A New Approach towards Solving the Location Discovery Problem in Wireless Sensor Networks

Transcription

1 A New Approach towards Solvng the Locaton Dscovery Problem n Wreless Sensor Networks (Techncal Report CS-TR-4551 and UMIACS-TR ) Gang Han, Shaoxong Ha and Gang Q Department of Electrcal and Compter Engneerng Unversty of Maryland, College Park, MD, 074 {hangang, sha, gangq}@gle.md.ed December 1, 003 ABSTRACT Locaton dscovery n wreless sensor network (WSN) s the process that sensor nodes collaborate to determne the poston for nknown sensor nodes. Anchors, sensors that know ther locatons, are expensve bt are reqred to be deployed nto the WSN to solve ths problem. Ths t s desrable to mnmze the nmber of anchors for ths prpose. In ths paper, we propose an anchor deployment scheme and a novel blateraton locatonng algorthm to acheve ths goal. The basc dea of anchor deployment method s to have three anchors deployed as a grop, and locate sensors arond them expansvely. The novelty of or blateraton algorthm s that t n general reqres only two neghbor sensors to determne a node s locaton. Comparng wth the state-of-the-art locaton dscovery approaches, or algorthm gves locaton estmaton wth hgh accracy, low commncaton cost and very small anchor percentage. We condct theoretcal analyss abot locaton estmaton error and extensve smlaton shows that or algorthm can derve sensor locaton wthn 4% locaton error and mch less commncaton cost compared wth other algorthms. ) relatvely long network lfetme de to low power and battery technology advances, 3) the more and more comptaton power the sensor nodes have. Bascally, each sensor wll montor ts local envronment and they collaborate as a whole to provde nformaton abot the sensor feld. Knowng ts own physcal locaton wth great accracy and precson s vtal for the sensng nt n a WSN to provde ts envronmental nformaton. It s possble to acqre a sensor node s locaton from the rather relable Global Postonng System (GPS). Sch sensor nodes that know ther locatons wll be referred to as anchor nodes or beacons. However, GPS reqres expensve nfrastrctre and costs abot $100/recever. Therefore, many locaton dscovery systems have been proposed recently to compte the sensors postons from lmted nmber of anchors [1, 7, 1, 17, 19], or even wthot anchors [1]. A good locatonng algorthm shold provde hgh accracy on locaton estmates and hgh scalablty wth low energy and commncaton cost. In ths paper, we propose a locatonng algorthm that acheves these goals based on the blateraton prmtve and a delberate anchor deployment scheme. Categores and Sbject Descrptors C..3 [Compter Systems Organzaton]: Network Operatons General Terms Algorthms, Performance Keywords Weghted mltlateraton, Blateraton prmtve, Locaton estmaton, Wreless Sensor Networks 1. INTRODUCTION Wreless sensor networks (WSNs) have fond a lot of mltary and cvl applcatons recently. Some of the reasons that WSN gans ts poplarty nclde 1) the low cost of deployment manly becase of MEMS technology, Fg. 1. Node dstrbton

2 A Motvatonal Example We llstrate the basc dea of or approach by the followng example. Consder the 15 sensor nodes as shown n Fgre 1, the dstance between any two neghborng nodes s eqal to the commncaton range of the sensor node. Ths each node can only commncate wth ts neghbors; for example, node 1 can talk to only nodes and 3, whle nodes,3,4,6,8,9 are all n the commncaton range of node 5. Sppose that we have three anchors, the problem s where shold we deploy these three anchors sch that we can locate the other sensor nodes accrately and effcently. Crrent approaches sch as those n [8] and [14] place the three anchors along the permeter of the area n order to redce locaton estmaton error. In ths scenaro, nodes 1, 11 and 15 wll be chosen as anchors. Both DV-hop and DV-dstance, two state-of-the-art locatonng algorthms [11], wll gve the same locaton estmaton fro the rest of the sensor nodes. If the standard devaton of the range error s 5% (normalzed to the commncaton range). Based on these two approaches, the average locaton error of these sensor nodes wll be abot 35% (normalzed to the commncaton range). However, f we place the three anchors at nodes 5, 8 and 9, node 6 wll be able to locate tself. Ths s becase that althogh node 6 has only two anchors 5 and 9 wthn ts commncaton range, ts mrror mage (poston where node 8 locates) can be elmnated by sng anchor 8 as a reference: node 6 cannot talk to node 8, bt ts mrror mage can. Smlarly we can locate nodes 4 and 13. Once these three nodes are dscovered, they behave as anchors and we can teratvely determne the locaton of other nodes by sch blateraton calclaton. In ths approach, the average locaton error s abot 5% for nodes 4,6 and 13; 10% for nodes,3,7,10,1 and 14; and 15% for nodes 1, 11, and 15. The overall locaton error wll be abot 10%, mch more accrate than the 35% by the tradtonal approaches. Ths example llstrates or approach: deployng three anchors close to each other and locate other sensor nodes arond them expansvely. We expect small locaton errors becase dstance measrement s performed only between drect (one-hop) neghbors. The proposed algorthm wll also have a low commncaton and energy cost becase there s no need to flood anchor s poston to the entre network or to refne locaton. Frthermore, less anchors wll be sed to dscover a gven sensor feld becase three anchors are pt together as a grop and the sensors are located expansvely. These expectatons are valdated by theoretcal locaton estmaton error analyss and extensve smlatons. Paper Organzaton The rest of the paper s organzed as follows. In Secton, we srvey the exstng locaton dscovery algorthms. We then formlate the problem n Secton 3. We elaborate the anchor deployment scheme and the blateraton prmtve n Secton 4. In Secton 5, we analyze the performance of or proposed approach n terms of commncaton cost, comptaton cost, coverage, and locaton accracy. The smlaton reslts are presented n Secton 6 before we conclde n Secton 7.. RELATED WORK Many algorthms are proposed to solve the sensor localzaton problem n the past several years. We can grop these algorthms accordng to dfferent crtera: nfrastrctre-based or nfrastrctre free (ad-hoc); centralzed or dstrbted; range-based or range-free; teratve or non-teratve; etc. The algorthms n the same grop have smlar problems, so we can actally analyze the common characterstc of these grops n order to have a global pctre of these locatonng algorthms. Infrastrctre-based algorthms [, 1] rely on an external nfrastrctre to locate sensors. Ths knd of approach may prodce good reslts, bt t s not a favorable method for ad-hoc wreless sensor networks. Centralzed algorthms [4,18] can prodce hgh accracy reslts, bt t reqres sgnfcant comptaton and commncaton. Snce sensor nodes are often lmted n power and comptatonal capactes, ths s not a good choce ether. The recent research work [6] by He et al. dvdes the locatonng algorthms nto two categores: range-free and range-based. They propose APIT (All Pont-In- Tranglaton Test), whch calclates the center of gravty of the ntersecton of all of the trangles n whch a sensor resdes to determne ts locaton. However, ther algorthm reqres large anchor range that s 10 tmes the normal sensor range. Ths assmpton may ncrease the nstallaton cost of sensor networks. Another range-free algorthm [18] by Shang et al. only needs three anchors to locate all the sensor nodes n the network. Bt ths approach s ntrnscally a centralzed approach wth sgnfcant comptaton (O( n 3 ) ) and commncaton cost. In general, althogh range-free approach [, 6, 18] does not rely on range estmaton, t can only prodce coarse-graned reslts that may not be sed n applcatons wth strngent accracy reqrements. Langendoen and Rejers present a qanttatve comparson between three range-based locatonng schemes [8]. Ths detaled comparson gves s some basc dea on the common characterstc of the crrent locatonng algorthms. As stated n the paper, three phases

3 are nclded n the algorthm: anchor dstance estmaton; ntal locaton estmaton and locaton refnement. However, a general drawback of these algorthms s bg anchor dstance estmaton error ntrodced by the frst phase of these algorthms. When dstance nformaton s propagated over several hops, the fnal dstance estmaton error becomes sgnfcant, especally for sparse and rreglar networks. Ths large dstance estmaton error leads to large locaton error of ntal locaton estmaton, whch n trn reslts n a heavy brden for refnement phase. As mentoned n [8], most commncaton cost s consmed n the refnement phase and system coverage s also decreased snce some locaton estmatons cannot be mproved to be acceptable. Iteratve mltlateraton algorthm [16] by Savvdes et al., localzed locaton dscovery algorthm [9] by Megerdchan et al., and poston dssemnaton algorthm by Albowcz [0] et al. descrbe the se of teratve algorthm to locate sensor nodes n wreless sensor networks. The appealng featre of teratve algorthm s free of anchor dstance estmaton and locaton refnement. However, t reqres hgh connectvty or hgh percentage of anchors that sgnfcantly ncrease the nstallaton cost. In addton, error accmlaton shold also be consdered de to the se of nknown nodes as anchors. The expansve locatonng algorthm proposed n ths paper s ntrnscally a range-based teratve algorthm. Compared wth other teratve algorthms, we can acheve smlar locaton accracy wthot hgh connectvty or hgh anchor percentage. The detaled descrpton of the algorthm wll be gven n Secton PROBLEM FORMULATION Consder a sensor feld S wth nknown nmber of sensors and no sensor knows ts locaton. However, sensors wthn the commncaton range R can talk to each other and therefore measre the dstance between them. We assme that sch measred dstance carres an error that s modeled as an ndependent Gassan random varable wth zero mean and varance σ. That s, f node and