Cooperative localization in wireless networks

Transcription

1 Cooperatve localzaton n wreless networks The MIT Faculty has made ths artcle openly avalable. Please share how ths access benefts you. Your story matters. Ctaton As Publshed Publsher Wymeersch, H., J. Len, and M.Z. Wn. Cooperatve Localzaton n Wreless Networks. Proceedngs of the IEEE 97. (009): IEEE. Insttute of Electrcal and Electroncs Engneers Verson Fnal publshed verson Accessed Wed Apr 17 00:46:1 EDT 019 Ctable Lnk Terms of Use Detaled Terms Artcle s made avalable n accordance wth the publsher's polcy and may be subect to US copyrght law. Please refer to the publsher's ste for terms of use.

2 INVITED PAPER Cooperatve Localzaton n Wreless Networks In harsh envronments where geographc postonng fals, communcaton between wreless nodes can be used to mprove the accuracy of locaton nformaton. By Henk Wymeersch, Member IEEE, Jame Len, Member IEEE, andmoez.wn,fellow IEEE ABSTRACT Locaton-aware technologes wll revolutonze many aspects of commercal, publc servce, and mltary sectors, and are expected to spawn numerous unforeseen applcatons. A new era of hghly accurate ubqutous locatonawareness s on the horzon, enabled by a paradgm of cooperaton between nodes. In ths paper, we gve an overvew of cooperatve localzaton approaches and apply them to ultrawde bandwdth (UWB) wreless networks. UWB transmsson technology s partcularly attractve for short- to medumrange localzaton, especally n GPS-dened envronments: wde transmsson bandwdths enable robust communcaton n dense multpath scenaros, and the ablty to resolve subnanosecond delays results n centmeter-level dstance resoluton. We wll descrbe several cooperatve localzaton algorthms and quantfy ther performance, based on realstc UWB rangng models developed through an extensve measurement campagn usng FCC-complant UWB rados. We wll also present a powerful localzaton algorthm by mappng a graphcal model for statstcal nference onto the network topology, whch results n a net-factor graph, and by developng a sutable net-message passng schedule. The resultng algorthm (SPAWN) s fully dstrbuted, can cope wth a wde varety of scenaros, and requres lttle communcaton overhead to acheve accurate and robust localzaton. Manuscrpt receved November 3, 006; revsed February 1, 008. Current verson publshed March 18, 009. Ths work was supported n part by the Belgan Amercan Educatonal Foundaton, the JPL Strategc Unversty Research Partnershps, and the Natonal Scence Foundaton under Grants ECCS and ANI H. Wymeersch and M. Z. Wn are wth the Laboratory for Informaton and Decson Systems, Massachusetts Insttute of Technology, Cambrdge, MA 0139 USA (e-mal: hwymeers@mt.edu; moewn@mt.edu). J. Len was wth the Laboratory for Informaton and Decson Systems, Massachusetts Insttute of Technology, Cambrdge, MA 0139 USA. She s now wth the Jet Propulson Laboratory, Pasadena, CA USA (e-mal: Jame.Len@pl.nasa.gov). Dgtal Obect Identfer: /JPROC /$5.00 Ó009 IEEE KEYWORDS Cooperatve processng; factor graphs; localzaton; sum-product algorthm; ultrawde bandwdth transmsson I. INTRODUCTION Locaton awareness s rapdly becomng an essental feature of many commercal, publc servce, and mltary wreless networks [1], []. Informaton collected or communcated by a wreless node s often meanngful only n conuncton wth knowledge of the node s locaton. For example, sensor networks used for detectng spatal varatons n envronmental condtons, such as temperature or polluton, requre knowledge of each sensor s locaton [3] [5]. Locaton nformaton also facltates a node s nteractons wth ts surroundngs and neghbors, enablng pervasve computng and socal networkng applcatons [6]. Locaton-aware technologes can enable or beneft a vast array of addtonal applcatons, ncludng ntruder detecton [7], blue force trackng [8], fndng frends or landmarks [9], healthcare montorng [10], asset trackng [11], and emergency 911 servces [1], [13]. Network nodes must have the capablty to self-localze n scenaros where nodes cannot be manually postoned or located by a central system admnstrator [14], [15]. The goal of self-localzaton s for every node to know ts own state. A state usually ncludes the two- or three-dmensonal poston, and possbly other propertes such as the velocty and the orentaton of the node [16] [18]. The concept of state depends on the applcaton and may also vary from node to node. In our exposton, we wll use the terms state, poston, and locaton nterchangeably, whle n our examples we wll narrow the scope of state to two-dmensonal geographcal coordnates. We wll dstngush between two types of nodes: agents, whch have a pror unknown states, and anchors, whch have known states at all tmes. Both agents and anchors may be moble. The localzaton process typcally conssts of two phases [1]. The frst phase s the measurement phase, durng whch Vol. 97, No., February 009 Proceedngs of the IEEE 47

3 agents measure nternal state nformaton (e.g., usng an nertal measurement unt (IMU)) and estmate sgnal metrcs 1 basedondrectcommuncatonwthneghborng agents and/or anchors. The second phase s the locatonupdate phase, durng whch agents nfer ther own state based on the nternal state measurements, estmated sgnal metrcs, and state nformaton of neghborng nodes. For nstance, when an agent obtans dstance estmates wth respect to three anchors, the agent can nfer ts own poston through trlateraton, provded the agent knows the postons of the anchors. The measurement phase s affected by uncertanty due to sources such as nose, multpath, blockages, nterference, clock drfts, and envronmental effects [1], [19] [4]. The underlyng transmsson technology s a crtcal factor n how these sources affect the measurements []. For nstance, the poor sgnal penetraton capabltes of the wdely used Global Postonng System (GPS) prevent consumer-grade GPS recevers from makng relable measurements ndoors, under forest canopes, and n certan urban settngs, leadng to nadequate postonal nformaton [5]. In challengng envronments such as these, ultrawde bandwdth (UWB) transmsson technology [6] [8] s a promsng alternatve for localzaton [9] [31]. UWB systems are nherently well suted for localzaton snce the use of extremely large transmsson bandwdths results n desrable capabltes such as 1) accurate rangng due to fne delay resoluton; ) smple mplementaton for multple-access communcatons; and 3) obstacle penetraton capabltes [3] [38]. For more nformaton on the fundamentals of UWB, we refer the reader to [6] [8], and [39] [45] and references theren. Gven an underlyng transmsson technology, localzaton performance s also dependent on the specfc algorthm used n the locaton-update phase. An emergng paradgm s cooperatve localzaton, n whch nodes help each other to determne ther locatons [46]. Cooperatve localzaton has receved extensve nterest from the robotcs, optmzaton, and wreless communcatons communtes (see [14], [1], [30], [31], and [47] [61] and references theren). A smple comparson of conventonal and cooperatve localzaton s depcted n Fg. 1: whle each agent (moble unt) cannot ndependently determne ts own poston based on dstance estmates wth respect to the anchors (base statons), they can cooperatvely fnd ther postons. In general, cooperatve localzaton can dramatcally ncrease localzaton performance n terms of both accuracy and coverage. 1 Sgnal metrcs nclude any property of the receved sgnal that depends on the relatve postons of the transmtter and the recever. Examples nclude the tme of flght, the angle of arrval, and the receved sgnal strength. Coverage s the fracton of nodes that have an accurate locaton estmate. Fg. 1. The beneft of cooperatve localzaton: usng only dstance estmates wth respect to the anchors (nodes 1,, and 5), agent nodes and 4 are unable to determne ther respectve postons wthout ambguty. Observe that node cannot communcate wth node 5, and node 4 cannot communcate wth node 1. When agent nodes and 4 communcate and range drectly (as depcted by the red arrow), they can cooperate to unambguously determne ther postons. In ths paper, we provde an overvew of cooperatve localzaton algorthms n wreless networks. We consder large-scale dynamc heterogeneous networks and examne how cooperaton can be used to mprove localzaton accuracy and coverage wth respect to noncooperatve technques. We focus on algorthms based on the prncples of estmaton theory and statstcal nference [6], [63] and outlne a framework for the systematc desgn of nference algorthms, usng the theory of factor graphs (FGs) [64] [66]. We develop a localzaton algorthm by mappng a FG onto the tme-varyng network topology and by employng a spatotemporal message schedule, resultng n a network FG (Net-FG) and network message passng (Net-MP). The proposed algorthm, called SPAWN (sum-product algorthm over a wreless network), s fully dstrbuted and cooperatve. SPAWN also accounts for dfferent state types among nodes, for node moblty, and for any uncertantes assocated wth both the measurement and locaton-update phases. We show how SPAWN generalzes prevously proposed localzaton algorthms, revertng to Bayesan flterng n the case of a sngle agent [18] and to nonparametrc belef propagaton localzaton [14] n the case of a homogeneous network wth statc nodes. Ths paper s organzed as follows. In Secton II, we provde an overvew of methods used for the two phases of localzaton. We descrbe and compare varous sgnal metrcs and localzaton approaches, emphaszng the advantages of UWB as an underlyng transmsson technology. In Secton III, we provde a concse overvew of general purpose estmaton technques and factor graphs. Secton IV deals wth non-bayesan and Bayesan 48 Proceedngs of the IEEE Vol.97,No.,February009

