GPS-free positioning in mobile Ad-Hoc networks

Transcription

1 Proceedngs of the 34th Hawa Internatonal Conference on Sstem Scences GPS-free postonng n moble Ad-Hoc networs Srdan Čapun, Maher Hamd, Jean-Perre Hubau EPFL - Communcaton Sstems Department (DSC) Ecole Poltechnque Federale de Lausanne (EPFL) CH-1015 Lausanne, Swtzerland {srdan.capun, maher.hamd, jean-perre.hubau}@epfl.ch Abstract In ths paper we consder the problem of node postonng n ad-hoc networs. We propose a dstrbuted, nfrastructure-free postonng algorthm that does not rel on Global Postonng Sstem (GPS). The algorthm uses the dstances between the nodes to buld a relatve coordnate sstem n whch the node postons are computed n two dmensons. The man contrbuton of ths wor s to defne and compute relatve postons of the nodes n an ad-hoc networ wthout usng GPS. We further eplan how the proposed approach can be appled to wde area ad-hoc networs. 1 Introducton The presented wor s a part of the Termnode project, a 10 ear ongong project at Swss Federal Insttute of Technolog (EPFL). The project nvestgates large area, wreless, moble networs referred to as moble ad-hoc wde area networs [1]. The man desgn ponts of the project are to elmnate an nfrastructure and to buld a decentralzed, self-organzed and scalable networ where nodes perform all networng functons (tradtonall mplemented n bacbone swtches/routers and servers). In ths paper we propose an algorthm for GPS-free postonng of the nodes n an ad-hoc networ. Our goal s to show that n the scenaros where an nfrastructure does not est and GPS cannot be used, there s a wa to obtan postons of the nodes b dstrbuted processng. GPS-free postonng s desrable, notabl when the GPS sgnal s too wea (e.g., ndoor), or when t s jammed, or f for cost or ntegraton reasons the ncluson of a GPS recever has to be avoded. We ntroduce a dstrbuted algorthm that enables the nodes to fnd ther postons wthn the networ area usng onl ther local nformaton. The algorthm s referred to as the Self-Postonng Algorthm (SPA). It uses range measurements between the nodes to buld a networ coordnate sstem. The Tme of Arrval (TOA) method s used to obtan the range between two moble devces. We show that despte the range measurement errors, and the moton of the nodes, the algorthm provdes enough stablt and locaton accurac to sustan basc networ functons. The basc dea of the algorthm s llustrated n Fg. 1. The nodes n the ad-hoc networs are not usuall aware of ther geographcal postons. As GPS s not used n our algorthm, the algorthm provdes relatve postons of the nodes n respect to the networ topolog. The algorthm can be useful n the scenaros where the locaton nformaton s used to support basc networ functons. Two eamples of ths nd of applcaton scenaro are Locaton Aded Routng [8] and Geodesc Pacet Forwardng [7]. In Geodesc Pacet Forwardng algorthm, the source sends pacets n the phscal drecton of the destnaton node. When the omn-drectonal antennas are used, Self-Postonng Algorthm provdes enough nformaton to ever node to support Geodesc Pacet Forwardng. Gven that the node nows ts poston and the postons of the destnaton node n the relatve coordnate sstem, t s able to compute n whch drecton (to whch net hop node) to send pacets. Ths paper s organzed as follows. Secton 2 descrbes the related wor n the feld of rado-locaton technques. In secton 3 we present the algorthm for buldng a local coordnate sstem at each node. In secton 4 we descrbe the means to defne the center and the drecton of the networ coordnate sstem. In secton 5 we dscus the nfluence of the range errors on the accurac of the poston estmaton. We present the smulaton results n secton 6. 2 Related wor Followng the US FCC regulatons for locatng E911 callers, postonng servces n moble sstems have /01 $10.00 (c) 2001 IEEE 1

