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# Rob ust Distrib uted Netw ork Localization with Noisy Rang Measurements Da vid Moore John Leonard Daniela Rus Seth eller MIT Computer Science and Ar ticial Intelligence Labor ator The Stata Center a PDF document - DocSlides

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### Presentations text content in Rob ust Distrib uted Netw ork Localization with Noisy Rang Measurements Da vid Moore John Leonard Daniela Rus Seth eller MIT Computer Science and Ar ticial Intelligence Labor ator The Stata Center a

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Rob ust Distrib uted Netw ork Localization with Noisy Rang Measurements Da vid Moore John Leonard Daniela Rus Seth eller MIT Computer Science and Ar tiﬁcial Intelligence Labor ator The Stata Center 32 assar Street Cambr idge MA 02139 dcm, jleonard, us teller @csail.mit.edu ABSTRA CT This pap er describ es distributed, linear-time algorithm for lo calizing sensor net ork no des in the presence of range mea- suremen noise and demonstrates the algorithm on ph ysi- cal net ork. in tro duce the probabilistic notion of obust quadrilater als as to oid ip am biguities that other- wise corrupt lo calization computations. form ulate the lo calization problem as o-dimensional graph realization problem: giv en planar graph with appro ximately kno wn edge lengths, reco er the Euclidean osition of eac ertex up to global rotation and translation. This form ulation is applicable to the lo calization of sensor net orks in whic eac no de can estimate the distance to eac of its neigh ors, but no absolute osition reference suc as GPS or xed an- hor no des is ailable. implemen ted the algorithm on ph ysical sensor net- ork and empirically assessed its accuracy and erformance. Also, in sim ulation, demonstrate that the algorithm scales to large net orks and handles real-w orld deplo ymen geome- tries. Finally sho ho the algorithm supp orts lo caliza- tion of mobile no des. Categories and Subject Descriptors C.2.4 Computer-Comm unications Net orks ]: Distrib- uted Systems; C.3 Sp ecial-Purp ose and Application- Based Systems ]: Real-time and em edded systems General erms Algorithms, Design, Exp erimen tation eyw ords Sensor net orks, lo calization, mobilit lo cation-a areness, trac king, erv asiv computing Permission to mak digital or hard copies of all or part of this ork for personal or classroom use is granted without fee pro vided that copies are not made or distrib uted for proﬁt or commercial adv antage and that copies bear this notice and the full citation on the ﬁrst page. cop otherwise, to republish, to post on serv ers or to redistrib ute to lists, requires prior speciﬁc permission and/or fee. SenSys’04, No ember 3–5, 2004, Baltimore, Maryland, USA. Cop yright 2004 CM 1-58113-879-2/04/0011 ... 5.00. 1. INTR ODUCTION This pap er describ es distributed algorithm for lo calizing no des in sensor net ork in whic no des ha the abilit to estimate distance to nearb no des, but suc measure- men ts are corrupted noise. Lo calization is an essen tial to ol for the dev elopmen of lo w-cost sensor net orks for use in lo cation-a are applications and ubiquitous net orking [5, 23]. Distributed computation and robustness in the pres- ence of measuremen noise are ey ingredien ts for practi- cal lo calization algorithm that will giv reliable results er large scale net ork. form ulate this as the follo wing o-dimensional graph realization problem: giv en planar graph with edges of kno wn length, reco er the Euclidean osition of eac ertex up to global rotation and transla- tion. This is dicult problem for sev eral reasons. First, there is often insucien data to compute unique osition assignmen for all no des. Second, distance measuremen ts are noisy comp ounding the eects of insucien data and creating additional uncertain Another problem is lac of absolute reference oin ts or anc hor no des whic could pro vide starting oin for lo calization. Finally it is di- cult to devise algorithms that scale linearly with the size of the net ork, esp ecially if data ust broadcast through the limited comm unications capacit of wireless hannel. presen distributed lo calization algorithm that gets around these diculties