Cooperative Localization in Wireless Networks

Transcription

1 Cooeratve Localzaton n Wreless Networks Henk Wymeersch, Member, IEEE, Jame Len, Member, IEEE, and Moe Z. Wn, Fellow, IEEE Invted Paer Abstract Locaton-aware technologes wll revolutonze many asects of commercal, ublc servce, and mltary sectors and are exected to sawn numerous unforeseen alcatons. A new era of hghly accurate ubqutous locaton-awareness s on the horzon, enabled by a aradgm of cooeraton between nodes. In ths aer, we gve an overvew of cooeratve localzaton aroaches and aly them to ultra-wde bandwdth UWB wreless networks. UWB transmsson technology s artcularly attractve for short- to medum-range localzaton, esecally n GPS-dened envronments; wde transmsson bandwdths enable robust communcaton n dense mult-ath scenaros, and the ablty to resolve sub-nanosecond delays results n centmeterlevel dstance resoluton. We wll descrbe several cooeratve localzaton algorthms and quantfy ther erformance, based on realstc UWB rangng models develoed through an extensve measurement camagn usng FCC-comlant UWB rados. We wll also resent a owerful localzaton algorthm by mang a grahcal model for statstcal nference onto the network toology, whch results n a net-factor grah, and by develong a sutable net-message assng schedule. The resultng algorthm SPAWN s fully dstrbuted, can coe wth a wde varety of scenaros, and requres lttle communcaton overhead to acheve accurate and robust localzaton. Index Terms Localzaton, ultra-wde bandwdth transmsson, cooeratve rocessng, factor grahs, sum-roduct algorthm. I. INTRODUCTION LOCATION-AWARENESS s radly becomng an essental feature of many commercal, ublc servce, and mltary wreless networks [], []. Informaton collected or communcated by a wreless node s often meanngful only n conuncton wth knowledge of the node s locaton. For examle, sensor networks used for detectng satal varatons n envronmental condtons, such as temerature or olluton, requre knowledge of each sensor s locaton [3] []. Locaton nformaton also facltates a node s nteractons wth ts surroundngs and neghbors, enablng ervasve comutng and socal networkng alcatons [6]. Locatonaware technologes can enable or beneft a vast array of addtonal alcatons, ncludng ntruder detecton [7], blue force trackng [8], fndng frends or landmarks [9], healthcare montorng [0], asset trackng [], and emergency 9 servces [], [3]. Ths research was suorted, n art, by the Belgan Amercan Educatonal Foundaton, the JPL Strategc Unversty Research Partnershs, 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 LIDS, Massachusetts Insttute of Technology, Room 3-D674, 77 Massachusetts Avenue, Cambrdge, MA 039 USA e-mal: hwymeers,moewn@mt.edu. J. Len was wth the Laboratory for Informaton and Decson Systems, Massachusetts Insttute of Technology, and s now wth the Jet Proulson Laboratory, 4800 Oak Grove Drve, Pasadena, CA 909 e-mal: Jame.Len@l.nasa.gov. The caablty of network nodes to self-localze s needed n scenaros where nodes cannot be manually ostoned or located by a central system admnstrator [4], []. The goal of self-localzaton s for every node to know ts own state. A state usually ncludes the two- or three-dmensonal oston, and ossbly other roertes such as the velocty and the orentaton of the node [6] [8]. The concet of state deends on the alcaton and may also vary from node to node. In our exoston, we wll use the terms state, oston, and locaton nterchangeably, whle n our examles we wll narrow the scoe of state to two-dmensonal geograhcal coordnates. We wll dstngush between two tyes of nodes: agents, whch have a ror unknown states, and anchors, whch have known states at all tmes. Both agents and anchors may be moble. The localzaton rocess tycally conssts of two hases []. The frst hase s the measurement hase, durng whch agents measure nternal state nformaton e.g., usng an nertal measurement unt IMU and estmate sgnal metrcs based on drect communcaton wth neghborng agents and/or anchors. The second hase s the locaton-udate hase, 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 resect to three anchors, the agent can nfer ts own oston through trlateraton, rovded the agent knows the ostons of the anchors. The measurement hase s affected by uncertanty due to sources such as nose, multath, blockages, nterference, clock drfts, and envronmental effects [], [9] [4]. The underlyng transmsson technology s a crtcal factor n how these sources affect the measurements []. For nstance, the oor sgnal enetraton caabltes of the wdely used Global Postonng System GPS revent consumer-grade GPS recevers from makng relable measurements ndoors, under forest canoes, and n certan urban settngs, leadng to nadequate oston nformaton []. In challengng envronments such as these, ultra-wde bandwdth UWB transmsson technology [6] [8] s a romsng alternatve for localzaton [9] [3]. UWB systems are nherently well-suted for localzaton snce the use of extremely large transmsson bandwdths results n desrable caabltes such as accurate rangng due to fne delay resoluton; smle mlementaton for multleaccess communcatons; and 3 obstacle enetraton caabltes [3] [37]. For more nformaton on the fundamentals of UWB, we refer the reader to [6] [8], [38] [44] and references theren. Gven an underlyng transmsson technology, localzaton Sgnal metrcs nclude any roerty of the receved sgnal that deends on the relatve ostons of the transmtter and the recever. Examles nclude the tme of flght, the angle of arrval, and the receved sgnal strength.

