos. Since the distance between nodes is independent fom each othe, the anging eo may undemine geometic topology elationship e.g. the tilateation measuements make the length of both sides less than the thid one, which causes failue of the localization algoithm. This pape poposes a wieless senso netwok localization algoithm DGC-TWCL which utilizes Cayley-Menge deteminant in two-dimension eal space to povide distance geomety constaints(dgc, makes optimization and coection to the anging eos and obtains the final position of the unknown node combined with the twice-weighted centoid localization(twcl calculation.. CAYLEY-MENGER DETERMINANT AND OPTI- MIZATION OF DISTANCE GEOMETRY CON- STRAINTS.1. Cayley-Menge Deteminant In the theoy of distance geomety, Cayley-Menge deteminant can be used to handle the geomety poblem of Eule distance in invaiant space [9]. Two n-points sequences {p 1, p,..., p n } and {q 1, q,..., q n } in m-dimension eal space fom the Cayley-Menge matix which is defined as: / Bentham Open

2 Reseach on Ranging Optimization and Twice-weighted Centoid Calculation The Open Automation and Contol Systems Jounal, 013, Volume M( p 1 ;q 1,...,q n d ( p 1,q 1 d ( p 1,q d ( p 1,q n 1 d ( p,q 1 d ( p,q d ( p,q n 1, d ( p n,q 1 d ( p n,q d ( p n,q n whee d( p i,q j, i, j {1,.., n} is the Euclidean distance between the point p i and q j. The bi-deteminant of Cayley- Menge composed by the two n-point sequences is defined as: D( p 1 ;q 1,...,q n det M( p 1 ;q 1,...,q n. ( When the two sequences of points ae the same, M( p 1 ; p 1 and D( p 1 ; p 1 can be defined simply as M( p 1 and D( p 1. D( p 1 is called Cayley-Menge deteminant which has the following classic conclusion: (Theoem 1[10] consideing n-tuple constituted by p 1 in m-dimension space, the ank of Cayley-Menge matix M( p 1 is at most m +1. Accoding to the above theoem, it's easy to know that in two-dimension eal space, D( p 0, p = 0. p 0 can be egaded as the unknown node duing the positioning pocess and it makes the distance measuements with the thee known ancho nodes p 1, p, which is shown in (Fig. 1. Fig. (1. Fou-tuple geometic elationship in two-dimension space. The value of Cayley-Menge deteminant compised by d 01, d 0, d 0 and the distances d 1, d 1, d among the ancho nodes is 0, that is: (1 When Eq.(4 is taken into Eq.(3, the following elationship can be obtained: T A + T b + c = 0, (5 whee = [ 1,, ] T, the matix A, the vecto b and c ae descibed by the distances d 1, d 1, d among the known ancho nodes and measued distances d 01, d 0, d 0... Ranging Optimization of Distance Geomety Constaints If the unknown node p 0 eceives the noise distance measuement infomation fom ancho nodes p 1,..., in accodance with the collections of nodes {p 0, p, p 3 },{p 0, p, p 4 },...{p 0, p }, - independent quadatic distance constaint equations can be obtained combined with Eq.(3, Eq.(4 and Eq.(5. Each equation can be expessed as f i ( 1,, i = 0,i = 3,4,...,, whee 1,,..., ae the eos occued among the unknown node and ancho nodes duing the measuement pocess. By getting the minimized sum of squaed eos unde the distance constaints limits, the optimization poblem can be expessed as: min J = f 1 ( 1,, 3 = 0 f s.t. ( 1,, 4 = 0 f ( 1,, = 0 Using numeical analysis methods, the optimization solution * 1, *,..., * can be get. Combined with the Eq.(4, the position estimated between the unknown node and ancho node can be obtained: ˆd0i = d0i * i,i = 1,,...,. 3. TWICE-WEIGHTED CENTROID LOCALIZATION COMPUTATION Though distance infomation with the thee known ancho nodes, the definite position coodinates of the unknown node in two-dimension space theoetically can be obtained by tilateation measuement method, i.e., thee ancho nodes as the centes, the elevant distance infomation as adiuses of the cicles that intesect at one point, as shown in (Fig.. (6 D( p 0, p = 0 d 01 d 01 d 0 d 0 d 0 0 d 1 d 1 d 1 d 0 d 1 0 d 1 d = 0. Due to noise influence of the envionment and equipment, eo inevitably occus duing the actual measuement. It's assumed that the accuate distance d 0i and measued distance d 0i between the unknown node and the ancho node have the following elationship: (3 d 0i = d 0i + i,i = 1,,. (4 Fig. (. Ideal model of tilateation measuement.