lo calizing no des in regions con- structed from obust quadrilater als term that formally dene in Section 2. Lo calization based on robust quads attempts to prev en incorrect realizations of ip am bigui- ties that ould otherwise corrupt lo calization computations. urthermore, sho that the criteria for quadrilateral ro- bustness can adjusted to cop with arbitrary amoun ts of measuremen noise in the system. The dra wbac of our approac is that under conditions of lo no de connectivit or high measuremen noise, the algorithm ma unable to lo calize useful um er of no des. Ho ev er, for man applications, missing lo calization information for kno wn set of no des is preferen tial to incorrect information for an unkno wn set. general result of our sim ulations is that ev en as noise go es to zero, no des in large net orks ust ha degree 10 or more on erage to ac hiev 100% lo calization. high lev el, our net ork lo calization algorithm orks use the term anchor no de to refer to no de that has prior kno wledge of its absolute osition, either man ual initialization or an outside reference suc as GPS. This yp of no de is also called ac on

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PSfrag replacemen ts Figure 1: An example run of our algorithm to es- timate the relativ ositions of no de A’s neigh ors. No des ABCD form obust quad ecause their real- ization is unam biguous ev en in the presence of noise. select as the origin of lo cal co ordinate system and ho ose ositions for B, C, and that satisfy the six distance constrain ts. In the next step, no de is lo calized relativ to the kno wn ositions of ABD us- ing trilateration. This lo calization is unam biguous ecause ABDE also forms robust quadrilateral. Con tin uing, the same pro cedure is used to lo calize no de whic is part of the robust quad ADFE. as follo ws. Eac no de measures distances to neigh oring no des, then shares these measuremen ts with the neigh ors. This \one-hop" information is sucien for eac no de to lo- calize itself and its neigh ors, whic call cluster, in some lo cal co ordinate system. Co ordinate transforms can then computed et een erlapping clusters to stitc them in to global co ordinate system. Suc stitc hing can done in an on-line fashion as messages are routed through the net ork rather than attempting to solv for the full lo calization up fron t. Similar cluster-based approac hes ha een prop osed efore, but often suer from gross lo calization errors due to graph ips that can comp ound er larger distances. Our no el use of robust quadrilaterals ensures that cluster-based lo calization do es not suer from suc errors. Figure depicts an illustrativ run of our algorithm. nd all sets of four no des that are fully connected distance measuremen ts and are \w ell-spaced" suc that ev en in the presence of measuremen noise, their relativ ositions are unam biguous. adopt this quadrilateral as the smallest subgraph that can denitiv ely realized, and dene it as obust quad Additional robust quads can \c hained" to the original quad if they ha no des in common with it. This approac allo ws eac hained quad to lo calize its fourth no de based on the kno wn ositions common to the quads using the standard tec hnique of trilater ation [4, 12]. This use of robust quadrilaterals enables our algorithm to tolerate noise computing unique realizations for graphs that migh otherwise am biguous. Our algorithm has the follo wing haracteristics: 1. It supp orts noisy distance measuremen ts, and is de- signed sp ecically to robust under suc conditions. 2. It is fully distributed, requiring no eacons or anc hors. 3. It lo calizes eac no de correctly with high probabilit or not at all. Th us, rather than pro duce net ork with an incorrect la out, an no des with am biguous lo cations are not used as building blo ks for further lo calization. 