node j are located at postons (x,y ) and (x j,y j ) respectvely, then ther measred dstance satsfes the followng nmber of anchors to determne the locatons of the sensors n feld S precsely and effcently? The solton to ths problem contans two parts. Frst, t has to specfy the anchor deployment scheme; second, once all the anchors are deployed, we have to develop an estmaton algorthm. The accracy s measred by the locaton error. The effcency s measred by the energy and commncaton cost. 4. ALOGORITHM DESCRIPTIONS The algorthm flow dagram s shown n Fg. below. The npt of the algorthm s the locatons of the anchors. Then a sensor has three or more neghborng anchors wll be located by sng weghted mltlateraton method, f both angle test (sed to avod large locaton error) and reference test (sed to avod wrong locaton solton) are passed. And the located sensor wll become an anchor. Otherwse, a sensor wthn the commncaton range of only two anchors wll be located throgh blateraton prmtve, f both angle test and reference test are passed. And the located sensor wll become an anchor. Then we wll locate the next sensor. The man contrbton of ths paper s to pt three anchors as a grop and to locate nknown sensors expansvely. Qte dfferent from other algorthms that pt anchors separately n the feld, we sggest pttng three anchors together and locatng nknown sensors arond them teratvely. When the nknown sensors arond these three ntal anchors are located, they become anchors and they help to locate the sensors arond themselves, whch cold be two hops away from the ntal anchors. Sch a process contnes, and more and more sensors wll be located. The deployment of the ntal three anchors wll be dscssed n secton 4.1. d ˆ = d + e = ( x x ) + ( y y ) + e (1) j j j j j j where d j s the real dstance between nodes and j n the two-dmensonal feld S and e j s the estmaton error. We now deploy M anchor nodes nto the same feld S at locatons (X,Y ) for =1,,,M. Becase the anchor nodes know ther physcal locatons, ther neghbors (the nodes wthn the the commncaton range R from anchor nodes) can obtan locaton nformaton from them and then locally estmate ther own locatons. We consder the followng problem: for a gven sensor feld S, how to se the mnmal Fg.. Algorthm flow dagram

4 In random deployed wreless sensor network, t s possble that sensor deployments n some drectons of the network are sparse and rreglar. Then n those drectons we may not have three anchors to locate the nknown sensor, so we propose blateraton algorthm that can locate a sensor wth jst two anchors. Ths blateraton prmtve needs at least an addtonal anchor two hops away from the nknown sensor to make a decson between two possble soltons prodced by two one-hop anchors. Dfferent from other approaches that an addtonal anchor may not be avalable, the sensors are located expansvely from the ntal anchors n or algorthm, ths we always have addtonal two-hop anchors on the opposte drecton of system expanson. The detals of locatng sensors n the next rond, both weghted mltlateraton and blateraton prmtve, wll be descrbed n secton 4.. In addton, reference test, whch s sed to choose one solton from blateraton prmtve or solve collnear anchors problem ; and angle test, whch s sed to lmt error propagaton speed are presented n secton Deployment of Intal Anchors In ths locaton dscovery algorthm, we sggest pttng three anchors as a grop and several sensors randomly near these three anchors. These sensors wll be located at the begnnng and they wll act as anchors to assst locatng more sensors. There are two problems that shold be consdered when pttng ntal three anchors. Frst, how far away are these anchors from each other? If no sensor s delberately placed near these anchors, the dstances between anchors shold be at least eqal to the commncaton range or t s possble that no sensor can be located at the begnnng. Meanwhle, they stll shold be smaller than two tmes the commncaton range even f some sensors are delberately deployed. However, the dstances cannot be too small or fewer sensors wll be benefted by the ntal anchors and locaton estmaton error wll be larger. In or algorthm, dstances between anchors are set to be 1.5 tmes the commncaton range. Second, how many sensors do we need to be delberately deployed? It s good to deploy more sensor nodes snce more sensors wll act as anchors at the begnnng. Bt too many addtonal sensors wll ncrease the nstallaton cost wth lttle nflence on the locaton accracy. In or algorthm, the nmber of addtonal sensors s set to be 7. Pttng three anchors together can be easly realzed when anchors are deployed manally or by robots. However, when anchors are dropped from an arplane, some technqes are needed to lmt the dstances between the anchors. A possble approach s sggested here: Three anchors, several sensors and a tmer are bond together by several sprngs. The anchors and sensors wll be sprng ot when they are close to grond. (Ths can be controlled by the tmer) Three anchors wll be away from each other wthn approprate dstances and the sensors wll be scattered arond the anchors. 