3 156 The Open Automation and Contol Systems Jounal, 013, Volume 5 Gao and Peng Howeve, in the actual application envionments, the andom vaiables caused by the pesence of noise duing the distance measuement pocess make the unknown node located in a egion (see Fig. 3, the centoid of which can be viewed as the unknown node position. Fig. (3. The centoid of tilateation measuement in the actual application envionments. The weighted centoid localization idea is to intoduce the distance between the unknown node and ancho node as a weighted facto, and the distance is invesely popotional to the facto, which eflects influence degee of the ancho node to the position of the centoid. When the unknown node eceives the k(k>=3 ancho node signals, they can constitute up to c 3 k tiangles and appoximate c 3 k coodinates of the unknown node can be calculated by using the weighted centoid calculation. The final position of the unknown node can be obtained though the second weighted centoid computation The Weighted Centoid Computation in Tilateation Measuement Method It s assumed that the unknown node p 0 caies out tilateation measuement by eceiving the infomation fom the ancho nodes p 1, p, as shown in (Fig. 4. distance d c1, pi between c 1 and the distance dc1 between c' 1 ',p i ae calculated espectively, compaed with the anging between p 0, the close degee fo dc1, pi and dc1 to ' d0i is detemined, i.e., c,p 1 (x 1,y 1 coesponding to i the smalle value of 1 d p0 /d,p i c1 < 1 d,p i p0 /d can be taken as,p ' i c1,p i the appoximate position of the unknown node. Similaly, by anging fom p 1 to p 0 as well as poximity judgment pinciple, the second estimated location c of p 0 can be obtained; and by anging fom p to p 0 as well as poximity judgment pinciple, the thid estimated location c 3 of p 0 can be obtained. Taking advantage of the thee estimated positions c 1 ( ˆx 1, ŷ 1,c ( ˆx, ŷ,c 3 ( ˆx 3, ŷ 3 of p 0, the weighted factos eflecting the influence degee of ancho nodes ae intoduced: 1/( d w 1 = 01 + d 0 1/( d 01 + d 0 +1/( d 01 + d 0i +1/( d 0 + d 0i 1/( d w = 01 + d 0i 1/( d 01 + d 0 +1/( d 01 + d 0i +1/( d 0 + d 0i 1/( d w 3 = 0 + d 0i (9 1/( d 01 + d 0 +1/( d 01 + d 0i +1/( d 0 + d 0i Using the weighted centoid calculation though the ancho nodes p 1, p, the final estimated position of the unknown node p 0 can be obtained: ˆx i = w 1ˆx 1 + w ˆx + w 3 ˆx 3, ŷ i = w 1 ŷ 1 + w ŷ + w 3 ŷ 3. When solving the weighted factos in DGC-TWCL algoithm, the estimated value ˆd0i is adopted afte optimization and coection. 3.. Twice-weighted Centoid Computation (7 (8 Fig. (4. The Weighted Centoid Computation in Tilateation Measuement. Accoding to the distance d 01 (measued between p 0 and p 1 and the distance d 0 (measued between p 0 and p, combined with the coodinates of p 1 and p, two positions coodinates( c 1 and c' 1 of p0 can be obtained. Afte the As descibed in Section., when the unknown node p 0 has eceived anging infomation fom p 1,..., the distances of the estimated position ˆd 01, ˆd 0,..., ˆd 0 between the unknown node and ancho nodes can be obtained by distance geomety constaints optimization of Cayley-Menge deteminant. Conside the following set {p 1, p, p 3 },{p 1, p, p 4 },...,{p 1, p } in which each element consists of thee ancho nodes, the weighted centoids collection {O 1 ( ˆx 1, ŷ 1,O ( ˆx, ŷ,...,o ( ˆx, ŷ } can be calculated espectively though the tilateation measuement method and the final unknown node position can be obtained by using the second weighted centoid computation as below: ˆx = [ ˆx k /(ˆd 01 + ˆd 0 ] [1 / ( ˆd 01 + ˆd 0 ] (10

4 Reseach on Ranging Optimization and Twice-weighted Centoid Calculation The Open Automation and Contol Systems Jounal, 013, Volume ŷ = [ŷ k /(ˆd 01 + ˆd 0 ] [1 / ( ˆd 01 + ˆd 0 ] (11 4. DGC-TWCL ALGORITHM DESCRIPTION AND EXPERIMENT ANALYSIS 4.1. Algoithm Desciption The DGC-TWCL algoithm includes two pats, one is solving the anging optimization value povided by distance geomety constaints(dgc of Cayley-Menge deteminant, and the othe is twice weighted centoid (TWC computation. The DGC potion is descibed as follows: Step 1: Afte the unknown node p 0 has eceived anging infomation fom the known ancho nodes p 1,..., the sequence d01 < d 0 < < d 0 with descending ode is set. Let the anging eos be 1,,..., ; Step : Taking p 0 