2 Proceedngs of the 34th Hawa Internatonal Conference on Sstem Scences d a m d l c b l n d lj j => ( 3,1) ( 3, 3) ( 2, 3) ( 2, 1) ( 1, 1) ( 1, 2.5) (1, 3) (4,1) (3, 0.5) (0.5, 1.5) Fgure 1: The algorthm uses the dstances between the nodes and bulds the relatve coordnate sstem. The algorthm s referred to as the Resdual Weghtng Algorthm (Rwgh). In [6] Rose and Yates gve a theoretcal framewor of the moblt and locaton tracng n moble sstems. The present a stud of moblt tracng based on user/servce/host locaton probablt dstrbuton. Recentl, some poston-based routng and pacet forwardng algorthms for the ad-hoc networs have been proposed n [8, 7, 14, 15]. In all these algorthms t s assumed that the postons of the nodes are obtaned through GPS. To the best of our nowledge, no algorthms have been proposed for postonng of the nodes wthout GPSnad-hocnetwors. 3 Local coordnate sstem drawn much attenton recentl. The new regulatons ntroduce strngent demands on the accurac of moble phone locaton. The FCC requres that b October 2001, the wreless operators locate the poston of emergenc callers wth a root mean square error below 125m [9]. Several rado-locaton methods are proposed for locatng the Moble Statons (MSs) n cellular sstems [2]: the Sgnal Strength method, the Angle of Arrval (AOA) method, the Tme of Arrval (TOA) and Tme Dfference of Arrval (TDOA) methods. The Tme of Arrval and Sgnal Strength methods use range measurements from the moble devce to several base statons to obtan ts poston. Thus, the accurac of the estmated poston depends on the accurac of the range measurements. Dstance measurements are corrupted b two tpes of errors: Non-Lne of Sght (NLOS) error and measurng error. The measurements n cellular sstems, taen b Noa [3], show that NLOS error domnates the standard measurement nose, and tends to be the man cause of the error n range estmaton. The also show that the locaton estmaton error lnearl ncreases wth the dstance error. Followng these measurements, Wle and Holtzman propose a method for the detecton and correcton of NLOS errors [5]. The show that t s possble to detect a NLOS envronment b usng the standard devaton of the measurement nose and the hstor of the range measurements. The propose a method for LOS reconstructon and the show that the correcton s onl possble f the standard measurement nose domnates the NLOS error. A dfferent approach s presented n [4] b Chen. Chen shows that f the NLOS measurements are unrecognzable, t s stll possble to correct the locaton estmaton errors, f the number of range measurements s greater than the mnmum requred. In ths secton we show how ever node bulds ts local coordnate sstem. The node becomes the center of ts own coordnate sstem wth the poston (0, 0) and the postons of ts neghbors are computed accordngl. The node j s called a one-hop neghbor of node f nodes and j can communcate drectl (n one hop). We defne N, asetofnodesk such that j K,j, j s a one hop neghbor of, wheren s a setofallthenodesnthenetwor.wecallk the set of one-hop neghbors of node. We defne N the set D as a set of dstances measured from the node to the nodes j K. The neghbors can be detected b usng beacons. After the absence of certan number of successve beacons, t s concluded that the node s no longer a neghbor. The dstances between the nodes are measured b some means, e.g. the Tme of Arrval method. The followng procedure s performed at ever node : detect one-hop neghbors (K ) measure the dstances to one-hop neghbors (D ) send the K and D to all one-hop neghbors Thus, ever node nows ts two-hop neghbors and some of the dstances between ts one-hop and two-hop neghbors. A number of dstances cannot be obtaned due to the power range lmtatons or the obstacles between the nodes. Fg. 2 shows node and ts one-hop neghbors. Contnuous lnes represent the nown dstances between the nodes, whle dashed lnes represent the dstances that can not be obtaned. B choosng two nodes p, q K such that the dstance between p and q (d pq ) s nown and larger than /01 $10.00 (c) 2001 IEEE 2

