(1, 3) ( 2, 3) ( 3,1) (4,1) ( 1, 1) (3, 0.5) (0.5, 1.5) ( 2, 1) ( 1, 2.5) ( 3, 3)

Transcription

1 Preprnt (21)?{? 1 GPS-free postonng n moble ad hoc networs Srdjan Capun, Maher Hamd, Jean-Perre Hubau Insttute for computer Communcaton and Applcatons (ICA) Communcaton Sstems Department (DSC) Swss Federal Insttute of Technolog Lausanne (EPFL) CH-115 Lausanne, Swtzerland (srdan.capun, maher.hamd, jean-perre.hubau)@ep.ch We consder the problem of node postonng n ad hoc networs. We propose a dstrbuted, nfrastructurefree postonng algorthm that does not rel on GPS (Global Postonng Sstem). Instead, the algorthm uses the dstances between the nodes to buld a relatve coordnate sstem n whch the node postons are computed n two dmensons. Despte the dstance measurement errors and the moton of the nodes, the algorthm provdes sucent locaton nformaton and accurac to support basc networ functons. Eamples of applcatons where ths algorthm can be used nclude Locaton Aded Routng [8] and Geodesc Pacet Forwardng [7]. Another eample are sensor networs, where moblt s less of a problem. The man contrbuton of ths wor s to dene 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. Kewords: ad hoc, moble, dstrbuted, wreless networs, personal communcaton sstem, self-locatng, rado-locaton, GPS-free, self-organzaton. 1. Introducton The presented wor s a part of the Termnode project, a 1 ear ongong project that nvestgates large area, wreless, moble ad hoc networs [1,19,2]. 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 descrbe an algorthm for the postonng of nodes n an ad hoc networ that does not use GPS. The algorthm provdes a poston nformaton to the nodes n the scenaros where an nfrastructure does not est and GPS cannot be used. GPS-free postonng s also desrable, when the GPS sgnal s too wea (e.g., ndoor), when t s jammed, or when a GPS recever has to be avoded for cost or ntegraton reasons. We ntroduce a dstrbuted algorthm that enables the nodes to nd 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 (Fg 1.). The Tme of Arrval (TOA) method s used to obtan the range between two moble devces. Despte the dstance measurement errors and the moton of the nodes, the algorthm provdes sucent locaton nformaton and accurac to support basc networ functons. Eamples of applcatons where ths algorthm can be used nclude Locaton Aded Routng [8] and Geodesc Pacet Forwardng [7], both n ad hoc and sensor networs. In the Geodesc Pacet Forwardng algorthm, the source sends pacets n the phscal drecton of the destnaton node. Gven that the node nows ts poston and the postons of the destnaton node n the relatve coordnate sstem, t s able to com-

2 2 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs 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,.5) (.5, 1.5) Fgure 1. The algorthm uses the dstances between the nodes and bulds the relatve coordnate sstem. pute n whch drecton (to whch net hop node) to send pacets. The nodes n the ad hoc networs are usuall not aware of ther geographcal postons. As GPS s not used n our algorthm, we provde relatve postons of the nodes wth respect to the networ topolog. For the sae of smplct, we present the algorthm n two dmensons, but t can be easl etended to provde poston nformaton n three dmensons. The paper s organzed as follows. In secton 2 related wor n the eld of rado-locaton technques s descrbed. In secton 3 we present the algorthm for buldng a Local Coordnate Sstem at each node. In secton 4 we descrbe the means to dene the center and the drecton of the Networ Coordnate Sstem. In secton 5 we dscuss the nuence of the range errors on the accurac of the poston estmaton. We present the smulaton results n secton Related wor Followng the release of US FCC regulatons for locatng E911 callers, postonng servces n moble sstems have drawn much attenton recentl. The new regulatons ntroduce strngent demands on the accurac of moble phone locaton. The FCC requres that b October 21, 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 Derence 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. The algorthm s referred to as the Resdual Weghng 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. A sgncant amount ofwor has been reported consderng contet-aware computer sstems. In these sstems, moble computers automatcall congure themselves based on what was happen-