4. Locatng Sensor n the Next Rond 4..1 Weghted Mltlateraton Algorthm It has been dscssed extensvely that a sensor can be located when t s wthn the commncaton range of three or more than three anchors [9, 11, 14, 15, 16, 0]. A weghted mltlateraton algorthm s sed to mnmze the resdal of locaton estmaton. Actally, we are tryng to mnmze f x, y) = ω ( ( x x) + ( y y) d ) () one hop anchors ( Where ( x, y) refers to nknown sensor locaton, ( x, y ) = 1,, 3... n refers to the locaton of the th onehop anchor, dˆ refers to the measred dstance between nknown sensor and the th one-hop anchor and ω refers to the weght assgned to the th one-hop anchor. The hgher certanty of the locaton of the one-hop anchor, the larger the weght. The detaled weght assgnment scheme wll be dscssed n secton 5.4. Rather than lnearzng ths nonlnear least sqare problem [15, 16], we prefer to solve ths nonlnear least sqare problem drectly n order to ncrease locaton accracy. In addton, locaton estmaton wth bg resdal wll not be accepted. However, nonlnear least sqare comptaton reqres more comptaton cost. We wll talk abot ths comptaton vs. accracy tradeoff n secton Blateraton Prmtve When the nknown sensor can only talk to two anchors, we wll try to fnd ts locaton wth only two anchors. As seen from Fg.3, gven the locaton of anchor M ( x, y ) and N ( x, y ), and the dstance to the nknown sensor, r and N r N, the locaton of the nknown sensor can be compted based on geometry relaton between sensor and two anchors. Ths the key pont to locate a sensor wth two anchors s to choose between two soltons (A and B, as shown n Fg.3). In order to solve ths problem, reference test s presented n secton In addton, reference test can be sed to solve collnear anchors problem [17, 0], whch refers to the staton that all the anchors (>) are on the same lne Reference Test In or algorthm, the one-hop anchors wll send a specfc nmber of addtonal anchors locaton as well as range measrement nformaton to the nknown sensor. (We lmt M ˆ M 1

5 Fg.3. Locate a sensor wth two anchors the nmber of addtonal reference anchors becase we want to lmt the commncaton cost) So the wrong solton wll be recognzed f t s wthn the commncaton range of these addtonal reference anchors. Reference test can be sed n two statons: choose the rght solton from blateraton prmtve and fnd the locaton when anchors (>) are on the same lne. The detals of reference test are smmarzed as psedo codes n Fg.4 and Fg.5 for both statons: I. When the nknown sensor talks to two neghborng anchors: Inpts: A(Solton A), B(solton B), commncaton range R, other reference anchors Otpts: Correct solton Correct_solton (A, B, R, reference anchors) N = 0 A N B = 0 for each anchor C n the reference anchors d AC = ( x A xc ) + ( y A yc d = ( x x ) + ( y y ) f BC B c B c dac < R NA = NA + 1 Fg.4. Reference test for solton selecton In the psedo code from Fg.4, N and N refer to nmber of addtonal anchors that can be heard for two soltons A and B. If the nmber of addtonal anchors can be heard s greater than one, ths solton wll be dscarded. A ) end f dbc < R NB = NB + 1 end end f N A >= 1 & N B >= 1 retrn No solton else f N A >= 1 retrn B else f N B >= 1 retrn A else retrn No solton end B If the nmber of addtonal anchors can be heard s jst one, we wll check the nmber of heard addtonal anchors of another solton to decde whch solton s correct. Ths addtonal check s de to the possblty that an addtonal anchor s moved closer to the nknown sensor nder the nflence of locaton estmaton error. When No solton s retrned, two-anchor locatonng wll not be sed. II. When the nknown sensor talks to more than two neghborng anchors: When the nknown sensor talks to more than two anchors, we wll stll check the reslt sng addtonal anchors to avod collnear anchors problem. The reslt wll not be accepted f t s wthn the commncaton range of any addtonal anchor. Then we can pck two one-hop anchors wth the angle of the nknown sensor closest to 90 0 to perform blateraton calclaton n order to get the reslt. Inpts: A(Solton A), commncaton range R, other reference anchors Otpts: Correct solton Correct_solton (A, R, reference anchors) N A = 0 for each anchor C n the reference anchors d f = x x y AC ( A c ) ( A c dac < R NA = NA y ) end end f N A >= 1 retrn No solton else retrn A end Fg.5. Reference test for collnear anchors problem 4..4 Angle Test Geographc dlton of precson (GDOP) can be sed to characterze the nflence on sensor locaton estmaton from geometry of anchors. A clear representaton of GDOP s gven n [1]: N GDOP = (3) ΣΣ, sn( θ ) j j> j Where N s the nmber of reference anchors and θj s the angle between anchor and anchor j. Eqaton (3) shows that the locaton error wll be very large f all the angles between pars of anchors are very small or all are close