4. Cluster-based lo calization supp orts dynamic no de in- sertion and mobilit 1.1 Related ork Eren et al. in [4] pro vide theoretical foundation for net- ork lo calization in terms of graph rigidit theory They sho that net ork has unique lo calization if and only if its underlying graph is generic al ly glob al ly rigid In ad- dition, they sho that certain sub class of globally rigid graphs, trilater ation gr aphs can constructed and lo cal- ized in linear time. tak global rigidit and trilateration graphs one step further with robust quadrilaterals that pro- vide unam biguous lo calizations and tolerate measuremen noise. In [18], Sa vvides et al. deriv the Cram er-Rao lo er ound (CRLB) for net ork lo calization, expressing the exp ected error haracteristics for an ideal algorithm, and compare it to the actual error in an algorithm based on ultilateration. They dra the imp ortan conclusion that error in tro duced the algorithm is just as imp ortan as measuremen error in assessing end-to-end lo calization accuracy In [13] and [12], Niculescu and Nath also apply the CRLB to few gen- eral classes of lo calization algorithms. Their \Euclide an metho is similar to our metho of cluster lo calization in that it dep ends on the trilateration primitiv e. They also state the relev ance of four-no de quadrilaterals. In their case, the quads are constrained with v distance measuremen ts the sixth is computed based on the rst v e. Flip am bi- guities are resolv ed using additional information from neigh- oring no des. Their \D V-c or dinate" propagation metho presen ted in [12] is similar to our metho in that clusters consisting of no de and its one-hop neigh ors are rst lo- calized in lo cal co ordinate systems. Registration is then used to compute the transformations et een neigh oring co or- dinate systems. This idea of lo cal clusters as also prop osed Capkun et al. in [2]. Ho ev er, neither algorithm consid- ers ho measuremen noise can cause incorrect realization of ip am biguit ariet of other researc attempts to solv the lo cal- ization problem using some form of global optimization. Dohert et al. describ ed metho using connectivit con- strain ts and con ex optimization when some um er of ea- con no des are initialized with kno wn ositions [3]. Ji and Zha use ultidimensional scaling (MDS) to erform dis- tributed optimization that is more toleran of anisotropic net ork top ology and complex terrain [9]. Priy an tha et al. eliminate the dep endence on anc hor no des using comm u- nication hops to estimate the net ork’s global la out, then using force-based relaxation to optimize this la out [15]. Other previous ork is based on propagation of lo cation information from kno wn reference no des. Bulusu et al. and Simic et al. prop ose distributed algorithms for lo calization of lo er devices based on connectivit [1, 21]. Other tec hniques use distributed propagation of lo cation informa- tion using ultilateration [11, 19]. Sa arese et al. use o-phase approac using connectivit for initial osition estimates and trilateration for osition renemen [17]. at- ari et al. use one-hop ultilateration from reference no des in ph ysical exp erimen using oth receiv ed signal strength (RSS) and time of arriv al (T oA) [14]. Grab wski and Khosla maximize lik eliho estimator to lo calize small team of rob ots, ac hieving some robustness including motion mo del in their optimization [6]. In this pap er mak sev eral departures from previous researc h. Most imp ortan tly no previous algorithm consid- ers the ossibilit of ip am biguities during trilateration

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(a) Ground truth (b) Alternate realization 3.0 3.0 er 37 er 34 Figure 2: Example graph sho wing (a) true ertex ositions and (b) an alternate realization of the graph in whic in ter-v ertex distances, depicted as lines, are preserv ed almost exactly The error met- ric er is sho wn elo eac realization, with in ter- ertex distances generated from Gaussian distri- bution with mean of the true distance and 35 Th us, in this example, an incorrect realization of the graph ts the constrain ts etter than the ground truth, sho casing wh net ork lo calization is di- cult. due to measuremen noise. Although the requiremen of global rigidit as means to oid ips has een ell estab- lished [4], the eects of measuremen noise on global rigidit are not ell understo d. Our notion of robust quadrilaterals minimizes the probabilit of realizing ip am biguit incor- rectly due to measuremen noise. Error propagation during trilateration is deriv ed in [13], but the oten tial for signi- can error due to ips is not