3 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs 3 ng n the envronment around them. Eamples of these sstems nclude Actve Badge [14] and RADAR [15]. Recentl, some poston-based routng and pacet forwardng algorthms for the ad hoc networs have been proposed n [7,8,16{18]. 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 GPS n ad hoc networs. 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 3. Local Coordnate Sstem We mae the followng assumptons on our sstem model: the observed networ s an nfrastructurefree networ of moble and wreless devces all the devces (nodes) have the same techncal characterstcs all the wreless lns between the nodes are bdrectonal the nodes use omndrectonal antennae the mamum speed of the movement of nodes s lmted to 2 m/s 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 ( ) and the postons of ts neghbors are computed accordngl. In secton 4 we further show how the agreement on a common coordnate sstem (Networ Coordnate Sstem) center and drecton s acheved. If a node j can communcate drectl (n one hop) wth node, j s called a one-hop neghbor of. Let N be the set of all the nodes n the networ. 8 2 N, we dene K as the set of one-hop neghbors of. Lewse, 8 2 N, we dene D, as a set of dstances between and each nodej 2 K. The neghbors can be detected b usng beacons. After the absence of a certan number of successve beacons, t s concluded that the node s no longer Thus, ever node nows ts two-hop neghbors and some of the dstances between ts one-hop and twohop 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 cannot be obtaned. B choosng two nodes p q 2 K such that the dstance between p and q (d pq )snown and larger than zero and such that the nodes p and q do not le on the same lne, node denes ts Local Coordnate Sstem. The latter s dened suchthat 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 dened as a functon of p and q. Thus, the coordnates of the nodes, p and q are: = = p = d p p = q = d q cos q = d q sn (1) where s the angle 6 (p q) and t s obtaned b usng a cosnes rule for trangles: = arccos d2 q +d2 p ;d2 pq 2d q d p (2) The postons of the nodes j 2 K, j 6= p q, for whch the dstances d j d qj andd pj are nown, are

4 4 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs q q q d q d pq p γ α j β j j d p p d q j d j d p j j j Fgure 2. The coordnate sstem of node s dened b choosng nodes p and q. Fgure 3. An eample llustratng the wa to obtan the poston of node j n the coordnate sstem of node. computed b trangulsaton. Therefore, we obtan j = d j cos j f j = j j ; j => j = d j sn j else => j = ;d j sn j (3) where j s the angle 6 (p j), and j s the angle 6 (j q). In practce, j wll never be eactl equal to j j ; j due to the errors n dstance measurements. The purpose of ths eercse s to nd on wtch sde of the as node j s located and some derence between these two values needs to be tolerated. We obtan the values of j and j b usng the cosne rule j = arccos d2 j +d2 p ;d2 pj 2d j d p (4) j = arccos d2 q +d2 j ;d2 qj 2d q d j (5) The angles j, j and are placed wthn the trangles (p,,j), (j,,q) and (p,,q), respectvel and thus we observe just ther absolute values and not ther drectons. Fg. 3 shows the eample of ths computaton for node j. The postons of the nodes 2 K, 6= p q, whch are 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 dene the Local Vew Set (LVS) for node as the set of nodes LV S (p q) K such that 8j 2 LV S, node can compute the locaton of the node j, n the Local Coordnate Sstem of node. Out of jk j neghbors, node can choose 2 ; jk j 2 derent couples of ps andqs. We denote the set of all possble combnatons of p and q for node as a set C. C = f(p q) 2 K such that p 2 K q g (6) jc j2 ; jk j 2 B choosng derent ps and qs for the same node, we obtan jc j derent Local Vew Sets, where jc j s the cardnalt oftheset C. The choce of p and q should mamze the number of the nodes for whch we can compute the poston: (p q) =arg ma jlv S (p q )j (7) (p q )2C 4. Networ Coordnate Sstem After the nodes buld ther Local Coordnate Sstems, ther postons are set to (,) and ther coordnate sstems have derent 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 drecton for all the