6 toπ. Ths we compte the angles between pars of anchors based on the ranges and anchors locaton, and to see f any of them s wthn the angle lmtaton range[ α, π α]. Locaton estmaton wll not be performed when none of the angles s n ths range, whch means the anchors are clstered to each other ( < α) or almost on the same lne ( > α). The reason why we se angle test nstead of sng GDOP s that angle test has smlar fncton as GDOP and we jst want to screen ot very bad topology rather than to know the exact nflence from the anchor geometry. Actally, whereas GDOP only consders range error, we can estmate the sensor locaton error that consders both range error and anchor locaton error. The detals of locaton estmaton are gven n secton 5.4. The angle lmtaton range wll ncrease when connectvty ncreases or range error becomes smaller. In or smlaton, α s taken as 5 0, when connectvty s 9 and range error s 1%. 5. PERFORMANCE ANALYSES 5.1 Commncaton Cost Smlar to other localzaton approaches[8, 14, 16], we sppose the commncatons between the neghborng nodes are n the form of broadcasts. Then the commncaton cost wll depend on the nmber of broadcast messages a node transmts and receves. In or algorthm, t s clear that each node wll jst broadcast one packet when ts locaton s avalable. And the average nmber of receved broadcast packets by one node wll be two, three or for. (Snce we set the commncaton lmt to be for, only nder very poor geometry of anchors do we need more than for anchors) Ths makes n total at most fve messages per node. Becase no poston refnement s needed n ths algorthm, commncaton cost wll be sgnfcantly redced. 5. Comptaton Cost The comptaton performed at each node nvolves the comptaton of nonlnear least-sqares estmaton. Actally the comptaton cost of nonlnear least sqare comptaton can be translated to n tmes lnear least sqare comptaton, here n refers to the nmber of teratons nvolved n the nonlnear least sqare comptaton. (Average of n s abot 10 from smlaton reslts). Ths the comptaton cost of or algorthm wll be hgher than other algorthms when only one locaton comptaton on one node s compared. However, or overall comptaton cost wll stll be less snce other algorthms need to perform lnear least sqare comptaton a lot of tmes [8] n the locaton refnement phase. Generally, a common processor can handle sch comptaton and the energy consmpton wll be manly reslted from commncaton cost. 5.3 Coverage Sppose that an error threshold s set to be the maxmm acceptable error for a located sensor. We can see that as long as a sensor s not far away from ntal anchors, t wll fnd ts locaton n or algorthm. Whle on the other hand, the tradtonal approaches may not locate a sensor becase comptaton performed at the sensor cannot converge. In addton, wth smaller range error or ncreased network densty, more sensors wll be located wth three ntal anchors. From smlaton reslts n secton 6, sensors that can be located wll be extended far away from the ntal anchors when range error s 1%. And sch coverage wll be mproved frther f range error s smaller. We can later learn from secton 6 that the coverage of or algorthm s mch hgher than other algorthms, becase other algorthms may not converge at some sensor nodes de to the large dstance estmaton error. If the whole feld wth many three anchor grops s consdered, the coverage area wll depend on how three anchor grop are deployed. If they are deployed n a grdlke manner, the coverage wll be the largest. If they are deployed randomly, there wll be overlaps between the felds that can be covered by several anchor grops. Then the coverage wll be smaller or we may need more anchor grops to mantan the coverage. If anchor grops are nformly dstrbted n a random manner, two tmes anchor grops are needed to mantan the coverage. Accordng to the smlaton reslts n secton 6, even f two tmes anchor grops are needed, the anchor percentage wll stll be very low. 5.4 Accracy In ths secton, we descrbe how to estmate locaton error of each sensor and how to assgn weght to each anchor n order to mnmze the locaton estmaton error. The basc steps can be lsted as follows: 1. Estmate approxmate locaton of the sensor. Weght assgned to each neghborng anchor 3. Estmate fnal locaton of the sensor 4. Estmate locaton error of the sensor In addton, we present a geometry explanaton of locaton error to show how anchor locaton error affects sensor locaton error Locaton Error Estmaton In ths expansve locatonng algorthm, error propagaton can be a seros problem. The farther the nknown pont s from the ntal three anchors, the larger the locaton error. Wthot carefl control of the error propagaton, the locaton estmaton several hops away from the ntal

