Location Tracking in Mobile Ad Hoc Networks using Particle Filter

Transcription

1 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer Rui Huang and Gergely V. Záruba Compuer Science and Engineering Deparmen The Universiy of Texas a Arlingon 46 Yaes, 3NH, Arlingon, TX rxh725@omega.ua.edu, zaruba@ua.edu January, 26 Absrac Mobile ad hoc neworks (MANET) are dynamic neworks formed on-he-fly as mobile nodes move in and ou of each ohers ransmission ranges. In general, he mobile ad hoc neworking model makes no assumpion ha nodes know heir own locaions. However, recen research shows ha locaion-awareness can be beneficial o fundamenal asks such as rouing and energy-conservaion. On he oher hand, he cos and limied energy resources associaed wih common, lowcos mobile nodes prohibis hem from carrying relaively expensive and power-hungry locaion-sensing devices such as GPS. This paper proposes a mechanism ha allows non-gps-equipped nodes in he nework o derive heir approximaed locaions from a limied number of GPS-equipped nodes. In our mehod, all nodes periodically broadcas heir esimaed locaion, in erm of a compressed paricle filer disribuion. Non-GPS nodes esimae he disance o heir neighbors by measuring he received signal srengh of incoming messages. A paricle filer is hen used o esimae he approximaed locaion, along wih a measure of confidence, from he sequence of disance esimaes. Simulaion sudies show ha our soluion is capable of producing good esimaes equal or beer han he exising localizaion mehods such as APS-Euclidean for he more difficul scenario when he nework conneciviy is low. Inroducion Mobile ad hoc neworks (MANET) are consruced on he fly as he nework nodes move in and ou of he ransmission range of each oher. A major challenge in proocol design for his ype of neworks is o provide mechanisms ha deal wih he dynamical opology change. Consan opology change makes i more difficul for fundamenal asks such as rouing since he rouing algorihm canno simply rely on is previous knowledge of he nework opology. Furhermore, even afer a roue has been successfully esablished, i can sill be disruped a any ime due o he movemen of he inermediae nodes. For his reason, mos proocols originally designed for saic neworks canno be adoped o ad hoc neworks wihou significan change. Many proocols have o be redesigned for ad hoc neworks in order o cope wih he opology change. Sudies have shown ha innovaive algorihms can aid mobile ad hoc nework (MANET) proocols if he nodes in he nework are capable of obaining heir own as well as oher nodes locaion informaion. For insance, algorihms such as LAR [8], GRID [], and GOAFR+ [] rely on he locaion informaion o provide more sable roues during unicas roue discovery. The locaion informaion is also applied o geocas (mulicas based on geographic informaion) [7] for algorihms such as LBM [9], GeoGRID [2] and PBM [4]. To minimize he power consumpion, The GAF algorihm [23] uses he locaion informaion o effecively modify he nework densiy by urning off cerain nodes a paricular insances. The algorihms lised earlier all rely on he availabiliy of reasonably accurae locaion informaion. This assumpion is valid for neworks in which some locaion sensing devices, such as GPS receivers, are available a all nodes. However, in realiy his is rarely he case; alhough GPS receivers are increasingly cheaper o produce and becoming more widely available, hey are sill relaively expensive and power-hungry. GPS receivers also require line-of-sigh o saellie, which precludes indoor usage. Therefore, i is oo general o assume ha hey will be applicable o every node in he ad hoc

2 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer 2 neworks. For his reason differen algorihms have been proposed o derive approximaed locaions of all nodes based on he relaxed assumpion ha direc locaion sensing devices (such as GPS) are available o only a subse of he nodes. This paper presens a soluion o he locaion racking problem based on paricle filers. Given an ad hoc nework wih limied number of locaion-aware nodes, our soluion esimaes he locaions of all oher nodes by measuring sensory daa, in his paricular case he received signal srengh indicaion (RSSI), from neighbors. For each node, he esimaed locaion is viewed as a probabilisic disribuion mainained by a paricle filer. Unlike oher locaion racking mehods, our soluion has low overhead because i is purely based on local broadcasing and does no require flooding of he locaion informaion over he enire nework. Simulaion sudies show ha even wihou flooding, our soluion can sill generae good esimaes comparable o oher exising mehods, given ha he percenage of GPS nodes is no exremely low. In addiion when conneciviy is low, our algorihm is sill able o derive locaion informaion which is no he case wih mos of he oher approaches. While mos algorihms eiher aemp o increase he accuracy of he esimae or o increase he coverage, our algorihm recognizes he radeoff beween he wo and provides a quanaive measure for boh. From he implemenaion poin of view, our algorihm can be easily implemened in disribued manner for boh saionary and mobile neworks. Mos imporanly, our algorihm provides a probabilisic framework in which oher sensory daa (such as angle of arrival) can be naurally incorporaed in he fuure. 2 Relaed Works Given a nework graph G = (V,E) in which he number of locaion-aware nodes (also called anchor nodes) V gps V, he objecive of he locaion racking algorihm is o find he locaions of non-anchor nodes {V} {V gps }. In his secion we survey he previous work on he locaion racking problem in ad hoc neworks. Generally speaking, here are wo caegories of disribued localizaion mehods depending on wheher sensory daa are used. The mehods ha do no use sensory daa are simpler bu end o perform poorly especially when anchor raio is low or he nework is sparse. The mehods ha do use sensory daa generally perform beer bu end o be significanly more complex. The performance in he laer case is also largely affeced by he noise inroduced o he sensory daa which ends o aggregae rapidly as sensory daa is propagaed hrough he nework. The Cenroid mehod [2] provides he mos sraigh-forward soluion ha does no use sensory daa. Assuming ha a non-anchor node is capable of receiving he locaion informaion from muliple anchor nodes, he Cenroid mehod derives he locaion of a non-anchor node as he average of is neighboring anchor nodes locaions. The mehod is simple and efficien, bu i requires he anchor nodes o redundanly cover large areas for an accepable performance. The APIT mehod [5] esimaes he node locaion by isolaing he area using various riangles formed by anchor nodes. The locaion of he node is narrowed down by analyzing overlapping riangles o deermine wheher he node is conained wihin he riangles. Boh he Cenroid mehod and he APIT mehod require he ransmission range of anchors o be much greaer han non-anchors (by an order of magniude [5]) in order for nodes o obain reasonable locaion esimaes. The DV-Hop mehod [8] allows he locaion informaion from anchor nodes o propagae hrough muliple hops. The locaions of anchors are periodically flooded hroughou he nework much like he rouing packes in a disance vecor rouing proocol. The locaions of non-anchor nodes are derived geomerically by performing rilaeraion of he disance esimaes from a leas hree anchor nodes. Here he disance esimaes are obained by muliplying he number of hops o he anchor node o a predefined average-disance-per-hop value. The DV-Hop mehod does no require a greaer ransmission range of anchors, and i works well even when he raio beween anchor and non-anchor nodes is low. However, he message complexiy is raher high due o he flooding of he locaion informaion. Furhermore, because he average-disance-perhop is an esimaed value over he enire nework, he accuracy of he locaion esimaion suffers when he nodes are no uniformly placed over he nework. Oher, significan locaion racking mehods make use of addiional sensors. In [3], he locaion, velociy and acceleraion of mobile nodes are esimaed by measuring he received signal srengh indicaor (RSSI) from muliple base saions in a cellular nework. The measured power levels are fed ino a Kalman filer o smooh ou (filer) he erraic readings and hus be able o derive he disance. Since base saion locaions are assumed o be well-known in a cellular nework, mobile nodes can use hem as reference poins for locaion esimaion. In [7], he auhors assume ha non-anchor nodes are equipped wih devices ha measure he incoming signal direcions. The direcional informaion allows he receivers o obain he angle of arrival (AoA) of he signal hus allowing more accurae locaion esimaes han he pure DV-Hop

