Disribued Online Localizaion in Sensor Neworks Using a Moving Targe Aram Galsyan 1, Bhaskar Krishnamachari 2, Krisina Lerman 1, and Sundeep Paem 2 1 Informaion Sciences Insiue 2 Deparmen of Elecrical Engineering-Sysems Universiy of Souhern California Los Angeles, California {galsyan, lerman}@isi.edu, {bkrishna, paem}@usc.edu Absrac. We describe a novel mehod for node localizaion in a sensor nework where here are a fracion of reference nodes wih known locaions. For applicaion-specific sensor neworks, we argue ha i makes sense o rea localizaion hrough online disribued learning and inegrae i wih an applicaion ask such as arge racking. We propose disribued online algorihm in which sensor nodes use geomeric consrains induced by boh radio conneciviy and sensing o decrease he uncerainy of heir posiion. The sensing consrains, which are caused by a commonly sensed moving arge, are usually igher han conneciviy based consrains and lead o a decrease in average localizaion error over ime. Differen sensing models, such as radial binary deecion and disance-bound esimaion, are considered. Firs, we demonsrae our approach by sudying a simple scenario in which a moving beacon broadcass is own coordinaes o he nodes in is viciniy. We hen generalize his o he case when insead of a beacon, here is a moving arge wih a-priori unknown coordinaes. The algorihms presened are fully disribued and assume only local informaion exchange beween neighboring nodes. Our resuls indicae ha he proposed mehod can be used o significanly enhance he accuracy in posiion esimaion, even when he fracion of reference nodes is small. We compare he efficiency of he disribued algorihms o he case when node posiions are esimaed using cenralized (convex) programming. Finally, simulaions using he TinyOS-Nido plaform are used o sudy he performance in more realisic scenarios. 1 Inroducion Wireless sensor neworks (WSN) hold a promise o dwarf previous revoluions in he informaion revoluion [1]. Fuure WSN are envisioned o consis of hundreds o housands of sensor nodes communicaing over a wireless channel, performing disribued sensing and collaboraive daa processing asks for a variey of vial miliary and civilian applicaions. Such ubiquious sensor neworks will improve he safey of our buildings and highways, enhance he viabiliy of wildlife habias, dramaically shoren disaser response imes, reduce commue imes, and conribue o many oher vial funcions as par of he embedded, everywhere vision. Node localizaion is a fundamenal problem in sensor neworks. Boh a he applicaion layers as well as for he underlying rouing infrasrucure, i is ofen useful o know he locaions of he consiuen nodes wih high accuracy. There have been a number of recen effors o develop localizaion algorihms for wireless sensor neworks, mos of which are based on using saic reference beacons, signal-srengh esimaion or acousic ranging. Common characerisics in hese effors have been (i) a view of localizaion as a one-sep process o be performed a deploymen ime and (ii) he separaion of localizaion from he applicaion asks being performed. For applicaion-specific sensor neworks, we argue ha i makes sense o rea localizaion as an online disribued problem and inegrae i wih he applicaion. Our approach explois addiional informaion gahered by he nework over he course of running an applicaion o significanly improve localizaion performance. The applicaion we consider in his paper is he single-arge
racking problem. Targe racking is a canonical applicaion for wireless sensor neworks, no only because of is relevance o inelligence gahering and environmenal monioring, bu also because i combines he basic challenges of sensor neworks: disribued sensing in power consrained, dynamic environmens wih likely node failures. In order o achieve good accuracy in arge localizaion ask, he nodes hemselves have o be well localized. Moreover, he localizaion algorihm should be efficien and scalable. The main conribuion of his paper is a simple disribued online scheme for simulaneous arge and node localizaion in neworks wih a small fracion of reference nodes (e.g., nodes whose exac posiions are known a priori eiher hrough planned placemen or hrough a Global Posiioning Sysems (GPS) receiver). In he algorihm we propose, sensor nodes use online observaions of a moving arge o simulaneously improve he esimaes of boh he arge s and heir own posiions. Each observaion adds a geomeric consrain on he posiion of sensor nodes and over ime leads o a dramaic improvemen in heir posiion esimaes. The approach is generalizable o differen sensing models. Firs we consider a simplified version of he localizaion ask in which a friendly beacon broadcass is coordinaes as i moves hrough he nework. We show ha his scenario leads o quick convergence o good posiion esimaes. Nex, we exend he approach o he more ineresing siuaion in which he arge s posiion is no known a priori. In his case, a boosrapping mechanism provides for he ieraive improvemen of esimaes of boh he arge s locaion and ha of he sensor nodes. The sensor nework is able o rack he arge more accuraely over ime by learning beer esimaes of boh node and arge posiions hrough sensor observaions. In addiion o using consrains imposed by arge observaion, we describe how a node can also exploi informaion abou a arge in is viciniy ha i did no observe o se igher bounds on is own posiion. The algorihms we propose are designed o be compleely disribued, and require he exchange of informaion only beween neighboring nodes. To our knowledge, his is he firs proposed disribued online localizaion mehod ha explois a sensing applicaion o improve is performance over ime. We underake a horough se of numeric experimens o analyze he performance of he proposed localizaion echniques. The imporan parameers ha we consider include he size of he nework, he fracion of known nodes, he radio range and he sensing modaliy/range. The performance of his learning-based localizaion scheme is shown as a funcion of ime, showing how he locaion errors decrease wih addiional observaions. We compare his disribued localizaion echnique wih a cenralized semi-definie programming which yields a globally opimal soluion. Finally, we presen resuls from simulaions using he TinyOS-Nido plaform o sudy he perforance in he presence of packe losses and anisoropy in communicaion and sensing. 