2 Fgure. The beneft of cooeratve localzaton: usng only dstance estmates wth resect to the anchors nodes, and, agent nodes and 4 are unable to determne ther resectve ostons wthout ambguty. Observe that node cannot communcate wth node, and node 4 cannot communcate wth node. When agent nodes and 4 communcate and range drectly as dected by the red arrow, they can cooerate to unambguously determne ther ostons. erformance s also deendent on the secfc algorthm used n the locaton-udate hase. An emergng aradgm s cooeratve localzaton, n whch nodes hel each other to determne ther locatons [4]. Cooeratve localzaton has receved extensve nterest from the robotcs, otmzaton and wreless communcatons communtes see [4], [], [30], [3], [46] [60], and references theren. A smle comarson of conventonal and cooeratve localzaton s dected n Fg. : whle each agent moble unt cannot ndeendently determne ts own oston based on dstance estmates wth resect to the anchors base statons, they can cooeratvely fnd ther ostons. In general, cooeratve localzaton can dramatcally ncrease localzaton erformance n terms of both accuracy and coverage. In ths aer, we rovde an overvew of cooeratve localzaton algorthms n wreless networks. In artcular: We consder large-scale dynamc heterogeneous networks and examne how cooeraton can be used to mrove localzaton accuracy and coverage wth resect to noncooeratve technques. We focus on algorthms based on the rncles of estmaton theory and statstcal nference [6], [6] and outlne a framework for the systematc desgn of nference algorthms, usng the theory of factor grahs FGs [63] [6]. We develo a localzaton algorthm by mang a FG onto the tme-varyng network toology and by emloyng a sato-temoral message schedule, resultng n a network FG Net-FG and network message assng Net-MP. The roosed algorthm s called, SPAWN sum-roduct algorthm over a wreless network, s fully dstrbuted and cooeratve. SPAWN also accounts for dfferent state tyes among nodes, node moblty, and any uncertantes assocated wth both the measurement and locaton-udate hases. We show how SPAWN generalzes revously roosed localzaton algorthms, revertng to Bayesan flterng n Coverage s the fracton of nodes that have an accurate locaton estmate. the case of a sngle agent [8] and to non-arametrc belef roagaton localzaton [4] n the case of a homogeneous network wth statc nodes. Ths aer s organzed as follows. In Secton II, we rovde an overvew of methods used for the two hases of localzaton. We descrbe and comare varous sgnal metrcs and localzaton aroaches, emhaszng the advantages of UWB as an underlyng transmsson technology. In Secton III we rovde a concse overvew of general urose estmaton technques and factor grahs. Secton IV deals wth non-bayesan and Bayesan cooeratve localzaton strateges. Secton V detals the results of an extensve range measurement camagn usng FCC-comlant UWB rados. We then resent a case study for ndoor localzaton n large UWB networks n Secton VI and quantfy the erformance of dfferent cooeratve and noncooeratve localzaton algorthms n terms of accuracy and avalablty usng exermental data. In Secton VII, we draw our conclusons and resent avenues for further research n ths area. Notaton. Throughout ths aer we wll use the followng notaton. The state of node at tme t wll be denoted by x t. The state-sequence of node from tme t to t wll be denoted by x t:t. Random varables wll be catalzed and vectors wrtten n bold, unless there s no ambguty. Dstrbutons such as X x wll at tmes be abbrevated by x. II. LOCALIZATION APPROACHES FOR WIRELESS NETWORKS In ths secton we descrbe dfferent tyes of sgnal metrcs and classfy dfferent tyes of localzaton algorthms. We aly ths classfcaton to well-known localzaton systems, consderng both ndoor and outdoor scenaros. A. Measurement Phase In the frst hase of localzaton, ackets are exchanged between neghborng nodes n the network say, nodes A and B. From the hyscal waveforms corresondng to these ackets, 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, 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 exlots the relaton between ower loss and the dstance between sender and recever [66]. The ablty of node B to receve ackets from node A, known as connectvty, can constran the dstance between node A and node B to the communcaton range of node A [67], [68]. Dstance measurements of fner resoluton can be obtaned by estmatng the roagaton tme of the wreless sgnals. Ths s the bass of tme of arrval TOA, tme dfference of arrval TDOA, and round-tr tme of arrval RTOA [3], [46]. RTOA s the most ractcal scheme n a decentralzed settng, as t does not requre a common tme reference between nodes [69]. For examle, node B sends a

3 3 acket to node A at tme t B,send n ts own tme reference. Node A receves the acket at tme t A accordng to ts own clock and resonds wth a acket at t A +, where s a tme nterval that s ether redetermned or communcated n the resonse acket. Node B receves the acket from A at tme t B,rec and can then determne the dstance d AB through the relaton t B,rec t B,send v = d AB, where v s the sgnal roagaton seed. Relatve orentatons can be determned through angle of arrval AOA estmaton when a node s equed wth drectonal or multle antennas [46]. For nstance, n a lnear array wth satal antenna searaton δ, the dfference n arrval tme between any two successve antenna elements s gven by t = δ/ v cos φ AB, where φab s the angle between the mngng sgnal and the antenna array [0]. Beyond dstance and angle, one can estmate other roertes such as the velocty of a node by measurng Doler shfts [70]. Informaton about the state of the node can also be measured nternally; for examle, dstances traveled usng an odometer or edometer, acceleraton usng an accelerometer, and orentaton usng an IMU. More roblem-secfc 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 multath. The connectvty metrc tends to roduce coarse locaton nformaton, esecally 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 suscetble to errors due to obstructons between the transmtter and the recever. These obstructons, leadng to so-called non-lne-of-sght NLOS condtons, can cause a ostve bas n the dstance estmate. Nose, nterference, multath, 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 roertes of the sensors. For addtonal nformaton regardng sources of error, the reader s referred to [], [], [3], [], [46], [7], [73]. B. Locaton-Udate Phase In the second hase, measurements are aggregated and used as nuts to a localzaton algorthm. A ossble taxonomy of localzaton algorthms s the followng see also [46], [3], [74] and references theren. Centralzed versus dstrbuted. In centralzed localzaton, the ostons of all agents are determned by a central rocessor. Ths rocessor gathers measurements from anchors as well as agents and comutes the ostons of all the agents. Centralzed algorthms are usually not scalable and thus mractcal for large networks. In dstrbuted localzaton, such as GPS, there s no central controller and every agent nfers ts own oston 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. Absolute versus relatve. Absolute localzaton refers to localzaton n a sngle re-determned coordnate system []. Relatve localzaton refers to localzaton n the context of one s neghbors or local envronment [3], [7]; hence, the coordnate system can vary from node to node. Non-cooeratve versus cooeratve. In a non-cooeratve aroach, there s no communcaton between agents, only between agents and anchors. Every agent needs to communcate wth multle anchors, requrng ether a hgh densty of anchors or long-range anchor transmssons. In cooeratve localzaton, nter-agent communcaton removes the need for all agents to be wthn communcaton range of multle 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, cooeratve localzaton can offer ncreased accuracy and coverage. We wll quantfy these erformance mrovements n Secton VI. C. Outdoor and Indoor Localzaton Outdoor localzaton: Examles of outdoor systems nclude GPS, LORAN-C, and rado-locaton n cellular networks. GPS s a dstrbuted, absolute, and non-cooeratve localzaton aroach [], relyng on TOA estmates from at least four anchors GPS satelltes to solve a four-dmensonal non-lnear roblem 3 satal dmensons and tme dmenson, snce the agent s not synchronzed to the anchors. Asssted GPS s a centralzed verson of GPS, reducng the comutatonal burden on the agents []. LORAN-C s a terrestral redecessor of GPS [76], whch offers centralzed, absolute, and non-cooeratve localzaton servces based on TDOA. Cell hone rado-locaton servces such as E9 commonly emloy TDOA and are centralzed, absolute, and non-cooeratve [77], [78]. Indoor localzaton: Exstng and emergng ndoor localzaton [79], [80] methods nclude WF, RFID, and UWB localzaton. RADAR, based on WF fngerrntng at multle anchors [8]; PlaceLab, usng connectvty from 80. access onts; and GSM base statons [8] emloy centralzed, absolute, and non-cooeratve aroaches. Passve RFID tags can be used n conuncton wth RFID readers to rovde connectvty-based localzaton [83] that s centralzed, relatve, and non-cooeratve. Both WF and RFID systems suffer from oor 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 [84] [86]. The fne delay resoluton of UWB sgnals s well-suted for estmatng roagaton tmes e.g., for RTOA or AOA, snce the erformance of delay estmaton algorthms scales wth the transmsson bandwdth [3]. Moreover, the wde bandwdth allows multath comonents to be resolved and enables sueror sgnal enetraton through obstacles [7], [3] [37]. Hence, robust communcatons n dense multath envronments and rangng n NLOS condtons can be acheved [3] [3]. The enetraton caabltes of UWB sgnals also make them useful for detectng and otentally comensatng for the effects of