3 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer 3 mehod. The DV-Disance mehod [8] is similar o he DV-Hop mehod bu uses he esimaed disance insead of he hop coun during rilaeraion. In [2] afer obaining he iniial locaion esimaes from he DV mehod, he nodes obain he esimaed locaions from he neighbors via local broadcas. The RSSI readings also provide he disance esimaes from he neighbors. Using he disance esimaes along wih he esimaed locaions from he neighbors, he nodes can refine heir iniial locaion esimaes via rilaeraion. Hardware-wise, sensors ha measure RSSI are widely available o mobile devices. Indeed, mos off-he-shelf echnologies implicily provide such informaion (e.g., mos Wi/Fi cards provide wih RSSI). Based on RSSI and an underlying signal propagaion model, he disance o he sender can be esimaed. Because of he noise caused by mulipah fading and far field scaering during he signal ransmission, he disance esimaes derived from RSSI suffer accordingly, especially when a large number of obsacles presen. However, a number of mechanisms have been proposed o improve he accuracy of such esimaes, such as he ones ha use a more robus acousic ranging sysem [3], device calibraion on he RSSI sensors [22], and Kalman filers o smooh ou he odd readings from he sensors [6]. Experimens have shown ha he disance esimaion error can be drasically reduced by using hose mehods. Thus, he RSSI-based mehods are becoming more pracical soluions o he locaion racking problem in ad hoc neworks. 3 Paricle Filer Soluion Geomerically speaking, in order o find he locaion of a node in a 2-dimensional space, he disances and locaions of a leas hree anchors need o be known (as each of hese anchors define a circle where he arge node could be). In a nework where he percenage of anchors is low, he major challenge is o obain he disances and locaions of anchors when he node is several hops away from he anchors. Previous works resolve his problem by eiher ) assuming a greaer ransmission range of anchors [2, 5] (hus, anchors are always -hop away), or 2) broadcasing he anchor locaions hop-by-hop over he enire nework [8, 7, 2]. The assumpion made in he firs soluion requires he nework o be heerogeneous in he node ypes (in which anchors radios are considered differen han hose of non-anchors) and requires homogeneiy (uniformiy) for anchor nodes locaion over he area. The flooding of he locaion packes in he second soluion requires exra overhead. This overhead can be especially heavy when nodes are mobile, where locaion packes need o be re-broadcased repeaedly by nodes. Furhermore, mos mehods in [8, 7, 2] requires muliple phases of operaions such as a phase of iniial locaion discovery followed by a phase of refinemen. However, in a more general nework model in which nodes are can come online and go offline a differen ime, i becomes more difficul o define he sar and he end of a phase. Lasly, due o he geomeric and algorihmic limiaions, mos exising mehods produce he locaion esimaes for a limied percenage of nodes. Bu, heir esimaes lack a measure ha qualifies he esimaes. In oher words, one canno ell how good hose esimaes are. Recognizing various shorcomings of previous approaches, we propose a differen locaion racking mehod ha is based on Bayesian filers using Mone Carlo sampling (also known as paricle filers) inroduced in [4]. Our mehod can be considered as a probabilisic approach in which he esimaed locaion of each node is regarded as a probabiliy disribuion capured by samples, hus he erm paricles. The disribuion of paricles (he probabiliy disribuion of a node s locaion over he area) is coninuously updaed as he node receives locaion esimaes from is neighbors along wih he disance esimaes from RSSI reading. Essenially, he nodes esimae heir own locaions by inerchanging he locaion disribuions wih heir neighbors. Our mehod has he following advanages over mos exising localizaion mehods:. Provide a measure of esimaion qualiy. DV based algorihms can generae locaion esimaes o a subse of nodes. The coverage of he esimaes depends on he naure of he algorihm. There is always radeoff beween he coverage and he qualiy of he esimaes. Some algorihms (such as DV-Hop) give beer coverage, while oher (such as Euclidean) gives beer esimaes. Our mehod, however, generaes locaion esimaes for all nodes in he nework. Each esimae is qualified by a variance, which serves as he qualiy measure. Thus, he coverage of our esimaes is no a fixed value bu a funcion of he variances. In pracice, cerain applicaions migh desire beer esimaion qualiy while oher migh desire beer coverage. Previously, differen localizaion mehods need o be applied separaely o accomplish he wo objecives. Our mehod, however, produces he resul saisfies boh scenarios all in same probabilisic framework.