2 Consrain-based Localizaion Dohery e al. [5] inroduce an approach o posiion esimaion in a sensor nework using convex opimizaion. If a node can communicae wih anoher node, is posiion is resriced by he conneciviy consrains o be in some region relaive o he oher nodes. Many such conneciviy or proximiy consrains define he se of feasible node posiions in a nework. These consrains can be encoded as a Linear Marix Inequaliies (LMI-s) and solved using convex opimizaion echniques o obain posiion esimaes. Dohery e al. considered a cenralized mehod where each node relays is connecion saisics o a cenralized auhoriy which hen compues he global soluion. However, a cenralized approach scales poorly wih he size of he nework. Simić and Sasry [7] presen a disribued version of he localizaion algorihm based on conneciviy consrains. They consider a discree nework model and derive probabilisic error and complexiy bounds. The disribued algorihm has much beer scaling properies han a cenralized soluion and a lower communicaion cos, because he nodes are no required o relay informaion; herefore, disribued soluions are more aracive for large neworks conaining housands of nodes. We propose an online disribued algorihm in which nodes improve heir locaion esimaes by incorporaing conneciviy consrains as well as consrains imposed by a moving arge. Nodes arrive a an iniial esimae of heir posiion using conneciviy consrains. Nodes hen use deecion
(or non-deecion) of a moving arge o updae heir posiion esimaes. Because he sensing range is usually smaller han he communicaion range, he igher consrains imposed by he arge help localize nodes more accuraely. Moreover, repeaed observaions of a arge by differen subses of he nework cause he mean nework localizaion error o decrease over ime. 3 Disribued Online Localizaion Consider an ad hoc wireless sensor nework of N nodes randomly deployed in an L L square area, communicaing over an RF channel. We make he simplifying assumpion of a roaionally symmeric communicaion range whereby each node communicaes wih neighboring nodes ha fall wihin he disk of radius r cenered on he node. The communicaion range, r, is assumed o be he same for all nodes; herefore, conneciviy via RF channel is symmeric. Alhough he radial communicaion model may no be a realisic descripion of wireless sensor neworks in physical environmens, i is a valid saring poin for modeling purposes, and has been sudied by various groups in he pas. A fracion f of nodes know heir posiions, for example, because of an on-board GPS device or because hey are affixed o known landmarks in he environmen. Le (x i, y i ) be he acual posiion of he i h node. For each node he uncerainy in is locaion is given by a bounding recangle Q i = [x min i, x max i ] [yi min, yi max ], x min i x i x max i, yi min y i. Iniially, we se Q i = [0, L] [0, L]. The esimaed posiion of he node is given by he cenroid y max i of he bounding box. The error is he difference beween he acual and esimaed posiions. 3.1 Sensing Models Our approach is general and direcly applicable o wo differen disance sensing models: radial binary deecion and disance-bound esimaion. In he radial binary deecion sensor modal, each sensor can deec wheher or no here is a arge wihin range s of he sensor. In he disancebound esimaion model, each sensor can esimae a bound on he disance wihin which he arge mus be presen. This model implicily allows he possibiliy of sensor noise in paricular, if a signal-srengh esimae is noisy, hen i may be easier o provide an upper bound on he disance esimae. Our approach also works in he special case where exac disance informaion is available. 3.2 Localizaion Using a Moving Beacon Firs we consider a simple scenario where a moving beacon whose posiion is known is used o dynamically self-configure a nework. As he beacon moves hrough he nework, i broadcass is coordinaes o he nodes which are a mos a sensing disance s away from i. Noe ha since communicaion beween he nodes is no relevan, he problem reduces o sudying he behavior of a single node. Every ime a node senses he beacon, i generaes a new quadraic consrain ha i uses o furher reduce he uncerainy in is posiion. This is illusraed in Fig. 1. Afer he firs observaion of he beacon (Fig. 1(a)), an unknown node s posiion is limied o he circle of radius s around he beacon. The second observaion of a moving beacon (Fig. 1(b)) furher consrains possible node posiions, reducing he uncerainy of is posiion (shaded region). Repeaed observaions of a moving beacon improve node localizaion over ime. We approximae he sensing region by a recangular bounding box, hereby replacing he quadraic consrain by a weaker bu simpler linearized consrain (see Fig. 1(b)). Alhough his approximaion overesimaes he uncerainy of he node s posiion, i guaranees ha he acual posiion of he node is always wihin he bounding box and considerably simplifies he compuaion of he bounding box. 3 More precisely, le Q be he bounding box for a node. Then, he node will updae is bounding box using he consrain imposed by he beacon a posiion (x b, y b ) according o he following rule: 3 Noe ha we can find a igher recangle ha bounds he region of inersecion of he wo circles; however, he exra compuaional effor does no buy us much beer final localizaion abiliy.