and {p1, p },i = 3,..., into the Cayley- Menge deteminant D( p 0, p = 0, the function f i ( 1,, i = 0 is detemined whose vaiables ae 1,, i ; {b 1,b,b 3 } ae placed in the points of (4, 8, (0, 0, (8, 0 (see Fig. 5, the testing nodes set is {p 1,p,...,p }. The fou ancho nodes configuation is shown in (Fig. 6 and the testing nodes set is {p 1,p,p 6,p 7,p 8,p 10,p 11,p 13,p 14,p 16,p 17,p 19,p 0 }. The five ancho nodes configuation is shown in (Fig. 7, and the test nodes set is {p 1,p,p 4,p 6,p 7,p 10,p 1 }. Fig. (5. Expeiment simulation scene with thee ancho nodes in DGC-TWC. Step 3: Taking the minimum value of J = as objective function, equations fom Step as constaints conditions, nonlinea optimization solutions * 1, * *,..., ae obtained; Step 4: Accoding to ˆd0i = d0i * i,i = 1,,...,, the estimated position of the unknown node to the ancho node can be obtained; The TWC pat is descibed as follows: Step 5: by using {p 1, p },i = 3,..., to constitute a collection of tiangles, in accodance with the method in Section 3, thee position coodinates can be calculated by tilateation measuement and poximity pinciple in each tiangle, and then the weighted centoid position coodinates can be obtained by making use of the factos w 1, w, w i invesely popotional to the estimated distance value ˆd 0i within each tiangle; Step 6: By adopting the method in Section 3. and intoducing the second centoid weighted factos, the final estimated position coodinate p 0 ( ˆx, ŷ of the unknown node p 0 can be obtained. 4.. Expeiment Analysis Against the DGC-TWCL positioning method poposed in this pape, the positioning eo, positioning eo sample vaiance, the scalability and obustness of the algoithm have been evaluated. Expeiment simulation scene has been set in a ectangula aea of 8m * 8m. Conside positioning of the unknown node p i located at the gid point unde the configuation cases of thee, fou and five ancho nodes espectively. Suppose the anging eo is within10% andom value of the tue distance. When thee ancho nodes Fig. (6. Expeiment simulation scene with fou ancho nodes in DGC-TWC. Fig. (7. Expeiment simulation scene with five ancho nodes in DGC-TWC. Besides, the positioning states of the linea least squaes(lls estimation, mean centoid and weighted centoid location ae calculated unde the same anging condition. The calculated indicatos include the biggest eo(be, the smallest eo(se, the mean eo(me and the eo sample vaiance(sv, whee ei = ( ˆx i x + ( ŷ i y,i = 1,,..., n is one point eo in the testing nodes set, and BE = max{ei }, SE = min{e i }, n ME =, ei / n SV = 1 i=1 n 1 n i=1 (e i e. e is the aveaged

5 158 The Open Automation and Contol Systems Jounal, 013, Volume 5 Gao and Peng measuement eo of the testing nodes set. Ranging eo optimization pocess takes advantage of fsolve in Matlab optimization toolbox. The expeiment esults with thee, fou and five ancho nodes ae shown in (Table 1, Table and Table 3 espectively. The coesponding gaphic plotting accoding to the data fom (Table 1 to Table 3 is shown in (Fig. 8. As it can be seen fom Table 1 to Table 3 and Fig. (8, unde the thee ancho nodes configuation, the smalle smallest eo(se can be obtained in DGC-TWCL with espect to the othe positioning methods. On the aspect of the mean eo (ME, except to obtain a value slightly lage than the mean centoid unde five ancho nodes configuation, the othe values ae lowe than the othe thee positioning algoithm. Compaed with the indicatos of the biggest eo(be and sample vaiance(sv, the values in DGC-TWCL ae medium, which shows that distance fluctuation degee between the positioning esult and tue position of the algoithm is modeate. In addition, unde the thee ancho node configuation pogam, the pefomance of the above fou statistical indicatos is stable in DGC-TWCL, which indicates the scalability and obustness of the algoithm. On the othe hand, DGC-TWCL algoithm inceases the calculation amount and time because of the anging calibation and the twice weighted factos computation pocess, which is moe suitable fo the positioning occasions of centalized singlehop and medium ancho nodes density. Table 4 shows the compaison of the seveal positioning algoithms in computation ovehead, positioning accuacy, stability and scalability. Wheein is the numbe of ancho nodes, T inv is the time-consuming fo calculating