4 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer 4 2. Single phase operaion. Many algorihms employ muliple phases during he localizaion process. For insance, DV- Hop requires a firs phase o calculae per-hop disance and a second phase o propagae he resul. The mulilaeraion mehods [2] conains hree phrases of iniial esimaion, grouping and refinemen. Our mehod, however, has he advanage of a single phase operaion. From he implemenaion poin of view, our algorihm can be easily implemened in disribued fashion because nodes do no have o collecively mainain he sae informaion of which phase are we in? From he funcional poin of view, he probabilisic naure of our mehod simplifies he algorihm by eliminaing he need for muliple phases. In mulilaeral mehods, an iniial esimae is obained based on a cerain measure (disance or hops) o GPS nodes followed by phase of furher refinemen. The iniial locaion esimae suffers because infomaion from non-gps nodes are no used. The refinemen phase is needed so ha informaion from non-gps nodes can be incorporaed ino he esimaes. Our mehod does no need separae phases, as he informaion from non-gps nodes is auomaically applied as soon as i becomes available. In paricular, as non-gps nodes becomes more aware of heir locaions, heir variances decrease, which allows heir esimaes o be used by neighboring nodes. 3. Simple communicaion model and fas convergence. Our mehod employs a simple compuaion and communicaion model which relies solely on local broadcas (broadcas o neighbors only). This allows our mehod o be naurally inegraed he periodical Hello messages used by mobile nodes in ad hoc neworks o declear heir exisence. No new ype of conrol messages is needed. Furhermore, our simulaion shows ha comparing o exising mehod such as APS, our mehod generally converges wih less message overhead. 4. Mobile ready. Because of our algorihm eliminaes muliple phases and uses a simple communicaion model, i can be applied direcly o mobile neworks wihou any modificaion. While previous works do no generally provide simulaion resul for mobile scenarios, we demonsrae via simulaion ha our mehod can be effecively used in mobile ad hoc neworks. 5. Exensibiliy. Peering away he dependency o he RSSI signal readings, he core of our algorihm is a probabilisic framework based on paricle filering ha is exremely versaile. The framework can be easily exended o differen signal and nework models. For insance, unlike DV-Hop, our mehod does no assume ha all nodes have he same ransmission range. Unlike Ceneriod or APIT, our mehod does no require a greaer range for GPS nodes, which allows i o work in homogeneous neworks. Furhermore, he framework is no ied o a paricular signal propagaion model or a paricular sensory daa. Alhough we have no implemened i, we expec oher sensory daa such as angle of arrival (AoA) can be used in place of RSSI as he inpu o our algorihm. More ineresingly, he same probabilisic framework will allow muliple sensory daa working ogeher o localize he nework. In oher words, a subse of nodes is capable of AoA readings while anoher subse is capable of RSSI readings. The framework provided by our algorihm can be adaped o solve such problem. A similar Bayesian based approach has been proposed in [24] for he in-door locaion racking problem. In [24], because of he differen obsacles (walls, windows and doors) presened in he in-door floor-plan, a signal srengh (RSSI) map needs o be obained via measuremen ahead of ime. The locaion racking problem hen becomes a decision-making problem. The problem can be solved using a measuremen model ha compares RSSI wih he signal srengh map o find he locaion in he map ha conains he larges probabiliy of maching he curren RSSI characerisics. While similar, our soluion is designed for our-door environmen in which obsacles are assumed o be minimum, and fairly reliable disance esimaes can be obained from RSSI readings and he signal propagaion model. Based on hose assumpions, our soluion does no require he RSSI map. The probabiliy disribuions of locaion esimaes are updaed solely from he disance and locaion esimaes from neighbors. Fig. demonsraes how our mehod solve he localizaion problem in a simple scenario. Here, node 2, 3 and 4 are GPS nodes, and node and are non-gps nodes. Of he non-gps nodes, node can receive signal from and 4 only, and node can receive signal from node, 2 and 3 only. The probabiliy disribuion of he esimaed locaion is represened by he paricles (dos) in he graph. In (a), node can only receive signal from node 4. Thus, as he paricle disribuion indicae, he probabiliy disribuion where node locaes a is a circle around node 4. In (b), node can receive signal from node 2 and 3. Thus, he probabiliy where node locaes ceners around wo areas where circles around node 2 and 3 inersec. Inuiively, in order o localize iself a node-gps node needs o receive locaion informaion from a minimum hree GPS-nodes eiher direcly or indirecly. In boh case (a) and case (b), he exac locaion of he non-gps nodes and

5 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer 5 canno be deduced because hey do no receive locaion informaion from all hree GPS nodes. In (c) and (d), node and are able o communicae o each oher and exchange heir probabiliy disribuions. Thus, heir exac locaions are idenified even hough neiher node receives locaion informaion from he all hree GPS nodes direcly. 3. Classic Mone Carlo Sampling-Based Bayesian Filering This secion describes he heoreical background behind Bayesian filering and how i can be applied o locaion esimaion using RSSI. Le us envision a grid sysem superimposed over he enire racking area, and le he sae s be he locaion of he node o be racked in he grid sysem a he ime. Our goal is o esimae he poserior probabiliy disribuion, p(s d,...,d ), of poenial saes - s, using he RSSI measuremens, d,...,d. The calculaion of he disribuion is performed recursively using a Bayes filer: p(s d,...,d ) = p(d s ) p(s d,...,d ) p(d d,...,d ) Assuming ha he Markov assumpion holds, i.e., p(s s,...,s,d,...,d ) = p(s s ), he above equaion can be ransformed ino he recursive form: p(s d,...,d ) = p(d s ) R p(s s ) p(s d,...,d )ds, p(d d,...,d ) where p(d d,...,d ) is a normalizaion consan. In he case of he localizaion of a mobile node from RSSI measuremens, he Markov assumpion requires ha he sae conains all available informaion ha could assis in predicing he nex sae and hus, an esimae of he non-random moion parameers of he nodes is required as par of he sae descripion. Saring wih an iniial, prior probabiliy disribuion, p(s ), a sysem model, p(s s ), represening he moion of he mobile node (he mobiliy model), and he measuremen model, p(d s), i is hen possible o drive new esimaes of he probabiliy disribuion over ime, inegraing one new measuremen a a ime. Each recursive updae of he filer can be broken ino wo sages: Predicion: Use he sysem model o predic he sae disribuion based on previous readings Z p(s d,...,d ) = p(s s ) p(s d,...,d )ds Updae: Use he measuremen model o updae he esimae p(s d,...,d ) = p(d s ) p(d d,...,d ) p(s d,...,d ) To address he complexiy of he inegraion sep and he problem of represening and updaing a probabiliy funcion defined on a coninuous sae space (which herefore has an infinie number of saes), he approach presened here uses a sequenial Mone Carlo filer o perform Bayesian filering on a sample represenaion. The disribuion is represened by a se of weighed random samples and all filering seps are performed using Mone Carlo sampling operaions. Since we have no prior knowledge of he sae we are in, he iniial sample disribuion, p N (s ), is represened by a se of uniformly disribued samples wih equal weighs, {(s (i),w(i) ) i [,N],w(i) = /N} and he filering seps are performed as follows: Predicion: For each sample, (s (i),w(i) ), in he sample se, randomly generae a replacemen sample according o he sysem (mobiliy) model p(s s ). This resuls in a new se of samples corresponding o p(s d,...,d ): Updae: For each sample, ( s (i),w (i) ), se he imporance weigh o he measuremen probabiliy of he acual measure- =., and draw N random samples for he sample se men, w (i) {( s (i),w (i) ) i [,N],w (i) = /N} = p(d s (i) ) i [, N]} according o he normalized weigh disribuion. Se he weighs of he new samples o /N, re- {( s (i),η w (i) suling in a new se of samples {(s (i) ). Normalize he weighs such ha i η w (i),w (i) ) i [,N],w (i) = /N} corresponding o he poserior disribuion p(s d,...,d ).