5 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs 5 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. α β rot β => rot π α => 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 perform the angle correcton operaton, two condtons have to be met 2 LV S and 2 LV S 9j 6= such that j 2 LV S and j 2 LV S There are two possble stuatons. In the rst stuaton, the drectons of the coordnate sstems of and are suchthattohave 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 stuaton, the rotaton of the coordnate sstem of s not enough to have the same drecton of the coordnate sstems n addton, the coordnate sstem of needs to be mrrored around one of ts aes after the rotaton. These two stuatons are llustrated n Fg. 4. The correcton angle for node n the rst stuaton s ; +. In the second scenaro, the correcton angle for node s +,andthemrrorng s done wth respect to the aes. Here, s the angle of the vector ~ n the coordnate sstem of the node and s the angle of the vector ~ n the coordnate sstem of the node. All the rotatons 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 stuaton that t s n. Node j s used (a) α β rot β => rot α => m rror => (b) Fgure 4. An eample llustratng two possble stuatons of correcton of the coordnate sstem of node. α α j j Fgure 5. Poston of the node j n the Local Coordnate Sstems of and. for ths detecton. f j ; < and j ; > or j ; > and j ; < => mrrorng s not necessar => the correcton angle s ; + f j ; < and j ; < or j ; > and j ; > => mrrorng s necessar => the correcton angle s + 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 j and the angle of the vector ~ j n the coordnate sstem of s j. Ths s llustrated n Fg. 5. If the coordnate β β j

6 6 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs j α j α (a) β β j l l = +l l j β j β α j α 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 whether ts coordnate sstems needs to be rotated (a) or rotated and mrrored (b). sstems of nodes and are rotated, b and respectvel, the angles of the vectors ~ j and ~ j change to j ; and j ;. We observe that the poston of the node j maes t possble to detect whether the mrrorng s necessar or not. Ths s llustrated n Fg. 6. Once node has rotated ts local coordnate sstem b the correcton angle and has mrrored t f necessar, nodes and have the same drecton of ther Local Coordnate Sstems. The same procedure can be repeated for all the nodes n the networ, n an approprate order Poston computng We showed that the nodes can compute ther local coordnate sstems, and are able to adjust ther coordnate sstems wth respect to ther neghbors. Our goal s that all the nodes n the networ compute ther poston n the Networ Coordnate Sstem. So far we have not dened the Networ Coordnate Sstem. For now, we wll choose the Networ Coordnate Sstem as the Local Coordnate Sstem of one of the nodes n the networ (.e. node ). If the Networ Coordnate Sstem s cho- sennthswa, 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 ths secton 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 postons, as t s computed drectl b node. Therefore, node nows ts poston n the coordnate sstem of node. We now observe nodel, whch satwo-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

7 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs 7 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 Networ Coordnate Sstem b trangulsaton 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 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 dsconnecton of the nodes s not epected. A more stable approach, albet epensve n terms of echanged messages 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. The algorthms proposed hereafter can also be used n the networs where some ed nodes are ntroduced and then used to stablze the Networ Coordnate Sstem. Ths would ease the assumptons on moblt of all the nodes n the networ. However, n the sequel we assume that all nodes are potentall moble. We propose the followng approach. We dene a set of nodes called the Locaton Reference Group LRG N chosen to be stable and less lel to dsappear from the networ. For eample, we choose t such that ts denst of nodes 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 own 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, but the more dcult t becomes to mantan and the more costl to compute the center and the drecton of the Networ Coordnate Sstem Locaton Reference Group ntalzaton We want to dentf the nodes belongng to the LRG. For ths purpose, ever node of N 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 = j m c = j 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

8 8 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs compute the denst factor as a functon of the number of nodes and the dstances to the nodes n ts n-hop neghborhood Once the node has computed 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 the 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 group. The sze of the LRG can be moded accordng to the epected networ sze Locaton Reference Group mantenance Because of the moblt of the nodes, the LRG members wll change ther poston and then 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 has the hghest number of the LRG nodes 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. The LRG mantenance procedure s repeated perodcall ever ed tme perod. Fgure 8. The Locaton Reference Group. Ever node that was n the LRG and no longer has the LRG master n ts n-hop neghborhood, starts an ntalzaton tmer. If wthn a certan tme the node does not receve the 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 ed 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 1-hop LRG Networ Coordnate Sstem drecton We have shown how the nodes buld ther Local Coordnate Sstems and how the Networ Coordnate Sstem s bult. Furthermore, we have shown the means of stablzng the center of the coordnate sstem. In ths secton we showhowto stablze the drecton of the Networ Coordnate Sstem. 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