4 cooperatve localzaton strateges. Secton V detals the results of an extensve range measurement campagn usng FCC-complant UWB rados. We then present a case study for ndoor localzaton n large UWB networks n Secton VI and quantfy the performance of dfferent cooperatve and noncooperatve localzaton algorthms n terms of accuracy and avalablty usng expermental data. In Secton VII, we draw our conclusons and present avenues for further research n ths area. Notaton: Throughout ths paper, we wll use the followng notaton. The state of node at tme t wll be denoted by x ðtþ.thestatesequenceofnodefrom tme t 1 to t wll be denoted by x ðt 1:t Þ. Random varables wll be captalzed and vectors wrtten n bold unless there s no ambguty. Dstrbutons such as p X ðxþ wll at tmes be abbrevated by pðxþ. II. LOCALIZATION APPROACHES FOR WIRELESS NETWORKS In ths secton, we descrbe dfferent types of sgnal metrcs and classfy dfferent types of localzaton algorthms. We apply ths classfcaton to well-known localzaton systems, consderng both ndoor and outdoor scenaros. A. Measurement Phase In the frst phase of localzaton, packets are exchanged between neghborng nodes n the network (say, nodes A and B). From the physcal waveforms correspondng to these packets, the recever (node B) can extract nformaton regardng ts locaton relatve to the locaton of the transmtter (node A) by measurng or estmatng one or more sgnal metrcs. The nherent uncertanty n localzaton arses from these sgnal metrcs, whch are subect to varous sources of error. Below, we brefly lst several common measurements and descrbe ther use n localzaton and ther sources of error. The dstance between nodes can be measured through a varety of metrcs. Receved sgnal strength (RSS) explots the relaton between power loss and the dstance between sender and recever [67]. The ablty of node B to receve packets from node A, known as connectvty, can constran the dstance between node A and node B to the communcaton range of node A [68], [69]. Dstance measurements of fner resoluton can be obtaned by estmatng the propagaton tme of the wreless sgnals. Ths s the bass of tme of arrval (TOA), tme dfference of arrval (TDOA), and round-trp tme of arrval (RTOA) [3], [47]. RTOA s the most practcal scheme n a decentralzed settng, as t does not requre a common tme reference between nodes [70]. For example, node B sends a packet to node A at tme t B;send n ts own tme reference. Node A receves the packet at tme t A accordng to ts own clock and responds wth a packet at t A þ, where s a tme nterval that s ether predetermned or communcated n the response packet. Node B receves the packet from A at tme t B;rec and can then determne the dstance d AB through the relaton ðt B;rec t B;send Þv ¼ d AB, where v s the sgnal propagaton speed. Relatve orentatons can be determned through angle of arrval (AOA) estmaton when a node s equpped wth drectonal or multple antennas [47]. For nstance, n a lnear array wth spatal antenna separaton, the dfference n arrval tme between any two successve antenna elements s gven by t ¼ =ðv cos AB Þ, where AB s the angle between the mpngng sgnal and the antenna array [0]. Beyond dstance and angle, one can estmate other propertes such as the velocty of a node by measurng Doppler shfts [71]. Informaton about the state of the node can also be measured nternally; for example, dstances traveled usng an odometer or pedometer, acceleraton usng an accelerometer, and orentaton usng an IMU. More problem-specfc technques such as vsual odometry [7] may also be consdered as sgnal metrcs. Measurements are subect to estmaton errors. For nstance, RSS estmators may exhbt large errors due to shadowng and multpath. The connectvty metrc tends to produce coarse locaton nformaton, especally when the communcaton range s large or the connectvty of the network s low. Tme delay-based sgnal metrcs (such as TOA, TDOA, RTOA, and AOA) are susceptble to errors due to obstructons between the transmtter and the recever. These obstructons, leadng to so-called non-lneof-sght (NLOS) condtons, can cause a postve bas n the dstance estmate. Nose, nterference, multpath, clock drfts, and other sources may also ntroduce errors n estmatng arrval tmes. Fnally, nternal measurement devces such as IMUs and odometers may accumulate errors over tme due to nherent propertes of the sensors. For addtonal nformaton regardng sources of error, the reader s referred to [1], [], [3], [5], [47], [73], and [74]. B. Locaton-Update Phase In the second phase, measurements are aggregated and used as nputs to a localzaton algorthm. A possble taxonomy of localzaton algorthms s the followng (see also [47], [54], and [75] and references theren). 1) Centralzed Versus Dstrbuted: In centralzed localzaton, the postons of all agents are determned by a central processor. Ths processor gathers measurements fromanchorsaswellasagentsandcomputesthepostons of all the agents. Centralzed algorthms are usually not scalable and thus mpractcal for large networks. In dstrbuted localzaton, such as GPS, there s no central controller and every agent nfers ts own poston based only on locally collected nformaton. Dstrbuted algorthms are scalable and thus attractve for large networks. 3 3 Dstrbuted localzaton s sometmes referred to as self-localzaton, whle centralzed localzaton s sometmes referred to as remote localzaton. Vol. 97, No., February 009 Proceedngs of the IEEE 49

5 ) Absolute Versus Relatve: Absolute localzaton refers to localzaton n a sngle predetermned coordnate system [5]. Relatve localzaton refers to localzaton n the context of one s neghbors or local envronment [54], [76]; hence, the coordnate system can vary from node to node. 3) Noncooperatve Versus Cooperatve: In a noncooperatve approach, there s no communcaton between agents, only between agents and anchors. Every agent needs to communcatewthmultpleanchors,requrngethera hgh densty of anchors or long-range anchor transmssons. In cooperatve localzaton, nteragent communcaton removes the need for all agents to be wthn communcaton range of multple anchors; thus hgh anchor densty or long-range anchor transmssons are no longer requred. Snce agents can obtan nformaton from both anchors and other agents wthn communcaton range, cooperatve localzaton can offer ncreased accuracy and coverage. We wll quantfy these performance mprovements n Secton VI. C. Outdoor and Indoor Localzaton 1) Outdoor Localzaton: Examples of outdoor systems nclude GPS, LORAN-C, and rado-locaton n cellular networks. GPS s a dstrbuted, absolute, and noncooperatve localzaton approach [5], relyng on TOA estmates from at least four anchors (GPS satelltes) to solve a fourdmensonal nonlnear problem (three spatal dmensons and one tme dmenson, snce the agent s not synchronzed to the anchors). Asssted GPS s a centralzed verson of GPS, reducng the computatonal burden on the agents [5]. LORAN-C s a terrestral predecessor of GPS [77], whch offers centralzed, absolute, and noncooperatve localzaton servces based on TDOA. Cell phone rado-locaton servces such as E911 commonly employ TDOA and are centralzed, absolute, and noncooperatve [78], [79]. ) Indoor Localzaton: Exstng and emergng ndoor localzaton methods nclude WF, rado-frequency dentfcaton (RFID), and UWB localzaton [80], [81]. RADAR, based on WF fngerprntng at multple anchors [8]; PlaceLab, usng connectvty from access ponts; and GSM base statons [83] employ centralzed, absolute, and noncooperatve approaches. Passve RFID tags can be used n conuncton wth RFID readers to provde connectvty-based localzaton [84] that s centralzed, relatve, and noncooperatve. Both WF and RFID systems suffer from poor accuracy due to coarse measurements. On the other hand, UWB sgnals have a number of characterstcs that make them more attractve for ndoor localzaton, as well as for ndoor communcaton n general [85] [87]. The fne delay resoluton of UWB sgnals s well suted for estmatng propagaton tmes (e.g., for RTOA or AOA), snce the performance of delay estmaton algorthms mproves wth ncreasng transmsson bandwdth [3]. Moreover, the wde bandwdth allows multpath components to be resolved and enables superor sgnal penetraton through obstacles [7], [3] [38]. Hence, robust communcatons n dense multpath envronments and rangng n NLOS condtons can be acheved [3] [36]. The penetraton capabltes of UWB sgnals also make them useful for detectng and potentally compensatng for the effects of obstacles and NLOS condtons [88], [89]. In addton, UWB transmtters are low complexty, low cost devces, practcal for dense and rapd deployment [90]. Snce the power s spread over a large bandwdth, UWB communcaton systems are covert and power-effcent, and cause mnmal nterference to other systems [6] [8], [34]. UWB sgnals have the unque advantage of smultaneously accomplshng robust communcaton and precson rangng. Nodes can therefore extract nformaton about ther relatve postons from sgnals already used for communcaton wthout any addtonal overhead. The recently completed IEEE a standard [91] wll lkely spawn numerous practcal systems and applcatons n ths sphere. III. BACKGROUND ON INFERENCE Before presentng cooperatve localzaton algorthms, we frst gve a bref overvew of mportant technques from estmaton theory and statstcal nference, whch can be appled to the localzaton problem. There are a number of approaches for estmatng a parameter x from an observaton z. Apart from ad hoc technques, we generally categorze these as Bayesan or non-bayesan, dependng on whether or not we consder x as a realzaton of a random varable [6]. In ths secton, we descrbe both approaches. Wthn the context of Bayesan technques, we then consder approxmate nference, factor graphs, and sequental estmaton. A. Non-Bayesan Estmaton Two common non-bayesan estmators, whch treat x as an unknown determnstc parameter, are the least squares (LS) estmator and the maxmum lkelhood (ML) estmator. The LS estmator assumes that z IR N and z ¼ fðxþþn, where fðþ s a known functon and n s a measurement error. The LS estmate of x s obtaned by solvng the followng optmzaton problem: ^x LS ¼ arg mnkz fðxþk : (1) x The LS estmator does not explot any knowledge regardng the statstcs of n. 430 Proceedngs of the IEEE Vol.97,No.,February009

6 The ML estmator accounts for the statstcs of nose sources and maxmzes the lkelhood functon ^x ML ¼ arg max p ZX ðzxþ: () x B. Bayesan Estmaton Two common Bayesan estmators, whch treat x as a realzaton of a random varable X wth an a pror dstrbuton p X ðxþ, are the mnmum mean squared error (MMSE) estmator and the maxmum a posteror (MAP) estmator. The MMSE estmator fnds the mean of the a posteror dstrbuton Z ^x MMSE ¼ xp XZ ðxzþdx: (3) The MAP estmator fnds the mode of the a posteror dstrbuton ^x MAP ¼ arg max p XZ ðxzþ: (4) x If p XZ ðzþ s dffcult to determne, or f the dmensonalty of X s hgh so that the ntegraton n (3) or the maxmzaton n (4) become ntractable, we can resort to MMSE or MAP estmates of the components of X rather than the entre vector. For example, when X ¼½X 1 ;...; X N Š, then the MMSE (respectvely, MAP) estmate of X k s gven by the mean (respectvely, mode) of the margnal a posteror dstrbuton p Xk ZðzÞ of the varable X k. C. Approxmate Inference In many nference problems, the a posteror dstrbuton p XZ ðzþ s dffcult to descrbe, and obtanng ts mean, mode, or margnals s a very hard problem. In such stuatons, one can turn to an alternatve dstrbuton, say, b X ðþ, belongng to a certan class C. Inferences can then be drawn based on a partcular b X ðþ that s close to p XZ ðzþ. A common measure of closeness between dstrbutons s the Kullback Lebler dvergence (KLD) [9], defned as Z Dðb X kp XZ Þ¼ b X ðxþ ln b XðxÞ dx: (5) p XZ ðxzþ It s easy to verfy that Dðb X kp XZ Þ0 and that Dðb X kp XZ Þ¼0 f and only f b X ¼ p XZ.Foragven Fg.. Mnmzng the KLD can lead to dfferent Bayesan nference algorthms, ncludng sum-product algorthm (SPA), MMSE, mean-feld (MF), expectaton-propagaton (EP), expectaton-maxmzaton (EM), varatonal EM (VEM), and MAP. class C, we try to fnd the dstrbuton b X that mnmzes the KLD b X ¼ arg mn b X C Dðb Xkp XZ Þ: (6) Often b X cannot be found n closed form; we can only determne a descrpton (e.g., a lst of necessary condtons mposed on b X ) of the statonary ponts 4 of Dðb X kp XZ Þ. Usng ths descrpton, one can then develop teratve algorthms to fnd those statonary ponts. Dfferent classes C lead to dfferent solutons, ncludng mean-feld, expectaton-propagaton, expectaton-maxmzaton, and the sum-product algorthm (see Fg. ). For a detaled exposton, see [93] and [94]. These solutons can be found through message passng on an FG, as detaled n the next secton. D. Factor Graphs and the Sum-Product Algorthm We cover some basc concepts of FGs and the SPA; for a detaled treatment, the reader s referred to [64] [66], and [94]. In many nference problems, the a posteror dstrbuton can be factorzed, wth every factor k ðþ dependng only on a small subset of varables x k x: p XZ ðxzþ ¼ 1 Q Y M k¼1 k ðx k Þ (7) where M sthenumberoffactorsandq s a (possbly unknown) normalzaton constant. 1) Factor Graphs: A factor graph 5 s a way to graphcally represent a factorzaton such as (7). For every factor, say, ðþ, we create a vertex (drawn as a crcle or square) and label t B.[ 4 A statonary pont can be a mnmum, a maxmum, or a saddle-pont. 5 We focus on Forney-style FGs, also known as normal graphs [95]. Vol. 97, No., February 009 Proceedngs of the IEEE 431