3 Proceedngs of the 34th Hawa Internatonal Conference on Sstem Scences zero and such that the nodes, p and q do not le on the same lne, node defnes ts local coordnate sstem. The latter s defned such that node p les on the postve as of the coordnate sstem and node q has a postve q component. In ths wa the local coordnate sstem of s unquel defned as a functon of, p and q. The coordnates of the nodes, p and q are: =0; =0; p = d p ; p =0; (1) q = d q cos γ; q = d q sn γ, where γ s the angle (p,, q) n the trangle (p,, q) and t s obtaned b usng a cosnes rule for trangles: γ = arccos d2 q +d2 p d2 pq (2) 2d qd p The postons of the nodes j K, j p, q, forwhch the dstances d j,d qj,d pj are nown, are computed b trangulsaton. Therefore, we obtan q Fgure 2: The coordnate sstem of node s defned b choosng nodes p and q. q q p j = d j cos α j f β j = α j γ => j = d j sn α j else => j = d j sn α j, (3) where, α j s the angle (p,, j) n the trangle (p,, j), β j s the angle (j,, q) n the trangle (j,, q) andγ s the angle (p,, q) n the trangle (p,, q). We obtan the values of α j and β j b usng the cosne rule j d q γ α j d j β j d q j j j d pq d p d p j p α = arccos d2 j +d2 p d2 pj 2d jd p (4) β = arccos d2 q +d2 j d2 qj (5) 2d qd j Fg. 3 shows the eample of ths computaton for node j. The postons of the nodes K, j p, q,whchare not the neghbors of nodes p and q, can be computed b usng the postons of the node and at least two other nodes for whch the postons are alread obtaned, f the dstance from the node to these nodes s nown. Lmted power ranges of the nodes reduce the number of one-hop neghbors for whch the node s able to compute the poston. We defne a Local Vew Set (LVS) for node as a set of nodes LV S (p, q) K such that j LV S,nodecan compute the locaton of the node j, n the local coordnate sstem of node. Outof K neghbors, node can choose mamall 2 ( K ) 2 dfferent pars of ps andqs, where K s the cardnalt of the set K. We denote the set of all possble combnatons of p and q for node as a set C. C = {(p, q) K such that p K q } 0 C 2 ( K ) (6) 2 Fgure 3: Eample llustrates the wa to obtan the poston of node j n the coordnate sstem of node. B choosng dfferent ps and qs for the same node, we obtan mamall C dfferent Local Vew Sets, where C s the cardnalt of the set C. The choce of p and q should mamze the number of the nodes for whch we can compute the poston. Therefore, we have: (p, q) =arg ma (p,q ) C LV S (p,q ) (7) 4 Networ coordnate sstem After the nodes buld ther local coordnate sstems, ther postons are set to 0,0 and ther coordnate sstems have dfferent drectons. We sa that two coordnate sstems have the same drecton f the drectons of ther and aes are the same. In ths secton we descrbe how to adjust the drectons of the local coordnate sstems of the nodes to obtan the same d /01 $10.00 (c) 2001 IEEE 3

4 Proceedngs of the 34th Hawa Internatonal Conference on Sstem Scences recton for all the nodes n the networ. We call ths drecton, the drecton of the networ coordnate sstem. We further eplan the algorthm for electng the center of the networ coordnate sstem. Fnall, we show the wa to compute the postons of the nodes n the networ coordnate sstem. 4.1 Coordnate sstem drecton We observe two nodes, and. To adjust the drecton of the coordnate sstem of the node to have the same drecton as the coordnate sstem of the node, node has to rotate and possbl mrror ts coordnate sstem. We denote ths rotaton angle as the correcton angle for the node. To obtan the correcton angle two condtons have to be met LV S and LV S j, such that j LV S and j LV S We dffer two possble scenaros. In frst scenaro, the drectons of the coordnate sstems of and are such, that to have the coordnate sstem of equall drected as the coordnate sstem of, the coordnate sstem of needs to be rotated b some rotaton angle. In the second scenaro the rotaton of the coordnate sstem of s not enough to have the same drecton of the coordnate sstems, but the coordnate sstem of needs to be mrrored around one of ts aes after the rotaton. These two scenaros are llustrated n Fg. 4. The correcton angle for node n the frst scenaro s β α + π. In the second scenaro, the correcton angle for node s β + α, and the mrrorng s done n respect to the aes. Here, α s the angle of the vector n the coordnate sstem of the node and β s the angle of the vector n the coordnate sstem of the node. All the rotatons done at node are n the postve drecton of ts local coordnate sstem. Before the correcton of