An Analytical Method for Centroid Computing and Its Application in Wireless Localization

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1 An Analytcal Method for Centrod Computng and Its Applcaton n Wreless Localzaton Xue Jun L School of Engneerng Auckland Unversty of Technology, New Zealand Emal: xuejun.l@aut.ac.nz Abstract Ths paper presents an analytcal method for one to compute the centrod of the overlappng area of two ntersectng crcles wth arbtrary orentaton. We also extend the soluton to estmate the centrod of the overlappng area of three ntersectng crcles. Furthermore, based on the analytcal soluton of centrod computng, we propose a cell-based localzaton technque, namely Centrod Based Locaton (CBL) for wreless sensor networks. CBL works by fndng the centrod of ntersecton of any two crcles. In partcular, we study the effect of power level msmatch among anchors. Smulaton results show that CBL can sgnfcantly mprove the accuracy whle reducng the transmsson power of anchors. Keywords Wreless sensor network, localzaton algorthm, centrod computng, power level msmatch. I. INTRODUCTION Coverage of a wreless transmtter s usually modeled as a crcle. Despte of multpath fadng, ths model s farly useful when the transmtter s equpped wth an omndrectonal antenna []. In order to maxmze network connectvty, omndrectonal antennae are usually adopted n wreless sensor networks (WSNs), whch become ncreasngly popular due to ther low cost and wde applcatons []. As we know, a WSN s bascally a collecton of small sze, low power and lmted computng capablty nodes, whch sense phenomenon such as lght, temperature, movement or humdty. A node transmts ts sensed data towards a snk node probably wth mult-hoppng as each node has lmted rado transmsson range. Then, the snk node wll send the syntheszed data, perhaps through satellte network, towards the end-user. Thanks to advances n modern rado frequency ntegrated crcuts (RFIC), the cost of manufacturng a sensor node has been reduced sgnfcantly. Meanwhle, researchers and engneers start to embed more sophstcated computng capablty onto a sensor node, and ths further promotes the research actvtes n the area of wreless sensor networks. Along wth the nformaton generated from sensor output, many WSN applcatons also requre locaton nformaton of sensor nodes to provde meanngful data. Interestngly, some WSN routng/clusterng protocols could make use of locaton nformaton to realze ntellgent routng and clusterng [3]. Therefore, localzaton,.e., how to obtan the locaton nformaton becomes a crtcal ssue n WSNs. In the past few years, many localzaton technques have been proposed and they can be classfed as range-based and range-free localzaton technques [4]. Range-based localzaton technques utlze specal hardware to measure dstance or angle nformaton among sensor nodes. For example, dstance can be estmated by tme of arrval (TOA) [5], tme dfference of arrval (TDOA) [6] and receved sgnal strength ndcator (RSSI) [7]. Angle can be measured by angle of arrval (AOA) [8]. On the contrary, range-free localzaton technques that do not requre specal hardware to be equpped on sensor nodes and estmate locaton of a sensor based on the connectvty nformaton among nodes [9]. Some range-free localzaton technques frst allow a sensor node to determne the regon t resdes n based on the connectvty to other sensor nodes, and then select a pont n that regon as ts locaton estmaton []. Ths type of localzaton technques are usually referred to as Regon-Based Localzaton (RBL). In partcular, as ponted out n [], t wll mnmze average locaton error usng the centrod of a regon as poston estmaton. Exstng algorthms nclude DV-Hop [], APIT [], Monte-Carlo Localzaton (MCL) [3] and Monte-Carlo Box (MCB) [4]. In general, range-based localzaton technques generally provde better accuracy than range-free localzaton technques at the extra expense on specal hardware, although the hgh cost may prevent them from beng mplemented for large-scale WSNs. For nfrastructure-based WSNs, we often need to