considered. Secondly lik [2] and [12], do not require anc hor no des, enabling lo caliza- tion of net orks without absolute osition information. This haracteristic is imp ortan for lo calization in homogeneous ad-ho net orks, where an no de ma ecome mobile. ur- thermore, man ual eacon initialization can error-prone or imp ossible, for example, in sensor net ork deplo ed mobile rob ot. 1.2 Challenges of netw ork localization The diculties inheren in lo calization can easily demon- strated with an example. Consider the follo wing metric that haracterizes the error for giv en lo calization, er =1 (1) where is the um er of distance measuremen ts, is eac distance computed from the lo calized ositions, and is eac measured distance. Without ground truth, er tells us ho ell computed lo calization ts the constrain ts. Fig- ure sho ws wh minimizing this error metric is insucien for lo calization. In this example, ossible realizations sho wn for the net ork ha similar alues for er but the ground truth actually has higher error than an incorrect realization. This demonstrates the need for an algorithm that appropriately handles no des with am biguous ositions refusing to assign osition to an no de that has more than one ossible lo cus for its osition. no formally address wh an algorithm based primarily on umerical op- timization of the distance constrain ts fails. In graph theory the problem of nding Euclidean osi- tions for the ertices of graph is kno wn as the gr aph e- alization pr oblem Saxe sho ed that nding realization (a) Flip am biguit (b) Discon tin uous ex am biguit PSfrag replacemen ts Figure 3: (a) Flip am biguit ertex can re- ected across the line connecting and with no hange in the distance constrain ts. (b) Discon tin- uous ex am biguit If edge AD is remo ed, then reinserted, the graph can ex in the direction of the arro w, taking on dieren conguration but ex- actly preserving all distance constrain ts. is strongly NP-hard for the o-dimensional case or higher [20]. Ho ev er, kno wing the length of eac graph edge do es not guaran tee unique realization, ecause deformations can exist in the graph structure that preserv edge lengths but hange ertex ositions. Rigidit theory distinguishes et een non-rigid and rigid graphs. Non-rigid graphs can con tin uously deformed to pro duce an innite um er of dieren realizations, while rigid graphs cannot. Ho ev er, in rigid graphs, there are yp es of disc ontinuous defor- mations that can prev en realization from eing unique [7]: 1. Flip ambiguities (Figure 3a) ccur for graph in dimensional space when the ositions of all neigh ors of some ertex span 1)-dimensional subspace. In this case, the neigh ors create mirror through whic the ertex can reected. 2. Disc ontinuous ex ambiguities (Figure 3b) ccur when the remo al of one edge will allo part of the graph to exed to dieren conguration and the remo ed edge reinserted with the same length. This yp of deformation is distinct from ontinuous ex ambigui- ties whic are presen only in non-rigid graphs. In the remainder of this pap er, use \ex am biguit y" to mean the discon tin uous yp e. Graph theory suggests ys of testing if giv en graph has unique realization determining whether or not ip or ex am biguities are presen in sp ecic graph. Ho ev er, neither these sp cic tests nor the gener al principles of gr aph the or etic rigidity apply to the gr aph alization pr oblem when distanc me asur ements ar noisy Since realizations of the graph will rarely satisfy the distance constrain ts exactly al- ternativ realizations can exist that satisfy the constrain ts as ell as or etter than the correct realization, ev en when the graph is rigid in the graph theoretic sense. In this situ- ation, assuming can mo del the measuremen noise as random pro cess, it is desirable to lo calize only those ertices that ha small pr ob ability of eing sub ject to ip or ex am biguit Our algorithm uses robust quadrilaterals as building blo for lo calization, adding an additional constrain e- ond graph rigidit This constrain ermits lo calization of only those no des whic ha high lik eliho of unam bigu- ous realization. presen the algorithm itself in the next section, then justify it deriving the orst-case error lik e- liho in Section 3.