the drecton of ts coordnate sstem, node uses the followng procedure to detect the scenaro that t s n. Node j s used for ths detecton. f α j α <πand β j β <π or α j α >πand β j β >π => mrrorng s necessar => the correcton angle = β + α f α j α <πand β j β >π or α j α >πand β j β <π => mrrorng s not necessar => the correcton angle = β α + π α α β β rot β => (a) rot β => (b) rot π α => rot α => m rror Fgure 4: Eample llustratng two possble scenaros of correcton of the coordnate sstem of node. Ths procedure s eplaned as follows. We observe the poston of the node j n the coordnate sstems of the nodes and. The angle of the vector j n the coordnate sstem of s α j and the angle of the vector j n the coordnate sstem of s β j.ths s llustrated n Fg. 5. If the coordnate sstems of nodes and are rotated, b α and β respectvel, the angles of the vectors j and j change to α j α and β j β. We observe that the poston of the node j wll detect f the mrrorng s necessar or not. Ths s llustrated n Fg. 6. Once that node rotates ts local coordnate sstem b correcton angle and mrrors t f necessar, nodes α α j j Fgure 5: Poston of the node j n the local coordnate sstems of and. β β j => /01 $10.00 (c) 2001 IEEE 4

5 Proceedngs of the 34th Hawa Internatonal Conference on Sstem Scences j l α j α β j β l l = +l (a) α j α j β j β Fgure 7: Poston computng when the local coordnate sstems have the same drecton. (b) Fgure 6: The poston of the node j n the local coordnate sstems of and gves to node the nformaton f ts coordnate sstems needs to be rotated (a) or rotated and mrrored (b). and have the same drecton of ther local coordnate sstems. The same procedure can be repeated for all the nodes n the networ. 4.2 Poston computng Our goal s to acheve that all the nodes obtan ther postons wthn one coordnate sstem. If the coordnate sstem of node s chosen to be the referent coordnate sstem, all the nodes n the networ have to adjust the drectons of ther coordnate sstems to the drecton of the coordnate sstem of node and ever node has to compute ts poston n the coordnate sstem of the node. In the prevous secton, we eplaned how a node can adjust ts coordnate sstem to have the same drecton as one of ts neghbors. Here we eplan how nodes can compute ther postons n the coordnate sstem of node. All the nodes that belong to the local vew set of the node now ther poston, as t s computed drectl b node. Therefore, node nows ts poston n the coordnate sstem of node. Wenowobservenodel, whchsatwo-hop neghbor of the node and belongs to the local vew set of node. Node nows ts poston n the coordnate sstem of node, and nows the poston of node n the coordnate sstem of node. As the coordnate sstems of nodes and have the same drectons, the poston of the node l n the coordnate sstem of the node s smpl obtaned as a sum of two vectors. l = + l (8) Ths s llustrated n Fg. 7. The same s appled to the 3-hop neghbors of node that belong to the local vew set of node l, f the coordnate sstem of l has the same drecton as the coordnate sstems of and. These nodes wll receve the poston of node l n the coordnate sstem of node and add ths vector to ther vector n the coordnate sstem of node l. In ths wa the nodes obtan ther poston n the coordnate sstem of node. The procedure s repeated for all the nodes n the networ, and all the nodes n the networ wll compute ther postons n the coordnate sstem of node. The nodes that are not able to buld ther local coordnate sstem, but communcate wth three nodes that alread computed ther postons n the referent coordnate sstem, can obtan ther poston n the referent coordnate sstem b trangulsaton. 4.3 Locaton Reference Group As descrbed n the prevous secton, the local coordnate sstem of node becomes the networ coordnate sstem and all the nodes adjust the drectons of ther coordnate sstems to the drecton of the coordnate sstem of node and compute ther poston n the coordnate sstem of node. However, the moton of node wll cause that all the nodes have to recompute ther /01 $10.00 (c) 2001 IEEE 5