S (a) (b) Fig. 1. (a) Sensing consrains limi he se of possible posiions of an unknown node (open circle) o he shaded region. Black circle corresponds o a moving beacon wih known coordinaes. (b) When he node deecs a beacon he second ime, he uncerainy is collapsed by he observaion even. We approximae he sensing regions by squares o simplify compuaion. Q Q [x b s, x b + s] [y b s, y b + s]. As illusraed in Fig. 2, his simple ieraion scheme leads o an accurae localizaion of he node for boh models of sensing. The ime-evoluion of localizaion error can be calculaed by means of order saisics. Indeed, consider, for insance, he one-dimensional localizaion problem wih binary sensing. Le {x 1, x 2,...x } be he se of beacon locaions sensed up o ime by a node locaed a x = 0, and le x max and x min me he righmos and lefmos posiions in he se respecively. The localizaion error A() = 2S (x max x min ) is hen a random variable wih probabiliy densiy funcion P (A; ) = ( 1) (2S) 1 (2S ( 1) A) 2 (2S) (2S A) 1, 0 A 2S, (1) and he average localizaion error goes o zero as A() = 4S/( + 1). The wo-dimensional case can be reaed similarly, alhough he calculaion are involved and we do no presen he resuls here due o space limiaions. One can insead use simple geomerical argumens o deduce ha he asympoic behavior of he average localizaion is A() 1/ 2/3 for binary sensing, and A() 1/ for he disance bound model. 2.0 0.5 1.5 Binary Disance-bound esimaion 0.4 Binary Disance-bound esimaion 0.3 A() 1.0 E() 0.5 50 50 Fig. 2. (a)average size of he bounding box A and (b) mean square error in node s posiion E vs ime. Average over 500 runs has been aken. 3.3 Localizaion Using a Moving Targe Now le us assume ha insead of a beacon wih known coordinaes a arge wih a-priori unknown coordinaes is observed. The single-node approach of he previous secion does no work. However,
if here are enough nodes wih known posiions (or ha are relaively well localized) in he viciniy of a given node, hen he arge can be localized and his informaion can be used o impose new consrains on he posiion of he node. A number of echniques exis for arge localizaion, riangulaion being he mos popular. However, riangulaion requires ha he arge be observed by hree sensors wih known locaions (in 2D), somehing which canno be guaraneed in a mixed sensor nework wih a small enough f. Here we describe an alernaive approach o arge localizaion. Le Q T be a bounding recangle for he arge (iniialized o [0, L] [0, L]), and le K be he se of nodes ha sense he arge. Then he bounding box for he arge is esablished as follows: Q T Q T [x k min S k, x k max + S k ] [ymin k S k, ymax k + S k ], (2) k K where x k min, xk max, y k min and yk max give he coordinaes of he bounding box for he posiion of he k h node. For he radial binary sensing model, S k = s k is he sensing range (assumed o be he same for all nodes), whereas for he disance esimaion model, S k = d k is he esimaed upper bound on he disance beween he k h node and arge. Noe ha his approach guaranees ha he acual arge posiion is always inside he bounding box and ha he uncerainy region remains convex afer he updae. s (a) (b) Fig. 3. (a) Sensing consrains limi he se of possible posiions of he sensor node (open circle) o he shaded region. Black circle is a moving arge whose posiion is no known bu esimaed o be a he cener of he shaded region. (b) Observaion of he arge shrinks uncerainy of he node s posiion. We approximae he sensing region by a square. The node can use consrains imposed by he arge in wo ways. I can use he corners of he bounding box Q T o impose consrains on is own posiion (Eq. 3). Alernaively, we can neglec he uncerainy in he arge posiion, and assume ha i is locaed a he cener of Q T (Eq. 4). These updae rules are specified below. Q i Q i [x T min S k, x T max + S k ] [y T min S k, y T max + S k ], (3) Q i Q i [x T es S i, x T es + S i ] [y T es S i, y T es + S i ], (4) where x T min, xt max, y T min and yt max specify he arge bounding box, and x T es and y T es give he arge s esimaed posiion (cenroid of he bounding box). S i is he sensing range for he radial binary sensing model, whereas for he disance-bound model, i is he esimaed disance beween he i h node and arge. The cenroid consrain, Eq. 4, is much sronger. We find ha despie loosing he guaranee ha he unknown nodes remain inside heir bounding boxes, for some regimes we can significanly reduce he oal nework localizaion error by using his scheme. This is especially useful for he radial binary sensing model