7 anchors wll be nacceptable. An anchor wth large locaton error wll adversely affect the locaton error of ts neghborng sensors. In order to prevent sch anchors from propagatng ther errors to the whole network, we propose a novel weght assgnment scheme to decrease the nflence from large error anchors. Qte dfferent from other weght assgnment scheme [14] before, or weght assgnment does not only depend on anchors locaton error, bt also range error and the relatve locaton between the nknown sensor and anchors. Whenever a sensor s located and becomes an anchor, we wll estmate ts locaton error. And when ths anchor s sed to locate other sensors, we wll assgn weght to ths anchor based on ts locaton error estmaton, range error and ts relatve locaton to the nknown sensor. We need one lnear least sqare comptaton to get the approxmate locaton of the nknown sensor (sed to assgn weght to anchors) and one weghted nonlnear least sqare comptaton (the approxmate locaton s sed as the ntal vale) to get the fnal locaton estmaton. Sppose the nknown sensor ( x, y ) s wthn the commncaton range of several reference anchors ( x y ) = 1,,3,4,... n., Fg. 6. Sensor locaton estmaton wth anchors As shown n Fg.6, we sppose the estmated nknown sensor to be x, y ) and estmated reference anchors to ( r be ( x, y ) x = 1,,3,4,... n. x y y d = + j s d d r the anchor dstance vector, e = x + j ys the anchor r locaton error vector and e = x + j y s the nknown sensor locaton error vector. (Note that and j do not mean the ndex of the anchor. It s the nt vector on x and y drecton. In addton, error vectors are bdrectonal de to ther random natre) Dstance between nknown sensor and reference anchor: d = x x + y y ( ) ( ) (4) Dstance between estmated nknown sensor and estmated reference anchor: d x x y y = ( ) + ( ) (5) By sng Taylor s seres arond the estmated anchor locaton and estmated sensor locaton, we can obtan: x x x x y y y y d = x x + y d d d d where d = d d = 1,,3, 4,... n y (6) d conssts of two error components: range error and least sqares estmaton error. We gnore the latter component snce t s relatvely small compared wth the range error. (Bg resdal reslt wll not be accepted n least sqares estmaton) From eqaton [], we may notce that weght s assgned to anchor nodes to ensre each anchor node has an approprate level of nflence on locaton estmaton. A hgh qalty anchor nflences the reslt more than a low qalty anchor. Optmal reslts, whch mnmze the ncertanty n the locaton estmaton, are obtaned when the weghts, ω, assgned to each anchor are nversely proportonal to the varances at each combnaton of predctve varable vales of each anchor.[] Ths n eqaton (), ω shold be nversely proportonal to var( ( x x) ( y y) d) +, whch s approxmately eqal to x var x y y ( d + x + y ), note that d d nknown sensor s locaton ( x, y ) shold be consdered accrate when calclate the varance. By lnearzng eqaton (), we get the approxmate sensor locaton estmaton throgh lnear least sqare comptaton [14, 16]. And we assgn weght based on ths approxmate sensor locaton reslt. After weght assgnment part, eqaton () s sed to solve the fnal sensor locaton.(throgh nonlnear least sqare comptaton) In order to estmate the error of ths fnal solton, we wrte (6) n a compact matrx formlaton as: x x1 y y 1 x x1 y y 1 d1+ x1+ y1 d1 d1 d1 d1 x x y y x x y y (7) d+ x+ y x W d W d = d d y x xn y y n x xn y y n dn+ xn+ yn dn dn dn dn

8 where W s the weght matrx, a dagonal matrx n whch w s the weght of the th anchor. Sppose, d ' = d x + x d x + then we get smplfed form of (7): y y d y = 1,,3... n W d ' = WA X (8) Solve poston error throgh weghted least sqares method and take covarance of both sdes, t s not hard to get cov( X WA W d W WA ) ' T T ) = ( ) cov( ) (( ) (9) T T where ( WA ) 1 = (( WA) WA) ( WA) (psedo nverse matrx of WA ). We assme that the locaton errors of dfferent sensors are ncorrelated. (It s not exactly tre snce one sensor may derve ts locaton based on another sensor s locaton nformaton. Bt the covarance s qte ' ' small compared wth the varance of ) Then co v( ) can be approxmated as a dagonal matrx and the dagonal element s d, d ' x x y y var( d ) = var( d ) + var( x + y ) (10) d d where the frst component s the varance of the range error, and the second component can be derved from the locaton error of the th anchor, whch s compted n the last rond. Ths we can estmate the locaton error of each sensor node by eqaton [9]. And locaton accracy wll be mproved snce or weght assgnment scheme not only consders locaton error, bt also range error and relatve locaton between nknown sensor and anchors Geometry Interpretaton of Locaton Error rr Gven the defnton of d, e, eqaton (7) can be smplfed as: (Here refers to nner prodct) r r r r d1 + d1, e 1 d1, e r r r r d + d, e d, e = r r r r dn + dn, e n dn, e (11) The left sde of eqaton (11) conssts of two parts d and r r d, e (shadow vector of e r on d r ). Actally, d + d r, e r can be vewed as eqvalent range error when anchor locaton error s consdered. Ths the nflence on the locaton estmaton of the nknown sensor from both anchor locaton error and range error becomes clear. We can see from eqaton (11) that when anchor locaton error ( e ) s