6 Proceedngs of the 34th Hawa Internatonal Conference on Sstem Scences postons n the networ coordnate sstem. Ths wll cause a large nconsstenc between the real and computed postons of the nodes. Ths approach can be used n small area networs where the nodes have low moblt and where the dsconnecton of the nodes s not epected. A more stable approach, but n terms of a message broadcast, a ver costl approach s to compute the center of the coordnate sstem as a functon of the postons of all the nodes n the networ. In ths case, the networ coordnate sstem center would be the geometrcal center of the networ topolog and the drecton of the coordnate sstem would be the mean value of the drectons of the local coordnate sstems of the nodes. We propose the followng approach. We defne a set of nodes called Locaton Reference Group LRG N chosen to be stable and less lel to dsappear from the networ. For eample, we chose t such that the denst of the nodes n the LRG s the hghest n the networ. The networ center s not a partcular node, but a relatve poston dependent on the topolog of the Locaton Reference Group. Wthn the Locaton Reference Group, a broadcast s used to obtan ts topolog. When the nodes are movng, the LRG center s recomputed accordngl. We epect the average speed of the LRG center to be much smaller than the average speed of the nodes. In ths wa, we stablze the center of the networ and reduce the nconsstenc. The drecton of the networ coordnate sstem s computed as the mean value of the drectons of the local coordnate sstems of the nodes n the LRG. The larger the LRG, the more stable t s, more dffcult t becomes to mantan and more costl to compute the center and the drecton of the networ coordnate sstem Locaton Reference Group ntalzaton Ever node performs the followng operatons: broadcast the hello pacet to ts n-hop neghborhood to obtan the node IDs, ther mutual dstances and the drectons of ther coordnate sstems compute the postons of the n-hop neghbors n ts local coordnate sstem compute the n-hop neghborhood center as: c = Σj m c = Σj m, (9) where m s the number of nodes n the n-hop neghborhood and j and j are the and coordnates of the nodes, respectvel. compute the n-hop neghborhood drecton as the average of the local coordnate sstem drectons of the nodes that belong to ts n-hop neghborhood, and for whch t can obtan the postons compute the denst factor as a functon of the number of nodes and the dstances to the nodes n ts n-hop neghborhood. The denst factor s the rato between the number of nodes and the sze of the observed area. Once the node computes these parameters, t broadcasts the denst factor, the nformaton about the center and the drecton of the n-hop neghborhood to ts neghbors. The nodes wth the lower denst factor wll be slaved b the nodes wth a hgher denst factor and wll adjust the drectons of ther coordnate sstems accordngl. The nodes n the networ wll then compute ther postons n the coordnate sstem of the n-hop neghborhood of the node wth a hghest denst factor. The node wth the hghest denst factor n the networ s called the ntal locaton reference group master and the nodes n ts n-hop neghborhood for whch t can obtan ther postons are called the ntal locaton reference group. The nodes belongng to the Locaton Reference Group mantan the lst of nodes n the ths group. The sze of the LRG can be modfed b changng the n factor. Ths factor can be tuned accordng to the networ sze Locaton Reference Group mantenance Due to the moblt of the nodes, the LRG members wll change ther poston and the center of the LRG wll change. To update ths change regularl, we ntroduce the followng algorthm performed b the members of the LRG: broadcast the hello pacet to ts n-hop neghborhood to obtan the node IDs, ther mutual dstances and the drectons of ther coordnate sstems compare the n-hop neghbors lst wth the lst of the LRG members The node that s a n-hop neghbor of the LRG master and the hghest number of the LRG nodes stll n ts n- hop neghborhood s elected to be the new LRG master and ts n-hop neghbors for whch t can obtan the poston nformaton become the new Locaton Reference Group. IfthenodedoesnothavetheLRGmasternts n-hop neghborhood, t starts an ntalzaton tmer. If wthn certan tme the node does not receve the /01 $10.00 (c) 2001 IEEE 6