3.4 Negaive Informaion In some cases if a node does no deec a arge whereas is neighbor nodes do, i may be able o use his informaion o reduce is posiion uncerainy. This siuaion is depiced in Fig. 4(a) where, for illusraive purposes, we assume ha he arge s exac posiion is known. If he node s acual posiion was in he shaded region wihin he circle, i would have deeced he arge. Hence, he region Q neg can be excluded from he node s bounding box. Noe, ha we use his negaive informaion consrain (i.e. a consrain ha is obained by he ac of no sensing a arge) only when Q Q neg is convex, where Q is he bounding recangle for he node. Again, we prefer working wih recangles raher han calculae he exac inersecion poins. This simplificaion is shown in Fig. 4(b) where we approximae he circle by an inner square of side R neg = 2 2 S. Le Q neg = [x T R neg /2, x T + R neg /2] [y T R neg /2, y T + R neg /2]. Then he updaing rule can be wrien as follows: Q Q Q Q neg if Q Q Q neg is convex Q Q oherwise (5) We also sudied he case when he circle is approximaed by a (larger) square of side R neg = 2Sγ, where 1 2 < γ < 1 (Fig. 4(c)). Alhough in his case here is a probabiliy he excluded region may in fac conain he node, his approach has an advanage ha now he convexiy condiion on (Q Q neg ) is more likely o hold. In fac, as we will show below, he model wih γ = 0.9 produces he bes resuls. Noe also, ha if he node keeps rack of he negaive regions for differen arge posiions, hese consrains can be combined. Hence, even hough each of he consrains individually were no iniially useful due o he convexiy condiion, he combinaion of boh can be used, as illusraed in Fig. 4(d), where he shaded region is he combined Q neg. Alhough we did no employ his approach in he presen paper, we believe ha is use can grealy improve he qualiy of localizaion. The generalizaion of he negaive consrain rule o he case when he arge is wihin a recangle [x T min, xt max] [ymin T, yt max] is sraighforward. The updaing rule is exacly he same, and he only difference is he region Q neg, which in his case is consruced as follows: le C i be a square of side R neg and cenered on he i h corner of he arge s bounding recangle, i = 1,..4. Then Q neg is he inersecion of hese squares, Q neg = C 1 C 2 C 3 C 4. Noe ha if x T max x T min > R neg or ymax T ymin T > R neg, his inersecion will be empy, Q neg =. T 1 S S R neg = 2 2 S T 2 Qneg Qneg (a) (b) (c) (d) Fig. 4. Differen ways in which he node can use negaive informaion resuling from he failure o deec a nearby arge 3.5 Algorihm Each node uses an algorihm oulined in Fig. 5 o improve is posiion esimae. There are hree disinc procedures a node uses o updae is bounding box Q i : (i) using consrains imposed by
conneciviy requiremens (Eq. 6), as well as using sensing consrains imposed by he arge boh when (ii) he node deecs a arge (Eq. 3 4) and when (iii) no arge is deeced by a node whose neighbor deecs he arge (using negaive informaion consrains, see Sec. 3.4). For compleeness, we specify he updae rule ha uses conneciviy based consrains: Q i Q i [x k min r, x k max + r] [ymin k r, ymax k + r], (6) k K where x k min, xk max, y k min and yk max specify he bounding box of node k and r is he radio conneciviy range. iniialize Q i = [L, L] updae Q i using conneciviy consrains (Eq. 6) ierae if T is deeced updae Q T (Eq. 2) if T is deeced by i updae Q i using arge informaion (Eq. 3 4) else updae Q i using negaive informaion (Eq. 5 end if end if for each neighbor k if Q k changes updae Q i wih conneciviy consrains (Eq. 6) end if end for loop Fig. 5. Pseudocode of he node localizaion algorihm 4 Resuls We carried ou exensive numerical simulaions of our models described in Sec. 3.3. In all he cases he nodes were disribued randomly and uniformly in an L L square area. All he lenghs are given in unis of L so ha we can se L = 1. We sudied scenarios oulined above for a range of parameers s, r, f and N. For each se of parameers we generaed 10 20 differen neworks, and averaged he resuls. In all simulaions we used only one arge ha randomly changes is posiion. Each ime a arge moves o a new locaion, τ ieraions are performed o allow he consrain imposed by he arge o propagae o he nodes. We choose τ = 5 for he resuls presened here. The oal ime (couned in number of ieraions) of he simulaions was T = 10000 (hence, 2000 arge locaions). Of course, real arge movemens are no random and nodes may be able o learn correlaion in arge movemen and exploi hem o arrive a even beer posiion esimaes. In his paper, however, we limi he sudy o he wors case scenario of uncorrelaed arge movemen. We use wo merics o evaluae he accuracy of localizaion. Mean error E is he roo mean square (rms) of he disance beween he nodes acual and esimaed (cener of he bounding box) posiions, averaged over all unknown nodes. The second meric we use is he average size of he bounding recangle A. Noe ha hese wo merics are relaed hrough a consan coefficien for he case when he node s acual posiion always says inside he bounding box. This is guaraneed when he nodes use consrains imposed by he bounding box Q T (hough i is no guaraneed when cenroid-based consrains are used). We should poin ou ha he exisence of such a relaionship beween average size of he bounding box and posiion error is useful for online sensor nework