9 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs 9 nodes n the Locaton Reference Group. We propose the followng algorthm to stablze the drecton of the Networ Coordnate Sstem, whch s performed at each node that belongs to the Locaton Reference Group: C The node ntall chooses the drecton of ts coordnate sstem, bchoosng ts (p q) par. We denote ths coordnate sstem as C 1. When rerunnng the Local Coordnate Sstem algorthm, the node chooses the new (p q) par. We denote 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. The node 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 cannot 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 tontroduce 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 the 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. 9. C 1 Fgure 9. An eample llustratng the reconstructon of the coordnate sstem C 1 n the coordnate sstem C Local Vew Set connectvt If the node cannot echange the nformaton about ts coordnate sstem wth other nodes, t cannot compute ts poston n the Networ Coordnate Sstem. We sa that the nodes are connected n terms of a coordnate sstem connectvt, f the nformaton about the Networ Coordnate Sstem can be propagated to all the nodes n N. We refer to the coordnate sstem connectvt as the Local Vew Set connectvt. We observe two graphs: The connectvt graph G(N E) s the graph wth set of vertces N and edges E where 8 j 2 N such that e j 2 E nodes and j can communcate drectl (n one hop). The Networ Coordnate Sstem graph G NCS (N E NCS ) s the graph wth set of vertces N and edges E NCS where 8 j 2 N such that e j 2 E NCS, node 2 LV S j, node j 2 LV S and 9m 2 N such that m 2 LV S \LV S j. The connectvt graph therefore shows between whch pars of nodes the adjustment of the Local Coordnate Sstem can be made and thus the Networ Coordnate Sstem propagated. The edge connectvt (G) s the mnmum number of edges whose deleton from graph G dsconnects G [1]. Not all neghbors are able to propagate the nformaton about the Networ Coordnate Sstem and therefore LV S K, 8 2 N. For ths reason the edge connectvt ofg NCS wll

10 1 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs (a) (a) (b) (b) (c) Fgure 1. An eample of node connectvt for derent power ranges (a) 9 m, (b) 1 m, (c) 11 m. The node denst s 4 nodes/m (c) Fgure 11. An eample of Local Vew Set connectvt for derent power ranges (a) 9 m, (b) 1 m, (c) 11 m. The node denst s 4 nodes/m 2.

11 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs 11 alwas be less or equal than the edge connectvt of G. (G NCS ) (G) (1) Ths relaton shows that n order to acheve a hgh LVS connectvt, we need to have a hgh node connectvt. The connectvt of both graphs depends on the power ranges of the nodes and on the denst of nodes n the regon. The hgh connectvt ensures that when the nodes are movng, the nodes wll reman LVS connected. Fg. 1 and 11 show eamples of node and Local Vew Set connectvt on the same topolog of the nodes. (4 nodes unforml dstrbuted over a 1 m 2 plane). 5. 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 nuence poston accurac. In rado-locaton methods for cellular sstems, two methods are provded that can be used for dstance measurements: Tme of Arrval and Sgnal Strength measurements. These measurements are corrupted b two tpes of errors: measurng errors and Non-Lne of Sght (NLOS) errors. Several models have been proposed to model both measurng error [13] and NLOS error [11] [12]. To the best of our nowledge, no measurements have been publshed to gve the overall dstrbuton of the range error n moble ad hoc networs. As a rst appromaton we wll assume that the errors are smlar to the ones encountered n cellular sstems 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 an ad hoc moble envronment, where there are no base statons to rel on. 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: the 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 neghbors 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 the mnmum 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 arelatvel small number of base statons whereas n ad hoc networs, the average number of neghbors can be hgher. However, n cellular sstems, base statons have ed postons, and ther mutual dstances do not ntroduce an error Error propagaton In ths secton we observe the nuence of error cumulaton on the node poston estmaton accurac. We observe how the poston estmaton error ncreases wth the dstance of the node from the Networ Coordnate Sstem center. If the node s n-hop dstant from the center of the coordnate sstem, ts poston estmaton wll be the sum of the poston vectors along the path from the center to the observed node. Therefore, the poston estmaton error wll be a sum of all error vectors along the same path. We assume that the dstrbuton of the drecton of the error vectors s unform, as the range estmaton errors can produce