7 Proceedngs of the 34th Hawa Internatonal Conference on Sstem Scences C C 1 Fgure 9: An eample llustratng the reconstructon of the coordnate sstem C 1 n the coordnate sstem C 2. Fgure 8: The Locaton Reference Group. new poston nformaton ssued b the LRG master, t starts the ntalzaton procedure descrbed above. Ths procedure can be ntalzed b an node, f after some fed tme perod, the node does not receve the nformaton about the new Locaton Reference Group master. Usng the LRG mantenance algorthm, the networ center moves at a much smaller speed than the nodes n the networ. In ths wa the nconsstenc due to the movement of the center s reduced. Fg. 8 shows the eample of the 2-hop LRG n a networ of 400 nodes Networ coordnate sstem drecton The choce of the nodes p and q and thus the drectons of the local coordnate sstems s random. Ths maes the drecton of the networ coordnate sstem random, as t depends onl on the drectons of local coordnate sstems of the nodes n the Locaton Reference Group. We propose the followng algorthm, whch s performed at each node that belongs to the Locaton Reference Group: The node ntall chooses the drecton of the coordnate sstem, b choosng ts (p, q) par. We note ths coordnate sstem as C 1. When rerunnng the local coordnate sstem algorthm, the node chooses the new (p, q) par. We note ths coordnate sstem as C 2. The postons of the nodes changed due to ther moton, and the choce of p and q ma change. It compares the postons of the nodes n two coordnate sstems and searches for the mamum set of nodes (at least 3) that have the same topolog n both C 1 and C 2. From ths, we conclude that the nodes belongng to ths set dd not move durng the tme between two runs of the algorthm. Ths concluson s not certan, but t has a ver hgh probablt of beng true. The node uses ths set of nodes and ther dstances to reconstruct the center of C 1 n the coordnate sstem C 2. Ths allows the node to adjust the drecton of C 2 to the drecton of C 1. If the node can not reconstruct the coordnate sstem C 1, then t eeps the drecton of C 2 as the drecton of ts new local coordnate sstem. Ths algorthm allows ever node that belongs to the LRG to ntroduce drecton stablt n ts local coordnate sstem. The LRG master computes the drecton of the networ coordnate sstem as the average drecton of the nodes n the LRG. Therefore, ths algorthm stablzes the drecton of the networ coordnate sstem. In a hgh denst area, such as LRG, we epect to have a low moblt set that wll enable ths algorthm to be used. The eample of the coordnate sstem reconstructon s shown n Fg Locaton estmaton error The algorthms descrbed n the prevous sectons use the ranges between nodes to buld a global coordnate sstem. Therefore, the accurac of the range measurements wll nfluence poston accurac. In radolocaton methods for cellular sstems, two methods are provded that can be used for dstance measurements: Tme of Arrval and Sgnal Strength measurements. These measurement are corrupted b two tpes of errors: measurng errors and Non-Lne of Sght (NLOS) /01 $10.00 (c) 2001 IEEE 7

8 Proceedngs of the 34th Hawa Internatonal Conference on Sstem Scences errors. Several models have been proposed to model both measurng error [13] and NLOS error [11] [12]. We beleve that a range estmaton errors wll be corrupted b the same tpes of errors n moble ad-hoc networs as n cellular sstems. To the best of our nowledge, no measurements have been made to gve the overall dstrbuton of the range error n moble adhoc networs. 