Localization in mobile networks via virtual convex hulls

Transcription

1 Localzaton n moble networs va vrtual convex hulls Sam Safav, Student Member, IEEE, and Usman A. Khan, Senor Member, IEEE arxv:.7v [cs.sy] Jan 7 Abstract In ths paper, we develop a dstrbuted algorthm to localze an arbtrary number of agents movng n a bounded regon of nterest. We assume that the networ contans at least one agent wth nown locaton (herenafter referred to as an anchor), and each agent measures a nosy verson of ts moton and the dstances to the nearby agents. We provde a geometrc approach, whch allows each agent to: () contnually update the dstances to the locatons where t has exchanged nformaton wth the other nodes n the past; and () measure the dstance between a neghbor and any such locatons. Based on ths approach, we provde a lnear update to fnd the locatons of an arbtrary number of moble agents when they follow some convexty n ther deployment and moton. Snce the agents are moble, they may not be able to fnd nearby nodes (agents and/or anchors) to mplement a dstrbuted algorthm. To address ths ssue, we ntroduce the noton of a vrtual convex hull wth the help of the aforementoned geometrc approach. In partcular, each agent eeps trac of a vrtual convex hull of other nodes, whch may not physcally exst, and updates ts locaton wth respect to ts neghbors n the vrtual hull. We show that the correspondng localzaton algorthm, n the absence of nose, can be abstracted as a Lnear Tme- Varyng (LTV) system, wth non-determnstc system matrces, whch asymptotcally tracs the true locatons of the agents. We provde smulatons to verfy the analytcal results and evaluate the performance of the algorthm n the presence of nose on the moton as well as on the dstance measurements. I. INTRODUCTION Localzaton s a well-studed problem, whch refers to the collecton of algorthms that estmate the locaton of nodes n a networ. Relevant applcatons nclude traffc control, ndustral automaton, robotcs, and envronment montorng [] []. In terms of the nformaton used for estmatng the locatons, localzaton schemes n Wreless Sensor Networs (WSN) can be classfed as range-based, [] [7], and rangefree, [8] []. Whle the former depends on measurng the dstance and/or angle between the nodes, the latter maes no assumptons about the avalablty of such nformaton and reles on the connectvty of the networ. The lterature on localzaton also conssts of centralzed and dstrbuted approaches. Centralzed algorthms, [] [], despte ther benefts, are mpractcal n large networs where each node has lmted power and communcaton capablty. Some notable dstrbuted localzaton technques nclude successve refnements, [] [8], multlateraton, [9] [], estmaton based methods, multdmentonal scalng [], and graph-theoretcal methods, [] []. The authors are wth the Department of Electrcal and Computer Engneerng, Tufts Unversty, College Ave, Medford, MA, {sam.safav@,han@ece.}tufts.edu. Ths wor s partally supported by an NSF Career award: CCF #. When rangng data s nosy, the followng estmaton-based localzaton technques are wdely used. Maxmum Lelhood Estmaton (MLE) methods estmate the most probable locaton of a moble node based on pror statstcal models. MLE only requres (current) measured data and does not ntegrate pror data. Sequental Bayesan Estmaton (SBE) methods use the recursve Bayes rule to estmate the lelhood of an agent s locaton. The soluton to SBE s generally ntractable and cannot be determned analytcally. An alternatve approach s Kalman-based technques, whch are only optmal when the uncertantes are Gaussan and the system dynamcs are lnear, [], [7]. However, localzaton has always been consdered as a nonlnear problem and hence the optmalty of Kalman-based solutons are not guaranteed. To address the nonlnear nature of localzaton problems, other suboptmal solutons to approxmate the optmal Bayesan estmaton nclude Extended Kalman Flter (EKF) and Partcle Flters (PF), [7]. In partcular, Sequental Monte Carlo (SMC) method s a PF that explots posteror probablty to determne the future locaton of an agent. Localzaton algorthms specfcally desgned for moble networs have been proposed n [8] []. Ref. [8] ntroduces the Monte Carlo Localzaton (MCL) method, whch explots moblty to mprove the accuracy of localzaton. Inspred by [8], the authors n [] propose Moble and Statc sensor networ Localzaton (MSL*) that extends MCL to the case where some or all nodes are statc or moble. Ref [] proposes varatons of MCL, namely dual and mxture MCL to ncrease the accuracy of the orgnal MCL algorthm. On the other hand, Ref [9] provdes Range-based Sequental Monte Carlo Localzaton method (Range-based SMCL), whch combnes range-free and range-based nformaton to mprove localzaton performance n moble sensor networs. In ths paper, we consder localzaton of a moble networ, assumng that each agent measures a nosy verson of ts dstances to the nodes (agents and/or anchors) n ts communcaton radus, and the moton t undertoo, e.g., by usng an accelerometer. We focus on scenaros where GPS (or related postonng system) s compromsed and/or unavalable, and develop a collaboratve algorthm for a networ of agents/robots to fnd ther locatons n order to mplement a locaton-aware tas or a msson. As an example, consder a networ of moble robots wth no central or local coordnator and wth lmted communcaton, whose tas s to transport goods n an ndoor faclty where GPS sgnals are not aval- In wreless networs, the dstance may be estmated wth, e.g., Receved Sgnal Strength (RSS), Tme of Arrval (ToA), Tme Dfference of Arrval (TDoA) measurements, [8]. We further note that the dstance nformaton can also be obtaned usng a camera at each agent, [9].