4 4 obstacles and NLOS condtons [87], [88]. In addton, UWB transmtters are low comlexty, low cost devces, ractcal for dense and rad deloyment [89]. Snce the ower s sread over a large bandwdth, UWB communcaton systems are covert, ower-effcent, and cause mnmal nterference to other systems [6], [7], [90], [9]. UWB sgnals have the unque advantage of smultaneously accomlshng robust communcaton and recson rangng. Node can therefore extract nformaton about ther relatve ostons from sgnals already used for communcaton wthout any addtonal overhead. The recently comleted IEEE 80..4a standard [9] wll lkely sawn numerous ractcal systems and alcatons n ths shere. III. BACKGROUND ON INFERENCE Before resentng cooeratve localzaton algorthms, we frst gve a bref overvew of mortant technques from estmaton theory and statstcal nference, whch can be aled to the localzaton roblem. There are a number of aroaches for estmatng a arameter x from an observaton z. Aart from ad-hoc technques, we generally categorze these as Bayesan or non-bayesan, deendng on whether or not we consder x as a realzaton of a random varable [6]. In ths secton, we descrbe both aroaches. Wthn the context of Bayesan technques, we then consder aroxmate nference, factor grahs, and sequental estmaton. A. Non-Bayesan Estmaton Two common non-bayesan estmators, whch treat x as an unknown determnstc arameter, are the least-squares LS estmator and the maxmum lkelhood ML estmator. The LS estmator assumes that z R N and z = f x + n, where f s a known functon and n s a measurement error. The LS estmate of x s obtaned by solvng the followng otmzaton roblem: ˆx LS = arg mn x z f x. The LS estmator does not exlot any knowledge regardng the statstcs of n. The ML estmator accounts for the statstcs of nose sources and maxmzes the lkelhood functon: B. Bayesan Estmaton ˆx ML = arg max x Z X z x. Two common Bayesan estmators, whch treat x as a realzaton of a random varable X wth an a ror dstrbuton X x, are the mnmum mean squared error MMSE estmator and the maxmum a osteror MAP estmator. The MMSE estmator fnds the mean of the a osteror dstrbuton: ˆx MMSE = x X Z x z dx. 3 The MAP estmator fnds the mode of the a osteror dstrbuton: ˆx MAP = arg max x X Z x z. 4 X Z mn KLD class C SPA b X MAP MMSE VEM Fgure. Mnmzng the KLD can lead to dfferent Bayesan nference algorthms, ncludng sum-roduct algorthm SPA, MMSE, mean-feld MF, exectaton-roagaton EP, exectaton-maxmzaton EM, varatonal EM VEM, and MAP. If X Z z 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 comonents of X rather than the entre vector. For examle, when X = [X,...,X N ], then the MMSE res. MAP estmate of X k s gven by the mean res. mode of the margnal a osteror dstrbuton Xk Z z of the varable X k. C. Aroxmate Inference In many nference roblems, the a osteror dstrbuton X Z z s dffcult to descrbe, and obtanng ts mean, mode, or margnals s a very hard roblem. In such stuatons, one can turn to an alternatve dstrbuton, say b X, belongng to a certan class C. Inferences can then be drawn based on a artcular b X that s close to X Z z. A common measure of closeness between dstrbutons s the Kullback- Lebler dvergence KLD [93], defned as D b X b X x X Z = b X x ln dx. X Z x z It s easy to verfy that D b X X Z 0, and that D b X Z X = 0 f and only f bx = X Z. For a gven class C, we try to fnd the dstrbuton b X that mnmzes the KLD: MF EP EM b X = arg mn b X C D b X X Z. 6 Often b X cannot be found n closed form; we can only determne a descrton e.g., a lst of necessary condtons mosed on b X of the statonary onts 4 of D b X Z X. Usng ths descrton, one can then develo teratve algorthms to fnd those statonary onts. Dfferent classes C lead to dfferent solutons, ncludng mean-feld, exectatonroagaton, exectaton-maxmzaton, and the sum-roduct algorthm see Fg.. For a detaled exoston, see [94], [9]. These solutons can be found through message assng on a FG, as detaled n the next secton. D. Factor Grahs and the Sum-Product Algorthm We cover some basc concets of FGs and the SPA; for a detaled treatment, the reader s referred to [63] [6], [9]. 4 That s, a mnmum, a maxmum, or a saddle-ont.