5.1 NLOS mtgaton In [4], the mtgaton was performed n the nfrastructure-based envronment, where the postons of the base statons are nown. We beleve that error mtgaton s stll possble n the ad-hoc moble envronment, where there are no base statons to rel on. Ths s due to the fact that t s relevant whether one staton s movng or both as ther relatve movement wll produce some errors. Addtonall, the errors wll be caused b the envroment, whch does not depend on the moton of the statons. We observe the locaton accurac wthn the local coordnate sstem. The node locaton model s formulated as an estmaton model. To estmate the poston of the node, the followng algorthm s used: postons of the nodes n the local coordnate sstem are computed wthout usng the observed node the poston of the observed node s estmated usng the postons of at least three of ts neghbours the resdual weghtng algorthm s appled to mtgate the error The detaled descrpton of the resdual weghng algorthm can be found n [4]. The analog between the error mtgaton n cellular sstems, and the error mtgaton n ad-hoc networs ests because, n both cases, the range measurements are used to obtan the postons of the nodes and f the number of dstances s larger than mnmal requred, the error can be mtgated. In cellular sstems, we epect a smaller number of range measurements than n ad-hoc networs because the moble staton s usuall covered b a relatvel small number of base statons, whereas n ad hoc networs, the average number of neghbors s much hgher. However, n cellular sstem base statons have fed postons, and ther mutual dstances do not ntroduce an error. 6 Smulaton results In ths secton we present the smulaton results and we show the performance of the algorthm. The results are dvded nto two parts. In the frst part we show the nfluence of the power range on node and LVS connectvt. In the second part we present the results that llustrate the moton of the center and the changes n the drecton of the networ coordnate sstem due to the moblt of the nodes. Thesstemmodelsthefollowng. Wemodelthe postons of the nodes accordng to the Possonan dstrbuton: When a set of nodes s generated, the ponts are dstrbuted from a center pont on the plane, the dstances between the nodes are dstrbuted accordng to the eponental dstrbuton, and the angle s dstrbuted unforml. The moton of the nodes s random. The nodes choose randoml a pont on the plane, and the speed requred to arrve to that pont. The mamum and the mnmum travelng speed s defned. When the nodes arrve at the chosen pont, the wat for a fed tme, and then another random par (speed, pont) s chosen. We assume that all the nodes have the same power range. The performance of the algorthm s observed wth respect to the power range. 6.1 Local Vew Set connectvt In ths secton we present the results regardng the connectvt of the nodes and the LVS connectvt. Fg. 10 shows that the average number of the nodes for whch the poston can be obtaned, LV S, salwas lower than the average number of neghbors K. Fg. 11 shows that as the power range ncreases, the dfference between the neghbor set K and the set LV S becomes smaller. Ths convergence s due to the ncreasng node connectvt and LV S connectvt as the power range ncreases. The edge connectvt n respect to the power range s shown n Fg. 11. Ths llustrates that the full LV S connectvt wll be acheved at 260 m power range, whle the node connectvt s acheved at 170 m. Therefore, the postons wll be computed for all the nodes f the nodes have 260 m power range. For 225 m, the postons of the nodes wll be computed wthn the domnant set that contans 90 percent of the nodes. 6.2 Center and drecton stablt In ths secton we llustrate the movement of the center and the change n the drecton of the networ coordnate sstem. Fg. 12 shows that f we choose a larger (3-hop) neghborhood nstead of a 2-hop neghborhood, /01 $10.00 (c) 2001 IEEE 8

9 Proceedngs of the 34th Hawa Internatonal Conference on Sstem Scences average number of nodes K edge connectvt node connectvt 10 5 LVS 5 LVS connectvt power range (m) power range (m) Fgure 10: Averagenumber ofneghbors K (sold lne) and average number of the nodes for whch the postons can be obtaned LV S (dashed lne). 0.4 Fgure 11: Node and LV S connectvt. the moblt of the center of the networ decreases accordngl. Fg. 13 llustrates the nfluence of the average node speed ncrease on the changes of the networ coordnate sstem drecton. 