7 mportant dstncton. Only when the FG does not have cycles,.e., t s a tree, t can be shown that the a posteror dstrbuton can be expressed as [94] Q M k¼1 p XZ ðxzþ ¼ p X k Zðx k zþ Q N l¼1 p dl (10) 1 X l Zðx l zþ Fg. 3. FG of A ðx 1 Þ B ðx 1 ; x Þ C ðx 1 ; x Þ D ðx ; x 3 Þ,ontheleft. Groupng B ðþ and C ðþ nto ðþ yelds the FG on the rght. For every varable X, wecreateanedge(drawnas a lne) and label t BX.[ When a varable X appears n a factor ðþ, weconnecttheedgex to the vertex. Snce edges can be connected to at most two vertces, we must treat varables that appear n more than two factors as a specal case. For a varable X that appears n D > factors, we create a so-called equalty vertex and label t B¼.[ We also create D edges and connect every edge to theequaltyvertexandoneofthed factors. The edges are labeled wth a dummy name of the varable (e.g., X 0 and X 00 for X). The equalty vertex represents a Drac delta functon. For nstance, an equalty vertex wth edges X, X 0, and X 00 corresponds to a functon ðx x 0 Þðx x 00 Þ.Fornotatonal convenence, we often label all the edges connected to an equalty vertex wth the same label (X, n ths case). Let us examne a smple example, where X ¼½X 1 ; X ; X 3 Š has an a posteror dstrbuton that can be factorzed as pðx 1 ; x ; x 3 zþ ¼ 1 Q Aðx 1 Þ B ðx 1 ; x Þ C ðx 1 ; x Þ D ðx ; x 3 Þ (8) where Q s an unknown constant. Factorzatons are by no means unque: by groupng B ðþ and C ðþ nto a new factor, say, ðþ, a dfferent factorzaton s obtaned: where M s the total number of factors, N s the total number of varables, and d l sthenumberoffactorsn(7) where the varable x l appears. 7 For nstance, when usng the factorzaton (9), we can express pðx 1 ; x ; x 3 zþ as (wth d 1 ¼ d 3 ¼ 1, d ¼ ) 8 pðx 1 ; x ; x 3 zþ ¼ pðx 1; x zþpðx ; x 3 zþ pðx 1 zþ 0 pðx zþ 1 pðx 3 zþ 0 : (11) Gven a factorzaton of the form (7), we can ntroduce aclassc SPA of functons b X ðþ of the followng form: Q M k¼1 b X ðxþ ¼ b X k ðx k Þ Q N l¼1½ b X l ðx l ÞŠ d (1) l 1 P P subect to b Xk ðx k Þ0, b Xl ðx l Þ0, x k b Xk ðx k Þ¼ x l b Xl ðx l Þ¼1, P x k nl b X k ðx k Þ¼b Xl ðx l Þ, 8k; l : x l x k.the descrpton of the statonary ponts of Dðb X kp XZ Þ s expressed n terms of the functons b Xk ðþ and b Xl ðþ. Comparng(10)and(1),weseethatwhentheFGof the factorzaton of p XZ ðzþ has no cycles, the optmzaton problem (6) wth C¼C SPA has a unque global mnmzer b X, wth correspondng KLD equal to zero. Furthermore, the a posteror dstrbuton p XZ ðzþ can be recovered based solely on b X k ðþ and b X l ðþ, 8k; l, sncethe unqueness of the soluton mples that b X k ðx k Þ¼p Xk Zðx k zþ; 8k; x k (13) b X l ðx l Þ¼p Xl Zðx l zþ; 8l; x l : (14) pðx 1 ; x ; x 3 zþ ¼ 1 Q Aðx 1 Þ ðx 1 ; x Þ D ðx ; x 3 Þ: (9) FGs correspondng to (8) and (9) are depcted n Fg. 3. ) Trees and Cyclc GraphsVPart 1: The factorzaton (8) results n a cyclc FG, 6 whle (9) does not. Ths s an 6 The FG of (8) has a cycle gven by edges X 0 1 ; X00 1 ; X ; X 00. 3) The Sum-Product Algorthm: The SPA s a message passng algorthm on a cycle-free FG that effcently computes b X k ðþ and b X l ðþ, 8k; l,whenc¼c SPA.TheSPA operates by computng messages nsde the vertces and sendng those messages over the edges. 7 For mathematcal convenence, we need to assume that x k contans at least two varables. We group varables together where necessary. See also [94]. 8 Snce A has only one varable, we have grouped A and. 43 Proceedngs of the IEEE Vol.97,No.,February009

8 A message over an edge X s a functon of the correspondng varable and s denoted X! ðþ or!x ðþ, where s a vertex adacent to edge X. Gvenafactor ðx 1 ;...; x D Þ and ncomng messages Xk!ðÞ8k, the outgong message over edge X s gven by Z!X ðx Þ/ ðx 1 ;...; x D Þ Y Xk!ðx k Þdx 1... k6¼ dx 1 dx þ1...dx D (15) where the proportonalty symbol ndcates that the message!x ðþ s normalzed, such that R!X ðx Þdx ¼ 1. For example, n Fg. 3, the message from to X s gven by Z!X ðx Þ/ ðx 1 ; x Þ X1! ðx 1 Þdx 1 : (16) Messages start wth the half-edges (sendng a constant message) and the vertces of degree 1 (sendng the correspondng factor). For example, n Fg. 3, X3! D ðx 3 Þ/1 and A!X 1 ðx 1 Þ¼ A ðx 1 Þ. For equalty vertces, t can be shown that an outgong message s smply the pont-wse product of the ncomng messages. For example, n Fg. 3, the message from X 0 1 to B s gven by X 0 1! B x 0 1 / A!X 1 x 0 1 C!X1 00 x 0 1 : (17) The margnal of a certan varable s obtaned by pontwse multplcaton of the two messages passed over the correspondng edge. In our example from Fg. 3, b X ðx Þ/ X! D ðx Þ D!X ðx Þ: (18) The margnal of a cluster of varables x k s obtaned by multplyng the ncomng messages wth the correspondng factor. For nstance, n the example from Fg. 3, b X 1 ;X ðx 1 ; x Þ/ ðx 1 ; x Þ X1! ðx 1 Þ X! ðx Þ: (19) 4) Trees and Cyclc GraphsVPart : The SPA provdes us wth exact margnals for FGs wthout cycles (.e., trees). For FGs wth cycles, we can easly extend the SPA, whch becomes teratve. However, the computed functons b X k ðþ and b X l ðþ are no longer exactly equal to the correspondng margnal a posteror dstrbutons, and b X s not necessarly a dstrbuton [96]. Furthermore, n an FG wth cycles, there are many possble orders n whch messages are computed (also known as the message schedule), and each schedule may lead to dfferent functons b X k ðþ and b X l ðþ. The ablty to choose schedules wll turn out to be mportant n dervng a cooperatve localzaton algorthm. E. Sequental Estmaton In some scenaros, varables may change over tme. Sequental or onlne estmaton deals wth such scenaros by estmatng varables at a certan tme, say, varable x ðtþ at tme t, takng nto account ndependent observatons taken up to and ncludng t, say,z ð1:tþ ¼½z ð1þ ;...; z ðtþ Š [17], [97], [98]. We rely on the followng Markovan assumptons: pðx ðtþ x ð0:t 1Þ Þ¼pðx ðtþ x ðt 1Þ Þ and pðz ðtþ x ð0:tþ Þ¼ pðz ðtþ x ðtþ Þ. It can then easly be shown that where Z p x ðtþ z ð1:tþ ¼ Z p x ðtþ z ð1:t 1Þ ¼ p x ðtþ ; x ðt 1Þ z ð1:tþ dx ðt 1Þ (0) / p z ðtþ x ðtþ p x ðtþ z ð1:t 1Þ (1) p x ðtþ x ðt 1Þ p x ðt 1Þ z ð1:t 1Þ dx ðt 1Þ : () Ths mples that, gven pðx ðt 1Þ z ð1:t 1Þ Þ, we can determne pðx ðtþ z ð1:tþ Þ as follows: ) a predcton operaton, durng whch we determne the dstrbuton pðx ðtþ z ð1:t 1Þ Þ,gven all observatons before tme t, accordng to the ntegral n (); and ) a correcton operaton, n whch we account for the new observaton z ðtþ to calculate pðx ðtþ z ð1:tþ Þ, accordng to (1). Hence, at every tme t, wehavethe a posteror dstrbuton pðx ðtþ z ð1:tþ Þ of the varable x ðtþ, gven all the observatons up to and ncludng tme t. We can determne the mean or the mode of ths a posteror dstrbuton, gvng us the MMSE estmate or MAP estmate of x ðtþ, respectvely. The entre procedure s ntalzed by pðx ðtþ z ð1:tþ Þ t¼0 ¼ pðx ð0þ Þ. Sequental Estmaton and FGs: Sequental estmaton can be obtaned by creatng an FG of pðx ð0:tþ z ð1:tþ Þ and then applyng the SPA. Usng the Markovan assumptons and the fact that the measurements are ndependent, we easly fnd that p x ð0:tþ z ð1:tþ Y T / p x ð0þ t¼1 p x ðtþ x ðt 1Þ p z ðtþ x ðtþ (3) Vol. 97, No., February 009 Proceedngs of the IEEE 433