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2. APPR CH or simplicit describ o-dimensional lo calization. Ho ev er, our algorithm extends straigh tforw ardly to three dimensions. dene no de’s neighb ors to those no des that ha direct bidirectional comm unications and ranging capabilit to it. Dep ending on the yp of ranging mec ha- nism used the net ork, these conditions ma alw ys satised together. cluster is no de and its set of neigh- ors. The algorithm can brok en do wn in to three main phases. The rst phase lo calizes clusters in to lo cal co ordinate sys- tems. The optional second phase renes the lo calization of the clusters. The third phase computes co ordinate transfor- mations et een these lo cal co ordinate systems. When all three phases are complete, an lo cal co ordinate system can reconciled in to unique global co ordinate system. Alter- nativ ely the transformation et een an connected pair of clusters can computed on-line haining the individual cluster transformations as messages are passed through the net ork. The three phases of the algorithm are as follo ws: Phase I. Cluster Localiza tion Eac no de ecomes the cen ter of cluster and estimates the relativ lo cation of its neigh ors whic can unam biguously lo calized. call this pro cess cluster lo alization or eac clus- ter, iden tify all robust quadrilaterals and nd the largest subgraph comp osed solely of erlapping ro- bust quads. This subgraph is also trilateration graph as in [4]; our restriction to robust quads pro vides an additional constrain that minimizes the probabilit of realizing ip am biguit osition estimates within the cluster can then computed incremen tally follo wing the hain of quadrilaterals and trilaterating along the as in Figure 1. Phase I. Cluster Optimiza tion (optional) Rene the osition estimates for eac cluster using umerical op- timization suc as spring relaxation or Newton-Raphson with the full set of measured distance constrain ts. This phase reduces and redistributes an accum ulated er- ror that results from the incremen tal approac used in the rst phase. It can omitted when maxim um e- ciency is desired. Note that this optimization imp oses no comm unications erhead since it is erformed er cluster and not the net ork as whole. Phase I. Cluster Transf orma tion Compute transfor- mations et een the lo cal co ordinate systems of neigh- oring clusters nding the set of no des in com- mon et een clusters and solving for the rotation, translation, and ossible reection that est aligns the clusters. This cluster-based approac has the adv an tage that eac no de has lo cal co ordinate system with itself as the ori- gin. The algorithm is easily distributed ecause clusters are lo calized using only distance measuremen ts to immediate neigh ors and et een neigh ors. urthermore, if one no de in the net ork mo es, only the (1) clusters con taining that no de ust up date their osition information. The follo wing sections describ the phases of the algorithm in more detail. 2.1 Cluster Localization The goal of cluster lo calization is to compute the osi- tion of cluster’s no des in lo cal co ordinate system up to (a) (b) PSfrag replacemen ts Figure 4: (a) The robust four-v ertex quadrilateral. The haracteristic features of this subgraph are that eac ertex is connected to ev ery other distance measuremen and that kno wing the lo cations of an three ertices is sucien to compute the lo cation of the fourth using trilateration. (b) Decomp osition of the robust quadrilateral in to four triangles. global rotation and ossible reection. An no des that are not part of the largest subgraph of robust quads in the cluster will not lo calized. Ho ev er, after Phases I-I are complete, the ositions of man of these unlo calized no des can computed using more error prone metho ds that do not rely on robust quads. do not use suc metho ds in this phase since inaccurate osition estimates will com- ounded later phases of the algorithm. Our cluster-based lo calization strategy is similar to that prop osed in [2] except that our use of robust quads sp ecically oids ip am bigu- ities. Quadrilaterals are relev an to lo calization ecause they are the smallest ossible subgraph that can unam bigu- ously lo calized in isolation. Consider the no de subgraph in Figure 4, fully-connected distance measuremen ts. Assuming no three no des are collinear, these distance con- strain ts giv the quadrilateral the follo wing prop erties: 1. The relativ ositions of the four no des are