smaller than range error ( d ), sensor locaton error s decded manly by range error; whereas when anchor locaton error s larger, sensor locaton error wll be manly decded by anchor locaton error. rr Frthermore, we notce from (11) that d, e s confned r r by d + d, e, whch means that the length of the shadow vector of e on d r wll be approxmately eqal to the eqvalent range error. Ths tells s that sensor locaton error wll be confned n the drectons of d r = 1,,3, 4,...n. However, locaton error can be r very large n the drectons that are away from d. Ths more anchors wll not help to get accrate estmaton f they are on the almost same drecton of the nknown sensor. Frthermore, snce these vectors are b-drectonal, we can get accrate locaton estmaton even f anchors are on the one sde of the nknown sensor, whch s the staton for or algorthm. Althogh the above argment s not a mathematcally rgoros proof, t does contan the key dea that anchor locaton error affects sensor locaton error throgh ts shadow on the dstance vector; and the sensor locaton error wll be confned n each drecton of the dstance vector. We shold not assgn weghts jst based on the ncertanty of the anchors; range error and the relatve locaton between an anchor and the nknown sensor shold also be consdered. 6. SIMULATION RESULTS The goal of the smlaton performed n ths secton s to nvestgate the characterstcs of the followng parameters: 1. Locaton estmaton error vs. 1) Commncaton lmt ) Range error 3) Commncaton range (connectvty) 4) Sensor deployment pattern 5) Intal anchor deployment pattern

9 . Coverage (located sensor percentage) 3. Commncaton cost In or experments, we ran ths expansve locatonng algorthm on varos topologes of networks n Matlab. The reslt s the average of 500 topologes. Three anchors are placed as an eqlateral trangle. And 7 (Ths s a herstc nmber as we mentoned n secton 4.1) sensors are randomly placed nsde the crcle crcmscrbng the trangle. Two sensor placement models are consdered: 1) nform placement, n whch sensors are placed nformly wthn a crcle area ) grd placement, n whch sensors are placed n a grd-lke poston wthn a crcle area. The grd length s normalzed to 1. The sensors are placed on grd ponts wth placement error modeled as Gassan noses N(0,0.5 ). The center of the crcle area s set to be (0,0) and the rads s set to be 5 n order to mantan a smlar anchor percentage wth other algorthms. Node densty n ths crcle area s eqal to one. We change the commncaton range from 1.6 to.6. The range error s modeled as N (0, er ), n whch er changes from to 0.05 (normalzed to commncaton range R). Anchor percentage s abot 3.8% based on ths setp. An nknown sensor can receve a message from each of ts one-hop neghbor anchors. However, recevng more messages wll ncrease the commncaton cost. So we set a commncaton lmt to lmt the nmber of messages an nknown sensor can receve. To get a tradeoff between accracy and effcency, we nvestgate how the average locaton error wll change wth commncaton lmt that lmts the nmber of messages ths nknown sensor can receve. As shown n Fg. 7, the smlaton reslts sggest that the locaton error s very large when commncaton lmt s set to 3. However, there s a sharp decrease between 3 and 4. And when commncaton range s larger than 4, the locaton error gradally decreases bt the dfference s rather small. Ths, n or experment, the commncaton lmt s set at 4 to get a better tradeoff between locaton accracy and commncaton cost. Jst as what we dscssed n secton 5.4, the reason why locaton error decreases when commncaton lmt ncreases s we have more drectons. Consderng the random choce of neghborng anchors, the satraton phenomenon s becase we already have enogh drectons to lmt the locaton error of the sensor. Fg 8 and Fg 9 show the average locaton error nder nform deployment pattern and grd deployment pattern. Based on the fgres, locaton error becomes smaller when commncaton range (connectvty) ncreases. Ths s becase that the nknown sensor can talk to more anchors, ths locaton error s confned n more drectons. In addton, grd deployment prodces better reslts than nform deployment becase grd deployment does not have bad sensor topologes whereas nform deployment may have bad sensor topologes that wll reslt n large locaton error. When range error s smaller than 1%, the average locaton error wll be always smaller than 4% (both normalzed to commncaton range). Fg.10 and Fg.11 show the changes of the system coverage nder dfferent sensor deployment patterns. When connectvty s 7, range error s 1%, the coverage wll be more than 78% nder nform deployment and 98% nder grd deployment. However, other approaches [8] cannot acheve hgh coverage and small locaton error at the same tme. As mentoned n [8], coverage of these tradtonal approaches wll drop by 50% when locaton refnement phase s performed Bt when refnement phase s not performed, the locaton error of these approaches wll be large (more than 5%) even