the genealized invese matix in LLS algoithm, T cl is the time ovehead fo seeking the centoid of one tilateation measuement, T w is the time consumption fo computing the weighted factos and T g is the time spent on the measuement coection. By inceasing a cetain amount of calculation seen fom Table 4, the positioning accuacy and stability in DGC-TWCL can be impoved. Table 1. Indicatos Statistics of Expeiment Results Unde 3 Ancho Nodes Localization Algoithm 3 Ancho Nodes BE SE ME SV LLS Mean Centoid Weighted Centoid DGC- TWCL Table. Indicatos Statistics of Expeiment Results Unde 4 Ancho Nodes Localization Algoithm 4 Ancho Nodes BE SE ME SV LLS *10-3 Mean Centoid *10-3 Weighted Centoid *10-3 DGC- TWCL *10-3 Table 3. Indicatos Statistics of Expeiment Results Unde 5 Ancho Nodes Localization Algoithm 4 Ancho Nodes BE SE ME SV LLS Mean Centoid Weighted Centoid DGC- TWCL

6 Reseach on Ranging Optimization and Twice-weighted Centoid Calculation The Open Automation and Contol Systems Jounal, 013, Volume Fig. (8. The coesponding gaphic plotting to expeiment indicatos statitics. Table 4. Pefomance Compaison of Seveal Positioning Algoithm Computation Ovehead LLS Mean Centoid Weighted Centoid DGC-TWCL T inv (-*T cl (-*T cl+t w (-*T cl+t w+t g Positioning Accuacy Lowe Lowe Lowe Bette Positioning Stability Lowe Lowe Lowe Bette Positioning Scalability Bette Bette Bette Bette 5. CONCLUSIONS Fo node localization in wieless senso netwok, because of its low-cost featue, it is impossible fo each node to have the ability to diectly access to the location infomation such as GPS global positioning system. Most nodes localization fistly makes distance measuements with ancho nodes, and due to the pesence of measuement noise and estimation eo, the anging value is non-pecise. In this pape, the DGC-TWCL algoithm poposed uses geometical elationship among the nodes in wieless senso netwok fomed by Cayley-Menge deteminant, takes the minimum of anging squaed eo sum as the optimization taget and coects the non-exact distance measuement values. Combined with the twice weighted centoid calculation, the high accuacy and obustness positioning esults can be achieved. CONFLICT OF INTEREST The autho(s confim that this aticle content has no conflicts of inteest. ACKNOWLEDGEMENT Declaed none. REFERENCES [1] J. Blumenthal, R. Gossmann, F. Golatowski, and D. Timmemann, "Weighted Centoid Localization in Zigbee-based Senso Netwoks", =444758&ul=http%3A%F%Fieeexploe.ieee.og%Fxpls %Fabs_all.jsp%3Fanumbe%3D The date of access: [] Y. Chen, J. Fancisco, W. Tappe, and R.P. Matin, "A pactical appoach to landmak deployment fo indoo localization", IEEE SECON Poceedings, pp , 006.

7 160 The Open Automation and Contol Systems Jounal, 013, Volume 5 Gao and Peng [3] N.B Piyantha, A. Chakaboty, H. Balakishnan, "The cicket location-suppot system", Poceedings of the 6th Annual Intenational Confeence on Mobile Computing and Netwoking, Boston, pp. 3-43, 000. [4] L. Dohety, K.S.J Piste, and L.E Ghaoui, "Convex position estimation in wieless senso netwoks", Poceedings of the IEEE INFO- COM, vol. 3, pp , 001. [5] D. Nicolescu and B. Nath, "Ad-Hoc positioning systems", Poceedings of the 001 IEEE Global Telecommunications Confeence, vol. 5, pp , 001. [6] D. Niculescu, B. Nath, "DV based positioning in ad hoc netwoks", Jounal of Telecommunication Systems, vol., no. 1/4, pp , 003. [7] D. Koks, "Numeical calculations fo passive geolocation scenaios", Technical Repot, DSTO-RR-0000, 005. [8] W.A. Gadne, C.K. Chen, "Signal-selective time-diffeence-ofaival estimation fo passive location of man-made signal souces in highly couptive envionments", IEEE Tansactions on Signal Pocessing, vol. 40, no. 5, pp , 199. [9] M. Cao, B.D.O. Andesonb, and A.S. Mosea, "Senso netwok localization with impecise distances", System and Contol Lettes, vol. 55, no. 5, pp , 006. [10] G. Mao, B. Fidan, and B.D.O. Andeson, "Wieless senso netwok localization techniques", Compute Netwoks, vol. 51, no.1, pp , 007. Received: August 13, 013 Revised: August 8, 013 Accepted: August 8, 013 Gao and Peng; Licensee Bentham Open. This is an open access aticle licensed unde the tems of the Ceative Commons Attibution Non-Commecial License ( which pemits unesticted, non-commecial use, distibution and epoduction in any medium, povided the wok is popely cited.