deploy some sensor nodes wth known postons, and these nodes are referred as anchors [4]. Vewng from ths aspect, localzaton technques can also be classfed as anchor-based and anchorfree technques. The former can map a sensor node to a global coordnate system whle the latter can only provde relatve node postons [5]. Generally, anchor-based localzaton technques can provde better locaton accuracy but are less flexble n the sense that anchors are fxed at known postons. Interested readers are referred to [4] for more on classfcaton of localzaton technques. Wth nfrastructure, Chu and Jan recently proposed an outdoor localzaton technque n [6] and the dea was based on cell-overlappng [7]. In bref, a sensor node determnes ts locaton by analyzng the beacon messages receved from anchors. However, the localzaton algorthm works based on the assumpton that all anchors have the same transmsson range, whch s unlkely n practce. In a later publcaton, they extended the locaton algorthm [6] to use multple power-level approach [8], assumng that the coverage rad of anchors can be controlled by settng dfferent transmsson power levels. Nevertheless, ths s extremely challengng to mantan the same transmsson power levels, especally when anchors operate on batteres. Furthermore, they dd not compute the centrod of the overlappng area of any two ntersectng crcles. The approxmated method used n [8] wll certanly lead to fnte unknown error. We notced that t s trval to compute the overlappng area

2 of two ntersectng crcles and one can fnd several analytcal solutons from textbooks on geometry. However, to date, we are not aware any analytcal soluton can be found on how to compute the centrod of that area. Ths paper makes two major contrbutons to localzaton technques for WSNs. Frst, we managed to compute the centrod of the overlappng area of any two crcles (regardless of ther respectve rad) and proposed a centrod based localzaton (CBL) algorthm smlar to the one n [6]. Furthermore, we study the effect of power level msmatch n anchors. Smulaton results show that CBL s able to mprove the localzaton accuracy whle reducng the transmsson power level of anchors. II. RELATED WORK Cell-based localzaton technque [7] was orgnally proposed for localzaton n cellular networks, where base statons (BSs) are deployed n hexagonal layout or mesh layout. For example, Fgure shows a hexagonal layout where the locaton of BSs can be pre-determned. A cell-based locaton algorthm for WSNs was proposed n [6], where BSs were replaced by reference ponts (RPs),.e., anchors. The area covered by a RP s referred to as a cell. Wth proper celloverlappng, the area of nterest can be dvded nto three types of regon: regon type A, regon type B and regon type C, as ndcated n Fgure, whch are referred to as Type I, Type II and Type III regons, respectvely. It s not dffcult to notce that type A regon, type B regon and type C regon s covered by one, two and three RPs sgnal, respectvely. Should the RPs have the same transmsson range, the centrod ( x, y ) of each localzaton regon can be found by n n ( x, y) ( x, y) () n n where (x, y ) s the th vertex of the regon. It was assumed that each RP knows all centrods of ts localzaton regons. Next, the beacon format contans the layout type, the centrods of dfferent localzaton regons. Consequently, a sensor node can estmated ts locaton by ntersectng the centrod sets contaned n the beacon frames from varous RPs. The accuracy of cell-based localzaton can be further mproved by usng multple power level approach [8]. In summary, each RP has a set of dscrete value for transmsson power level, whch n turn leads to a set of dscrete coverage rad. As a result, the area of nterest can be dvded nto concentrc rngs of regons. By estmatng centrods of dfferent ntersecton regons of dfferent rngs, a sensor node could better estmate ts locaton. Smlar method was proposed n [9], n whch grd scan algorthm was adopted to estmate the centrod of an ntersecton regon. We notced that the aforementoned works dd not provde