2 able. In order to perform a delvery tas, each moble robot has to now ts own locaton frst. In such settngs, we are nterested n developng an anchor-based dstrbuted algorthm that uses only the dstance and angle measurements to trac the agent locatons such that the convergence s nvarant to the ntal poston estmates. However, the mplementaton and analyss of such dstrbuted algorthm n moble networs s not straghtforward, because: () an agent may not be able to fnd nearby nodes at any gven tme to mplement a dstrbuted algorthm; () an agent may not be n the proxmty of an anchor at any gven tme; () the neghborhood at each agent s dynamc, resultng nto a tme-varyng dstrbuted algorthm. In ths context, the man contrbuton of ths wor s to develop a lnear framewor for localzaton that enables us to crcumvent the challenges posed by the predomnant nonlnear approaches to ths problem. Ths lnear framewor s not to be nterpreted as a lnearzaton of an exstng nonlnear algorthm. Instead, the nonlnearty from range to locaton s embedded n an alternate representaton provded by the barycentrc coordnates. However, forcng ths lnear update comes at a prce,.e., a locaton update s only possble when an agent les n a trangle (convex hull n R ) formed by three neghborng nodes. Snce such neghbors may never exst, see () above, we ntroduce the noton of a vrtual convex hull. We then show that localzaton n moble networs can be acheved wth exactly one anchor when the total number of nodes (agents and anchors) s at least n R. Ths s n star contrast to the requrement of at least anchors that s prevalent n the trangulaton-based localzaton lterature. See [] for a recent survey on localzaton approaches wth sngle vs. multple anchor nodes. As we wll dscuss n Secton V, we abstract the correspondng localzaton algorthm as a Lnear Tme-Varyng (LTV) system whose system matrces may be: dentty when no locaton update occurs; stochastc when there s no anchor among the neghbors of the updatng agent; and, strctly sub-stochastc when the neghborhood ncludes at least one anchor. Note that the order n whch the updates occur and the weghts assgned to the neghbors are not nown a pror. To address these challenges, we apply a novel method to study LTV convergence by parttonng the entre chan of system matrces nto slces and relatng the convergence rate to the slce lengths. In partcular, we show that the algorthm converges f the slce lengths do not grow faster than a certan exponental rate. Snce our localzaton scheme s based on the moton and the dstance measurements, t s meanngful to evaluate the performance of the algorthm when these parameters are corrupted by nose. Therefore, we study the mpact of nose on the convergence of the algorthm and provde modfcatons to counter the undesrable effects of nose. Owng to the lnear representaton, we are able to provde a few smple yet comprehensve modfcatons to the proposed approach, whch as shown n the smulatons lead to a bounded error n the locaton estmates. We now descrbe the rest of the paper. In Secton II, we formulate the problem. We ntroduce the geometrc approach to trac the dstances n Secton III. We then provde our localzaton algorthm that reles on the noton of vrtual convex hull n Secton IV, followed by the convergence analyss n Secton V. In Secton VI, we examne the effects of nose on the algorthm. We provde smulaton results and compare the performance of the algorthm wth MCL, MSL*, Dual MCL, and Range-based SMCL algorthms n Secton VII. Fnally, Secton VIII concludes the paper. II. PRELIMINARIES AND PROBLEM FORMULATION Consder a networ of N moble agents, n the set Ω, wth unnown locatons, and M anchor(s), n the set κ, wth fxed nown locatons, all located n R ; let Θ = Ω κ be the set of all nodes. Let x R be a row vector that denotes the true locaton of the -th agent, Ω, at tme, where s the dscrete-tme ndex. Regardless of what moton modalty (aeral, ground, nematcs) s employed by the agents, ther moton can be expressed as the devaton between the current and next locatons,.e., x + = x + x +, Ω, () where x s the true moton vector at tme. We assume that the agents move along straght lnes, or otherwse the moton traectores can be approxmated by a pecewse lnear model. Ths s a common assumpton n robotcs lterature [], e.g., a smple dfferental wheeled robot whose wheels run n the same drecton and speed, wll move n a straght lne. We also assume that agent measures a nosy verson, x, of ths moton, e.g., by usng an accelerometer: x = x + n, () where n s the accelerometer nose at tme. We assume that the moton s restrcted n a bounded regon n R. We defne the dstance between any two nodes, and, measured at the tme of communcaton,, as. We assume that the dstance measurement, d, at node s not perfect and ncludes nose,.e., d = + r, () n whch r s the nose n the dstance measurement at tme. The problem s to fnd the locatons of the moble agents n the set Ω gven any ntalzaton of the underlyng algorthm. We then consder the mnmal number of anchors requred for such a process to wor. Man dea: We now descrbe the man dea behnd our approach. In order to avod nonlnearty n the soluton, we consder a barycentrc-based lnear representaton of postons; the advantage of a lnear approach s that the convergence s ndependent of the ntal locaton estmates. Let Θ () denote a set of three agents such that agent les nsde ther convex hull, C(Θ ()), at tme. In the barycentrc-representaton, the locaton of node s descrbed by a lnear-convex combnaton of the nodes n Θ (),.e., x = a + a + a, Θ () = {,, } () When only dstances to at least m+ anchors are nown n R m, trlateraton, e.g., n R, requres anchors and solves crcle equatons, []. Clearly, trlateraton s nonlnear and coupled n the coordnates; whle an teratve procedure bult on these nonlnear updates does not converge n general.