5 φ A φ B X X X X φ C X φ A X ψ X Trees and Cyclc Grahs Part : The factorzaton 8 results n a FG wth cycles 6, whle 9 does not. Ths s an mortant dstncton. For FGs wthout cycles,.e., trees, t can be shown that the a osteror dstrbuton can be exressed as [9] M k= X Z x z = X k Z x k z [ Xl Z x l z ] d l, 0 N l= X 3 φ D X Fgure 3. FG of φ A x φ B x, x φ C x, x φ D x, x 3, on the left. Groung φ B and φ C nto ψ yelds the FG on the rght. In many nference roblems, the a osteror dstrbuton can be factorzed, wth every factor φ k deendng only on a small subset of varables x k x: X Z x z = M φ k x k, 7 Q k= where M s the number of factors and Q s a ossbly unknown normalzaton constant. Factor Grahs: A factor grah s a way to grahcally reresent a factorzaton such as 7: For every factor, say φ, we create a vertex drawn as a crcle or square and label t φ. For every varable X, we create an edge drawn as a lne and label t X. When a varable X aears n a factor φ, we connect the edge X to the vertex φ. Snce edges can be connected to at most two vertces, we must treat varables that aear n more than two factors as a secal case: For a varable X that aears n D > factors, we create a so-called equalty vertex and label t =. We also create D edges and connect every edge to the equalty vertex and one of the D factors. The edges are labeled wth a dummy name of the varable e.g., X and X for X. The equalty vertex reresents a Drac delta functon. For nstance, an equalty vertex wth edges X, X and X corresonds to a functon δx x δx x. For notatonal convenence, we often label all the edges connected to an equalty vertex wth the same label X, n ths case. Let us examne a smle examle, where X = [X,X,X 3 ] has an a osteror dstrbuton that can be factorzed as x,x,x 3 z = 8 Q φ A x φ B x,x φ C x,x φ D x,x 3, where Q s an unknown constant. Factorzatons are by no means unque: by groung φ B and φ C nto a new factor, say ψ, a dfferent factorzaton s obtaned: X 3 x,x,x 3 z = Q φ A x ψ x,x φ D x,x 3. 9 FGs corresondng to 8 and 9 are dected n Fg. 3. We focus on Forney-style FGs, also known as normal grahs [96]. φ D where d l s the number of factors n 7 where the varable x l aears. 7 For nstance, when usng the factorzaton 9, we can exress x,x,x 3 z as wth d = d 3 =, d = 8 x,x,x 3 z = x,x z x,x 3 z x z 0 x z x 3 z 0. Gven a factorzaton of the form 7, we can ntroduce a class C SPA of functons b X of the followng form: b X x = M k= b X k x k N l= [b X l x l ] d l, subect to b Xk x k 0, b Xl x l 0, x k b Xk x k = x l b Xl x l =, x k \l b X k x k = b Xl x l, k,l : x l x k. The descrton of the statonary onts of D b X Z X s exressed n terms of the functons b Xk and b Xl. Comarng 0 and, we see that when the FG of the factorzaton of X Z z has no cycles, the otmzaton roblem 6 wth C = C SPA has a unque global mnmzer b X, wth corresondng KLD equal to zero. Furthermore, the a osteror dstrbuton X Z z can be recovered based solely on b X k and b X l, k,l, snce the unqueness of the soluton mles that b X k x k = Xk Z x k z, k,x k, 3 b X l x l = Xl Z x l z, l,x l. 4 3 The Sum-Product Algorthm: The SPA s a message assng algorthm on a cycle-free FG that effcently comutes b X k and b X l, k,l when C = C SPA. The SPA oerates by comutng messages nsde the vertces and sendng those messages over the edges. A message over an edge X s a functon of the corresondng varable, and s denoted µ X φ or µ φ X, where φ s a vertex adacent to edge X. Gven a factor φx,...,x D and ncomng messages µ Xk φ k, the outgong message over edge X s gven by µ φ X x φx,...,x D k µ Xk φ x k dx...dx dx +...dx D, where the roortonalty symbol ndcates that the message µ φ X s normalzed, such that µ φ X x dx =. For examle, n Fg. 3, the message from ψ to X s gven by µ ψ X x ψ x,x µ X ψ x dx. 6 6 The FG of 8 has a cycle gven by edges X, X, X, X. 7 For mathematcal convenence, we need to assume that x k contans at least varables. We grou varables together where necessary. See also [9]. 8 Snce φ A has only one varable, we have groued φ A and ψ.

6 6 Messages start wth the half-edges sendng a constant message and the vertces of degree sendng the corresondng factor. For examle, n Fg. 3, µ X3 φ D x 3 and µ φa X x = φ A x. For equalty vertces, t can be shown that an outgong message s smly the ont-wse roduct of the ncomng messages. For examle, n Fg. 3, the message from X to φ B s gven by µ X φ B x µ φa X x µ φc X x. 7 The margnal of a certan varable s obtaned by ontwse multlcaton of the two messages assed over the corresondng edge. In our examle from Fg. 3, b X x µ X φ D x µ φd X x. 8 The margnal of a cluster of varables x k s obtaned by multlyng the ncomng messages wth the corresondng factor. For nstance, n the examle from Fg. 3: b X,X x,x ψ x,x µ X ψ x µ X ψ x. 9 4 Trees and Cyclc Grahs Part : The SPA rovdes 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 comuted functons b X k and b X l are no longer exactly equal to the corresondng margnal a osteror dstrbutons, and b X s not necessarly a dstrbuton [97]. Furthermore, n a FG wth cycles, there are many ossble orders n whch messages are comuted also known as the message schedule, and each schedule may lead to dfferent functons b X k and b X l. The ablty to choose schedules wll turn out to be mortant n dervng a cooeratve 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 t at tme t, takng nto account ndeendent observatons taken u to and ncludng t, say z :t = [ z,...,z t] [7], [98], [99]. We rely on the followng Markovan assumtons: x t x 0:t = x t x t and z t x 0:t = z t x t. It can then easly be shown that x t z :t = x t,x t z :t dx t 0 x t z :t, z t x t where x t z :t = x t x t x t z :t dx t. Ths mles that, gven x t z :t, we can determne x t z :t as follows: a redcton oeraton, durng whch we determne the dstrbuton x t z :t, gven all observatons before tme t, accordng to the ntegral n ; and a correcton oeraton, n whch we account for X 0 X X φ 0 φ φ tme ψ ψ Fgure 4. Sequental estmaton: a FG of `x 0: z :, where φ t `x t, x t s a shorthand for `x t x t, and ψ t `x t s a shorthand for `z t xt. the new observaton z t to calculate x t z :t, accordng to. Hence, at every tme t, we have the a osteror dstrbuton x t z :t of the varable x t, gven all the observatons untl and ncludng tme t. We can determne the mean or the mode of ths a osteror dstrbuton, gvng us the MMSE estmate or MAP estmate of x t, resectvely. The entre rocedure s ntalzed by x t z :t t=0 = x 0. Sequental estmaton and FGs: Sequental estmaton can be obtaned by creatng a FG of x 0:T z :T and then alyng the SPA. Usng the Markovan assumtons and the fact that the measurements are ndeendent, we easly fnd that x 0:T z :T 3 x 0 T t= x t x t z t x t, wth a corresondng FG dected n Fg. 4, for T =. The message schedule s shown wth arrows, startng wth the black messages from the leaves of the tree. At tme t =, the red message µ φ X s gven by µ φ X x 4 µ X 0 φ x 0 φ x,x 0 dx 0 = x 0 x x 0 dx 0 = x, whch corresonds to the redcton oeraton. Usng 7, the blue message µ X φ s gven by µ X φ x µ ψ X x µ φ X x = z x x, whch s exactly the correcton oeraton. At tme slot t =, we easly fnd that µ φ X x µx φ x x x dx and that µ X φ 3x z x µ φ X x. The sequence of redcton oeraton red arrows followed by correcton oeraton blue arrows contnues wth messages flowng from ast to resent to future. In rncle, messages can also be comuted from future to resent to ast, a rocess known as smoothng.