6 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer 6 (a) Paricle disribuion of node when node is no presened. (b) Paricle disribuion of node when node is no presened. (c) Paricle disribuion of node when node is presened. (d) Paricle disribuion of node when node is presened. Figure : Locaion Disribuion in Simple Scenarios.

7 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer Modified Paricle Filering for Locaion Esimaions The classical Mone Carlo mehod is ofen implemened using paricle filers. To apply he filer o he locaion racking problem a sysem model and a measuremen model mus be provided. We use a simple random placemen model as our sysem model (please noe ha his is he mobiliy model used in he filer which is differen from he mobiliy model used in he simulaions o enable node movemen). The model assumes ha a any poin in ime he node moves wih a random velociy drawn from a Normal disribuion wih a mean of m/s and a fixed sandard deviaion σ. No informaion abou he environmen is included in his model, and as a consequence, he filer permis he esimaes o move along arbirary pahs. Thus, our sysem model is simply p(s s ) = N(,σ), where N is a Normal disribuion. Noe ha while such sysem model should work well in saionary neworks, i s no bes suied for mobile neworks. In realiy, mobile nodes follow a cerain kind of movemen profile insead of random moion. The sysem model should closely resemble he curren movemen profile of he node. However, since i s difficul o obain a reliable movemen profile when he locaion is unknown, he assumpion of random movemen is probably he bes we can do a his sage. The measuremen daa are obained by observing he periodical locaion daa broadcas from neighbors. To minimize he impac of he measuremen error, we apply a simple Kalman filer o he RSSI sensor readings [6] before feeding he measuremen daa o he paricle filer. When a node u receives broadcas locaion daa from node v, he broadcas daa consis of he unique idenifier of v, and he probabiliy disribuion, X v, of he locaion esimae of v a ime. The X v disribuion is a compressed version of he acual paricle disribuion a v. The deail mehod of compressing and decompressing he paricle disribuion is he opic of he nex secion. For now, le us assume ha X v conains a se of sample paricles ha represens v s locaion. Along wih he RSSI reading of he broadcas, RSSI v, he complee measuremen merics d is herefore (id,x v,rssi v ). Afer he measuremen from he neighbor v is colleced, he paricle filer a node u is updaed. In he classic paricle filering, paricles are re-sampled based on weighs, which are in urn assigned based on he measuremen. More weighs are assigned o he paricle values ha are more consisen wih he measuremen reading. Afer re-sampling, he paricle disribuion becomes more consisen wih he curren measuremen. In our siuaion we have a unique scenario where he measuremen iself consiss of a paricle disribuion, X v. Furhermore, boh X u and X v are imprecise. Our ask during he updae sep is o modify he paricle disribuion X u so ha i becomes more consisen wih RSSI v while aking ino accoun he inheren impreciseness of X u and X v. Firs, we obain a disance esimae from he inverse of he signal propagaion model P: D (RSSI) = P (RSSI v ) Noe ha P can be arbirary as long as i depends on he disance from he sender o he receiver. Noise can be added o he model, bu we disregard i when calculaing he inverse and le i be filered ou by he paricle filering (noe, ha in he simulaions noise is indeed added o he RSSI measuremens). For each paricle x u in X u, we randomly selec a paricle x v in X v and calculae heir disance D (x u,x v ). We hen measure he difference beween D (x u,x v ) and D (RSSI), and selec a new locaion for re-sampling based on he difference as well as he variances of he paricle disribuion X v and X u. For insance, before he updae sep x u and x v are locaed a poin A and B, respecively. Thus, D (x u,x v ) = AB. Le A be he locaion of x u based on he RSSI reading on he same line, i.e., D (RSSI) = A B. Inuiively, if he locaion esimae given by he disribuion X v is accurae and he acual locaion for node v is indeed a x v, hen he new locaion for paricle x u should be a poin A. Conversely, if he locaion esimae of he disribuion X u is accurae, he new locaion for x u should say a A. Therefore, we selec he new locaion based on he perceived accuracy, i.e., he variances, of he disribuion X u and X v. Le he variance of a disribuion X be var(x). We selec he new locaion of x u, x u, along he line AA such ha Ax u x u A = var(x u) var(x v ) A new paricle is hen randomly re-sampled by a Normal disribuion cenered a x u wih he variance being he average of he variances of X u and X v. We consider he variances of boh X u and X v during re-sampling because he spread of boh disribuions affecs he spread of he updaed disribuion X u. Comparing o he re-sampling mehod of classic paricle filers, our mehod is differen in ha we do no use a weigh based re-sampling mehod. Insead, we re-sample by comparing he wo disribuions ogeher agains he measuremen reading. Bu, he concep is he same as we are updaing he disribuion o fi he measuremen readings. Our re-sampling