9 Fg. 4. Sequental estmaton: an FG of pðx ð0:þ z ð1:þ Þ,where ðtþ ðx ðtþ ; x ðt 1Þ Þ s a shorthand for pðx ðtþ x ðt 1Þ Þ and ðtþ ðx ðtþ Þ s a shorthand for pðz ðtþ x ðtþ Þ. wth a correspondng FG depcted n Fg. 4 for T ¼. The message schedule s shown wth arrows, startng wth the black messages from the leaves of the tree. At tme t ¼ 1, the red message ð1þ!xð1þðþ s gven by ð1þ!x ð1þ xð1þ Z / X ð0þ! ð1þ xð0þ ð1þ x ð1þ ; x ð0þ Z ¼ p x ð0þ p x ð1þ x ð0þ ¼ p x ð1þ dx ð0þ dx ð0þ (4) whch corresponds to the predcton operaton. Usng (17), the blue message X ð1þ!ðþðþ s gven by X ð1þ! ðþ xð1þ / ð1þ!xð1þ xð1þ ¼ p z ð1þ x ð1þ p x ð1þ ð1þ!x ð1þ xð1þ (5) whch s exactly the correcton operaton. At tme slot t ¼, we easly fnd that ðþ!x ðþðxðþ Þ/ R X ð1þ! ðþðxð1þ Þ pðx ðþ x ð1þ Þdx ð1þ and that X ðþ! ð3þðxðþ Þ/pðz ðþ x ðþ Þ ðþ!x ðþðxðþ Þ. The sequence of predcton operaton (red arrows) followed by correcton operaton (blue arrows) contnues wth messages flowng from past to present to future. In prncple, messages can also be computed from future to present to past, a process known as smoothng. IV. COOPERATIVE LOCALIZATION The concept of cooperaton n networks s farly new: t reles on drect communcaton between agents rather than through a fxed nfrastructure [99] [101]. Cooperaton has been successfully appled to wreless peer-to-peer communcaton, leadng to standards such as Bluetooth [10] and Zgbee [103], and s expected to be appled to cellular systems over the next few years [104]. In ths secton, we apply the cooperatve paradgm to a completely dfferent problem: localzaton. We present several fundamental cooperatve localzaton algorthms based on the methodologes from Secton III. Both non- Bayesan and Bayesan approaches wll be consdered. For the sake of the exposton, we focus on small-scale examples. In Secton VI, we wll present a case study for alargenetworkwthmorethan100nodes. A. Problem Formulaton We consder a wreless network wth N nodes n an envronment E. Tme s slotted, and nodes can move ndependently from postons at tme slot t 1tonew postons at tme slot t. Thestateofnodeat tme t s wrtten as x ðtþ.wedenotebys ðtþ! the set of nodes from whch node may receve sgnals durng tme slot t. Smlarly, we denote by S ðtþ! the set of nodes that may receve a sgnal from node durng tme slot t. At tme slot t, node may estmate local metrcs z ðtþ ;self based on nternal measurements (e.g., usng an IMU or an odometer). Based on the sgnal receved from node S ðtþ!, node can estmate sgnal metrcs z ðtþ!. We denote the collecton of all sgnal metrcs and all nternal measurements collected by all nodes at tme t as z ðtþ.note that we can break up z ðtþ nto z ðtþ self and z ðtþ rel,wherezðtþ self contans all the nternal measurements of all the nodes, whle z ðtþ rel contans all the relatve sgnal metrcs from all the nodes wth respect to ther neghbors. The goal of node s to estmate ts own state x ðtþ at tme t, gven only nformaton up to tme t. Ideally,the localzaton process should requre low complexty and communcaton overhead per node and ncur a low latency. B. Assumptons We make the followng assumptons, whch are reasonable n many practcal scenaros. a) The states of the nodes are aprorndependent: pðx ð0þ Þ¼ Q N ¼1 pðxð0þ Þ. b) Nodes move accordng to a memoryless walk: p x ð0:tþ Y T ¼ p x ð0þ t¼1 c) Nodes move ndependently: p x ðtþ x ðt 1Þ ¼ YN ¼1 p x ðtþ x ðt 1Þ : (6) p x ðtþ x ðt 1Þ : (7) d) Relatve measurements are ndependent of the nternal measurements, condtoned on the states 434 Proceedngs of the IEEE Vol.97,No.,February009

10 of the nodes: p z ð1:tþ rel x ð0:tþ ; z ð1:tþ self ¼ p z ð1:tþ rel x ð0:tþ : (8) e) Internal measurements are mutually ndependent and depend only on the current and prevous state: p z ð1:tþ self x ð0:tþ ¼ YT t¼1 p z ðtþ self xðt 1Þ ; x ðtþ : (9) f) Internal measurements at node depend only on the state of node : p z ðtþ self xðtþ ; x ðt 1Þ ¼ YN ¼1 p z ðtþ ;self xðt 1Þ ; x ðtþ : (30) g) Relatve measurements are ndependent from tme slot to tme slot, condtoned on the states of the nodes. Moreover, they depend only on the current states: p z ð1:tþ rel x ð0:tþ ¼ YT t¼1 p z ðtþ rel xðtþ : (31) h) Relatve measurements at any tme slot t are condtonally ndependent and depend only on the two nodes nvolved: p z ðtþ rel xðtþ ¼ YN ¼1 Y p z ðtþ S ðtþ!!x ðtþ ; x ðtþ : (3) We further assume that node knows the followng: ) the state dstrbuton pðx ð0þ Þ at tme t ¼ 0; ) ts own moblty model pðx ðtþ x ðt 1Þ Þ at any tme t; ) the nternal measurements z ðtþ ;self and the correspondng lkelhood functon pðz ðtþ ;self xðt 1Þ ; x ðtþ Þ at any tme t; v) the sgnal metrcs z ðtþ! and the lkelhood functon pðz ðtþ!x ðtþ ; x ðtþ Þ at any tme t. Snce ths nformaton s avalable to node at tme t, we call such nformaton local. For other nformaton to be obtaned by node, packets must be sent over the network. C. Non-Bayesan Cooperatve Localzaton In non-bayesan cooperatve localzaton, we treat the state of node at tme t as a nonrandom but unknown parameter. For an overvew of non-bayesan localzaton technques, the reader s referred to [75]. We focus on cooperatve ML and LS localzaton. The cooperatve LS algorthm forms the bass of [30], [47], [48], [50], [53], [54],[56] [58],and[61],aswellasvaratonssuchas weghted LS, where sgnal metrcs have an assocated weght reflectng the qualty of the estmate [105], and regularzed LS, where certan locatons are penalzed [106]. Based on cooperatve LS, a cooperatve ML algorthm was adopted n [51]. The ML and LS estmators mnmze a cost functon C ðtþ ðxþ wth respect to x ¼½x 1 ;...; x N Š at a partcular tme slot t. For the LS estmator, ths cost functon s gven by C ðtþ LSðxÞ ¼XN ¼1 X z ðtþ! fðx ; x Þ (33) S ðtþ! where fðx ; x Þ s a sutable functon based on the sgnal metrcs. For nstance, when x and x are the poston coordnates of nodes and,andwhenz ðtþ! s an estmate of the dstance between node and node, asestmatedby node,thenfðx ; x Þ¼kx x k. For the ML estmator, the cost functon becomes (see (3)) C ðtþ ML ðxþ ¼ log p zðtþ ¼ XN ¼1 X S ðtþ! rel x (34) log p z ðtþ!x ; x : (35) In general, for both the ML and LS estmators, the cost functon s of the form C ðtþ ðxþ¼ P N P ¼1 c! ðz ðtþ!; S ðtþ! x ; x Þ. To mnmze ths cost functon, we set the dervatve wth respect to x equal to zero, ðtþ ¼ X S z ðtþ!; x ; þ X ks z ðtþ!k ; x k; x : We can now apply gradent descent to teratvely mnmze C ðtþ ðxþ, startng from an ntal estmate at tme slot t, ^x ðt;0þ. A dstrbuted, cooperatve gradent descent algorthm s shown n Algorthm 1, where ðt;lþ represents a step sze that controls the convergence speed. Vol. 97, No., February 009 Proceedngs of the IEEE 435