12 12 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs the error vector wth the same probablt never drecton. We further assume that the length of the error vector and the drecton of the error vector are mutuall ndependent. The power ranges of the nodes are assumed to be the same for all the nodes and thus we epect the lengths of the error vectors to be equall dstrbuted. Therefore, the epected value of the total error vector s E( ~ X)=E(~ 1 )+ ::: +E(~ n;1 ) =(n ; 1)E(arg(~ 1 ))E(cos 1 + sn 1 ) = (11) where ~ 1 ::: ~ n;1 are the error vectors n the Local Coordnate Sstems used to compute the poston of the observed node, and 1 s the drecton of the error vector of the one hop neghbor of the Networ Coordnate Sstem center. Ths shows that the epected value of the error vector n the node that s n-hops dstant fromthe center wll be zero. Nevertheless, the standard devaton of the error vector s epected to ncrease when the node s more remote from the center of the networ. Ths ponts to a lmtaton n the scalablt of the algorthm. 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 rst part we show the nuence 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 mobltofthe nodes. The sstem model s the followng. We model the postons of the nodes accordng to the Possonan dstrbuton: When a set of nodes s generated (4 nodes n our smulaton), 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 dened. When the nodes arrve at the chosen pont, the wat for a ed 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 Local Vew Set connectvt In ths secton we present the results regardng the connectvt of the nodes and the LVS connectvt. Fg. 12 shows that the average number of the nodes for whch the poston can be obtaned, jlv Sj, s lower than the average number of neghbors jkj. Fg. 12 shows that as the power range ncreases, the derence 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 wth respect to the power range s shown n Fg. 13. Ths llustrates that the LV S connectvt larger than zero wll be acheved at 11 m power range, whle the node connectvt s larger than zero alread at 9 m power range. Therefore, the postons wll be computed n the Networ Coordnate Sstem for all the nodes f the nodes have 11 m power range 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. 14 shows that f we choose a larger (3-hop) neghborhood nstead of a 2-hop neghborhood, the moblt of the center of the networ decreases accordngl. Fg. 15 llustrates the nuence of the average node speed ncrease on the changes of the networ coordnate sstem drecton.

13 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs average number of nodes 15 1 K avegrage speed of the NCS center (m/s) hop LRG 3 hop LRG 5 LVS power range (m) average node speed (m/s) Fgure 12. Average number of neghbors jk j (sold lne) and average number of the nodes for whch the postons can be obtaned jlv Sj (dashed lne). Fgure 14. Speed of the Networ Coordnate Sstem center for 2-hop and 3-hop LRG edge connectvt node connectvt average change of the drecton of the NCS (rad/s) hop LRG 3 hop LRG 1 5 LVS connectvt power range (m) Fgure 13. Node and LV S connectvt average node speed (m/s) Fgure 15. Comparaton of speed of change of the drecton of the Networ Coordnate Sstem center for 2-hop and 3- hop Locaton Reference Groups Communcaton cost In ths secton we observe the algorthm n terms of number of messages that need to be echanged between the nodes to eep the algorthm runnng. Here t s mportant to remember that the nodes are usng omndrectonal antennae and that we consder each broadcast message as one message sent to all the neghbors and not as messages sent to neghbors. The average number of messages that needs to be sent per node n order to buld a Local Co- ordnate Sstems s n the order of, where s the average number of one-hop neghbors n the networ. messages are echanged to measure the dstances between the node and ts neghbors. One addtonal message s used b each node to broadcast ths nformaton to ts one-hop neghbors. Ths maes n total + 1 messages sent per node to buld Local Coordnate Sstems. After buldng ther Local Coordnate Sstems, the nodes compute ther denst factors (as a functon of the number of nodes and the dstances to