7 7 Algorthm Cooeratve LS and ML localzaton : gven ˆx 0, : for t = to T do {tme slot ndex} 3: set ˆx t,0 = ˆx t 4: for l = to N ter do {teraton ndex} : nodes = to N n arallel 6: broadcast current locaton estmate ˆx t,l 7: receve estmate from neghbors ˆx t,l 8: udate locaton estmate {only for agents} ˆx t,l = ˆx t,l + δ t,l ψ t,l 9: end arallel 0: end for : set ˆx t = ˆx t,nter : end for S t IV. COOPERATIVE LOCALIZATION, S t The concet of cooeraton n networks s farly new: t reles on drect communcaton between agents, rather than through a fxed nfrastructure [00] [0]. Cooeraton has been successfully aled to wreless eer-to-eer communcaton, leadng to standards such as Bluetooth [03] and Zgbee [04], and s exected to exand to cellular systems over the next few years [0]. In ths secton, we aly the cooeratve aradgm to a comletely dfferent roblem: localzaton. We resent several fundamental cooeratve localzaton algorthms based on the methodologes from Secton III. Both non-bayesan and Bayesan aroaches wll be consdered. For the sake of the exoston, we focus on small-scale examles. Later, n Secton VI, we wll resent a case study for a large network wth more than 00 nodes. A. Problem Formulaton We consder a wreless network wth N nodes n an envronment E. Tme s slotted, and nodes can move ndeendently from ostons at tme slot t to new ostons at tme slot t. The state of node at tme t s wrtten as x t. We denote by the set of nodes from whch node may receve sgnals durng tme slot t. Smlarly, we denote by S t the set of S t nodes whch may receve a sgnal from node durng tme slot t. At tme slot t, node may estmate local metrcs z t,self based on nternal measurements e.g., usng an IMU or an odometer. Based on the sgnal receved from node S t, node can estmate sgnal metrcs z t. We denote the collecton of all sgnal metrcs and all nternal measurements collected by all nodes at tme t as z t. Note that we can break u z t nto z t self and z t rel, where zt self contans all the nternal measurements of all the nodes, whle z t rel contans all the relatve sgnal metrcs from all the nodes wth resect to ther neghbors. The goal of node s to estmate ts own state x t at tme t, gven only nformaton u to tme t. Ideally, the localzaton rocess should requre low comlexty and communcaton overhead er node and ncur a low latency. B. Assumtons We make the followng assumtons, whch are reasonable n many ractcal scenaros: a The states of the nodes are a ror ndeendent: x 0 = N = x0. b Nodes move accordng to a memoryless walk: x 0:T = x 0 T c Nodes move ndeendently: x t x t = t= N = x t x t. 6 x t x t. 7 d Relatve measurements are ndeendent of the nternal measurements, condtoned on the states of the nodes: x 0:T,z :T = x 0:T. 8 z :T rel self z :T rel e Internal measurements are mutually ndeendent and deend only on the current and revous state: z :T self T x 0:T = t= z t self x t,x t. 9 f Internal measurements at node deend only on the state of node : z t self x t,x t = N = z t,self x t,x t. 30 g Relatve measurements are ndeendent from tme slot to tme slot, condtoned on the states of the nodes. Moreover, they deend only on the current states: z :T rel T x 0:T = t= z t rel x t. 3 h Relatve measurement at any tme slot t are condtonally ndeendent, and deend only on the two nodes nvolved: z t rel N x t = = S t z t x t,x t. 3 We further assume that node knows the followng: the state dstrbuton x 0 at tme t = 0; ts own moblty model at any tme t; the nternal x t x t measurements z t,self and the corresondng lkelhood functon z t,self x t,x t at any tme t; v the sgnal metrcs z t and the lkelhood functon z t x t,x t 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, ackets must be sent over the network.

8 node 80 node 70 node node 3 70 node node 3 [m] 60 node 4 [m] 60 node 4 0 node 0 node [m] a [m] b Fgure. Contour lot of the LS cost functon for agent node, for two ossble oston estmates of agent node 4. The true locatons of anchors and agents are dected by red squares and green crcles, resectvely. The estmated oston of agent 4 s marked as a cross, wth ˆx t,l 4 = [0, 60] n a, and ˆx t,l 4 = [40, 60] 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. C. Non-Bayesan Cooeratve Localzaton In non-bayesan cooeratve localzaton, we treat the state of node at tme t as a non-random but unknown arameter. For an overvew of non-bayesan localzaton technques, the reader s referred to [74]. We focus on cooeratve ML and LS localzaton. The cooeratve LS algorthm forms the bass of the works [30], [46], [47], [49], [], [3], [] [7], [60], as well as varatons such as weghted LS, where sgnal metrcs have an assocated weght reflectng the qualty of the estmate [06], and regularzed LS, where certan locatons are enalzed [07]. Based on cooeratve LS, a cooeratve ML algorthm was adoted n [0]. The ML and LS estmators mnmze a cost functon C t x wth resect to x = [x,...,x N ] at a artcular tme slot t. For the LS estmator, ths cost functon s gven by C t LS x = N = S t z t f x,x, 33 where f x,x s a sutable functon based on the sgnal metrcs. For nstance, when x and x are the oston coordnates of nodes and, and when z t s an estmate of the dstance between node and node, as estmated by node, then f x,x = x x. For the ML estmator, the cost functon becomes see 3 C t ML x = log z t rel x 34 = N = S t log z t x,x. 3 In general, for both the ML and LS estmators, the cost functon s of the form C t x = N = c S t z t ;x,x. To mnmze ths cost functon, we set the dervatve wth resect to x equal to zero, where C t x x = S t + k S t c z t ;x,x x c k z t k ;x k,x x. 36 We can now aly gradent descent to teratvely mnmze C t x, startng from an ntal estmate at tme slot t, ˆx t,0. A dstrbuted, cooeratve gradent descent algorthm s shown n Algorthm, where δ t,l reresents a ste sze that controls the convergence seed. For notatonal convenence we have ntroduced ψ t,l c z t ;x, ˆx t,l x x=ˆx t,l. 37 Here, l s the teraton ndex and ˆx t,l s the estmate of the x t at the lth teraton. Note that the term c k z t k ;x k,x k S t x n 36 s omtted n Algorthm, lne 8, snce the measurement z t k s not avalable at node. Observe also that Algorthm oerates n two tme scales: n the shorter tme scale, ndexed by l n lne 4, nodes teratvely udate 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