operaion, because he nodes can use he size of he bounding box o esimae heir posiion error a any insan (o our knowledge his connecion has no been indicaed in any prior work). 4.1 Radial Binary Sensing Model We assume a homogeneous and uniform sysem in which all nodes have he same sensing range s. For a given densiy of nodes, he communicaion range r was chosen so as o ensure ha he nework is fully conneced. Iniially, conneciviy based consrains are used o esablish bounding boxes for he nodes, hen consrains imposed by he arge are used o improve node localizaion. Figure 6 shows he error E and average size A of he bounding box (averaged over all unknown nodes) versus ime for differen numbers of he fracion of unknown nodes f. The plo is obained by averaging resuls of 20 independen rials o suppress flucuaions. In he resuls shown, we used s = 2 and r = 0.3. One can see ha here is a fas decrease in error, followed by sauraion. Noe also ha he curves shif down as he he number of known nodes increases (lower f). Alhough we don show resuls here, we also found ha he localizaion error decreases wih increasing densiy; his is because a higher node densiies, here are more nodes (including known ones) in he viciniy of a arge, which helps o beer localize he arge, and hence, impose sricer consrains on he unknown nodes ha sense i. The resuls in Fig. 6, which were obained by using he consrains imposed by he arge s bounding box, are raher modes: he average uncerainy in he nodes posiion can be larger han he sensing range s. The poor performance can be explained by he relaively weak consrains imposed by he bounding box Q T. This is more pronounced a higher fracions of unknown nodes. However, we can significanly improve node localizaion resuls by using consrains imposed by he cenroid of Q T (see Sec. 3.3). In some parameer range, his radeoff resuls in an improved performance. Figure 7 shows he average localizaion error for nodes using he exac (bounding box) approach and using he approximae (cenroid) approach as a funcion of boh ime and f. Noe ha he approximae approach reduces he final error by a facor of wo as compared o he exac approach. Figure 8 shows he dependence of he localizaion error on he communicaion and sensing range when he nodes use consrains imposed by he bounding box (Fig. 8(a)) and when he nodes use he cenroid approach (Fig. 8(b)). The bounding box approach seems o achieve beer localizaion for smaller values of r and s. We believe his is because he localizaion error is largely deermined by he conneciviy-imposed consrains, which become less effecive as r increases. In he limiing case where each node is conneced o all oher nodes, he uncerainy in nodes posiions is simply he size of he region L. Therefore, for a high fracion of unknown nodes, he arge-imposed consrains are weak due o he iniially large uncerainy. For he cenroid-based scenario, on he oher hand, he siuaion is differen. Even hough he uncerainy in he arge posiion can be large, is esimaed posiion will be more accurae as more nodes are sensing i. For larger conneciviy range r, here appears o be a value of sensing range ha minimizes he rms error. 4.2 Disance-Bound Sensing Model As we explained above, he poor performance of he binary sensing model is explained by he relaively weak consrains imposed by arge deecion on he node posiion. For he disancebound model, however, he siuaion is differen. Every ime a arge is a a close disance d from a known node, he uncerainy in he esimaed arge posiion will be a mos d. Moreover, he consrains imposed by such a arge on a given node will be sronger since he disance o he arge will be generally smaller han he maximum sensing range s. Hence, we expec he disance-bound sensing model o ouperform he radial binary sensing model. In Fig. 9 we plo he average error E versus ime for s =, r = 0.3, and for various values of he fracion of unknown nodes f. The nodes use consrains imposed by he arge s bounding box o updae he uncerainy in heir own posiion. Afer abou 100 arge observaions, he rms localizaion error is an order of magniude
5 f = f = 0.5 f = 0.8 0.4 0.3 E() A() 5 0 500 1000 1500 2000 0 500 1000 1500 2000 Fig. 6. Mean error E and uncerainy A vs ime for he radial binary sensing model Box-consrained Cenroid-consrained Box-consrained Cenroid-consrained E() 5 MSE (E( )) 5 a) b) 0 500 1000 1500 2000 0.3 0.4 0.5 0.6 0.7 0.8 0.9 f Fig. 7. Comparison of rigorous and approximae (cenroid based consrains) algorihms a) Time evoluion b) Value afer sauraion MSE 2 8 6 MSE 2 8 6 4 4 2 2 5 5 Sensing Range (a) 5 0.3 Radio Range 0.35 0.4 5 5 Sensing Range (b) 5 0.3 Radio Range 0.35 0.4 Fig. 8. Dependence of he final rms localizaion error (E( )) on he sensing and radio conneciviy ranges using (a) he bounding box approach and (b) using he cenroid approach