14 14 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs nodes n the n-hop neghborhood). The number of messages that need to be sent per node to compute the denst factor depends on the sze of the regon (neghborhood) for whch the denst factor s computed. The average number of messages sent per node to compute the denst factor s n the order of n;1, where s the average number of one-hop neghbors n the networ and n s the neghborhood sze for whch the node s computng the denst factor (n-hop neghborhood). The nal step of the algorthm supposes a broadcast to all the nodes n the networ. Ths operaton s ver costl n terms of number of messages. The number of messages sent per node s n the order of l;1, where l s the average number of hops needed to reach the border of the graph (averaged over all the nodes n the networ). Buldng a Local Coordnate Sstem and computng the denst factor are not ver costl operatons, but the are performed at each node frequentl, whereas buldng the Networ coordnate sstem s a ver costl operaton. How often the Networ Coordnate Sstem has to be rebuld depends on the algorthms for mantanng the center and the drecton of the NCS, and on the algorthms wthn the nodes whch stablze the LCS's of the nodes. 7. Conclusons We have proposed algorthms that: It s possble to acheve a relatve coordnate sstem b self-organzaton of the nodes. The power range should be chosen carefull to ensure hgh LV S connectvt and hgh algorthm coverage. The algorthm mposes low requrements on the node connectvt. The angle and the center of the Networ Coordnate Sstem can be stablzed usng smple heurstcs. The dstance measurement errors wll ntroduce an error n poston estmaton. Despte the dstance measurement errors and the moton of the nodes, the algorthm provdes sucent locaton nformaton and accurac to support basc networ functons. 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 ofthe center and the drecton of the Networ Coordnate Sstem. Ths wll reduce the nconsstenc between the computed and the real poston of the center. One major characterstc of ths approachsthat the nodes do not now thephscal 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 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. 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 centralzed nowledge of the networ topolog. Future wor n GPS-free postonng ncludes the mprovement of the accurac of the range measurements and therefore to reduce the poston error. Addtonall, to mprove the center and drecton stablt heurstcs and to optmze the algorthm and to etend t for three dmensonal models. Fnall we envson testng the algorthm and ts performance n real world applcatons.

15 Capun, Hamd, Hubau / GPS-free postonng n moble ad hoc networs 15 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. Buttan and M. Vojnovc Towards Moble Ad-Hoc WANs: Termnodes, IEEE WCNC, September 2. [2] J.J. Caer and G.L. Stuber, Overvew of Radolocaton CDMA Cellular Sstems, IEEE Communcatons Magazne, Aprl [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, The non-lne of sght problem n moble locaton estmaton, 5th IEEE Internatonal Conference on Unversal Personal Communcatons, [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 2/ISAS 2, Orlando, Jul 2. [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 [1] H. Whtne, Congruent Graphs and the Connectvt of Graphs, Amer. J. Math. 54, [11] R. Steele, Moble Rado Communcatons, Pentech- Press, [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] And Harter, And Hopper, Pete Steggles, And Ward and Paul Webster, The Anatom of a Contet- Aware Applcaton, Proceedngs of the Ffth Annual ACM/IEEE Internatonal Conference on Moble Computng and Networng, MOBICOM'99, Seattle, Washngton, USA, August 1999, pp [15] P. Bahl and V.N. Padmanabhan, RADAR: an nbuldng RF-based user locaton and tracng sstem, INFOCOM 2. Nneteenth Annual Jont Conference of the IEEE Computer and Communcatons Socetes. Proceedngs. IEEE, Volume: 2, 2. [16] E. Kranas, H. Sngh and J. Urruta, Compass routng n geometrc networs, Proc. Canadan Conference on Computatonal Geometr, [17] I. Stojmenovc and X. Ln, Aloop-free routng for wreless networs, Proc. IASTED Conf. On parallel and Dstrbuted sstems, [18] P.Bose, P. Morn, I. Stojmenovc and J. Urruta, Routng wth guaranteed delver n ad hoc wreless networs, Proc. of 3rd ACM Int. Worshop on Dscrete Algorthms and methods for monle computng and Communcatons DIAL M99, Seattle, August 2, 1999, [19] [2] J.-P. Hubau, T. Gross, J.-Y. Boudec and M. Vetterl Towards self-organzed moble ad hoc networs: the Termnode project, IEEE Communcatons Magazne, Januar 21