11 Algorthm 1 Cooperatve LS and ML Localzaton 1: gven ^x ð0þ, 8 : for t ¼ 1toT do {tme slot ndex} 3: set ^x ðt;0þ ¼ ^x ðt 1Þ 4: for l ¼ 1toN ter do {teraton ndex} 5: nodes ¼ 1toN n parallel 6: broadcast current locaton estmate ^x ðt;l 1Þ 7: receve estmate from neghbors ^x ðt;l 1Þ, S ðtþ! 8: update locaton estmate {only for agents} ^x ðt;lþ ¼ ^x ðt;l 1Þ þ ðt;lþ P ðt;l 1Þ! 9: end parallel 10: end for 11: set ^x ðtþ ¼ ^x ðt;n terþ 1: end for S ðtþ! For notatonal convenence we have ðt;l 1Þ! ¼! z ðtþ!; x ; ^x ðt;l x ¼^x ðt;l 1Þ : (37) Here, l s the teraton ndex and ^x ðt;lþ s the estmate of the P x ðtþ at the lth teraton. Note that the term ð@c ks ðtþ!k ðz ðtþ!k ; x k; x Þ=@x Þ n (36) s omtted n! Algorthm 1, lne 8, snce the measurement z ðtþ!k s not avalable at node. Observe also that Algorthm 1 operates n two tme scales: n the shorter tme scale, ndexed by l n lne 4, nodes teratvely update ther state estmate for fxed t. The movement of the nodes occurs n the longer tme scale, set by the tme slots and ndexed by t n lne. The global mnmum may not be reached through teratve descent, snce the cost functon C ðtþ ðxþ s usually not convex. Example: We wll now llustrate the behavor of cooperatve LS localzaton n a plane usng the example n Fg. 1, where z ðtþ! ¼ ^d ðtþ! s an estmate made by node regardng ts dstance to node and fðx ; x Þ¼kx x k. After some straghtforward manpulatons, lne 8 n Algorthm 1 becomes ^x ðt;lþ ¼ ^x ðt;l 1Þ where ~d ðt;l 1Þ! þ ðt;lþ ¼k^x ðt;l 1Þ X S ðtþ! ^d ðtþ! ~d ðt;l 1Þ! ^x ðt;l 1Þ k and e ðt;l 1Þ s a unt- and ^x ðt;l 1Þ : vector orented along the lne connectng ^x ðt;l 1Þ e ðt;l 1Þ ¼ ^x ðt;l 1Þ ^x ðt;l 1Þ e ðt;l 1Þ (38) ^x ðt;l 1Þ ^x ðt;l 1Þ : (39) Equaton (38) can be nterpreted as follows: every term n the summaton s zero when the dstance estmate ^d ðtþ! matches the dstance between the estmates ^x ðt;l 1Þ and ^x ðt;l 1Þ.When^d ðtþ! s smaller than the dstance k^x ðt;l 1Þ ^x ðt;l 1Þ k between the two estmates, the LS algorthm corrects ths by movng the estmated poston ^x ðt;lþ when ^d ðtþ! s larger than k^xðt;l 1Þ of node towards^x ðt;l 1Þ. Conversely, ^x ðt;l 1Þ k, thelsalgorthm corrects ths by movng ^x ðt;lþ away from ^x ðt;l 1Þ can be tuned by the postve scalar step sze ðt;lþ. The movement. Algorthm SPAWN 1: gven pðx ð0þ Þ, 8 : for t ¼ 1toT do {tme ndex} 3: nodes ¼ 1toN n parallel 4: predcton operaton, accordng to (15) h ðt 1Þ! x ðtþ Z / p x ðtþ x ðt 1Þ p z ðtþ ;self xðt 1Þ ; x ðtþ fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼h ðt 1Þ ðx ðt 1Þ ;x ðtþ Þ ðt 1Þ X x ðt 1Þ!h ðt 1Þ dx ðt 1Þ 5: end parallel 6: correcton operaton: see Algorthm 3 7: end for In Fg. 1, anchor nodes 1, 3, and 5 have perfect locaton nformaton, so that ^x ðt;lþ ¼ x ðt;lþ, 8l; t, f1; 3; 5g.Wealso know from Fg. 1 that agent node 4 suffers from a poston ambguty. Specfcally, f we place the network topology n a5050 m map (see Fg. 5), agent node 4 can constran ts locaton to ether ^x ðt;lþ 4 ¼½0; 60Š (the correct poston) or ^x ðt;lþ 4 ¼½40; 60Š (the ncorrect poston), under nose-free dstance estmates. If the correct poston estmate s broadcast by agent 4 and receved by agent (Fg. 5(a)), the LS cost functon for node has a global mnmum at ts true poston. Hence, the LS algorthm wll move the estmate of the poston of node towards thetrueposton,asthe teratons progress. If the nformaton from agent node 4 s ncorrect (Fg. 5(b)), the LS cost functon has a mnmum at a poston far away from ts true poston, and the LS algorthm wll move the estmate of the poston of node away fromthetrueposton. D. Bayesan Cooperatve Localzaton Bayesan approaches to localzaton have been used n robotcs for the noncooperatve moble sngle-agent case [18] and for the cooperatve moble multagent case [49], 436 Proceedngs of the IEEE Vol.97,No.,February009

12 Fg. 5. Contour plots of the LS cost functon for agent node, for two possble poston estmates of agent node 4. The true locatons of anchors and agents are depcted by red squares and green crcles, respectvely. The estmated poston of agent 4 s marked as a cross, wth ^x ðt;lþ 4 ¼½0; 60Š n (a) and ^x ðt;lþ 4 ¼½40; 60Š n (b). The mnmum of the LS cost functon for agent node s far away from the correct locaton when agent node 4 broadcasts an ncorrect locaton estmate (n (b)). [60], [107]. Bayesan cooperatve localzaton for networks wthout moblty was nvestgated n [14], [108], and [109]. In ths secton, we develop a general Bayesan framework for cooperatve localzaton n heterogeneous, moble networks. We frst create an FG of a factorzaton of pðx ð0:tþ z ð1:tþ Þ and map ths FG onto the tme-varyng network topology, resultng n a network FG (Net-FG). Ths mappng s one-to-one n the sense that every node n the network s assocated wth a unque subgraph of the FG. We then ntroduce a message schedule that accounts for both spatal and temporal constrants of the message flow, resultng n network message passng (Net-MP). In partcular, we can execute a sum-product algorthm on the Net-FG, gvng rse to the SPA over a wreless network (SPAWN). Below, we descrbe the steps n developng the Net-FG, Net-MP, and, fnally, SPAWN. Due to ndependent movement and ndependent nternal measurements, both pðx ðtþ x ðt 1Þ Þ and pðz ðtþ self xðt 1Þ ; x ðtþ Þ can be further factorzed accordng to (7) and (30). The FG of pðx ð0:þ z ð1:þ Þ correspondng to the example network n Fg. 1 has a structure as shown n Fg. 6. The vertces n blue correspond to the factors pðz ðtþ rel xðtþ Þ, each Step 1VFactorzaton of pðx ð0:tþ z ð1:tþ Þ: We frst create an FG of a factorzaton of pðx ð0:tþ z ð1:tþ Þ. Usng our complete statstcal descrpton, we factorze pðx ð0:tþ z ð1:tþ Þ as (see (8)) p x ð0:tþ z ð1:tþ / p x ð0:tþ ; z ð1:tþ self p z ð1:tþ rel x ð0:tþ : (40) Substtutng (6), (9), and (31) nto (40) then leads to p x ð0:tþ z ð1:tþ Y T / p x ð0þ t¼1 p z ðtþ self xðt 1Þ ; x ðtþ n p x ðtþ x ðt 1Þ p z ðtþ rel xðtþ o : (41) Fg. 6. FG of pðx ð0:þ z ð1:þ Þ, correspondng to the example network n Fg. 1. We use the followng abbrevatons: f ðx ð0þ Þ¼pðX ð0þ Þ and h ðt 1Þ ðx ðt 1Þ ; Þ¼pð X ðt 1Þ Þ pðz ðtþ ;self Xðt 1Þ ; Þ. Thearrows represent the temporal flow of the message (from past to present). Vol. 97, No., February 009 Proceedngs of the IEEE 437

13 Fg. 7. Correcton operaton: FG of pðz ðtþ rel XðtÞ Þ, wth ncomng messages ðt 1Þ h ðþ and outgong messages! ðtþ X ðþ.!h ðtþ The node! ð ; Þ¼pðz ðtþ! XðtÞ ; Þ. The structure of ths FG depends on the network topology at tme slot t. of whch can be further factorzed 9 as n (3) wth an FG shownnfg.7. Step VCreatng the Net-FG: The Net-FG nvolves mappng the FGs from Fgs. 6 and 7 onto the network topology accordng to the nformaton that s local to each node. From Fg. 6, we see that the vertces ðx ðt 1Þ ; Þ¼ pð X ðt 1Þ Þ pðz ðtþ ; Þ can h ðt 1Þ ;self Xðt 1Þ be mapped to node, as these vertces contan nformaton local to node. Ths s depcted for node ¼ n Fg. 6 wth a red box. The mappng of the FG n Fg. 7 onto the network topology s less obvous. We see from Fg. 7 that for every varable, there s an equalty vertex as well as a number of vertces labeled! for S ðtþ!. These latter vertces are a functon of z ðtþ!, local to node. A natural mappng would thus be to assocate the equalty vertex and the vertces labeled! to node. Ths assocaton s shown n Fg. 8. As an example, the box n red shows the vertces assocated wth node. Combnng all the vertces mapped to a sngle node, we observe that they form a tree subgraph of the overall FG correspondng to pðx ð0:tþ z ð1:tþ Þ, and that ths tree depends only on locally avalable measurements z ðtþ! for S ðtþ! and z ðtþ ;self, t¼1;...; T. Algorthm 3 SPAWNVCorrecton Operaton 1: nodes ¼ 1toN n parallel : ntalze b ð0þ ðþ ¼ ðt 1Þ h ðþ! 3: end parallel 4: for l ¼ 1toN ter do {teraton ndex} 5: nodes ¼ 1toN n parallel 6: broadcast b ðl 1Þ ðþ 7: receve b ðl 1Þ ðþ from neghbors S ðtþ 9 In FG parlance, ths s known as openng a vertex [66].! 8: convert b ðl 1Þ ðþ to a dstrbuton over usng (15) ðlþ!! x ðtþ Z / p z ðtþ!x ðtþ ; x ðtþ b ðl 1Þ 9: compute new message usng (15) b ðlþ x ðtþ / ðt 1Þ h! Y x ðtþ S ðtþ! 10: end parallel 11: end for 1: nodes ¼ 1toN n parallel 13: determne outgong message: ðþ ¼ b ðn terþ ðþ X ðtþ!h ðtþ 14: end parallel ðlþ!! x ðtþ dx ðtþ x ðtþ Step 3VCreatng the Net-MP and SPAWN: We ntroduce a message schedule that accounts for the tme-varyng network topology, leadng to the Net-MP. Net-MP conssts of two types of messages: messages nternal to subgraphs (ntranode messages, correspondng to messages computed nternally by a node n the network) and messages between subgraphs (nternode messages, correspondng to messages between nodes n the network). The former type of message nvolves computaton wthn a node, whle the latter s sent as a packet over the wreless lnk. We ntroduce a message schedule that takes nto account the spatotemporal constrants of the network: To account for temporal constrants, messages flow only forward n tme. Ths s shown by the arrows n Fg. 6. Messages from the present to the past are not computed, as the state nformaton would be outdated and network connectvty may have 438 Proceedngs of the IEEE Vol.97,No.,February009