f the range error s very small (eqal to zero) and connectvty s very hgh (eqal to 15). (Fg.11 n [8]) Ths or algorthm can acheve hgh accracy as well as hgh coverage. When anchors are dropped from the arplane, the relatve locaton between ntal three anchors cannot be precsely controlled. Snce ntal anchor deployment pattern may nflence the average locaton error, we nvestgate the followng fve ntal anchor deployment patterns: Pattern1: {(0, 1.731), (-1.5, ), (1.5, )} Pattern: {(0, ), (-1.3, ), (1.3, )} Pattern3: {(0, ), (-1.0, ), (1.0, )} Pattern4: {(0, ), (-1.6, ), (1.6, )} Pattern5: {(0, ), (-1.4, ), (1.4, )}. Each pattern refers to the locatons of three anchors, as shown n Fg. 1. The reslts ndcate that average locaton errors don t change mch wth deployment patterns and ths ths algorthm s robst to the anchor deployment pattern. In addton, the larger the dstance between the ntal anchors, the smaller the average locaton error. Ths can be clearly recognzed from the crves of pattern 1, and 3. Actally, when anchors are farther from each other, more sensors wll be benefted by the ntal anchors, ths locaton estmaton accracy s mproved. A comparson between MDS-MAP approach [18], Smdst approach [8], DV-hop approach [8, 11, 15] and or Expanson approach s gven n Table.1. The range error s 5% for all these approaches. Under smlar anchor percentage, connectvty and range error, or expansve approach can acheve better accracy, hgher coverage and sgnfcantly less commncaton cost. Detaled commncaton cost s not gven n MDS-MAP paper[18], bt ths approach s essentally a centralzed method whch reqres large commncaton cost. Or algorthm prodces very good reslts when range error becomes smaller, as shown n Table.. Table. descrbes the localzaton pper bond ( ) as a fncton of locaton L

10 error pper bond( E ) and range error ( e ). Here E refers to the largest locaton error that can be accepted by locaton-based applcatons, refers to the expected largest extendng dstance from ntal anchors gven error pper bond and range error. The smlaton s performed wth connectvty eqal to 11. When range error s 1% and locaton error pper bond s 40%, localzaton pper bond becomes Ths means nder smlar connectvty level wth other approaches, anchor percentage can be less than 0.1%! Ths reslt s mch smaller than other approaches that reqre anchor percentage to be at least 5%. Moreover, when range error s 0.1%, small locaton error can be acheved as well as low anchor percentage. When localzaton pper bond s fxed to be 50, range error at 0.1% and connectvty eqal to 11.6, less than 5% average locaton error can be acheved wth only 0.04% anchor percentage! The objectve for localzaton n wreless sensor networks s not only to mnmze locaton estmaton error, bt also to mnmze cost, whch ncldes nstallaton cost, comptaton cost and commncaton cost. In conclson, the advantage of or approach s to locate sensor wth less localzaton error, less nstallaton cost (mch less anchors needed) and less commncaton cost. Encoragng reslts are obtaned when range error s small. (<1%) L r Fg. 8. Average locaton error vs. commncaton range nder nform deployment Fg. 9. Average locaton error vs. commncaton range nder grd deployment Fg. 7. Average locaton error Vs commncaton lmts (Range error 0.01, Unform deployment) Fg. 10. Average coverage vs. commncaton range nder nform deployment

11 Table.1. Performance comparson MDS- MAP Smdst DVhop Expanson Anchor percentage 4% 5% 5% 3.8% Connectvty Locaton error 1% 17% 17% 13% Coverage 100% <60% <60% 89% Commncaton messages large <5 Table.. Localzaton pper bond L Fg. 11. Average coverage Vs commncaton range nder grd deployment e r E Fg. 1. Dfferent ntal anchor deployments 7. CONCLUSION In ths paper, we propose an expansve locatonng approach. The novelty of ths paper s to pt three anchors together, and locate sensors expansvely arond them. To the best of or knowledge, we are the frst to propose sch localzaton scheme. Extensve theoretcal error analyses are presented n order to control error propagaton. In addton, blateraton prmtve s sed to mprove coverage and collnear anchors problem s solved. Or smlaton reslts demonstrate that or algorthm can derve sensor locaton wthn 4% locaton error and mch less commncaton cost compared wth other locatonng algorthms. It provdes a sefl tool for locaton dscovery n WSNs. 8. REFERENCES Fg. 13. Average locaton error for dfferent patterns of ntal anchor deployment [1] Bahl, P., Padmanabhan. V.N., RADAR: An In-Bldng RF- Based User Locaton and Trackng System. IEEE INFOCOM. Vol.., pp , 000. [] Bls, N., Hedemann, J., Estrn, D., GPS-Less Low Cost Otdoor Localzaton for Very Small Devces, IEEE Personal Commncatons, Specal Isse on Smart Spaces and Envronments, Vol. 7(5), pp. 8-34, 000. [3] Capkn, S., Hamd, M., and Hbax, J.-P., GPS-free postonng n moble Ad-Hoc networks. Hawa Internatonal Conference on System Scences, pp , 001.

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