8 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer 8 mehod has a number of advanages over he radiional mehod. Firs, our mehod does no re-sample direcly from he original paricle locaion using a weigh based Gaussian disribuion. Insead, i re-samples from a more accurae locaion influenced by neighbor s disribuion. Thus, our mehod requires less amoun of random probing and converges more quickly. Secondly, since our mehod requires less amoun of random probing, a significanly smaller number of paricles are required. Wih less paricles, he paricle filer updae procedure compues more efficienly. 3.3 Compressing and Decompressing Paricle Filer Disribuion The previous secion makes he assumpion ha he complee locaion disribuion is received from he neighbor. Since he complee disribuion consiss of a large number of paricles wih heir locaion daa, doing so is obviously no very pracical due o he limied bandwidh of ad hoc neworks. Therefore, we propose a simple ye effecive compressing mechanism ha allows he paricle disribuion o be ransmied in a compac form. Given a paricle disribuion X, we locae he expeced value, ˆx, as he paricle in he disribuion ha has he minimum overall disance beween iself and oher paricles, i.e., ˆx = argmin x X ( y Y x y ). In oher words, ˆx is he mos represenaive paricle of he enire disribuion. From ˆx, we coun he number of paricles n wihin he predefined range r. We hen calculae he variance, σ 2 wihin hose n paricles. Thus, we obain a quadruple ( ˆx,r,n,σ 2 ). From here, we remove he n paricles in he previous quadruple from he disribuion and repea he process of finding he expeced value, a larger range (explained laer) and he variance. By coninuing he same process unil all paricles have been covered, we obain a sequences of quadruples ha approximaes he original paricle disribuion. When he quadruples are received by he receivers, a decompressing algorihm runs o reproduce he disribuion by randomly generaing paricles based on he expeced value, range, paricle number and variance for each quadruple. For each broadcas, a fixed number of aforemenioned quadruples are ransmied. The following algorihm is used o progressively increase he range r for each quadruple. Q := number of quadruples desired R := max range ha covers he enire area minquoa := X /Q rincremen := X 2/3 /R xcoun := r := currange := q := FOR q = o Q maxrange := q rincremen 3/2 WHILE currange < maxrange AND number of paricles in currange + xcoun < minquoa q DO currange := q rincremen 3/2 q := q + r q := currange xcoun := xcoun + number of paricles in currange The algorihm sars wih an iniial range of X/R 3/2 and a minimum quoa of paricle size X /Q for each quadruple. As each quadruple is defined, a running sum, xcoun, keeps rack of he he oal number of paricles covered hus far. A

9 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer 9 Figure 2: Compressing he paricle filer disribuion. each sep, he range is incremened exponenially a each quadruple by r := r 3/2, unless he running sum already exceeds he minimum quoa. The algorihm guaranees ha all paricles are covered by he predefined number of quadruple, and he overall rends of he original disribuion are mainained. Meanwhile, by using a quoa limi wih he exponenial range incremen, more heavily populaed areas are preserved wih finer deail. Our experimen has shown ha he compression mehod reduces he amoun of daa exchange by nearly 9 percen wihou a significan increase o he locaion esimaion error Fig. 2 shows he compressed disribuion, where he circles shows he ranges. 4 Simulaion Resuls We have conduced a number of experimens o validae he effeciveness of our paricle filer based soluion. Our experimens aemp o duplicae real world scenarios as closely as possible. In our simulaions we assume a nework in which all nodes have an idenical ransmission power, wih a cerain percenage of nodes (simulaion parameer) being GPS nodes. For a nework of fixed size, he conneciviy of he nework depends (almos solely) on he ransmission range. When a node is locaed wihin he ransmission range of anoher node, we assume ha i is capable of receiving signal from he sender when noise is no presen. The received signal srengh depends on he disance o he sender as well as a signal propagaion

10 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer model and a noise model. The signal propagaion model is given by P = c d 2, in which he power of he received signal P is inversely proporional o he second power of he disance d. Here, c is an arbirary consan. When he received signal power P is below a hreshold P min, i is considered oo weak o be capured by he receiver hus he link breaks. For our simulaions, we selec c = 6 and P min =. Noe he c and P min selecion does no affec he overall simulaion resuls, as long as he same values are used in he observaion model of he filers. In fac, he same can be said abou all oher signal propagaion models - all we require is a model ha represens he receiving power as a funcion of disance, and we le he filer o filer ou he noise. For he paricle filer iself, we use a oal number of 2 paricles a each node. We use wo ypes of neworks, isoropic and anisoropic, of randomly placed nodes. Wih isoropic neworks, nodes are randomly placed ino a square wih an average degree of 7.6. Wih anisoropic neworks, nodes are placed ino a C shape area wih an average degree of 7. Noise is added o he signal srengh calculaed via he signal propagaion model as a percenage of he calculaed signal srengh. For insance, a percen noise means ha he received signal srengh may vary wihin a plus-minus percen range of he calculaed signal srengh (uniformly disribued). Noe ha our nework configuraion and noise model is idenical o ha of he isoropic opology in [6], so ha we can effecively compare our mehod wih APS. We sar by running he simulaion on saionary neworks, which resembles sensor nework in he real world. 4. Filer Convergence Figure 3 shows how he esimaion error converges as more measuremen readings are processed in a saic nework. We are ineresed in how long and how many messages i akes for he error o reach an accepable level from which i only reduces marginally. We added a noise level of 5 percen o he measuremen readings. The esimaion error is calculaed as he difference beween he mos likely value given by he paricle disribuion and he acual locaion. The difference is hen measured in erm of he raio agains he maximum ransmission range. Thus, an esimaion error of. means ha difference beween he expeced value and acual locaion equals o he maximum ransmission range. The daa is colleced of enough simulaion runs o claim a 95 percen confidence, which shows as he verical scale a each daa poin; he error raio is he average of all non-gps nodes (i.e., he perfec esimaes of GPS nodes are no biasing he resuls). Two obvious facs can be observed from Figure 3: i) neworks wih higher GPS raio produce beer esimaions and ii) esimaion error reduces quicker wih higher GPS raio. Boh of hose observaions can be explained by he fac ha GPS raio deermines how fas and how accurae locaion informaion can be propagaed hrough he nework. Wih a higher GPS raio, non-gps nodes will be able o obain he necessary locaion informaion faser because non-gps nodes are closer (i.e., less number of hops) o GPS nodes. Also, since measuremen error is aggregaed a each hop, he locaion informaion will be more accurae wih higher GPS raio. GPS raio also affecs he confidence inerval. This is because when GPS raio is low, he esimaion error depends grealy on he posiion of he GPS nodes. When heir posiion does no spread ou evenly hrough he nework (for insance, GPS nodes is concenraed around one edge of he nework), i becomes more difficul for he he nodes furher away o obain good esimaes. As he GPS raio increase, he chances of bad posiions reduces, and hus he variance of he esimaion error reduces. Figure 3 also shows ha he esimaion error converges o he minimum beween 2 o 5 seconds depending on he GPS raio. Considering ha he locaion broadcas occurs every.5 seconds, i akes abou 4 o rounds of broadcass for he error o reach he minimum. Since he average degree of he nework is 7.5 wih a oal of nodes, each round of broadcas is equivalen o 75 messages. Therefore, i akes abou 3 o 75 messages o minimize he error depending on he GPS raio. Noe ha when even in he wors case where he GPS raio is low, error converges very quickly and is close he minimum afer 2 seconds. The resuls are a leas as good as hose of APS, where he DV-disance mehod uses 65 messages (when GPS raio is.) o 9 messages (when GPS raio is.9), and he Euclidean mehod akes from 3 o 85 messages (resuls aken from Figures 7 and of [6]). Noe ha our mehod converges quicker and akes less number of messages when he GPS raio is high. 4.2 Minimum Fig. 4 compares he esimaion error of our paricle filer mehod wih oher mehods. Again, he simulaion scenario is duplicaed from ha of he isoropic opology in APS [6], wih a raher dense nework of degree averages a 7.6. One