14 Fg. 8. Correcton operaton: mappng of subgraphs to nodes and schedulng of messages leads to a Net-MP. changed. Ths leads to the frst part of SPAWN, as descrbed n Algorthm. Observe that, smlar to Secton III-E, there s a predcton operaton, accountng for local moblty, and a correcton operaton, accountng for measurements between nodes. Durng the predcton operaton, node computes the message h ðt 1Þ! ðþ, based on the message ðt 1Þ X ðþ, on the local moblty!h ðt 1Þ model pðx ðtþ x ðt 1Þ Þ, and on the local lkelhood functon pðz ðtþ ;self xðt 1Þ ; x ðtþ Þ. Durng the correcton operaton, node determnes messages ðtþ X ðþ,!h ðtþ based on all the metrcs z ðtþ rel measured by all the nodes, as well as on all the messages ðt 1Þ h ðþ,! k k 8k. Ths mples that the correcton operaton requres exchange of nformaton between nodes. In other words, nodes need to cooperate. To account for the network topologcal constrants at afxedtmet, we choose a message flow shown n Fg. 8, wth the arrows showng the drecton of the messages. The bold red arrows represent nternode messages sent as packets over a wreless lnk. The blue arrows represent ntranode messages, computed nternally wthn a node. Accordng to ths schedule, messages only flow n one drecton over every edge. Ths mples that nternode messages do not depend on the recpent node. In other words, these messages can be broadcast. Theresultng SPAWN for the correcton operaton s gven n Algorthm 3. The nternode message broadcast by node s denoted by b ðlþ ðþ, wheret sthetmendexandls the teraton ndex. The overall SPAWN algorthm thus corresponds to Algorthm, wth the correcton operaton computed accordng to Algorthm 3. We wll name the message b ðlþ ðþ the belef of node at teraton l n tme slot t. Atanytme slot t, everynode can determne the MMSE or MAP estmate of ts own state by takng the mean or the mode of ts local message ðtþ X ðþ. Note that the tme slots at!h ðtþ dfferent nodes need not be synchronzed; the algorthm can be nterpreted as beng completely asynchronous. Example: Let us consder the correcton operaton for the example network n Fg. 1, where we perform localzaton n a m plane. Assume that the agent nodes and 4 begn wth no nformaton about ther poston, so that b ð0þ ðþ and b ð0þ ðþ n lne of Algorthm 3 4 are unform over the entre map. Anchor nodes 1, 3, and 5 have perfect locaton nformaton, so that b ð0þ ðþ, b ð0þ ðþ, 1 3 and b ð0þ ðþ are Drac delta functons. We focus on two 5 successve teratons of Algorthm 3 for agent node. Due to symmetry n the network, all the statements below can be appled to agent node 4, mutats mutands. 1) Iteraton l ¼ 1. All the nodes broadcast ther current belef. Agents can reman slent durng ths step, snce ther belefs contan no useful nformaton about ther locaton at ths teraton. Agent node has as neghbors S ð1þ! ¼f1; 3g and computes the message ð1þ ðþ (and ð1þ ðþ), usng 1!! 3!! lne 8 n Algorthm 3, based on the receved belef ðþ (and b ð0þ ðþ), as well as on range estmate b ð0þ 1 3 z ðtþ 1! (and z ðtþ 3! ). As expected, the messages ð1þ ðþ and ð1þ ðþ are roughly crcular 1!! 3!! dstrbutons around the postons of the two anchors (see Fg. 9(a)). Node now computes ts new belef (lne 9 n Algorthm 3) by multplyng ð1þ ðþ, ð1þ ðþ, and ts own (unform) 1!! 3!! belef b ð0þ ðþ. The result s depcted n Fg. 9(b), whch shows the contour plot of b ð1þ ðþ, a bmodal dstrbuton. Agent node 4 goes through smlar steps and determnes ts belef b ð1þ ðþ (whch of 4 course s also a bmodal dstrbuton). ) Iteraton l ¼. Agent nodes and 4 broadcast ther belefs to ther neghbors (lne 6). Agent Vol. 97, No., February 009 Proceedngs of the IEEE 439

15 Fg. 9. Consder the pont of vew of agent node for teraton l ¼ 1 (a)-(b) and teraton l ¼ (c)-(d). (a) At l ¼ 1, anchor nodes 1 and 3 broadcast ther belefs (lne 6 of Algorthm 3). Node receves the belefs and converts them based on range measurements (lne 8 of Algorthm 3). The messages ð1þ 1!! ðþ and ð1þ 3!! ðþ are shown as contour plots. (b) Agent node updates ts belef (lne 9 of Algorthm 3). Observe that the updated belef s bmodal, as ndcated n Fg. 1. (c) At teraton l ¼, node receves belefs from anchor nodes 1 and 3 and agent 4 (lne 7 of Algorthm 3) and converts them (lne 8 of Algorthm 3) based on range measurements. The messages ðþ 1!! ðþ, ðþ 3!! ðþ, and ðþ ðþ are shown as contour plots. Observe that ðþ ðþ s more spread out than ðþ ðþ and ðþ ðþ. Ths s because nformaton 4!!X 4!!X 1!!X 3!!X from anchors only has a sngle source of uncertanty (the range measurement), whle nformaton from agents has two sources of uncertanty (the range measurement and the uncertan locaton of the agent node 4). (d) Agent node updates ts belef (lne 9 of Algorthm 3) through multplcaton of the messages ðþ ðþ, ðþ ðþ,and ðþ ðþ. The updated belef s unmodal so that agent node can now determne ts 1!!X 3!!X 4!!X poston wthout ambguty. node receves belefs from anchor nodes 1 and 3 and from agent node 4 (lne 7). Agent node then computes messages ðþ ðþ, ðþ ðþ, 1!! 3!! and ðþ ðþ (lne 8). Contour plots of 4!! these three messages are shown n Fg. 9(c). Observe that the messages from the anchors are unchanged: ðþ ðþ¼ ð1þ ðþ, 1!! 1!! ðþ ðþ¼ ð1þ ðþ, and that the message 3!! 3!! correspondng to agent 4, ðþ 4! ðþ, s much broader than those correspondng to the anchors, due to the uncertanty that agent 4 has wth respect to ts own poston. Node computes ts new belef (lne 9) by multplyng ðþ ðþ, 1!! ðþ ðþ, ðþ ðþ, and ts own (unform) 4!! 4!! 440 Proceedngs of the IEEE Vol.97,No.,February009 belef b ð0þ ðþ. The result s depcted n Fg. 9(d), whch shows the contour plot of b ðþ ðþ havng a unmodal dstrbuton. Thus, agent node can unambguously estmate ts own poston by takng the mean or mode of b ðþ ðþ (for MMSE or MAP estmaton, respectvely). Smlarly, agent 4 can now determne ts poston wthout ambguty. In concluson, through cooperaton, both agents can self-localze. V. TRACTABLE AND REALISTIC UWB RANGING MODELS From the prevous secton, we know that cooperatve ML, MMSE, and MAP localzaton requres every node to

16 Table 1 Envronments Used for Measurement Campagn know the dstrbuton pðz ðtþ!x ðtþ ; x ðtþ Þ. Exstng rangng models, derved from expermental campagns, are based on hghly dealzed sgnals [110], [111] or sgnfcant postprocessng [11]. Such smplfcatons lead to unrealstc or mpractcal rangng models. Other UWB measurement campagns have been undertaken wth the goal of characterzng channel parameters such as path loss, fadng, and delay spread, ndependent of the effect of the measurement devce and methods [34], [36], [113]. Rangng models extracted from these channel models [3], [86] make mplct assumptons that may not hold n realstc envronments, whch n turn may lead to unrealstc predctons of localzaton performance. In order to obtan rangng models that closely reflect practcal operatng condtons, we have performed an extensve expermental campagn wth commercal UWB rados, performng RTOA dstance estmaton. In ths secton, we descrbe the expermental setup, our methodology to extract rangng models, and the resultng rangng models. At the end of ths secton, we show dfferent ways n whch cooperatve localzaton algorthms can cope wth NLOS condtons. A. Expermental Setup The experment conssted of commercal FCC-complant [114] UWB rados wth a bandwdth of approxmately 3. GHz centered at 4.7 GHz. Each rado was capable of transmttng and recevng packets through an omndrectonal antenna. To account for the nature of realstc localzaton networks, whch may be composed of off-theshelf parts, range measurements were collected as s, wthout makng any modfcatons to the hardware or embedded and host software n the UWB rados. A seres of fve campagns was performed n dfferent ndoor envronments around the MIT campus: two campagns at the Laboratory for Informaton and Decson Systems (LIDS), two campagns at the Computer Scence and Artfcal Intellgence Laboratory (CSAIL), and one campagn n a hangar of the Department of Aeronautcs and Astronautcs. Detals of the envronments are gven n Fg. 10. Expermental setup nvolvng two FCC-complant UWB rados. Table 1. Of these campagns, two were n NLOS condtons (CSAIL-NLOS and LIDS-NLOS), whle the remanng three were n lne-of-sght (LOS) condtons. In each envronment, the nodes were placed 89 cm above the ground (see Fg. 10). One rado was statc, whle the other moved n 5 cm ncrements towards the statc rado. At each separaton dstance d sep, 1000 rangng measurements were collected. We do not perform averagng of measurements, unlke [110] and [111]. Floor plans 10 for the LIDS and CSAIL expermental campagn are provded n Fg. 11. B. Rangng Models We observed that a hstogram of the 1000 rangng measurements collected at any dstance d sep typcally contans one large peak near d sep plus a small set of outlers on each sde of the peak (see Fg. 1). The outlers are consstently located at large dstances from the man peak, sometmes producng negatve range measurements. The fact that some measured ranges are sgnfcantly smaller or greater than the true dstance d sep ndcates that far-lyng outlers are lkely caused by the rangng algorthm (both postvely and negatvely based outlers) and multpath (postve outlers due to strong reflectons) rather than NLOS condtons. Further examnaton revealed that the tme of flght estmated 11 by the UWB nodes, usng an exstng propretary algorthm, exhbts hgh varance and possbly large errors. These fndngs ndcate that the measurement devces and rangng protocols are mportant factors to take nto consderaton when characterzng UWB range measurements. The operatng envronment has a sgnfcant effect on the dstrbuton of the measurements. The measurements collected n the CSAIL hallway (wth no clutter) have many fewer outlers than those collected n the LIDS hallway (wth adacent concrete pllars and walls) and n the hangar (wth large crates and other obects nearby). Measurements made n NLOS condtons tend to have more outlers than n LOS condtons. Addtonally, the nature of the blockage affects the measurements n NLOS condtons. The glass doors n CSAIL caused many fewer outlers than the concrete wall n LIDS. These fndngs corroborate the results n other UWB measurement campagns, e.g., [30] and [111]. Based on these hstograms, we concluded that a reasonable underlyng dstrbuton for the measurements collected n a gven envronment E at dstance d sep s a 10 Detaled floor plans are avalable at floorplans.mt.edu/pdfs/ 3_6.pdf (LIDS) and floorplans.mt.edu/pdfs/3_6.pdf (CSAIL). 11 That s, t B;rec t B;send n the notaton from Secton II-A. Vol. 97, No., February 009 Proceedngs of the IEEE 441