3 n whch x represents the true locaton of node at tme, and the coeffcents, a s, are the barycentrc coordnates, assocated to Möbus, [], gven by a = A Θ () {}\, () A Θ() where A Θ() {}\ denotes the area of the convex hull formed by the agents n the subscrpt set. To mplement ths update, agent has to le nsde the convex hull of the (three) neghbors n the set Θ (). Appendx F provdes a smple procedure, [], to test f an agent les nsde or outsde of the convex hull formed by the nearby nodes. Fnally, note that the barycentrc coordnates are always postve and they sum to. Challenges and our approach: To mplement the above barycentrc-based procedure, agent must acqure the mutual dstances among the nodes (n the set { Θ ()}). However, an agent may never fnd such a convex hull because the nodes are moble, n a regon possbly full of obstacles, wth lmted communcaton and/or vsual radus. To address ths challenge, we provde a geometrc approach that allows an agent to trac ts dstance to any locaton, where t has exchanged nformaton wth the other nodes n the past (Secton III-A). We further show that at the tme of communcaton an agent can compute the dstance between a neghbor and any such locatons (Secton III-B). Usng these methods, we ntroduce the noton of a vrtual convex hull, whch s a trangle whose vertces are located at the vrtual locatons where the agent exchanged nformaton n the past (Secton IV). The descrbed approach results n a rather opportunstc algorthm where an update occurs only when a certan set of real and vrtual condtons are satsfed. The subsequent analyss s to characterze the convergence of such LTV algorthms whose system matrces are non-determnstc. In ths paper, we rgorously develop the condtons of convergence and study the locatontracng performance of ths procedure (Secton V). We now enlst our assumptons: A: Each anchor, κ, nows ts locaton, x = x,. A: Each agent, Ω, has a nosy measurement, x, of ts moton vector, x, see Eq. (). A: Each agent, Ω, has a nosy measurement, d, of the dstances to nodes, Θ wthn a radus, r, see Eq. (). Under the above assumptons, we are nterested n fndng the true locatons of each agent n the set Ω, wthout the presence of any central coordnator. In the process, we study what s the mnmal number of anchors requred and n partcular: does moblty reduce the number of anchors from n R (Secton V). In the followng, we frst dscuss the deal scenaro when the moton and dstance measurements are not affected by nose. We then evaluate the performance of the algorthm n the presence of nose on the moton as well as on the dstance measurements, and present a modfed algorthm to counter the undesrable effects of nose (Secton VI). III. A GEOMETRIC APPROACH TOWARDS LOCALIZATION In ths secton, we consder the moton and dstance n the noseless case,.e., n = and r =, n Eqs. () and (). Let The mnmal number of anchors requred to solve a localzaton problem, wthout ambguty, gven only dstance measurements s m n R m, []. the true locaton of the -th agent, Ω n R, be decomposed as x = [x y ]. We then descrbe the moton of agent as: ( + ) x + = x + d + cos θ + + θ, y+ = y + d + sn = ( + ) θ + + θ, () = where d + and θ + denote the dstance and angle traveled by agent, between tme and +, and θ s agent s ntal orentaton, see Fg.. Let us also consder x n θ n x θ Fg.. Intal orentatons of N = agents; flled colored crcles and red trangle ndcate the ntal locatons of the agents,,, l, n and the anchor, A, respectvely; red crcle shows the communcaton radus of agent. a vson-based dstance measurement process, where an agent needs to see another agent wth ts camera n order to fnd the mutual dstance, as opposed to wreless-based dstance measurements that are prone to extremely large errors. When agent moves to a new locaton at tme, t frst scans the neghborhood,.e., rotates by an angle, β < π, n order to fnd (and mae vsual contact wth) another node (agent or anchor). If the agent does not fnd any neghbor at tme, then β =. If on the other hand, agent fnds another node wthn the communcaton radus, t exchanges nformaton wth that node, then maes another rotaton by an angle, α < π, whch s randomly chosen at θ x A x x l θ l each teraton, and travels the dstance of d + n the new drecton. Thus, the angle traveled by agent between tme and +, θ +, can be represented as θ + = α + β. (7) Before we proceed, note that under perfect moton and noseless dstance measurements, an agent cannot fnd ts poston by drect communcaton wth an anchor, unless t nows ts angle towards the anchor wth respect to a mutual frame; n contrast, the proposed framewor n ths paper assumes that only relatve angles are nown to each agent. For example, agent n Fg. ntally fnds anchor A n ts communcaton radus. However, n order to fnd ts exact locaton, agent needs to now θ, n addton to x A and. We assume that such angles are not avalable to the agents. In what follows, we frst explan the procedure to trac the dstance between an agent and the locatons where t has exchanged nformaton wth other nodes n the past. We Note that these ntal orentatons, θ s, are arbtrary and we nether assume any global synchronzaton nor we now what the true angles are. A

4 then show how an agent can compute the dstance between a neghbor and any such locaton at the tme of communcaton. A. Tracng the dstance after a drect communcaton Consder an agent, Ω, to fall wthn a dstance, r, of node, Θ, at some tme ; by Assumpton A, agent can measure ts dstance, to node, and receve -th node s locaton estmate, herenafter denoted as x. Once ths nformaton s acqured at tme, agent tracs the dstance to the true poston, x, for all, even when the two agents move apart. In other words, agent, has the followng nformaton for each node t has communcated wth: {,,, x },, Θ, (8) where s the nstant of the most-recent contact, and s the dstance between the true postons, x and x,. Ths procedure s llustrated n Fg.. Note that n order to (a) d β (b) α B. Fndng the dstance wth ndrect communcaton We now show how agent can use the dstance/angle nformaton to fnd the dstances between a neghbor and any vrtual locaton, where t has prevously communcated wth another node. Lemma. Suppose agent has prevously made contact wth node,, hence possesses the followng nformaton: {,,, x },, Suppose agent fnds a neghbor, say agent l, at tme l >. Agent can then fnd the dstance between x and x l l as follows: l l = ( l ) + ( l l ) l l l cos ( l, l l ), () n whch l ) s the angle between the two lnes connect- and x l and ( ng x l l l, to x l s the dstance between x l and x, + α + l. φ d + α φ d d l l d l (c) d + α d d + l + + (d) + β + x l + d + l φ l φ l d l β l Fg.. At tme l, agent fnds the dstance between a new neghbor, node l, and the vrtual locaton, x, where t exchanged nformaton wth node at tme < l. l l Fg.. Dstance tracng after a drect communcaton; agent s ndcated by red flled crcle; agents and l are represented by blue and green crcles, respectvely; red crcle ndcates the communcaton radus. mae contact wth agent, agent has changed ts orentaton by β at tme, see Fg. (a). After the two agents exchange nformaton, agent changes ts orentaton by α, Fg. (b), and travels the dstance of d + n the new drecton, Fg. (c). We now show how an agent tracs the dstance to a vrtual locaton, where t has exchanged nformaton wth another node n the past. Lemma. Consder agent, Ω, wth true poston, x. Suppose agent communcates wth node, Θ, at tme. Let be the dstance between and at the tme of communcaton. Suppose agent and move apart at tme +, where the moton s gven by d and + θ, + both nown to agent. The dstance between the true postons, x + and x, s ( + = ( ) + (d +) d + cos(α )). (9) See Appendx B for the proof. Note that at tme +, agent fnds agent l n ts communcaton radus, hence changes ts drecton by β + n order to mae contact wth agent l, see Fg (d). See Appendx C for the proof. In the next secton, we use these dstance tracng methods to descrbe the poston tracng algorthm. IV. DISTRIBUTED MOBILE LOCALIZATION: ALGORITHM Consder a networ of N agents wth unnown locatons and M anchors, accordng to the moton model ntroduced n Secton II. Let V () Θ be the -vsted set, defned as the set of dstnct nodes vsted by agent, Ω, up to tme ; and call an element n ths set as -vsted node. We start by ntroducng the noton of a vrtual convex hull. A. Vrtual convex hull Suppose agent communcates wth node at tme, and obtans the dstance, to, along wth s current locaton estmate, x,.e., V ( ). At any tme, >, agents, and, may move apart but agent now nows >, usng the geometrc framewor dscussed n Secton III-A. At some later tme, l >, agent maes contact wth another node, l, and thus obtans x l l l and l, and eeps l trac of, > l, thus, l V (), l. Usng the approach descrbed n Secton III-B, at tme l, agent

5 l q also computes l,.e., the dstance between x and x l l. Fnally, agent meets agent n at some n > l, and thus now possesses the locaton estmates: x, x l l, x n n ; and the dstances:, > n, wth q =, l, n (computed by Lemma ), l along wth the followng dstances: n, n nl n, and n (computed by Lemma ). At ths pont, l, n V (), n, and agent can use the dstances to perform the ncluson test, descrbed n Appendx F, to chec f the three vsted nodes forms a vrtual convex hull n whch agent les at tme. If the test s passed, the set, Θ () {, l, n} V (), forms the vrtual convex hull; otherwse, agent contnues to move and add nodes n V () untl some combnaton passes the convexty test. Note that the dstance between the agent and each of the vrtual locatons s updated every tme agent moves to a new locaton. However, the dstance between the vrtual locatons does not change unless agent revsts any of the prevously vsted nodes. Fg. (a) shows the traectores of four agents:,,,, over =,..., 9; the tme-ndces are mared nsde the agent symbols. From the perspectve of agent, see Fg. (b): t frst maes contact (communcates) wth agent, at tme =, and then they both move apart; next, t maes contact wth agents, at =, and at =. We have V () = { }, V () = {, }, and V () = {,, }, where a non-trval convex hull becomes avalable at =. However, agent does not le n the correspondng convex hull, C(V ()), and cannot update ts locaton estmate wth the past neghborng estmates: x, x, x. At ths pont, agent must wat untl t ether moves nsde the convex hull of,,, or fnds another agent wth whch the convexty condton s satsfed. Fg. llustrates ths process n four frames. The former s shown n Fg. (d), where agent has moved nsde C(V ()) at some later tme, = 9; we have Θ (9) = {,, }. The noton of a vrtual convex hull s evdent from ths dscusson: an agent may only communcate wth at most one agent at any gven tme; when the convexty condton s satsfed eventually, the updatng agent may not be n communcaton wth the correspondng nodes. Once Θ () s successfully formed, agent updates ts locaton lnearly usng the barycentrc representaton n Eq. (), where the coeffcents are computed usng the dstance equatons n Lemmas,, and the Cayley-Menger determnant to compute areas. After ths update, agent removes Θ () from V ( ), as the locaton estmates of the nodes n Θ () have been consumed. The followng result wll be useful n the sequel. Lemma. For each Ω, there exsts a set, Θ () V (), such that Θ () = and C(Θ ()), for nfntely many s. See Appendx D for the proof. We choose ths smple strategy to remove nformaton from V () for convenence. Another canddate strategy s to use a forgettng factor, whch chooses the past used nodes less frequently. B. Algorthm We now descrbe the localzaton algorthm n ths case accordng to the number, V (), of -vsted nodes n the - vsted set, V (). There are two dfferent update scenaros for any arbtrary agent, : () V () < : Agent,, does not update ts current locaton estmate. () V () : Agent,, performs the ncluson test; f the test s passed the locaton update s appled. Usng the above, consder the followng update: x + = α x + ( α ) Θ () a x + x +, () where x s the vector of the -th agent s coordnates at tme, x + s the moton vector, a s the barycentrc coordnate of node wth respect to the nodes Θ (), and α s such that {, Θ () =, α = () [β, ), Θ (), where β s a desgn parameter and Θ () s a vrtual convex hull. As we explan later, an updatng agent receves the valuable locaton nformaton only f t updates wth respect to an anchor, or another agent that has prevously communcated wth an anchor. The non-zero self-weghts assgned to the prevous state of the updatng agent guarantees that the agent does not completely forget the valuable nformaton after recevng them, e.g., by performng an update where none of the agents n the trangulaton set has prevously receved anchor nformaton. Note that Θ () = does not necessarly mply that agent has no neghbors at tme, but only that no set of neghbors meet the (vrtual) convexty. The above algorthm can be wrtten n matrx form as x + = P x + B u + x +, >, () where x s the vector of agent coordnates evaluated at tme, u s the vector of anchor coordnates at tme, and x + s the change n the locaton of agents at the begnnng of the th teraton accordng to the moton model. Also P and B are the system matrx and the nput matrx of the above LTV system. We denote the (, )-th element of the matrces, P and B, as (P ), and (B ),, respectvely. We can now rewrte Eq. () as x + = α x + ( α ) a, + ( α ) m Θ () κ Θ () Ω a m u m x + x +, () where u m denotes the nown coordnates of the m-th anchor at tme. Thus, (P ), = α as defned n Eq. (), and (P ), = ( α )a f Θ () Ω, (B ),m = ( α )a m f m Θ () κ, () Man steps of the algorthm are summarzed n Appendx E.