11 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer Convergence (saionary nework, isoropic, noise raio 5%) GPS Raio 5% GPS Raio % GPS Raio 2% GPS Raio 3% GPS Raio 5% Convergence (saionary nework, anisoropic, noise raio 5%) GPS Raio 5% GPS Raio % GPS Raio 2% GPS Raio 3% GPS Raio 5% Time (seconds) (a) isoropic Time (seconds) (b) anisoropic Figure 3: Filer Convergence. advanage of our mehod is ha our mehod produces he locaion esimae along wih a variance indicaing is qualiy. Thus, by varying he variance hreshold, we are able o conrol he effecive esimaion coverage. Fig. 4(a) plos he acual filer variances agains he esimaes for he scenario where GPS raio is o 5 percen. Here, a rend can be observed ha he variances increase linearly wih he esimaes. Fig. 4(b) shows relaionship beween he coverage and esimaion error when he GPS raio varies. Oher mehods such as DV-Hop, DV-Disance and Euclidean generae esimaes wih fixed coverage, and hus hey are ploed as single poins in he graph. Noe ha when GPS Raio is percen, our mehod generaes similar esimaion error as Euclidean when adjused o he same coverage, bu DV-Hop and DV-Disance give beer esimaion wih he same coverage. Fig. 4(c) hrough Fig. 4(e) shows he resul of a more deailed comparison agains DV-Hop, DV-Disance and Euclidean. We compare he esimaion error of our mehod wih oher mehods by obaining he average esimaion error when is corresponding coverage maches he oher mehod. The figures show ha DV-Hop and DV-Disance give lower esimaion error when he GPS raio is less han 2 percen. Wih a higher GPS raio, our mehod gives beer resul. Similar resul can be observed when comparing o Euclidean, bu he cu-off poin here is around percen GPS raio. The higher error a low GPS raio can be explained by he fac ha he paricle filer mehod prefers he scenarios where GPS nodes are locaed around all edges of he nework forming a near convex hull, in which case he locaion informaion from various GPS nodes can uilized more effecively. When GPS raio is low, such ideal scenarios are less likely o occur, and here will be nodes ouside he convex hull ha are more difficul o localize. DV based mehods, however, are affeced less by hose scenarios, since hey uses globally colleced daa such as disance-per-hop o perform he riangulaion. 4.3 Conneciviy Simulaion resuls in previous work are based on a raher dense nework wih an average degree of Similar neworks were used in [6], and hus allow a more sensible comparison. Fig. 5 shows he esimaion error of our paricle filer based localizaion mehod in more sparse neworks. Here, we vary he nework conneciviy by changing he ransmission range while mainaining he nework size ( nodes). Fig. 5(a) shows ha as expeced more error is inroduced in sparser neworks. Roughly speaking, he error halves when he nework conneciviy doubles. Fig. 5(b) hrough Fig. 5(d) shows he resul of direc comparison he esimaion error beween our mehod and ohers under he same coverage. When he nework is sparser, our mehod clearly ou-performs all oher hree mehods. In heory a node needs o receive signal readings from a minimum of hree neighbors in order o pinpoin is locaion. Thus, a nework wih degree of a leas hree will be needed o localize all is nodes. Wih our localizaion mehod, respecable esimaions are obained even wih very sparse neworks of degrees less han hree (no oher approaches are able o derive esimaes for such siuaions).

12 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer Variance (saionary nework, isoropic, GPS raio 5%, noise raio 5%).4.2 vs. Coverage (saionary nework, isoropic, noise raio 5%) GPS Raio 5% GPS Raio 3% GPS Raio 2% GPS Raio % GPS Raio 5% DV-Hop, GPS Raio % DV-Disance, GPS Raio % Euclidean, GPS Raio % Variance (a) vs. Filer Variance Coverage (. = %) (b) vs. Coverage.4.2 vs. Coverage (saionary nework, isoropic, Noise =.5) DV-Hop Paricle Filer.8.6 vs. Coverage (saionary nework, isoropic, Noise =.5) DV-Disance Paricle Filer GPS Raio (c) Paricle Filer vs. DV-Hop GPS Raio (d) Paricle Filer vs. DV-Disance. vs. Coverage (saionary nework, isoropic, Noise =.5) Euclidean Paricle Filer GPS Raio (e) Paricle Filer vs. Euclidean Figure 4: Effec of GPS Raio on.