9 9 be reached through teratve descent, snce the cost functon C t x s usually not convex. Examle: We wll now llustrate the behavor of cooeratve LS localzaton n a lane usng the examle n Fg., where z t t = ˆd s an estmate made by node regardng ts dstance to node and f x,x = x x. After some straghtforward manulatons, lne 8 n Algorthm becomes ˆx t,l = ˆx t,l + δ t,l ˆdt t,l d e t,l, e t,l = S t 38 s a unt- and ˆx t,l : t,l where d = ˆx t,l ˆx t,l and e t,l vector orented along the lne connectng ˆx t,l ˆx t,l ˆx t,l ˆx t,l ˆx t,l. 39 Equaton 38 can be nterreted as follows: every term n the t summaton s zero when the dstance estmate ˆd matches the dstance between the estmates ˆx t,l and ˆx t,l. When ˆd t s smaller than the dstance t,l ˆx ˆx t,l between the two estmates, the LS algorthm corrects ths by movng the estmated oston ˆx t,l of node towards ˆx t,l. Conversely, when ˆd s larger than t,l ˆx ˆx t,l t, the LS algorthm corrects ths by movng ˆx t,l away from ˆx t,l. The movement can be tuned by the ostve scalar ste sze δ t,l. In Fg., anchor nodes, 3 and have erfect locaton nformaton, so that ˆx t,l = x t,l, l,t, {,3,}. We also know from Fg. that agent node 4 suffers from a oston ambguty. Secfcally, f we lace the network toology n a 0 m 0 m ma see Fg., agent node 4 can constran ts locaton to ether ˆx t,l 4 = [0, 60] the correct oston or ˆx t,l 4 = [40, 60] the ncorrect oston, even wth nose-free dstance estmates. If the correct oston estmate s broadcast by agent 4 and receved by agent Fg. a, the LS cost functon for node has a global mnmum at ts true oston. Hence, the LS algorthm wll move the estmate of the oston of node towards the true oston, as the teratons rogress. When the nformaton from agent node 4 s ncorrect Fg. b, the LS cost functon has a mnmum at a oston far away from ts true oston, and the LS algorthm wll move the estmate of the oston of node away from the true oston. D. Bayesan Cooeratve Localzaton Bayesan aroaches to localzaton have been used n robotcs for the non-cooeratve moble sngle-agent case [8] and for the cooeratve moble mult-agent case [48], [9], [08]. Bayesan cooeratve localzaton for networks wthout moblty was nvestgated n [4], [09], [0]. In ths secton, we develo a general Bayesan framework for cooeratve localzaton n heterogeneous, moble networks. We frst create a FG of a factorzaton of x 0:T z :T and ma ths FG onto the tme-varyng network toology, resultng n a network FG Net-FG. Ths mang s a one-to-one mang n a sense that every node n the network s assocated wth a unque subgrah of the FG. We then ntroduce a message schedule that accounts for both satal and temoral constrants of the message flow, resultng n network message assng Net-MP. In artcular we can execute the sum-roduct algorthm on the Net-FG, gvng rse to the SPA over a wreless network SPAWN. Below, we descrbe the stes n develong the Net-FG, Net-MP and fnally, SPAWN. Ste - Factorzaton of x 0:T z :T : We frst create a FG of a factorzaton of x 0:T z :T. Usng our comlete statstcal descrton we factorze x 0:T z :T as see 8 x 0:T z :T x 0:T,z :T self z :T rel Substtutng 6, 9 and 3 nto 40 then leads to x 0:T. 40 x 0:T z :T x 0 4 T t= { x t x t z t self x t,x t z t rel } x t. Due to ndeendent movement and ndeendent nternal measurements, both x t x t and z t self x t,x t can be further factorzed accordng to 7 and 30. The FG of x 0: z : corresondng to the examle network n Fg. has a structure as shown n Fg. 6. The vertces n blue corresond to the factors z t rel x t, each of whch can be further factorzed 9 as n 3 wth a FG shown n Fg. 7. Ste - Creatng the Net-FG: The Net-FG nvolves mang the FGs from Fg. 6 and Fg. 7 onto the network toology accordng to the nformaton that s local to each node. From Fg. 6, we see that the vertces h t t X,X t = X t X t t z,self X t,x t can be maed to node, as these vertces contan nformaton local to node. Ths s dected for node = n Fg. 6 wth a red box. The mang of the FG n Fg. 7 onto the network toology 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 φ for S t. These latter vertces are a functon of z t, local to node. A natural mang would thus be to assocate the equalty vertex and the vertces labeled φ to node. Ths assocaton s shown n Fg. 8. As an examle, the box n red shows the vertces assocated wth node. Combnng all the vertces maed to a sngle node, we observe that they form a tree subgrah of the overall FG corresondng to x 0:T z :T, and ths tree deends X t only on locally avalable measurements z t for St and z t,self, t =,...,T. Ste 3 - Creatng the Net-MP and SPAWN: We ntroduce a message schedule that accounts for the tme-varyng network toology, leadng to the Net-MP. Net-MP conssts of two tyes of messages: messages nternal to subgrahs ntra-node messages, corresondng to messages comuted nternally by a 9 In FG arlance, ths s known as oenng a vertex [6].