5 f = f = 0.5 f = 0.8 0.3 E() A() 5 0 500 1000 1500 2000 0 500 1000 1500 2000 Fig. 9. Error vs ime for disance-bound sensing model smaller han using conneciviy-based consrains alone. Remarkably, excellen localizaion can be achieved even for values of f (fracion of unknown nodes) as high as f = 0.8. We aribue his dramaic improvemen in performance o he srong consrains imposed by arge deecion in his sensing model. Nex, we sudied he impac of using negaive informaion as was described in he Sec. 3.4. Inuiively, is effec is o push he nodes away, especially ones ha are locaed a he boundary and end o have a bounding recangle biased owards he cener. In Fig. 10 we compare he performance of localizaion algorihm wih γ = 0 ( negaive informaion is no used), γ = 2/2 (he inner recangle is used), and γ = 0.9. Clearly, inroducing negaive consrains significanly improves he accuracy of localizaion. Generally speaking, his improvemen depends on he oher parameers of he model. Noe ha for γ = 0.9 he propery ha he he node is always inside he box does no hold in general; however, he average error (expressed by he meric d) in his case decreases even furher compared o he oher wo safer approaches. E() 75 5 25 a) = 0 = 1/2 1/2 75 b) = 0.9 MSE (E( )) 5 25 = 0 = 1/2 1/2 = 0.9 0 500 1000 1500 2000 0.3 0.4 0.5 0.6 0.7 0.8 0.9 f Fig. 10. Effec of he negaive consrains on he algorihms a) Time evoluion b) The sauraed rms error value as a funcion of f. The parameers are N = 100, r = 0.3, s =
3 25 MSE 2 15 1 05 5 5 Sensing Range 0.4 0.35 0.3 Radio Range 5 Fig. 11. Mean squared error as a funcion of r and s for he disance-bound model In Fig.11 we show he dependence of he localizaion error on he radio conneciviy range r and maximum sensing range s. As one would expec, he bes localizaion is achieved for he larger values of r and s. Finally, a he end of his secion we reflec on how he arge localizaion iself is improved wih ime. Because of he noise, we had o ake an average over for 100 realizaions. The resul is shown in Fig.11. The error in arge localizaion drops as nodes become beer localized hemselves and sauraes a abou a enh of he maximum sensing range value. 5 4 8 E T () 3 2 A T () 6 4 1 2 0 500 1000 1500 2000 0 500 1000 1500 2000 Fig. 12. Time evoluion of he error and uncerainy of esimaed arge posiion 4.3 Comparison o a Cenralized Soluion To esimae he qualiy of our algorihms, we compared our resuls wih a baseline soluion using cenralized, semi-definie programming, he approach used for node localizaion using conneciviy based consrains [5]. To accoun for he consrains imposed by he arge, we used an alernaive
approach o he one considered in he previous secions. Namely, insead of esimaing he arge s posiion and hen consraining he node, we inroduce pair-wise consrains beween he nodes ha sense he same arge. Specifically, le us assume ha he arge is sensed by i h and k h nodes. Then heir posiions are consrained via he inequaliy x i x k s i + s k, where s i = s k = s for he binary sensing model, and s i = d i, s k = d k for he disance bound model. We compared he resuls of our and cenralized algorihms for differen values of he fracion of unknown nodes f. The resuls are presened in Fig. 13. For he binary-deecion model, he cenralized soluion is beer han he disribued one for smaller fracion of unknown nodes, and wo algorihms perform more or less he same as f is increases. For he disance-bound sensing model, he cenralized soluion is beer when γ = 0, i.e., when negaive consrains are no used. Our disribued algorihm wih γ = 0.9, on he oher hand, grealy ouperforms he cenralized soluion. 75 Cenralized a) Disribued 75 b) Cenralized Disribued ( = 0) Disribued ( = 0.9) MSE 5 MSE 5 25 25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 f 0.3 0.4 0.5 0.6 0.7 0.8 0.9 f Fig. 13. Performance of he cenralized (conneciviy consrains only) and disribued localizaion (including sensing consrains) algorihms (averaged over 10 realizaions) for (a) binary-deecion and (b) disancebound sensing models. The parameers are N = 100, r = 0.3, s = 2 for binary and s = for disance bound esimaion model 5 Simulaions using he TinyOS-Nido plaform In order o validae our algorihms under a more pracical seing, we implemened i o work wih TinyOS, he operaing sysem used by Berkely moes. We hen simulaed he performance of he algorihms using he Nido (Tossim) nework simulaor [11]. Nido works by replacing a small number of TinyOS componens and providing addiional componens like an exensible radio model, ADC and spaial models which ogeher allow users o execue programs ha can run on he acual moes. In our simulaions, we used one of he Nido nodes as a mobile arge. Since Nido canno simulae a sensing modaliy, we simulae sensing hrough communicaion packes from he arge node. To achieve his, a new spaial model was defined in he exensible radio model. In his model, nodes are placed a random in a given 2-D field and conneciviy among hem is compued given he communicaion radius. Boh he binary and disance bound sensing model algorihms were implemened as follows. Binary Sensing: A each arge posiion, he arge sends ou packes wih is unique nodeid number. All nodes wihin a sensing radius receive his packe and communicae o heir neighbors ha hey have sensed he arge(using a flag) and also he coordinaes of heir bounding box. Once a node ges his informaion from all nodes ha have sensed he arge, i can compue he arge bounding box locally and updae is own bounding box.