6 (a) (b) (c) (d) Fg.. Vrtual convex hull wth four agents:,,, ; wth respect to agent : (a) Agent traectores and tme-ndces; (b) at =, at =, at = ; crcles ndcate communcaton radus of agent ; (c) Vrtual convex hull of agents,,,, avalable at agent at = ; (d) Traectores at >, test passed at = 9. If the trangulaton set for agent at tme does not contan the -th agent or the m-th anchor, the correspondng elements n system and nput matrces, (P ), and (B ),m are zero. Note that Eq. () mmedately mples that the self-weght at each agent s always lower bounded,.e., < β (P ),,,. () Snce accurate nformaton s only nected va the anchors, t s reasonable (and necessary) to set a lower bound on the weghts assgned to the anchor states. In partcular, we mae the followng assumpton. A: Anchor contrbuton. For any update that nvolves an anchor,.e., for any (B ),m, we assume that < α (B ),m,, Ω, m Θ () κ, (7) where α s the mnmum anchor contrbuton, see Secton V-B. Assumpton A mples that f there s an anchor n the (vrtual) convex hull, t always contrbutes a certan amount of nformaton. In other words, the updatng agent has to le n an approprate poston nsde the convex hull. Assumpton A states that the weght assgned to the anchor (whch comes from barycentrc coordnates) should be at least α. Therefore, the area (n R ) of the trangle correspondng to the anchor must tae an adequate porton of the area of the whole convex hull trangle. Ths s llustrated n Fg., where the updatng agent les nsde a vrtual convex hull, consstng of an anchor (node ), and two other agents, m and l. Node has communcated wth the anchor and the two other agents, at tme nstants,, m, and l, respectvely. l x l x x m x m Fg.. (Left) At tme, agent s located on the threshold boundares, whch assgns the mnmum weght, α, to the anchor. (Rght) At tme agent s located n an napproprate locaton nsde the convex hull. l x l x x m x m Fg. (Left) llustrates the poston of agent at tme, x on the threshold boundares, such that (B ), = A Θ () {}\ A Θ() = α. (8) Fg. (Rght) on the other hand shows that f the agent les n x (or any other poston wthn the trangle of x, xl l, and x m m ), the left hand sde of Eq. (8) becomes less than α, and Eq. (7) does not hold. Snce the correspondng update does not provde enough valuable nformaton for agent, no update occurs n ths case. Wth the lower bounds on both the self-weghts, Eq. (), and the anchor weghts, Eq. (7), we note that at tme, the matrx of barycentrc coordnates wth respect to agents wth unnown locatons,.e., the system matrx P, s ether () dentty, when no update occurs; or, () dentty except a stochastc -th row, when there s no anchor n the vrtual convex hull,.e., Θ () κ = ; or, () dentty except a strctly sub-stochastc -th row, when there s at least one anchor n the vrtual convex hull 7. In the next secton, we provde suffcent condtons under whch the teratve localzaton algorthm, Eq. (), tracs the true agent locatons. Before we proceed, let us mae the followng defntons to clarfy what we mean by (strctly sub-) stochastcty throughout ths paper: Defnton. A non-negatve, stochastc matrx s such that all of ts row sums are one. A non-negatve, strctly sub-stochastc matrx s such that t has at least one row that sums to strctly less than one and every other row sums to at most one. V. DISTRIBUTED MOBILE LOCALIZATION: ANALYSIS In ths secton, we address the challenge on the analyss of LTV systems wth potentally non-determnstc system matrces. We borrow the followng result on the asymptotc stablty of LTV systems from []. A. Asymptotc stablty of LTV systems Consder an LTV system: x + = P x such that the system matrx, P, s tme-varyng and non-determnstc. The 7 Note that f an agent les nsde a vrtual convex hull of three anchors, then practcally by assgnng a zero weght on ts past, the agent can fnd ts exact locaton and tae the role of an anchor, whch could subsequently ncrease the convergence rate of the algorthm.

7 7 system matrx, P, represents at most one state update, say the -th state, for any,.e., at most one row,, of P s dfferent from dentty and can be ether stochastc or strctly sub-stochastc, not necessarly n any order and wth arbtrary elements as long as the bounds n (A), (B), and (B) n the followng are satsfed: B: If the updatng row,, n P sums to, then < β (P ),, β R, (9) B: If the updatng row,, n P does not sum to, then (P ), β <, β R. () To analyze the asymptotc behavor of an LTV system wth such system matrces, we utlze the noton of a slce, M, whch s the smallest product of consecutve system matrces, such that the entre chan of systems matrces s covered by non-overlappng slces,.e., t M t = P, and each slce has a subunt nfnty norm (maxmum row sum),.e., M t <, t. MM MM MM PP mm PP PP PP mm PP mm Fg.. Slce representaton. PP mm + PP mm + PP mm PP mm Slces are ntated by strctly sub-stochastc system matrces, and termnated after all row sums are strctly less than one. Slce representaton s depcted n Fg., where the rghtmost system matrces (encrcled n Fg. ) of each slce,.e., P, P m,..., P m..., are strctly sub-stochastc. The length of a slce s defned as the number of matrces formng the slce, and for the -th slce length we have M = m m, m =. Ref. [] also shows that the upper bound on the nfnty norm of a slce s further related to the length of the slce,.e., the number of matrces formng the slce. The followng theorem characterzes the asymptotc stablty of the above LTV system. Theorem. Wth Assumptons, B-B, the system, x + = P x, converges to zero,.e., lm x = N, f for every N, there exsts a subset, J, of slces such that ( ) M J : M ln (β ) ln e ( γ γ ) +, () β for some γ [, ], γ >. For any other slce, / J we have M <. The proof s avalable n our pror wor, []. Here, we explan the ntuton behnd the above theorem. The asymptotc stablty of the system s guaranteed f all (or and nfnte subset of) slces have bounded lengths. Theorem states that the system s asymptotcally stable even n the non-trval case, where there exst an nfnte subset of slces whose lengths are not bounded, but do not grow faster than the exponental growth n Eq. (). As detaled n [], f the lengths of an nfnte subset of slces follow Eq. (), the nfnte product of slces goes to a zero matrx, and the system s asymptotcally stable. Note that n the RHS of Eq. (), the frst ln s negatve; for the bound to reman meanngful, the second ln must also be negatve that requres β < e ( γ γ ), whch holds for any value of β [, ) by choosng an approprate γ >. Please see [] for a detaled dscusson on Eq. () and ts parameters. Wth the help of ths theorem, we now analyze the convergence of Eq. () n the followng secton. B. Convergence Analyss We now adapt the above LTV results to the dstrbuted localzaton setup descrbed n Secton II. Recall that the - th row of the system matrx, P n Secton V-A, collects the agent-to-agent