a method to compute the exact centrod of ntersecton of any two crcles [6]. Furthermore, they requre strct control on the transmsson power level of a RP, whch s not easy to realze ths n practce, especally when the RP s operatng on battery nstead of unlmted power supply. Enlghten by the celloverlappng localzaton method proposed n [6] [7] [8], we fgure out a method of fndng the centrod of overlappng area of any two ntersectng crcles. Subsequently, we proposed a new localzaton algorthm, CBL, whch computes the centrod of ntersecton of any two crcles and relaxes the transmsson power constrants of RPs. Hereafter, we use anchors nstead of RPs for consstency. P6 P5 C5 B6 B5 C6 C4 B P P A B4 P4 C C3 B B3 C Fgure. Layout of anchors. III. PROPOSED CBL ALGORITHM P P3 Regon Type A Regon Type B Regon Type C A. Centrod of Intersecton of Any Two Crcles Let us consder a common scenaro n WSNs, as shown n Fgure, where a sensor node, e.g. S wth unknown locaton s able to communcate wth other two sensor nodes, e.g., A and A wth known locatons. Mathematcally, we have ( xx ) ( y y ) r, ( xx ) ( y y ) r ( ) ( ), ( ) ( ) r r x x y y r r xs x ys y r xs x xs y r ( ) ( ) In order to estmate the locaton of S, we need to determne the locaton of the centrod, e.g., pont C of that partcular regon, as ndcated as shaded area n Fgure. As we know, the centrod of a regon R wth any shape can be found by xdxdy R xy,, dxdy R R ydxdy dxdy Wth a careful study, we notce that the centrod C wll always be on the lne connectng A and A, because the regon formed by the ntersecton of any two crcles s symmetrc respect to the lne connectng the centers of the two crcles. Therefore, we only need to fnd the value of x. Next, the equatons for the left boundary and rght boundary of the shaded regon can be determned as follows. f( x) r ( xx), for left boundary f( x) r ( xx), for rght boundary Consequently, we are able to fnd out the value of x : R () (3) (4)

3 x xr xr x r x r ( x x ) dx x r ( x x ) dx (5) r ( x x ) dx r ( x x ) dx x r Due to lmted space, here we only provde the fnal result for x, x PQ, where P and Q are gven by ( x x ) y r ( x x ) y r '' '' '' '' '' '' () r x P ( xr xr ) r x r x r x x r x x r x sn sn 3 6 r r (6) x x Q ( r r ) r x r x 4 r x r x sn sn r r For the scenaro shown n Fgure, y s equal to y (or y ). Then, we wll use ( x, y ) as the estmated poston for node S. Fgure. Intersecton of two crcles Vglant readers may notce that the scenaro shown n Fgure s a specal case as y s equal to y and the two ntersecton ponts are located on the y axs. Next, we consder a general scenaro of the ntersecton of two crcles n Fgure 3. In order to use the formula we have developed, we frst need to rotate both x and y axs ant-clockwse by a certan angle α, x' xcos ysn y y where tan y' ycos xsn x x Then, after axs rotaton, the lne A A wll be parallel to x axs. Next, we need to shft both x and y axs by certan amount as follows. x '' x' a y'' y' b where a and b are gven by r r x x a, b y ( ) ' ' ' ' x x Eventually we can obtan the scenaro shown n Fg., whch has two crcle equatons as follows. ' (7) (8) (9) Fgure 3. generalzed scenaro of ntersecton of two crcles. Fnally, we can use (8) to solve for the ( x '', y '' ) and then conduct the reverse procedures to obtan the respectve ( x, y ) n the orgnal coordnate system. B. Centrod-Based Localzaton Now we are able to fnd the centrod, C -j, of ntersecton of any two crcles P and P j. Mathematcally, C j :( x, y) F( x, y, xj, yj, r, rj) () where F s the logc mappng functon; (x, y ) and r are the centre and the radus of crcle, respectvely. The proposed localzaton technque s named as centrod-based localzaton (CBL) because t s based on fndng the centrod of ntersecton of multple crcles. CBL works n four steps: Step Anchors Deployment: Frst, anchors are deployed wth known locatons to cover an area of nterest. These anchors may not necessarly be deployed n a regular layout structure such as hexagonal or mesh structure used n [6]. Furthermore, these anchors are not requred to know