Disance Bound Sensing: A each arge posiion, he arge sends ou packes wih is exac locaion and unique nodeid number. All nodes wihin a sensing radius receive his packe, compue heir disance from he arge and use his o updae he local arge bounding box. All neighbors exchange his informaion and make he final updaes o heir bounding box as in he binary case. Afer all updaes, he spaial model allows for changing arge node posiion and recompuing he conneciviy marix. Fig. 14 shows he behavior of he localizaion error for boh models using Nido simulaions. I shows ha increasing he communicaion range resuls in more conneciviy consrains bu also weaker consrains. Hence, increasing r afer a cerain value acually resuls in higher localizaion error. And we also find here ha he performance of he algorihms improves as he fracion f of nodes wih unknown posiions decreases. 8 6 f =, s = 1 r = 1.5 r = 2 r = 3 r = 4 5 f = 0.9 f = 0.8 f = 0.7 f = 4 2 5 E() E() 8 6 4 5 2 0 0 100 200 300 400 500 600 700 800 900 1000 (a) 0 0 100 200 300 400 500 600 700 800 900 1000 (b) Fig. 14. Localizaion error vs ime obained for he disance bound sensing model via Nido nework simulaions for (a) varying communicaion ranges and (b) for varying fracion of nodes wih known posiions. In order o sudy he performance of he algorihm in more realisic scenarios, we simulaed packe loss due o bi errors and anisoropy in he communicaion range. Bi errors were inroduced by manipulaing he ransmi funcion for our new spaial model. Anisoropy in communicaion range can be modelled by using an addiive zero mean Gaussian while compuing he conneciviy marix. 5 45 4 f =, r = 2, s =1 e = 0 e = 10% e = 20% e = 40% 35 3 E() 25 2 15 1 05 0 100 200 300 400 500 600 700 800 900 1000 Fig. 15. Localizaion error vs ime obained via Nido nework simulaions for varying error raes (disance bound sensing)
5 4 3 f =, s = 1 r = 2 r = 3 r = 2 + N(0,1) r = 2 + N(0,2) f = A(), r = 2 A(), r = 2 + N(0,1) E(), r = 2 E(), r = 2 + N(0,1) 2 5 1 E() 9 8 7 5 6 5 0 100 200 300 400 500 600 700 800 900 1000 (a) 0 0 100 200 300 400 500 600 700 800 900 1000 (b) Fig. 16. Localizaion error E vs ime wih anisoropy in communicaion range for (a) disance bound sensing) and (b) binary sensing models (also showing he area of bounding box A) using Nido nework simulaions. Since each packe provides a consrain, we can expec ha losing packes leads o a slower rae of convergence. Fig. 15 shows he effec of packe loss(e % of all packes are los) due o bi errors on he disance bound model (he resuls are similar for he binary sensing model as well). I is observed ha he performance is no affeced very much wih up o 20% packe drops. Fig. 16 (a) shows he effec of anisoropy in communicaion range for disance bound sensing. Using r = 2 + N(0, 1), some nodes ha are separaed by disance more han r = 2 away are also conneced and hence provide sronger consrains(since disance beween he nodes is known). In binary sensing, all conneced nodes are assumed o be wihin r = 2. I can be seen from Fig. 16 (b) ha while he bounding box area A decreases rapidly, he nodes are no longer wihin he bounding box resuling in an increase in he localizaion error E. These resuls illusrae boh he robusness of he proposed algorihms, as well as he feasibiliy of implemening hese lighweigh algorihms on a pracical sensor nework plaform such as Moes running TinyOS. 6 Relaed Work Providing GPS or precise locaion informaion o all nodes may no be feasible in large scale wireless sensor neworks due o consideraions of cos, or adverse deploymen environmen (such as indoors or under foliage). A survey of alernaive localizaion echniques for a range of applicaions is provided in [10]. In paricular, prior work for sensor has examined he possibiliy of providing localizaion when a few reference or beacon nodes are available [8], [9], [6], [2], [5], [7]. If accurae ranging is available, i.e. he disance o he reference nodes can be measured perfecly hrough signal-srengh based measuremens, hen mulilaeraion echniques may be used for accurae localizaion [8], [9], hough here may be significan challenges in such an approach when fading and noise are aken ino accoun, as discussed in [6]. Bulusu e al. [3] showed ha a simpler cenroid-based approach can be used wih he reference beacons. Dohery e al. [5] formalized he localizaion problem as a convex non-linear program o be solved cenrally, using only radio conneciviy beween he sensor nodes as consrains. Simić and Sasry [7] presen a disribued version of he localizaion algorihm based on conneciviy consrains. They consider a discree nework model and derive probabilisic error and complexiy bounds. Savvides, Han and Srivasa in [8] have presened a disribued, ieraive echnique for mulilaeraion when signal-srengh measuremens are available. This is improved using a more compuaionally inensive approach for collaboraive mulilaeraion by Savvides, Park and Srivasava in [9].