17 Fg. 11. Floor plans of (a) LIDS and (b) CSAIL expermental campagns. The dots represent the ntal postons of the UWB rados. One rado s statc (the red dot), whle the other rado (the black dot) moves n 5 cm ncrements towards the statc rado. Gaussan mxture densty wth three components, labeled k ¼ 1; ; and 3 for the lower outlers, man component, and upper outlers, respectvely. Each component s parameterzed by a mean E;k ðd sep Þ, a varance E;k ðd sepþ, anda weght w E;k ðd sep Þ. Hence, the dstrbuton of range measurements collected at dstance d sep s gven by pð^dd sep ; EÞ¼ X3 k¼1 w E;k ðd sep ÞN ^d E;kðd sep Þ; E;k ðd sepþ (4) where N x ð; Þ denotes the Gaussan dstrbuton wth mean and varance, evaluated n x. In order to estmate the mean E;k ðd sep Þ, varance E;k ðd sepþ, and weght w E;k ðd sep Þ for each k and E, we appled the expectaton-maxmzaton (EM) algorthm [115], [116] to the set of 1000 measurements collected at each dscrete dstance d sep n envronment E. The estmated Gaussan parameters capture the features of the hstograms. The man component k ¼ typcally has a large weght, a small varance, and a mean wth small bas. Measurements made n NLOS condtons exhbt a larger postve bas than those n LOS condtons. Ths agrees wth other UWB measurement campagns and models [30], [117]. The lower and upper outlers, represented by components k ¼ 1 and k ¼ 3 respectvely, are characterzed by smaller weghts than k ¼ and larger varances. Unlke [111], we fnd that the varance of the measurements does not always ncrease wth dstance. Our results also demonstrate that the effect of the surroundng envronment may outwegh the effect of dstance and LOS/NLOS condtons. Fnally, we model the dstrbuton of the rangng error for the contnuous value of dstances d by fttng quadratc polynomals P E;;k ðdþ, P E; ;kðdþ,andp E;w;k ðdþ to E;k ðd sep Þ, E;k ðd sepþ, and w E;k ðd sep Þ, respectvely. Snce w E;k ðd sep Þ1fork ¼ 1andk ¼ 3, the outlers are easy to dentfy, and we only report n Table the coeffcents for the resultng polynomals of form d þ d þ for the man mode of the dstrbutons. Our fnal model of the UWB rangng measurements ^d as a functon of true dstance d n envronment E s a Gaussan densty descrbed by pð^dd; EÞ ¼ N ^d P E;;ðdÞ; P E; ;ðdþ : (43) Fg. 1. Hstogram of range measurements, envronment LIDS-LOS, at d sep ¼ 16:5 m. Outlers on ether sde of the man peak are vsble. Wth the polynomal coeffcents as ts only parameters, ths UWB rangng model s tractable and easly mplemented n 44 Proceedngs of the IEEE Vol.97,No.,February009

18 Table Polynomal Coeffcents of Rangng Models d þ d þ ) When LOS/NLOS detecton s not performed, but nodes have knowledge of the envronment E and the locatons of obstructons, then [30], [1] pð^d! x ; x Þ¼ X pð^d! x ; x ; Þpð x ; x Þ (46) where pð x ; x Þ s ether zero or one, dependng on whether there s an obstructon between the postons x and x. 3) When LOS/NLOS detecton s performed, t can be based on some underlyng statstcs! extracted from the receved sgnal [89], such as the mean excess delay. We can replace ^! by! n (44), n whch case nodes requre knowledge of pð! Þ rather than pð ^! Þ. localzaton systems. Moreover, t accurately descrbes a practcal rangng devce operatng n realstc envronments. C. NLOS Condtons Our measurements ndcate that rangng performance depends on the LOS/NLOS condtons. In a practcal settng, rados may be mplemented wth the capablty for NLOS detecton [88], [89], [118] [11]. Let us ntroduce the sgnal metrcs estmated by node as z ðtþ! ¼½^d ðtþ ðtþ! ; ^! Š, denotes the estmate of kxðtþ x ðtþ k by node, where ^d ðtþ! and ^ ðtþ! denotes the decson at node regardng the LOS/ NLOS condton ðtþ flos; NLOSg between nodes and at tme t. Thefunctonpðz ðtþ!x ðtþ ; x ðtþ Þ then becomes (omttng the tme ndex t) VI. INDOOR LOCALIZATION: ACASESTUDY In ths secton, we combne the algorthms from Secton IV wth the rangng models from Secton V. We frst descrbe the performance crteron and smulaton setup and then provde numercal results for both statc and dynamc locaton-aware networks. A. Performance Crteron and Smulaton Setup To evaluate localzaton performance, we employ the outage probablty crteron: for a certan scenaro (number of agents, number of anchors, anchor postons, tme ndex t, and LOS/NLOS condtons) and a certan allowable error e th (say, m), an agent s sad to be n outage when ts poston error kx ^Xk exceeds e th.theoutage probablty s then gven by pð^d! ; ^! x ; x Þ¼ X pð^d! x ; x ; Þ P out ðe th Þ¼IE II kx ^Xk > e th (47) pð ^! Þpð x ; x Þ: (44) Varatons of (44) nclude the followng. 1) When LOS/NLOS detecton s not performed, and nodes have no nformaton regardng ther envronment, then ^! s not computed, pð x ; x Þ¼pð Þ and pðz! x ; x Þ reverts to [51], [87] pð^d! x ; x Þ¼ X pð^d! x ; x ; Þpð Þ: (45) In ths case, pð Þ s assumed to be predetermned wthn the rado or smply set to 1/. where IIfPg s the ndcator functon, whch s zero when proposton P sfalseandoneotherwse. Theexpectaton n (47) s taken wth respect to the locatons of the agents. The outage probablty s mplctly condtoned on the scenaro. To characterze the outage performance of the dfferent cooperatve localzaton algorthms developed n Secton IV, we have performed computer smulatons usng on the rangng models descrbed n Secton V. We consdered localzaton n a 100 by 100 m envronment wth 13 anchors and 100 or 50 agents. Anchors are statc, whle agents may be moble. Every node can range wth neghbors wthn 0 m. Messages are represented by samples [14], [18], [13]: n our case, 500 samples for nternode messages and 000 samples for ntranode messages. Note that messages may be represented n Vol. 97, No., February 009 Proceedngs of the IEEE 443

19 Fg. 13. The beneft of cooperaton. The 13 squares are the anchors, the 100 blue dots are the agents, and the whte crcles are the estmated postons of the agents. The lnes represent localzaton errors, connectng the estmated poston and the true poston of every agent. (a) shows the performance of MMSE localzaton wthout cooperaton. (b) shows the performance of the cooperatve MMSE localzaton algorthm after four teratons. otherways,usngtechnquessuchastheextendedkalman flter [14], [15] or Rao Blackwellzaton [97], [16]. B. Statc Networks We now llustrate the performance of MMSE localzaton n LOS condtons. Intally, none of the agents has nformaton about ther poston, represented by a unform a pror dstrbuton. The agents then perform MMSE localzaton, the results of whch are shown n Fg. 13 for a partcular realzaton of the network. The lnes n Fg. 13 connectthemmseestmateandthetruepostonofthe agent. It can be seen n Fg. 13(a) that wthout cooperaton, only a few agents can be localzed accurately. By contrast, Fg. 13(b) shows that wth cooperaton, all agents but one are localzed wth hgh accuracy after only four teratons. The sngle agent that cannot be localzed s connected to only two other nodes (one agent and one anchor), and therefore unable to determne ts poston wthout ambguty. A dfferent vew s offered n Fg. 14, showng the evoluton of the localzaton error for 100 agents as a functon of the teraton ndex. For a systematc comparson of dfferent localzaton algorthms, we have evaluated the outage probablty for noncooperatve and cooperatve MMSE localzaton, as well as cooperatve LS localzaton. The cooperatve LS algorthm uses a fxed step sze and s ntalzed wth the noncooperatve MMSE poston estmates. As a benchmark, we have also ncluded a centralzed MMSE algorthm. 1 Fg. 15 shows the outage probablty as a functon of allowable error for networks wth 100 agents over an ensemble of 0 random network nstantatons. We observe that noncooperatve localzaton results n large outage probabltes, greater than 50% for allowable errors below 5 m. Cooperatve LS can reduce the outage probablty to some extent, whle cooperatve MMSE offers sgnfcant addtonal performance gans. In fact, the performance of cooperatve MMSE s close to centralzed MMSE. At an allowable error of e th ¼ 1m,about90%of the agents are n outage for noncooperatve localzaton. The outage probablty drops to 40% for cooperatve LS, and s further reduced to less than 1% for cooperatve MMSE. From our smulatons, we observe that for ths scenaro the cooperatve MMSE converges 13 n about four teratons, whereas cooperatve LS requres many more teratons to converge. Thus, whle cooperatve MMSE certanly requres more computatons per node, t Fg. 14. A contour plot of the localzaton error (logarthmc) for every agent, as a functon of the teraton ndex. 1 The centralzed algorthm s closely related to [14] and corresponds to an FG (see Fg. 7) where vertces! and! are merged nto a sngle vertex. 13 By convergence, we mean that the outage curve no longer changes notceably for subsequent teratons. 444 Proceedngs of the IEEE Vol.97,No.,February009

20 Fg. 15. Comparson of dfferent localzaton algorthms for 13 anchors and 100 agents. Cooperatve LS outperforms noncooperatve MMSE localzaton. The cooperatve MMSE approach offers sgnfcant performance gans and attans an outage performance close to that of the centralzed algorthm. Fg. 17. Outage probablty for 13 statc anchors and 100 moble agents. The agents move accordng to a Gaussan random walk. After 0 tme slots, noncooperatve MMSE localzaton results n hgh outages, whereas outages reman low for cooperatve MMSE localzaton. converges n far fewer teratons. Ths s a crtcal pont for network algorthm desgn, snce every teraton requres packets to be broadcast over the network, ncreasng nterference and reducng throughput. Moreover, more teratons corresponds to an ncreased delay n determnng one s poston. Ths makes cooperatve MMSE localzaton more sutable for real-tme applcatons. Smlar results are shown n Fg. 16 for a settng wth 50agents.BycomparngFgs.15and16,weobservethat the outage performance for noncooperatve localzaton attans smlar values, rrespectve of the number of agents, whle cooperatve localzaton mproves as the network densty ncreases. Ths was also observed n [17]. C. Dynamc Networks Let us now consder the case when agents are moble. As a worst case scenaro, every agent moves a dstance d ðtþ n a drecton ðtþ at every tme step t, wth ðtþ U½0; Þ and d ðtþ Nð0; 1Þ. 14 Agents move ndependently wth respect to one another and from tme slot to tme slot. At tme t ¼ 0, every node s assumed to have perfect knowledge of ts poston, so that the aprordstrbuton of every agent s a Drac delta functon. Agents do not know n whch drecton they move, but they do know the dstance they travel. 15 Hence, z ðtþ ;self ¼ ^d ðtþ ¼ d ðtþ and p z ðtþ ;self xðt 1Þ ; x ðtþ ¼ x ðtþ x ðt 1Þ z ðtþ ;self (48) so that p x ðtþ x ðt 1Þ p z ðtþ ;self xðt 1Þ ; x ðtþ / x ðtþ x ðt 1Þ z ðtþ ;self : (49) Fg. 16. Comparson of dfferent localzaton algorthms for 13 anchors and 50 agents. Cooperatve LS outperforms noncooperatve MMSE localzaton. The cooperatve MMSE approach offers sgnfcant performance gans and attans an outage performance close to that of the centralzed algorthm. For far comparson between noncooperatve and cooperatve localzaton, we reduce the complexty of Algorthm 3 14 Here U½a; bþ denotes the unform dstrbuton between a IR and b IR, a G b. 15 Agents could be equpped wth a pedometer, for nstance. Vol. 97, No., February 009 Proceedngs of the IEEE 445