13 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer vs. Coverage (saionary nework, isoropic, GPS raio = %, noise raio = 5%) Degree = 9.4 Degree = 7.66 Degree = 6.3 Degree = 4.6 Degree = 2.83 DV-Hop, Degree = 4.6 DV-Disance, Degree = 4.6 Euclidean, Degree = saionary node, noise =.5, GPS Raio = %, ran seconds DV-Hop Paricle Filer Coverage (. = %) (a) vs. Coverage Average Degree (b) Paricle Filer vs. DV-Hop saionary node, noise =.5, GPS raio = %, ran seconds DV-Dis Paricle Filer saionary node, noise =.5, GPS raio = %, ran seconds Euclidean Paricle Filer Average Degree (c) Paricle Filer vs. DV-Disance Average Degree (d) Paricle Filer vs. Euclidean Figure 5: Effec of Conneciviy on.

14 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer Convergence (saionary nework, isorophic, noise raio 5%) GPS Raio 5% GPS Raio % GPS Raio 2% GPS Raio 5% No Compression, GPS Raio 5% No Compression, GPS Raio % No Compression, GPS Raio 2% No Compression, GPS Raio 5%.4.2 vs. Coverage (saionary nework, isoropic, noise raio 5%) GPS Raio 5% GPS Raio 2% GPS Raio % GPS Raio 5% No Compression, GPS Raio 5% No Compression, GPS Raio 2% No Compression, GPS Raio % No Compression, GPS Raio 5% Time (seconds) Coverage (. = %) Figure 6: Effec of Compression on Filer Convergence. Figure 7: Effec of Compression on. 4.4 Compression vs. No Compression Fig. 6 and Fig. 7 demonsrae he effeciveness of our compression algorihm on ransmiing he paricle disribuion. The same scenario is repeaed when a complee paricle disribuion is ransmied insead of he compressed version. The resuls are compared side by side. While he localizaion algorihm works beer when he complee disribuion is sen, he differences are raher minimal. While he original paricle disribuion consiss of 2 paricles, each of which conains wo decimal numbers o designae he locaion, he compressed version consiss of quadruples, each of which conains five decimal numbers. The compression mehod archives a oal bandwidh saving of 87.5% a each locaion exchange. Given ha he nework bandwidh can be expensive, one can easily jusifies he minimal radeoff of performance using our compression scheme. 4.5 Resuls on Mobile Neworks Previous work on MANET localizaion generally do no conain exensive simulaion and analysis when he nework is indeed mobile (as he definiion of MANETs imply). As discussed earlier, many previous mehods are specifically designed o work in saionary sensor neworks, in which i is sufficien o complee one round of localizaion and here is no requiremen for furher adjusmen when opology changes. Thus, adaping hem o work in mobile neworks can be quie challenging. In he wors case, he enire localizaion scheme has o be rerun. Our mehod, however, are specifically designed o work in mobile neworks. This secion discusses simulaion resuls on running our he paricle filer localizaion mehod on mobile neworks. Again, we use a nework wih a populaion of nodes and average degree of 7.5. We use he epoch-based mobiliy model of [5] o simulae node movemen, which is widely acceped as a good mobiliy model for ad hoc neworks - more realisic han, e.g., simple Brownian moion models. The enire movemen pah of he node is defined by a sequence of epochs, i.e., (e,e 2,,e n ). The duraion of each epoch is I.I.D. exponenially disribued wih a mean of /λ. Wihin each epoch nodes move wih a consan velociy vecor. A he end of each epoch, nodes randomly selec a new velociy vecor. The direcion of he movemen is I.I.D. uniform beween and 2π. The absolue value of he velociy is I.I.D. normal wih a mean µ of and a variance of σ 2. Our simulaion uses a fixed mean and variance such ha µ= σ. The resul is obained by varying µ and σ from m/s o 4m/s. The expeced amoun of ime a node mainains is curren velociy is se o 5 seconds, i.e., λ = 5. Figure 8 shows he filer convergence on mobile neworks wih measuremen noise level se o 5%. Here, all nodes in he nework moves a m/s in average, i.e., µ = σ =. Comparing o he resuls of saionary neworks in Figure 3, he random movemen of he nodes causes he esimaion error o swing. However, he error variances are no very high once he nodes deermine heir iniial locaions afer he firs couple of seconds. This indicaes ha he filer is able o adap o he node movemen well enough o mainain is overall esimaion accuracy. Figure 9 shows he average esimaion error of mobile neworks. The error does increases gracefully as he speed increases. Considering ha neighbors exchange