ther localzaton regons and the correspondng centrod of any partcular localzaton regon, whch s the dstnctve feature that makes our CBL salently dfferent from the localzaton method proposed n [6]. Step Beacon Frame Broadcast: Next, anchors perodcally broadcast ther beacon frames to notfy all the sensor nodes resdng n ther coverage area. Smlar to the beacon frame n [8], the CBL beacon frame from a partcular anchor node A contans the followng data: I {( x, y), r} () where (x, y ) s the locaton coordnates of anchor node A and r s A s coverage radus. As mentoned n [8], the coverage radus may be obtaned usng the free space propagaton model. In partcular, f an anchor node s operated on battery, ts coverage radus wll change accordng to ts transmsson power level. Step 3 Beacon Frame Synthess: Subsequently, a sensor node wth unknown locaton wll lsten to the beacon broadcast from anchors nearby. As shown n Fgure 4, a sensor node S wth unknown locaton receves beacon frames from anchor node

4 A, A and A 3, thus S knows that t s resdng n the ntersecton regon of the three crcles. where A R denotes the area of regon R. Therefore, we can rewrte the precson as e() r P{ D r} f (, x y) dxdy A C C A r R (6) r where Cr ( x, y) ( xxest) ( y yest) r R. The smallest value of r that satsfes e(r) =. s the crtcal radus and the worst-case accuracy s the maxma among the three crtcal rad of the three types of localzaton regons. The average accuracy s calculated as Fgure 4. A general scenaro of CBL Step 4 Locaton Estmaton: After lstenng to the broadcast of anchors for suffcent long tme nterval (about beacon perod), a sensor node may receve beacon frames from N anchors. The sensor node computes the centrod of ntersecton regon of any two crcles. Then, t takes the centrod of these centrods as ts locaton estmatons. Take Fgure 4 as an example, sensor node S wll estmate ts locaton as follows. () S calculates C - usng the beacon frames from anchors A and A by applyng (). () Then, S calculates C -3 and C -3 n a smlar way. (3) Fnally, S estmates ts locaton as x x x y y y S':( xs', ys' ), 3 3 C C3 C3 C C3 C3 Specally, f a sensor node, S, only receves beacon frame from a sngle anchor node, A, then S takes A s locaton as ts locaton estmate. C. Postonng Same as the metrcs used n [6], we use absolute error dstance, cumulatve dstrbuton functon (CDF) of error dstance and average value of error dstance to measure the performance of our proposed CBL. For the sake of comparson, we revst the defnton of these parameters. Error dstance s mathematcally defned as D ( X x ) ( Y y ) (3) est est where (X, Y) s the actual locaton coordnate of a sensor node, and (x est, y est ) s the estmated locaton coordnate of the sensor node. The CDF of D s defned as er () PD { r} (4) Ths s also referred to as the precson. If the sensor nodes were unformly dstrbuted over a regon R, the probablty densty functon f(x, y) can be found as f ( xy, ) A R (5) 3 ED [ ] ped [ ] (7) where p s the probablty that a sensor node falls n the type localzaton regon, and E[D ] s calculated as ED [ ] ( x x ) ( y y ) f( xydxdy, ) (8) ( xy, ) R c c where R s the localzaton regon type and ( xc, y ) c s the centrod of R. IV. SIMULATION RESULTS We frst consder a hexagonal layout of anchors, where the dstance between two neghborng anchors s one dstance unt and the coverage radus of an anchor node s vared from / 3 to 3/ dstance unt [7]., sensor nodes are unformly dstrbuted over the area of nterest and we study the performance of CBL and the technque presented n [6] Transmsson Range Type I CBL Type II CBL Type III CBL Type I [5] Type II [5] Type III [5] Fgure 5. Worst-case vs. Transmsson Range As shown n Fgure 5, for type I and type III regons, our CBL algorthm has the same performance wth the one proposed n [6]. However, for the type II regon, our CBL requres hgher error tolerance level. Interestngly, the mnmum worst-case accuracy occurs at transmsson range d= at whch the worst-case accuracy s.886. Ths s better than the.887 at d=.7638 presented n [6].