Common feaures o hese prior effors is ha hey separae he localizaion problem from he applicaion and ha he localizaion is performed in a one-sho manner. Our work significanly exends hese prior approaches because our disribued approach permis he nodes in he nework o incorporae addiional consrains over ime hrough sensor measuremens of an unknown arge. 4 We exploi he applicaion-specific naure of sensor neworks o furher opimize for localizaion. Paper [4] is closes in spiri o our work, because i oo uilizes arge racking o improve node localizaion using DOA-based echniques. However, here are sill significan differences in deail. In [4] i is assumed ha he arge follows a simple linear rajecory wih consan velociy. Furher, [4] addresses he localizaion of nodes in a small-scale sensor array (no a nework of sensors) and also use a compuaionally inensive, cenralized exended Kalman filer for he localizaion, which is no a scalable soluion o he problem we consider in his paper. The relaed problem of placing reference nodes/beacons adapively o provide good coverage of he operaional region is discussed in [3]. Our work is complemenary and can be used wih such a beacon placemen echnique. 7 Discussion We have described an approach ha reas localizaion in sensor neworks as an online learning problem and presened a disribued algorihm for i. One novel aspec of our approach is ha we allow nodes o use applicaion-specific informaion, in his case online observaions of a arge, o improve esimae of boh heir own as well as a arge s posiion over ime. The nodes can use arge informaion in one of wo ways: (i) observaion of a arge imposes consrains on he node s posiion, and (ii) if a arge is in he viciniy of he node bu no deeced, he node is also able o use such negaive informaion o impose igher consrains on is own posiion. The mechanism for using negaive informaion is he second novel aspec of our work. We performed exensive numeric and nework level(nido) simulaions of he neworks for differen number of nodes wih known posiions, differen radio conneciviy and sensing ranges, packe drops and anisoropic communicaion range. Saring from an iniial configuraion wih a small fracion of nodes whose posiions are known, he nodes ieraively refined heir posiion esimaes, achieving dramaically improved localizaion for he nework as a whole as well as for he arge. For he disance-bound sensing model, localizaion was observed o be an order of magniude beer han using radio conneciviy consrains alone, even for relaively large fracions of unknown nodes, while for he binary sensing model we found more modes, hough sill significan, improvemen. Negaive informaion also subsanially helped on he localizaion ask. Localizaion gains depend on he ype of consrains being used, especially for he binary sensing model. Ineresingly, we found ha by sacrificing he guaranee ha he arge s acual posiion remain wihin he bounding box and using consrains imposed by he arge s esimaed posiion resuled in significanly beer nework localizaion. However, his radeoff may no always be beneficial, because using he more precise consrains imposed by he bounding box also guaranees a simple relaionship beween he area of he bounding box and mean posiion error. Such a relaionship is useful for online nework operaions, where nodes can esimae he (unknown) posiion error using he (known) area of he bounding box. A more general sensing model han hose considered here is one in which he sensor nodes can measure he disance o a arge, using signal srengh-based esimaion, bu he measuremen is corruped by noise. Geomerically, such a sensing model can be represened by an annulus of some radius and widh: i.e., he arge is a leas a disance s min away from he node and a mos a disance s max from i. The mehodology presened here is no longer direcly applicable, because 4 Anoher way o look a i is o divide he node localizaion process ino wo phases - pre-operaion and during operaion. Exising approaches, such as [6], [2], [5], [7], [8], [9] provide a way for pre-operaion localizaion of a sensor nework. Once he sensor nework is made operaional, our proposed echnique could be used o furher improve he localizaion by incorporaing consrains from he racking ask during nework operaion.
he approach no longer guaranees he bounding box remains convex in fac, observaions could spli he bounding box ino disjoin regions. However, we believe ha i is possible o exend our mehodology o his more general sensing model. References 1. D. Esrin e al. Embedded, Everywhere: A Research Agenda for Neworked Sysems of Embedded Compuers, Naional Research Council Repor, 2001. 2. Nirupama Bulusu, John Heidemann and Deborah Esrin, GPS-less Low Cos Oudoor Localizaion For Very Small Devices, IEEE Personal Communicaions, Special Issue on Smar Spaces and Environmens, Vol. 7, No. 5, pp. 28-34, Ocober 2000. 3. Nirupama Bulusu, John Heidemann and Deborah Esrin, Adapive Beacon Placemen, Proceedings of he Tweny Firs Inernaional Conference on Disribued Compuing Sysems (ICDCS-21), Phoenix, Arizona, April 2001. 4. V. Cevher and J. H. McClellan, Sensor array calibraion via racking wih he exended Kalman filer, 2001 IEEE Inernaional Conference on Acousics, Speech, and Signal Processing, Vol. 5, pp. 2817-2820, 2001. 5. L. Dohery, K. S. J. Piser, and L. El Ghaoui, Convex posiion esimaion in wireless sensor neworks, Infocom 2001, Anchorage, AK, 2001. 6. P. Bergamo, G. Mazzini, Localizaion in Sensor Neworks wih Fading and Mobiliy, IEEE PIMRC 2002, Lisbon, Porugal, Sepember 15-18, 2002. 7. Disribued Localizaion in Wireless Ad Hoc Neworks, Memo. No. UCB/ERL M02/26, 2002. Available online a (hp://cieseer.nj.nec.com/464015.hml). 8. Andreas Savvides, Chih-Chieh Han, Mani B. Srivasava, Dynamic fine-grained localizaion in Ad-Hoc neworks of sensors, Mobicom 2001. 9. A.Savvides, H.Park and M. Srivasava, The bis and flops of he N-hop mulilaeraion primiive for node localizaion problems, WSNA 02, 2002. 10. Jeffrey Highower and Gaeano Borriello, Locaion Sysems for Ubiquious Compuing, Compuer, vol. 34, no. 8, pp. 57-66, IEEE Compuer Sociey Press, Aug. 2001. 11. P. Levis e al. TOSSIM: Accurae and Scalable Simulaion of Enire TinyOS Applicaions, Proceedings of he Firs ACM Conference on Embedded Neworked Sensor Sysems (SenSys 2003), November 2003.