21 Fg. 18. Contour plots of the outage probablty as a functon of tme. (a) corresponds to noncooperatve MMSE localzaton, whle (b) corresponds to cooperatve MMSE localzaton. In the noncooperatve case, outages ncrease as tme progresses, whle for the cooperatve case, outages reman low for all tmes. by settng N ter ¼ 1 (see lne 4). Fg. 17 shows the outage probablty for tme slots t ¼ 1 and t ¼ 0, for both noncooperatve and cooperatve MMSE localzaton. Observe that after 0 tme slots, the noncooperatve approach exhbts sgnfcant performance degradaton, whle cooperatve localzaton mantans farly low outage probabltes. For a more detaled nvestgaton of the behavor over tme,werefertofg.18,whchshowstheoutagesasa functon of tme. In the noncooperatve case, performance clearly degrades wth tme. In the cooperatve case, the network quckly acheves a steady-state performance, and outages reman low. VII. CONCLUSION Locaton-awareness s a key feature of future-generaton wreless networks, enablng a multtude of applcatons n the mltary (e.g., blue force trackng), publc (e.g., searchand-rescue), and commercal (e.g., navgaton) sectors. Cooperaton among nodes has the potental to dramatcally mprove localzaton performance. In ths paper, we have gven an overvew of the man approaches to cooperatve localzaton from the vewpont of estmaton theory and factor graphs. We have shown how to create a network FG by mappng vertces of the FG onto the network topology. A network message passng algorthm can then be obtaned by approprate message schedulng, accountng for the tme-varyng network topology. The resultng algorthm (SPAWN) s a dstrbuted, cooperatve localzaton algorthm that outperforms many conventonal noncooperatve and cooperatve localzaton technques. We have performed an extensve UWB measurement campagn to determne tractable yet realstc rangng models. These models were used to valdate SPAWN n a large-scale network nvolvng 100 agents. Locaton-awareness has a great number of assocated research challenges, ncludng effcent, robust, and accurate rangng algorthms; low-complexty mplementatons of algorthms such as SPAWN; ntegraton of dfferent sgnal metrcs; LOS/NLOS detecton; nvestgaton of nterference effects, update rate, and message representatons; and the determnaton of fundamental performance bounds. A next phase to the development of ths feld nvolves the nteracton of SPAWN wth hgher level communcaton applcatons, such as locaton-aware routng [18], locaton-aware cryptography [19], locatonaware nformaton delvery [130], and locaton-aware computng [131], to name but a few. h Acknowledgment The authors would lke to thank Y. Shen, U. J. Ferner, G. Chrskos, A. Cont, and W. M. Gfford for ther careful readng of the manuscrpt and valuable suggestons. They are also grateful to Z. Botev for provdng them wth a fast kernel densty estmator, to U. Ferner for helpng wth the smulatons for moble networks, and to the TELIN department and P. Serbruyns for provdng them wth the computatonal resources to perform ther extensve smulatons. Fnally, they would lke to thank everyone nvolved n the UWB measurement campagn, n partcular S. J. Teller for the lvely dscussons. REFERENCES [1] F. Gustafsson and F. Gunnarsson, BMoble postonng usng wreless networks: Possbltes and fundamental lmtatons based on avalable wreless network measurements,[ IEEE Sgnal Process. Mag., vol., pp , Jul [] S. Gezc, Z. Tan, G. B. Gannaks, H. Kobayash, A. F. Molsch, H. V. Poor, and Z. Sahnoglu, BLocalzaton va ultra-wdeband rados: A look at postonng aspects for future sensor networks,[ IEEE Sgnal Process. Mag., vol., pp , Jul [3] D. Dardar, A. Cont, C. Buratt, and R. Verdone, BMathematcal evaluaton of envronmental montorng estmaton error through energy-effcent wreless sensor networks,[ IEEE Trans. Moble Comput., vol. 6, pp , Jul [4] A. Manwarng, D. Culler, J. Polastre, and R. S. J. Anderson, BWreless sensor networks for habtat montorng,[ n Proc. 1st ACM Int. Workshop Wreless Sensor Netw. Applcat. (WSNA 0), New York, 00, pp [5] N. Bulusu, J. Hedemann, and D. Estrn, BGPS-less low-cost outdoor localzaton for 446 Proceedngs of the IEEE Vol.97,No.,February009

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24 wreless sensor networks,[ n Proc. IEEE Int. Conf. Netw., Sep. 006, vol.. [104] R. Pabst, B. H. Walke, D. C. Schultz, P. Herhold, H. Yankomeroglu, S. Mukheree, H. Vswanathan, M. Lott, W. Zrwas, M. Dohler, H. Aghvam, D. D. Falconer, and G. P. Fettwes, BRelay-based deployment concepts for wreless and moble broadband rado,[ IEEE Commun. Mag., vol. 4, pp , Jan [105] C. Savarese, J. M. Rabaey, and K. Langendoen, BRobust postonng algorthms for dstrbuted ad-hoc wreless sensor networks,[ n Proc. General Track: 00 USENIX Annu. Tech. Conf., Berkeley, CA, Jun. 00, pp [106] J. Costa, N. Patwar, and A. O. Hero, III, BDstrbuted multdmensonal scalng wth adaptve weghtng for node localzaton n sensor networks,[ ACM Trans. Sensor Netw., vol., no. 1, pp , Feb [107] D. Fox, W. Burgard, H. Kruppa, and S. Thrun, BA Monte Carlo algorthm for mult-robot localzaton,[ Computer Scence Dept., Carnege Mellon Unv., Pttsburgh, PA, Tech. Rep. CMU-CS-99-10, [108] C. Chang, W. Snyder, and C. 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Mag., vol. 4, pp. 4 47, Oct ABOUT THE AUTHORS Henk Wymeersch (Member, IEEE) receved the Ph.D. degree n electrcal engneerng from Ghent Unversty, Belgum, n 005. He s a Postdoctoral Assocate wth the Laboratory for Informaton and Decson Systems, Massachusetts Insttute of Technology (MIT), Cambrdge. In , he was a Postdoctoral Fellow wth the Belgan Amercan Educatonal Foundaton, MIT. He s author of Iteratve Recever Desgn (Cambrdge, U.K.: Cambrdge Unversty Press, 007). Hs research nterests nclude algorthm desgn for wreless transmsson, statstcal nference, and teratve processng. Dr. Wymeersch s an Assocate Edtor of IEEE COMMUNICATION LETTERS. In 006, he receved the Alcatel Bell Scentfc Award for hs Ph.D. dssertaton. Jame Len (Member, IEEE) receved the bachelor s and master s degrees n electrcal engneerng and computer scence from the Massachusetts Insttute of Technology, Cambrdge, n 005 and 007, respectvely. Snce 007, she has been wth the Jet Propulson Laboratory, Pasadena, CA, as a Member of the Telecommuncatons Archtectures group. Her research nterests nclude wreless communcatons, localzaton, and cooperatve networks. Ms.Lenrecevedthe007DavdAdlerMemoralThessAwardforher research on localzaton n ultrawde bandwdth rado networks, conducted at the MIT Laboratory for Informaton and Decson Systems. Vol. 97, No., February 009 Proceedngs of the IEEE 449

25 Moe Z. Wn (Fellow, IEEE) receved both the Ph.D. n Electrcal Engneerng and M.S. n Appled Mathematcs as a Presdental Fellow at the Unversty of Southern Calforna (USC) n He receved an M.S. n Electrcal Engneerng from USC n 1989, and a B.S.(magna cum laude) n Electrcal Engneerng from Texas A&M Unversty n Dr. Wn s an Assocate Professor at the Massachusetts Insttute of Technology (MIT). Pror to onng MIT, he was at AT&T Research Laboratores for fve years and at the Jet Propulson Laboratory for seven years. Hs research encompasses developng fundamental theory, desgnng algorthms, and conductng expermentaton for a broad range of realworld problems. Hs current research topcs nclude locaton-aware networks, tme-varyng channels, multple antenna systems, ultra-wde bandwdth systems, optcal transmsson systems, and space communcatons systems. Professor Wn s an IEEE Dstngushed Lecturer and an elected Fellow of the IEEE, cted for Bcontrbutons to wdeband wreless transmsson.[ He was honored wth the IEEE Erc E. Sumner Award (006), an IEEE Techncal Feld Award for Bponeerng contrbutons to ultra-wde band communcatons scence and technology.[ Together wth students and colleagues, hs papers have receved several awards ncludng the IEEE Communcatons Socety_s Guglelmo Marcon Best Paper Award (008) and the IEEE Antennas and Propagaton Socety_s Serge A. Schelkunoff Transactons Prze Paper Award (003). Hs other recogntons nclude the Laurea Honors Causa from the Unversty of Ferrara, Italy (008), the Techncal Recognton Award of the IEEE ComSoc Rado Communcatons Commttee (008), Wreless Educator of the Year Award (007), the Fulbrght Foundaton Senor Scholar Lecturng and Research Fellowshp (004), the U.S. Presdental Early Career Award for Scentsts and Engneers (004), the AIAA Young Aerospace Engneer of the Year (004), and the Offce of Naval Research Young Investgator Award (003). Professor Wn has been actvely nvolved n organzng and charng a number of nternatonal conferences. He served as the Techncal Program Char for the IEEE Wreless Communcatons and Networkng Conference n 009, the IEEE Conference on Ultra Wdeband n 006, the IEEE Communcaton Theory Symposa of ICC-004 and Globecom- 000, and the IEEE Conference on Ultra Wdeband Systems and Technologes n 00; Techncal Program Vce-Char for the IEEE Internatonal Conference on Communcatons n 00; and the Tutoral Char for ICC-009 and the IEEE Semannual Internatonal Vehcular Technology Conference n Fall 001. He was the char ( ) and secretary (00 004) for the Rado Communcatons Commttee of the IEEE Communcatons Socety. Dr. Wn s currently an Edtor for IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. HeservedasAreaEdtorfor Modulaton and Sgnal Desgn ( ), Edtor for Wdeband Wreless and Dversty ( ), and Edtor for Equalzaton and Dversty ( ), all for the IEEE TRANSACTIONS ON COMMUNICATIONS. He was Guest-Edtor for the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS (Specal Issue on Ultra-Wdeband Rado n Multaccess Wreless Communcatons) n Proceedngs of the IEEE Vol.97,No.,February009