5 .5 CBL result [5] msmatch n the power levels of anchors wll affect the localzaton accuracy. The hgher the msmatch, the hgher the localzaton error wll be caused Transmsson Range Fgure 6. Average accuracy vs. Transmsson Range Next, we look at the average accuracy of CBL. As shown n Fgure 6, the average accuracy E(D) can be expressed as a convex functon of transmsson range d,.e. g(d). Furthermore, we found that the mnmum of g(d) has s.453 where d= Ths mnma of g(d) s lower than that found n [6], whch s.55 at d=.744. Importantly, our CBL can acheve better average accuracy whle usng smaller transmsson range that reduces transmsson power level. Ths s desred as power s a scarce resource n WSNs. Precson Perfect Model =. =. = Fgure 7. Effect of Power Level Msmatch on Average We have assumed that all the anchors have dentcal coverage rad n the prevous smulatons. However, n practce dfferent anchors probably have dfferent power levels and thus dfferent coverage rad. Therefore, we shall take ths effect nto account by modelng ther coverage rad as a Gaussan dstrbuted random varable wth mean value of d and standard devaton of. From above, we set the mean d to.7735 n order to acheve mnmzed average localzaton error. In ths smulaton,, sensor nodes are consdered and anchors are deployed wth hexagonal layout. The dstance between any two anchors s one dstance unt. Durng the localzaton process of each sensor node, the coverage radus of every anchor node s generated by Gaussan dstrbuton wth parameter (d, ) and CBL s used to estmate the poston of the sensor node. As shown n Fgure 7, we found that V. CONCLUSION We presented an analytcal method of computng the centrod of the overlappng area of any two ntersectng crcles, and then developed a localzaton technque, namely centrodbased localzaton (CBL) for wreless sensor networks. Smulaton results showed that CBL s able to mprove localzaton accuracy whle reducng the transmsson power level of anchors. Furthermore, we studed the effect of power level msmatch of anchors on the localzaton error of CBL. The performance of CBL degrades gracefully when the power level msmatch n anchors ncreases. REFERENCES [] A. Goldsmth, Wreless Communcatons: Cambrdge Unversty Press, 5. [] J. Yck, B. Mukherjee, and D. Ghosal, "Wreless sensor network survey," Computer Networks, vol. 5, pp. 9-33, 8. [3] K. Akkaya and M. Youns, "A survey on routng protocols for wreless sensor networks," Ad Hoc Networks, vol. 3, pp , 5. [4] G. Mao, B. Fdan, and B. D. O. Anderson, "Wreless sensor network localzaton technques," Computer Networks, vol. 5, pp , July 7. [5] B. H. Wellenhoff, H. Lchtenegger, and J. Collns, Global Postonng System: Theory and Practce, Fourth Edton ed.: Sprnger Verlag, 997. [6] B. P. Nssanka, C. Ant, and B. Har, "The Crcket locaton-support system," n ACM MobCom', Boston, MA,, pp [7] P. Bahl and V. N. Padmanabhan, "RADAR: an n-buldng RF-based user locaton and trackng system," n IEEE INFOCOM',, pp [8] D. Nculescu and N. Badr, "Ad hoc postonng system (APS) usng AOA," n IEEE INFOCOM'3, San Francsco, CA, 3, pp [9] N. Bulusu, J. Hedemann, and D. Estrn, "GPS-less low-cost outdoor localzaton for very small devces," IEEE Personal Communcatons, vol. 7, pp. 8-34, October. [] T. He, C. Huang, B. M. Blum, J. A. Standkovc, and T. Abdelzaher, "Range-free localzaton schemes for large scale sensor networks," n ACM MobHoc'3, San Dego, CL, 3, pp [] S. Zhang, J. Cao, L. Chen, and D. Chen, "On accuracy of Regon-Based Localzaton algorthms for Wreless Sensor Networks," n IEEE MASS'9, 9, pp [] D. Nculescu and B. Nath, "Ad hoc postonng system (APS)," n IEEE GLOBECOM ', San Antono, TX,, pp [3] L. Hu and D. Evans, "Localzaton for moble sensor networks," n ACM MobCom'4, Phladelpha, PA, 4, pp [4] A. Baggo and K. Langendoen, "Monte-Carlo localzaton for moble wreless sensor networks," Ad Hoc Networks, vol. 6, pp , January 8. [5] G. J. Jordt, R. O. Baldwn, J. F. Raquet, and B. E. Mullns, "Energy cost and error performance of range-aware, anchor-free localzaton algorthms," Ad Hoc Networks, vol. 6, pp , 8. [6] H.-C. Chu and R.-H. Jan, "A GPS-less, outdoor, self-postonng method for wreless sensor networks," Ad Hoc Networks, vol. 5, pp , 7. [7] H.-C. Chu and R.-H. Jan, "A cell-based locaton-sensng method for wreless networks," Wreless Communcaton and Moble Computng, vol. 3, pp , June 3. [8] J.-Y. Fang, H.-C. Chu, R.-H. Jan, and W. Yang, "A multple power-level approach for wreless sensor network postonng," Computer Networks, vol. 5, pp. 3-38, November 8. [9] C. Lu, K. Wu, and T. He, "Sensor localzaton wth rng overlappng based on comparson of receved sgnal strength ndcator," n IEEE MASS'4, 4, pp