barycentrc coordnates correspondng to agent and ts neghbors. We now relate the slce representaton n Theorem to the nformaton flow n the networ: Each slce, M, s ntated wth a strctly sub-stochastc update,.e., when one agent wth unnown locaton drectly receves nformaton from an anchor by havng ths anchor n ts vrtual convex hull. On the other hand, a slce, M, s termnated after the nformaton from the anchor(s) s propagated through the networ and reaches every agent ether drectly or ndrectly. Here, drectly means that an agent has an anchor n ts vrtual convex hull; whle ndrectly means that an agent has a neghbor n ts vrtual convex hull, whch has prevously receved the nformaton (ether drectly or ndrectly) from an anchor. Once the anchor nformaton reaches every agent n the networ, the slce noton and Theorem provde the condtons on the rate, Eq. (), at whch ths nformaton should propagate for convergence. We proceed wth the followng lemma. Lemma. Under the condtons A-A and no nose, the product of system matrces, P s, n the LTV system, Eq. (), converges to zero f the condton n Theorem holds. Proof. Frst, we need to show that the Assumptons B-B follow from A-A. Frst, note that B s mmedately verfed by Eq. (), assumng β = β. Next, we note that B s mpled by A. Ths s because f Θ κ s not empty, we can wrte (P ), = (B ),m, () Θ () Ω m Θ () κ where we used the fact that the barycentrc coordnates sum to one. When there s only one anchor among the nodes formng the vrtual convex hull, and the mnmum weght s assgned to ths anchor, Eq. (7), the rght hand sde of Eq. () s maxmzed. Ths provdes an upper bound on the -th row sum: (P ), α <, () Θ () Ω

8 8 whch ensures B wth β = α. In addton, Lemma ensures that each agent updates nfntely often wth dfferent neghbors. Subsequently, each slce s completed after all agents receve anchor nformaton (at least once) ether drectly or ndrectly, and the asymptotc convergence of Eq. () follows under the condtons n Theorem. Note that Assumpton B, whch s equvalent to Eq. (), mples that each agent remembers ts past nformaton. If a lower bound on the self-weghts s not assgned, an agent may lose valuable nformaton when t updates wth other agents that have not prevously updated (drectly or ndrectly) wth an anchor. On the other hand, Assumpton B restrcts the amount of unrelable nformaton added n the networ by the agents, when an anchor s nvolved n an update. The followng theorem completes the localzaton algorthm for moble mult-agent networs. Theorem. Under the Assumptons A-A and no nose, for any (random or determnstc) moton that satsfes Eq. (), Eq. () asymptotcally tracs the exact agent locatons. Proof. In order to show the convergence to the true locatons, we show that the error between the locaton estmate, x, and the true locaton,, goes to zero. To fnd the error dynamcs, note that the true agent locatons follow: + = P + B u + x +. () Subtractng Eq. () from Eq. (), we get the networ error e + + x + = P ( x ) = P e, () whch goes to zero when from Lemma. lm l= P l = N N, () Before we proceed, t s reasonable to comment on the choce of parameter α; The proposed localzaton algorthm s proved, both n theory and smulatons, to converge for any value of < α <. However, the convergence rate of the algorthm s affected by the choce of α as follows. Choosng α arbtrarly close to zero corresponds to an nfntely large upper-bound on the length of a slce (ths can be verfed by replacng α = β by zero on the rght hand sde of Eq. (). On the other hand, by settng α arbtrarly close to one, an agent has to get arbtrarly close to an anchor n order to perform an update wth respect to the anchor (see Fg. ), whch agan leads to arbtrary large number of teratons requred for each slce to complete. In summary, there s a trade-off n the choce of α, between recevng more nformaton from an anchor at the tme of an update, whch s the case for α values closer to, and ncreasng the chance of an update wth an anchor, whch requres for α to be closer to. A proper choce can be made by consderng the moton model, the communcaton protocol, and the number of avalable anchors n the networ. The followng theorem characterzes the number of anchors. Theorem. Under Assumptons A-A and no nose, Eq. () tracs the true agent locatons n R, when the number of agents, N, and the number of anchors, M, follow: M, (7) N + M. (8) Proof. Let us frst consder the requrement of at least one anchor. Wthout an anchor, a strctly sub-stochastc row never appears n P, mang P stochastc at each tme and hence the nfnte product of P s s also stochastc and not zero. Wth at least one anchor, strctly sub-stochastc rows, followng Eq. (), appear n P s, and zero convergence of the error dynamcs, Eq. (), follows. Next, exactly nodes (agents and/or anchors) are requred to form a (vrtual) convex hull n R. Thus, when the total number of nodes, N + M, s or less, no agent can fnd other nodes to fnd (vrtual) convex hulls. Usng the same argument as the one n the proof of Lemma, we can show that wth at least one anchor and at least total nodes, any agent wth unnown locaton nfntely fnds tself n arbtrary (vrtual) convex hulls. Thus, Theorem s applcable and the proof follows. VI. MOBILE LOCALIZATION UNDER IMPERFECT MEASUREMENTS The nose on dstance measurements and moton degrades the performance of the localzaton algorthm, as expected, and n certan cases the locaton error s as large as the regon of moton; ths s shown expermentally n Secton VII-B. In what follows, we dscuss the modfcatons, M-M, to the proposed algorthm to address the nose n case of moton, x, and on the dstance measurements, d. A. Agent contrbuton If an agent s located close to the boundares of a (vrtual) convex hull, nose on dstance measurements may lead to ncorrect ncluson test, Appendx F, results,.e., a set of vsted neghbors may fal the convexty test whle the agent s ndeed located nsde the convex hull, or vce versa. To avod such scenaros, we generalze Assumpton A to non-anchor agents. In partcular, we restrct an agent from performng an update, unless t s located n a proper poston nsde a canddate (vrtual) convex hull. We augment Assumpton A by the followng: M: For any agent, Ω, nvolved n an update,.e., for any (P ),, we assume that < α (P ),,, Ω, Θ () Ω, (9) where s the updatng agent s ndex, and α s the mnmum agent contrbuton. In other words, for agent to perform an update, the area of the correspondng nner trangle has to consttute a mnmum proporton, α, of the area of the convex hull trangle. Ths s shown n Fg. 7 (Left); At tme, agent s located n the vrtual convex hull of the nodes, m and l, such that the rato of the area of the shaded trangle to the area of the convex hull trangle s α. Agent can perform an update at tme. The upper sde of the shaded trapezod n