10 0 f f f 3 f 4 f X 0 X 0 X 0 3 X 0 4 X 0 h 0 h 0 h 0 3 h 0 4 h 0 X X X 3 X 4 X z rel X X X X 3 X 4 X h h h 3 h 4 h X X X 3 X 4 X z rel X X X X 3 X 4 X Fgure 6. FG of `X 0: z:, corresondng to the examle network n Fg.. We use the followng abbrevatons: f `X0 = `X0, and h t `Xt, X t = `Xt X t `zt,self X t, X t. The arrows reresent the temoral flow of the message from ast to resent. Algorthm SPAWN : gven x 0, : for t = to T do {tme ndex} 3: nodes = to N n arallel 4: redcton oeraton, accordng to µ t h X t t x : end arallel 6: correcton oeraton: see Algorthm 3 7: end for x t x t t z,self x t,x t }{{} =h t x t,x t µ X t h t x t dx t node n the network and messages between subgrahs nternode messages, corresondng to messages between nodes n the network. The former tye of message nvolves comutaton wthn a node, whle the latter s sent as a acket over the wreless lnk. We ntroduce a message schedule that takes nto account the sato-temoral constrants of the network: To account for temoral constrants, messages flow only forward n tme. Ths s shown by the arrows n Fg. 6. Messages from the resent to the ast are not comuted, as the state nformaton would be outdated and network connectvty may have changed. Ths leads to the frst art of SPAWN, as descrbed n Algorthm. Observe that, smlar to Secton III-E, there s a redcton oeraton, accountng for local moblty, and a correcton oeraton, accountng for measurements between nodes. Durng the redcton oeraton, node comutes the message µ t h, based on the message µ X t t X, h t on the local moblty model x t x t, and on the local lkelhood functon z t,self x t,x t. Durng the correcton oeraton, node determnes messages µ X t, based on all the metrcs z t h t rel measured by all the nodes, as well as on all the messages µ h t k X t k, k. Ths mles that the correcton o-

11 µ t h X t µ h t X t µ h t 3 X t 3 µ h t 4 X t 4 µ h t X t X t X t X t 3 X t 4 X t φ φ 3 φ 3 4 φ 4 φ 4 φ 4 Fgure 7. φ `Xt µ t X µ ht t X µ ht t X µ 3 ht t 3 X µ 4 ht t 4 X Correcton oeraton: FG of z t rel X, t wth ncomng messages µ t h X t, and outgong messages µ t X, X t = `zt, X t. The structure of ths FG deends on the network toology at tme slot t. X t µ t h X t µ h t X t µ h t 3 X t 3 µ h t 4 X t 4 µ h t ht X t X t X t 3 X t 4 X t h t X t. The node φ φ 3 φ 3 4 φ 4 φ 4 φ 4 Fgure 8. µ t X ht µ X t ht µ X t 3 ht 3 µ X t 4 ht 4 Correcton oeraton: mang of subgrahs to nodes, and schedulng of message leads to a Net-MP. µ X t ht eraton requres exchange of nformaton between nodes. In other words, nodes need to cooerate. To account for the network toologcal constrants at a fxed tme t, we choose a message flow shown n Fg. 8, wth the arrows showng the drecton of the messages. The bold red arrows reresent nter-node messages sent as ackets over a wreless lnk. The blue arrows reresent ntra-node messages, comuted nternally wthn a node. Accordng to ths schedule, messages only flow n one drecton over every edge. Ths mles that nter-node messages do not deend on the recent node. In other words, these messages can be broadcast. The resultng SPAWN for the correcton oeraton s gven n Algorthm 3. The nter-node message broadcast by node s denoted by b l, where t s the tme ndex and l s the teraton ndex. X t The overall SPAWN algorthm thus corresonds to Algorthm, wth the correcton oeraton comuted accordng to Algorthm 3. We wll name the message b l the belef of X t node at teraton l n tme slot t. At any tme slot t, every node can determne the MMSE or MAP estmate of ts own state by takng the mean or the mode of ts local message µ t X. Note that the tme slots at dfferent nodes need h t not be synchronzed; the algorthm can be nterreted as beng comletely asynchronous. Examle: Let us consder the correcton oeraton for the examle network n Fg., where we erform localzaton n a 0 m 0 m lane. Assume that the agent nodes and 4 begn wth no nformaton about ther oston, so that and b 0 n lne of Algorthm 3 are unform b 0 X t X t 4 over the entre ma. Anchor nodes, 3 and have erfect locaton nformaton, so that b 0, b 0 and b 0 are X t X t 3 X t Drac delta functons. We focus on two successve teratons of Algorthm 3 for agent node. Due to symmetry n the network, all the statements below can be aled to agent node 4, mutats mutands. Iteraton l =. All the nodes broadcast ther current belef. Agents can reman slent durng ths ste, snce ther belefs contan no useful nformaton about ther locaton at ths teraton. Agent node has a

12 node 80 node 70 node node 3 70 node node 3 y [m] y [m] 60 node 4 60 node 4 0 node 0 node x [m] 90 a x [m] 90 b 80 node 80 node 70 node node 3 70 node node 3 y [m] y [m] 60 node 4 60 node 4 0 node 0 node x [m] c x [m] d Fgure 9. Consder the ont of vew of agent node for teraton l = a-b and teraton l = c-d. a: Anchor nodes and 3 broadcast ther belefs lne 6 of Algorthm 3. Node receves the belefs lne 7 of Algorthm 3, and converts them based on range measurements lne 8 of Algorthm 3. The messages µ and µ are shown as contour lots. b: Agent node udates ts belef lne 9 of Algorthm 3. Observe φ X t φ 3 X t that the udated belef s b-modal, as ndcated n Fg.. c: Node receves belefs from anchor nodes and 3, and agent 4 lne 7 of Algorthm 3, and converts them lne 8 of Algorthm 3 based on range measurements. The messages µ, µ and µ are shown as contour lots. Observe that µ φ 4 X t s more sread out than µ φ X t φ X t and µ φ 3 X t φ 3 X t φ 4 X t. Ths s because nformaton 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 udates ts belef lne 9 of Algorthm 3 through multlcaton of the messages µ, µ and µ. The udated belef s un-modal so that agent node can now determne ts oston wthout ambguty. φ 3 X t φ 4 X t φ X t