15 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer Convergence (mobile nework, isoropic, noise raio 5%) GPS Raio 5% GPS Raio % GPS Raio 2% GPS Raio 3% GPS Raio 5% when Coverage = % (mobile nework, isoropic, noise raio 5%, ran 5 seconds).4 Speed = m/s Speed = 5m/s Speed = m/s.2 Speed = 2m/s Speed = 4m/s Time (seconds) Figure 8: Filer Convergence wih Mobile Neworks GPS Raio Figure 9: of Varying GPS Raio and Nework Mobiliy. locaion informaion every.5s, in a nework wih an average nodal speed of 4m/s nodes move an average of 2 meers per observaion; ye our mehod is capable of producing usable locaion esimaes. From he above resul, one can foresee ha i is possible o reduce he rae of localizaion exchanges beween he neighbors afer he iniial localizaion complees, while sill mainaining reasonable good esimaes. A challenge of fuure works is o find ou he ideal rae of exchanges for a given amoun of nework mobiliy. 5 Conclusions This paper described a novel soluion o he locaion racking problem for mobile ad hoc neworks ha uses a Mone Carlo sampling-based Bayesian filering (i.e., paricle filering) mehod. The esimaed locaion for nodes is regarded as a probabiliy disribuion represened by a collecion of sample poins. The locaion informaion from he GPS nodes is propagaed hrough he nework via local broadcasing of he locaion esimaes. When a node receives he locaion esimaes from neighbors, i updaes is locaion disribuion using he paricle filering mehod. Simulaion sudy has shown ha he paricle filer soluion is capable of producing good esimaes equal or beer han he exising localizaion mehods such as APS-Euclidean. Our soluion also performs quie well when he nework conneciviy is low. Sudy has also shown ha he soluion is resilien o nework opology change, making i suiable for ad hoc neworks wih significan mobiliy. Our paricle filer based localizaion mehod currenly uses RSSI as he sole measuremen. However, because our mehod is based on a raher generic algorihm of probabilisic filers, i can be easily exended o incorporae oher measuremen ypes such as angle of arrival (AoA). To do so, only he filer updae sep needs o be changed in order o meaningfully updae he filer according o he properies of he new measuremen, bu he basic algorihm remains he same. In fac, i is easy o implemen our mehod wih muliple ypes of measuremens coexising in he nework. The same paricle filer mehod can be used in a nework where an arbirary porion of nodes are capable of measuring RSSI, anoher par of he nodes are capable of measuring AoA, and some are capable of measuring boh. This makes our mehod ruly versaile and ideal for such heerogeneous neworks. References [] P. Bahland and V.N. Pamanabhan, RADAR: An in-building RF-Based user Locaion and Tracking Sysem, In Proceedings of he IEEE INFOCOM, March 2. [2] N. Bulusu, J. Heidemann, and D. Esrin, GPS-less low cos oudoor localizaion for very small devices, IEEE Personal Communicaions Magazine, vol. 7, no. 5, pp , Ocober 2.

16 Locaion Tracking in Mobile Ad Hoc Neworks using Paricle Filer 6 [3] L. Girod and D. Esrin, Robus Range Esimaion using Acousic and Mulimodal Sensing, In Proceedings of IROS, Maui, Hawaii, Ocober 2. [4] N. Gordon, Bayesian Mehods for Tracking, PhD hesis, Universiy of London, 993. [5] T. He, C. Huang, B. M. Blum,J. A. Sankovic,and T. F. Abdelzaher. Range-Free Localizaion Schemes in Large Scale Sensor Neworks, CS-TR23-6. Submi o MobiCom 23. [6] R. Huang, G.V. Záruba, and M. Huber, Link Longeviy Kalman-Esimaor for Ad Hoc Neworks, To appear in he Proceedings of he VTC23, IEEE 54h Vehicular Technology Conference, 23. [7] X. Jiang and T. Camp. Review of geocasing proocols for a mobile ad hoc nework. In Proceedings of he Grace Hopper Celebraion (GHC), 22. [8] Y. Ko and N. H. Vaidya, Locaion-Aided Rouing (LAR) Mobile Ad Hoc Neworks, MOBICOM 98, Dallas, TX, 998. [9] Y. Ko and N. Vaidya. Geocasing in Mobile Ad Hoc Neworks: Locaion-Based Mulicas Algorihms. In IEEE Workshop on Mobile Compuing Sysems and Applicaions (WMCSA), 999. [] F. Kuhn, R. Waenhofer, Y. Zhang and A. Zollinger: Geomeric ad-hoc rouing: of heory and pracice. PODC 23: [] W.-H. Liao, Y.-C. Tseng, and J.-P. Sheu. GRID: A fully locaion-aware rouing proocol for mobile ad hoc neworks. Telecommunicaion Sysems, 8():37 6, 2. [2] W.-H. Liao, Y.-C. Tseng, K.-L. Lo, and J.-P. Sheu. Geogrid: A geocasing proocol for mobile ad hoc neworks based on grid. Journal of Inerne Technology, (2):23 32, 2. [3] T. Liu, P. Bahl, and I. Chlamac, A hierarchical posiion-predicion algorihm for efficien managemen of resources in cellular neworks, Proceedings of he IEEE GLOBECOM 97, Phoenix, Arizona, November 997. [4] M. Mauve, H. Fuler, J. Widmer, and T. Lang. Posiion-Based Mulicas Rouing for Mobile Ad-Hoc Neworks. Technical Repor TR-3-4, Deparmen of Compuer Science, Universiy of Mannheim, 23. [5] A. B. McDonald and T. Znai, A mobiliy-based framework for adapive clusering in wireless ad-hoc neworks, IEEE Journal on Seleced Areas in Communicaion Special Issue on Wireless Ad-Hoc Neworks, vol. 7, no. 8, Augus 999. [6] D. Niculescu and B. Nah, Ad hoc posiioning sysem (APS), Proceedings of he IEEE GLOBECOM, San Anonio, 2. [7] D. Niculescu and B. Nah, Ad hoc posiioning sysem (APS) using AoA, Proceedings of he IEEE INFOCOM, San Francisco, 23. [8] D. Niculescu and B. nah, DV Based Posiioning in Adhoc Neworks, o appear in journal of Telecommunicaion Sysems, 23. [9] P.J. Nordlund, F. Gunnarsson, and F. Gusafsson, Paricle filers for posiioning in wireless neworks, Proceedings of he XI. European Signal Processing Conference (EUSIPCO), 2. [2] C. Savarese, J. Rabay and K. Langendoen, Robus Posiioning Algorihms for Disribued Ad-Hoc Wireless Sensor Neworks, USENIX Technical Annual Conference, Monerey, CA, June 22. [2] A. Savvides, H. Park and M. B. Srivasava, The n-hop Mulilaeraion Primiive for Node Localizaion Problems. Mobile Neworks and Applicaions 8(4): , Augus 23. [22] K. Whiehouse and D. Culler, Calibraion as Parameer Esimaion in Sensor Neworks, In Firs ACM Inernaional Workshop on Wireless Sensor Neworks and Applicaion, Alana GA, Sepember 22. [23] Y. Xu, J. Heidemann, and D. Esrin. Geography-informed energy conservaion for adhoc rouing. In Proc. sevenh Annual ACM/IEEE Inernaional Conference on Mobile Compuing and Neworking (MobiCom), pages 7 84, 2. [24] G. V. Záruba, M. Huber, and F. A. Karmangar, Mone Carlo Sampling Based In-Home Locaion Tracking wih Minimal RS Infrasrucure Requiremens, o be published, 23.