9 9 l x l x x m x m Fg. 7. (Left) Agent s located on the threshold boundares, whch assgns the mnmum weght, α, to the anchor. (Rght) Agent s located n an napproprate locaton nsde the convex hull. Fg. 7 (Rght) s the threshold boundary. If agent remans n the same vrtual convex hull, but moves nsde the shaded trapezod, e.g., to x at tme, Eq. (9) wll not hold and no update wll occur at that tme. Clearly, Assumpton M can be relaxed f the nose s neglgble. Otherwse, the value of α can be adusted accordng to the amount of nose on dstance measurements. B. Incluson test error Even f Assumpton M s satsfed, the ncluson test results may not be accurate due to the nose on the moton, whch corresponds to mperfect locaton updates at each and every teraton. To tacle ths ssue, we propose the followng modfcaton to the algorthm: M: If the ncluson test s passed at tme by a trangulaton set, Θ (), agent performs an update only f l x l ɛ = Θ A () Θ () {}\ A Θ() ɛ, A Θ() () n whch ɛ s the ncluson test relatve error for agent at tme, and ɛ s a desgn parameter. C. Convexty Fnally, recall that f the ncluson test s passed at tme, by a trangulaton set, Θ (), the updatng agent,, updates ts locaton estmate, x, as a convex combnaton of the locaton estmates of the nodes n Θ (). In order to guarantee the convexty n the updates n presence of nose, we consder the followng modfcaton to the algorthm: M: If the ncluson test s passed at tme by a trangulaton set, Θ (), and Eq. () holds at tme, agent (randomly) chooses one of the -vsted nodes from the trangulaton set Θ (), say agent, and fnds the correspondng weght, a, as follows: assumng that am + an <, a = am x x a n, {, m, n} Θ (). () VII. SIMULATIONS In ths secton, we frst provde smulaton n noseless scenaros and then examne the effects of nose. m x m A. Localzaton wthout nose We now consder the noseless scenaros,.e., we assume that n = and r =, n Eqs. () and (). In the begnnng, all nodes (agents and anchor(s)) are randomly deployed wthn the regon of x [ ], y [ ] n R. For the smulatons, we consder random waypont moblty model, [], for the moton n the networ, whch has been the predomnant moton model n the localzaton lterature over the past decade, see e.g., [7] [], [7]. We note that better performance may be acheved f the agents follow some determnstc (msson-related) moton model. However, our localzaton algorthm converges f the moton s such that the slce lengths grow slower than the exponental rate n Eq. (). For all moble nodes, Θ, we chose d + and θ + n Eq. () as random varables wth unform dstrbutons over the ntervals of [ ] and [ π], respectvely. Each agent can only communcate wth the nodes wthn ts communcaton radus, whch s set to r = for all smulatons. For each smulaton, we assume exactly one fxed anchor,.e., M =. In all smulatons, we set α =. to ensure that the agents do not completely forget the past nformaton and α =. to guarantee an adequate contrbuton from the anchor(s). Fg. 8 (Left) shows the random traectores of N = moble agents and M = anchor for the frst teratons. To characterze the convergence, we choose the -norm of the error vector normalzed wth respect to the dmensons of the regon as follows: ( e = (x ) + x ( y y ) ), () y n whch x = and y = denote the lengths of the regon, whch s a square n ths case. The dvson by x and y s done to obtan a normalzed error. It can be verfed that the maxmum error wth ths normalzaton s. The algorthm converges when e. As shown n Fg. 8 (Rght), the localzaton algorthm, Eq. (), tracs the true agent locatons. The smulaton results for a networ of y x e agents, anchor 8 Fg. 8. A networ of moble agents and anchor; (Left) Moton model; (Rght) Convergence; red trangle ndcates anchor and flled crcles show the ntal postons of the agents. and moble agents and one moble anchor s llustrated n Fg. 9 (Left) and (Rght), respectvely. Before we examne the effects of nose on the proposed localzaton algorthm, let us study the case where there s no anchor n the networ,.e., Eq (7) s not satsfed; In most

10 e agents, anchor e agents, anchor! normalzed localzaton error s reduced to less than % for one smulaton. e agents, anchor e..... agents, anchor Fg. 9. Convergence of a networ wth one anchor; (Left) moble agents (Rght) moble agents. 8 8 localzaton algorthms, m + anchors are requred to localze an agent wth unnown locaton n R m wthout ambguty. However, when the nodes are moble, exactly one anchor s suffcent to nect valuable nformaton and the moton of the agents provdes the remanng degrees of freedom. When there s no anchor n the networ, all updates are stochastc, and we get e + = P e, wth ρ(p ) =,, where ρ(.) denotes the spectral radus,.e., a neutrally stable system, whch leads to a bounded steady-state error n locaton estmates. In other words, relatve locatons are traced because the moton removes certan ambguty from the locatons. However, absolute tracng wthout any ambguty requres at least one anchor. Ths s llustrated n Fg. for a networ of M = agents and no anchor. Fg.. Effect of nose on the convergence of a networ of moble agents and moble anchor wth ±% nose on dstance measurements and ±% nose on the moton; (Left) Orgnal algorthm (Rght) Modfed algorthm. Fnally, n Fg. we provde Monte Carlo smulatons for a networ of M = moble anchor and N = and N = moble agents n presence of nose. max( e ) agents, anchor, Monte Carlo trals max( e ) agents, anchor, Monte Carlo trals e.... agents, anchor x Fg.. Monte Carlo trals; ±% nose on dstance measurements, ±% nose on the moton; (Left) N =, M = (Rght) N =, M =.. x 8 Fg.. Steady state error for a networ of moble agents and no anchor; (Left) -norm (Rght) Red and blue curves represent the exact and estmated values of x coordnate, for the last teratons of a smulaton. B. Localzaton wth nose To examne the effects of nose on the proposed localzaton algorthm, we let the nose on the moton of agent Θ at tme, n to be ±% of the magntude of the moton vector, x, and the nose on dstance measurements, r to be ±% of the actual dstance between nodes and, d. As shown n Fg. (Left) ths amount of nose leads to an unbounded error, whch s due to ncorrect ncluson test results and the contnuous locaton drfts because of the nose on the dstance measurements and the nose on moton, respectvely. However, by modfyng the algorthm accordng to Secton VI, t can be seen n Fg. (Rght) that the C. Performance evaluaton We now evaluate the performance of our algorthm n contrast wth some well-nown localzaton methods; MCL [8], MSL* [], Dual MCL [], and Range-based SMCL [9]. Please refer bac to Secton I for a bref descrpton of these methods. In Fg., we compare the localzaton error n the Vrtual Convex Hull (VCH) algorthm wth MCL, MSL*, and Range-based SMC. As shown n Table I, n these algorthms node densty, n d, and anchor (seed) densty, n s, denote as the average number of nodes and anchors n the neghborhood of an agent, respectvely. Total number of agents and anchors can therefore be determned by nowng these denstes as well as the area of the regon. We consder N = agents and M = anchors, and use the same metrc as used n [8] [],.e., the locaton error as a percentage of the communcaton range. Each data pont n Fg. s computed by averagng the results of smulaton experments. We eep the other parameters the same as descrbed earler n ths secton. Wth hgh measurement nose,.e., n the presence of % nose on the range measurements and % nose on the moton, our algorthm outperforms MCL, MSL* and Range-based