7 Conclusons In ths paper we showed that: avegrage speed of the NCS center (m/s) hop LRG 3 hop LRG It s possble to acheve a unque coordnate sstem b self-organzaton of the nodes. The algorthm mposes requrements on the node connectvt. The angle and the center of the networ coordnate sstem can be stablzed usng smple heurstcs. The algorthm provdes enough nformaton to support networ functons such as Locaton Aded Routng and Geodesc Pacet Forwardng. Several ssues need to be addressed when mplementng the algorthm. Frst, the power range must be large enough to ensure LVS connectvt (smple node connectvt does not guarantee that the postons of all the nodes wll be computed). Second, the sze of the Locaton Reference Group must be chosen such that t ncreases the stablt of the center and the drecton of the networ coordnate sstem, ths wll reduce average node speed (m/s) Fgure 12: Speed of the networ coordnate sstem center for 2-hop and 3-hop LRG. the nconsstenc between the computed and the real poston of the center. The presented algorthm provdes poston nformaton to the nodes, based onl on the local vew of each node and usng the local processng capabltes of the nodes. We showed that t s possble to buld a coordnate sstem wthout the centralzed nowledge about the networ topolog. One major drawbac of ths approach s that the nodes do not now the phscal drecton of the coordnate sstem. The nodes now where ther neghbors are placed n the coordnate sstem, but the have no wa to assocate the networ coordnate /01 $10.00 (c) 2001 IEEE 9

10 Proceedngs of the 34th Hawa Internatonal Conference on Sstem Scences average change of the drecton of the NCS (rad/s) hop LRG 3 hop LRG [3] M.I. Slventonen and T. Rantalanen, Moble Staton Emergenc Locatng n GSM, Personal Wreless Communcatons, IEEE Internatonal Conference on [4] P.-C. Chen, A non-lne-of-sght error mtgaton algorthm n locaton estmaton, IEEE Wreless Communcatons and Networng Conference, [5] M.P. Wle and J. Holtzman, Thenon-lneofsght problem n moble locaton estmaton, 5th IEEE Internatonal Conference on Unversal Personal Communcatons, average node speed (m/s) Fgure 13: Comparaton of speed of change of the drecton of the networ coordnate sstem center for 2-hop and 3-hop Locaton Reference Groups. sstem wth the geographc coordnate sstem. Ths s onl possble f the algorthm s used along wth some GPS-capable devces. However, the algorthm can be used wthout the use of GPS for geodesc pacet forwardng and locaton dependent routng. Future wor n GPS-free postonng n ad-hoc networs would be to mprove the accurac of the range measurements and therefore to reduce the poston error. An addtonal mprovement would be to mprove the center and drecton stablt heurstcs and to etend the algorthm for three dmensonal models. Fnall we envson to test the algorthm and ts performance n the real world applcaton. Acnowledgments We than Martn Vetterl, Ramond Knopp, John Farserotu and Mlan Vojnovc for ther valuable suggestons and the stmulatng dscussons. References [1] J.-P. Hubau, J.-Y. Le Boudec, S. Gordano, M. Hamd,Lj.Blazevc,L.ButtanandM.Vojnovc Towards Moble Ad-Hoc WANs: Termnodes, IEEE WCNC, September [6] C. Rose and R. Yates, Locaton Uncertant n Moble Networs: a theoretcal framewor, D. of Electrcal engneerng WINLAB, November [7] Lj. Blazevc, S. Gordano and J. Y. Le Boudec, Self-Organzng Wde-Area routng, SCI 2000/ISAS 2000, Orlando, Jul [8] Y.B. Ko and N.H. Vada, Locaton aded routng (LAR) n moble ad-hoc networs, MOBICOM, [9] FCC RM Revson of the commsson rules to ensure compatblt wth enhanced 911 emergenc callng sstem, RM 8143, FCC, October [10] H. Whtne, Congruent Graphs and the Connectvt of Graphs, Amer. J. Math. 54, [11] R. Steele, Moble Rado Communcatons, PentechPress, [12] W. C.Y. Lee, Moble Communcaton Engneerng, McGraw-Hall, [13] W. Zhang, J. Luo and N. Mandaam, Moble locaton estmaton usng tme of arrvals n cdma sstems, Techncal report 166, WINLAB, Rutgers Unverst, [14] I. Stojmenovc and X. Ln, A loop-free routng for wreless networs, Proc. IASTED Conf. On parallel and Dstrbuted sstems, [15] E. Kranas, H. Sngh and J. Urruta, Compass routng n geometrc networs, Proc. Canadan Conference on Computatonal Geometr, [2] J.J. Caffer and G.L. Stuber, Overvew of Radolocaton CDMA Cellular Sstems, IEEE Communcatons Magazne, Aprl /01 $10.00 (c) 2001 IEEE 10