13 3 Algorthm 3 SPAWN - Correcton Oeraton : nodes = to N n arallel : ntalze b 0 = µ t h X t X t 3: end arallel 4: for l = to N ter do {teraton ndex} : nodes = to N n arallel 6: broadcast b l X t 7: receve b l X t 8: convert b l X t µ l φ X t b l X t from neghbors S t to a dstrbuton over X t t x z t x t,x t b l X t 9: comute new message usng t x µ t h X t t x 0: end arallel : end for : nodes = to N n arallel 3: determne outgong message: µ t X h t 4: end arallel = b Nter X t S t t t x dx µ l φ X t usng t x neghbors S = {,3 } and comutes the message and µ, usng lne 8 n µ φ X t φ 3 X t Algorthm 3, based on the receved belef b 0 b 0 X t 3 z t 3 X t and, as well as on range estmate z t and. As exected, the messages µ µ φ 3 X t φ X t and are roughly crcular dstrbutons around the ostons of the two anchors see Fg. 9 a. Node now comutes ts new belef lne 9 n Algorthm 3 by multlyng µ, µ, and ts φ X t own unform belef b 0 X t φ 3 X t. The result s dected n Fg. 9 b, whch shows the contour lot of b X t, a b-modal dstrbuton. Agent node 4 goes through smlar stes, and determnes ts belef b whch of course X t 4 s also a b-modal dstrbuton. Iteraton l =. Agent nodes and 4 broadcast ther belefs to ther neghbors lne 6. Agent node receves belefs from anchor nodes and 3, and from agent node 4 lne 7. Agent node then comutes messages µ φ X t, µ φ 3 X t, and µ φ 4 X t lne 8. Contour lots of these 3 messages are shown n Fg. 9 c. Observe that the messages from the anchors are unchanged: µ = µ, φ X t φ X t µ φ 3 X t = µ φ 3 X t, and that the message corresondng to agent 4, µ φ 4, s much broader than those corresondng to the anchors, due to the uncertanty that agent 4 has wth resect to ts own oston. Node comutes ts new belef lne 9 by multlyng µ φ X t, µ φ 3 X t and ts own unform belef b 0 X t, µ φ 4 X t,. The result s dected n Fg. 9 d, whch shows the contour lot of b havng a un-modal dstrbuton. Thus, agent X t node can unambguously estmate ts own oston by takng the mean or mode of b for MMSE or MAP X t estmaton, resectvely. Smlarly, agent 4 can now determne ts oston wthout ambguty. In concluson, through cooeraton, both agents can self-localze. V. TRACTABLE AND REALISTIC UWB RANGING MODELS From the revous secton, we know that cooeratve ML, MMSE, and MAP localzaton requres every node to know the dstrbuton z t x t,x t. Exstng rangng models, derved from exermental camagns, are based on hghly dealzed sgnals [], [] or sgnfcant ost-rocessng [3]. Such smlfcatons lead to unrealstc or mractcal rangng models. Other UWB measurement camagns have been undertaken wth the goal of characterzng channel arameters such as ath loss, fadng, and delay sread, ndeendent of the effect of the measurement devce and methods [3], [90], [4]. Rangng models extracted from these channel models [3], [8] make mlct assumtons that may not hold n realstc envronments, whch n turn may lead to unrealstc redctons of localzaton erformance. In order to obtan rangng models whch closely reflect ractcal oeratng condtons, we have erformed an extensve exermental camagn wth commercal UWB rados, erformng RTOA dstance estmaton. In ths secton we descrbe the exermental setu, our methodology to extract rangng models, and the resultng rangng models. At the end of ths secton, we show dfferent ways n whch cooeratve localzaton algorthms can coe wth NLOS condtons. A. Exermental Setu The exerment conssts of commercal FCC-comlant [] UWB rados wth a bandwdth of aroxmately 3. GHz centered at 4.7 GHz. Every rado s able to transmt and receve ackets through an omn-drectonal antenna. To account for the nature of realstc localzaton networks, whch may be comosed of off-the-shelf arts, range measurements were collected as s, wthout makng any modfcatons to the hardware or embedded and host software n the UWB rados. Fgure cm cm Exermental setu nvolvng two FCC-comlant UWB rados.

14 4 Table I ENVIRONMENTS USED FOR MEASUREMENT CAMPAIGN Locaton Sgnal characterzaton Mn-Max searaton d se LIDS 6th-floor hallway LOS m LIDS 6th-floor offce and lobby NLOS concrete wall m CSAIL 3rd-floor hallway LOS m CSAIL 3rd-floor hallway NLOS glass doors m Aero/Astro hangar LOS m a LIDS b CSAIL Fgure. Floor lans of a LIDS and b CSAIL exermental camagns. The dots reresent the ntal ostons of the UWB rados. One rado s statc the red dot, whle the other rado the black dot moves n cm ncrements towards the statc rado. A seres of fve camagns was erformed n dfferent ndoor envronments around the MIT camus: two camagns at the Laboratory for Informaton and Decson Systems LIDS, two camagns at the Comuter Scence and Artfcal Intellgence Laboratory CSAIL, and one camagn n a hangar of the Deartment of Aeronautcs and Astronautcs. Detals of the envronments are gven n Table I. Of these camagns, 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 laced 89 centmeters above the ground see Fg. 0. One rado was statc, whle the other rado moved n cm ncrements towards the statc rado. At each searaton dstance d se, 000 rangng measurements were collected. We do not erform averagng of measurements, unlke [], []. Floor lans 0 for the LIDS and CSAIL exermental camagn are rovded n Fg.. 0 Detaled floor lans are avalable at floorlans.mt.edu/dfs/3 6.df LIDS and floorlans.mt.edu/dfs/3 6.df CSAIL. B. Rangng Models We observed that a hstogram of the 000 rangng measurements collected at any dstance d se tycally contaned one large eak near d se, lus a small set of outlers on each sde of the eak see Fg.. The outlers are consstently located at large dstances from the man eak, sometmes roducng negatve range measurements. The fact that some measured ranges are sgnfcantly smaller or greater than the true dstance d se ndcates that far-lyng outlers are lkely caused by the rangng algorthm both ostvely and negatvely based outlers and multath ostve outlers due to strong reflectons, rather than NLOS condtons. Further examnaton revealed that the tme of flght estmated by the UWB nodes, usng an exstng roretary algorthm, exhbts hgh varance and ossbly large errors. These fndngs ndcate that the measurement devces and rangng rotocols are mortant factors to take nto consderaton when characterzng UWB range measurements. The oeratng envronment has a sgnfcant effect on the That s, t B,rec t B,send n the notaton from Secton II-A.