11 Mean Error (%r). MSL* MCL Range-based SMCL Proposed VCH algorthm n presence of nose and provde modfcatons to the proposed algorthm to address nose on moton and on dstance measurements.. APPENDIX A TABLE OF PARAMETERS Table II summarzes the mportant notaton. Fg.. Accuracy comparson Tme SMCL after teratons. Clearly, the localzaton error n our algorthm decreases as the amount of nose decreases, and our algorthm converges to the exact agent locatons n the absence of nose. Table I summarzes the performance of the proposed VCH algorthm n comparson to the above methods. Localzaton algorthm Smulaton envronment, Networ sze, Communcaton range Average error %r MCL *, n d =, n s =, r =. -. Dual MCL *, n d =, n s =, r =. -. MSL* *, n d =, n s =, r =. -. Range-based SMCL *, n d =.9, n s =., r =.. -. Proposed VCH *, N=, M=, r= <. TABLE I COMPARATIVE PERFORMANCE OF LOCALIZATION ALGORITHMS VIII. CONCLUSIONS In ths paper, we provde a dstrbuted algorthm to localze a networ of moble agents movng n a bounded regon of nterest. Assumng that each agent nows a nosy verson of ts moton and the dstances to the nodes n ts communcaton radus, we provde a geometrc framewor, whch allows an agent to eep trac of the dstances to any prevously vsted nodes, and fnd the dstance between a neghbor and any vrtual locaton where t has exchanged nformaton wth other nodes n the past. Snce agents are moble, they may not be able to fnd neghbors to perform dstrbuted updates at any tme. To avod ths ssue, we ntroduce the noton of a vrtual convex hull, whch forms the bass of a range-based localzaton algorthm n moble networs. We abstract the algorthm as an LTV system wth (sub-) stochastc matrces, and show that t converges to the true locatons of agents under some mld regularty condtons on update weghts. We further show that exactly one anchor suffces to localze an arbtrary number of moble agents when each agent s able to fnd approprate (vrtual) convex hulls. We evaluate the performance of the APPENDIX B PROOF OF LEMMA Proof. Consder the trangle, whose vertces are at x, x + and x. To fnd the current dstance of agent, from the poston of agent at tme, we can use the law of cosnes, whch connects the length of an unnown sde of a trangle to the lengths of the two other sdes and the angle opposte to the unnown sde. Ths trangle s depcted n Fg. (c), n whch the two nown sde lengths are and d +. Thus, nowng the angle between these sdes of the trangle, α +, the length of the thrd sde can be determned accordng to the followng ( + ) = ( ) + (d + ) () d + cos(α ), whch corresponds to Eq. (9), and completes the proof. APPENDIX C PROOF OF LEMMA Proof. We llustrate ths procedure n Fg., where all nown/measured dstances and angles are dstngushed from the unnowns by bold lnes and bold arcs, respectvely. After agent maes contact and exchange nformaton wth node at tme, t moves the dstance of d + to the new locaton, x +. At ths pont, agent can (use the law of cosnes to) fnd + as a functon of, d, + and α, whch are all nown to ths agent,.e., + = f(, d, + α ). Knowng +, agent can then fnd φ = g(, d +, + ). Note that gven the three sdes of a trangle, each angle can be computed. Snce agent cannot fnd any neghbor at tme +, we have β = π. + Agent then changes ts drecton by α + and travels the dstance of d + l to the new locaton, x l, where l = +. At ths pont, agent can fnd l = f(, + d + l, φ ), note that φ = π α + φ +. Agent can then fnd φ = g( l, d + l, ) at tme + l. Snce agent fnds node l wthn the communcaton radus at tme l, t changes ts drecton by β l n order to mae a contact wth agent l. Fnally, nowng l l l, l, and, φ = ( l, l ) = π φ β l, agent can fnd the dstance between x and x l l accordng to Eq. ().

12 TABLE II PARAMETERS AND DESCRIPTIONS; RANDOM PARAMETERS ARE DISTINGUISHED BY. x l les nsde the correspondng vrtual trangle. The probablty of such event at any gven tme, >, s gven by P( C(x, x m, x l )) = area( (x, x m, x l ) n, whch corresponds to a non-zero probablty for agent to fnd a trangulaton set. Fg.. Exstence of a vrtual convex hull; Agent can perform an update f t les nsde the blue trangle at any tme >. APPENDIX D PROOF OF LEMMA Proof. Consder a networ of agents, {,, m, l}, and let us focus on fndng a trangulaton set for agent. Suppose all agents are movng n an n by n regon, and the communcaton radus for each agent s r. In order for agent to communcate wth another agent, say, at tme, agent has to le nsde a crcle wth radus r centered at agent s locaton, x, see Fg.. The probablty of such event at tme, s therefore gven by P( V ( )) = πr 8 n. Smlarly, the probablty that agent communcates wth agent m at a later tme, >, and wth agent l, at tme > > can be gven by P(m V ( )) = πr n and P(l V ( )) = πr n, respectvely. Note that the probablty of agent m beng exactly at x at tme s zero, and therefore we can draw a vrtual lne (the dotted lne n Fg. ) between the locatons at whch agent has communcated wth agents and m,.e., between x and x m. Smlarly the probablty of agent l lyng on the aforementoned lne at tme s zero. Thus, the locatons where agent can meet the other three agents at dfferent tmes wth nonzero probablty, form a vrtual convex hull. Therefore, the three agents, m and l can pass the ncluson test for agent at any gven tme >, f agent 8 Ths expresson assumes that agents are unformly dstrbuted over the regon at any gven tme. Ths assumpton can be ustfed by consderng random ntal deployment and moton. However, the proof follows as long as there s a nonzero probablty for each agent to communcate wth other agent(s) n the networ. APPENDIX E PSEUDOCODE Algorthm Localze N agents n R n the presence of M anchors wth precson p Requre: M and N + M x random ntal coordnates for = to N do V () = end for whle < termnaton crteron 9 do + for = to N do V () nodes n the communcaton radus of agent at tme f V () < then do not update else perform ncluson test on (all possble combnatons of) neghbors f no trangulaton set found then do not update else update locaton accordng to Eq. () end f end f end for end whle 9 The termnaton crteron can be desgned accordng to the number of teratons typcally needed gven the sze, moblty, models, and nose parameters, as evdent from the smulaton fgures n Secton VII.