unique up to global rotation, translation, and reection. In graph theory terms, the quadrilateral is glob al ly rigid 2. An globally rigid quadrilaterals sharing three er- tices form 5-v ertex subgraph that is also globally rigid. By induction, an um er of quadrilaterals hained in this manner form globally rigid graph. Despite these useful prop erties of the quadrilateral, global rigidit is not sucien to guaran tee unique graph realization when distance measuremen ts are noisy Th us, further restrict our quadrilateral to obust as follo ws. The quadrilateral sho wn in Figure 4a can decomp osed in to four triangles: ABC, ABD, A CD, and BCD, as sho wn in Figure 4b. If the smallest angle is near zero, there is risk that measuremen error, sa in edge AD, will cause ertex to reected er this sliv er of triangle as sho wn in Figure 5. Accordingly our algorithm iden ties only those triangles with sucien tly large minim um angle as robust. Sp ecically ho ose threshold min based on the measuremen noise and iden tify those triangles that satisfy sin min (2) where is the length of the shortest side and is the small- est angle, as robust. This equation ounds the orst-case probabilit of ip error for eac triangle. See Section for full deriv ation. dene obust triangle to triangle that satises Equation 2. urthermore, dene obust

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D PSfrag replacemen ts BD BD CD CD AD AD Figure 5: An example of ip am biguit realized due to measuremen noise. No de is trilaterated from the kno wn ositions of no des A, B, and C. Measured distances BD and CD constrain the o- sition of to the in tersections of the dashed circles. Kno wing AD disam biguates et een these ositions for D, but small error in AD (sho wn as AD selects the wrong lo cation for D. No de Graph Ov erlap Graph Figure 6: The dualit et een cluster ro oted at and graph of robust quads, whic call an over- lap gr aph In the erlap graph, eac robust quadri- lateral is ertex. Edges are presen et een quads whenev er they share three no des. Th us, if all four no de ositions are kno wn for some quad, an neigh oring quad in the erlap graph can use the three common no des to trilaterate the osition of the unkno wn no de. breadth-rst searc in to the erlap graph from some starting quad, trilaterat- ing along the lo calizes the cluster as describ ed Algorithm 2. Note that the erlap graph for cluster can ha distinct, unconnected subgraphs as sho wn in this example. No des that are unique to one subgraph cannot lo calized with resp ect to those of an unconnected subgraph. quadrilater al as fully-connected quadrilateral whose four sub-triangles are robust. ey feature of our algorithm is that use the ro- bust quadrilateral as starting oin t, and lo calize additional no des haining together connected robust quads. When- ev er quads ha three no des in common and the rst quad is fully lo calized, can lo calize the second quad trilaterating from the three kno wn ositions. natural rep- resen tation of the relationship et een robust quads is the overlap gr aph sho wn in Figure 6. Since three ertices in common mak it ossible to lo calize quads relativ to eac other, it is natural to represen the space as graph of robust quads. Lo calization then amoun ts to tra ersing the erlap graph with breadth-rst searc and trilaterating as go, linear time op eration as in [4]. The en tire algorithm for Phase I, cluster lo calization, is as follo ws: 1. Distance measuremen ts from eac one-hop neigh or are broadcast to the origin no de so that it has kno wl- edge of the et een-neigh or distances. 2. The complete set of robust quadrilaterals in the cluster is computed (Algorithm 1) and the erlap graph is generated. 3. osition estimates are computed for as man no des as ossible via breadth-rst searc in the erlap graph (Algorithm 2). the start of the graph searc h, ho ose ositions for the rst three no des to x the ar- bitrary translation, rotation, and reection. place the origin no de at (0 0) to sp ecify the global trans- lation, the rst neigh or on the -axis to sp ecify the global rotation, and the second neigh or in the osi- tiv direction to sp ecify the global reection. The remaining no des are trilaterated as they are encoun- tered. Algorithm Finds the set of robust quadrilaterals that con tain an origin no de Eac quad is stored as 4-tuple of its ertices and is returned in the set Quads assume that distance mea- suremen ts ha already een gathered as follo ws: Me as is set of ordered pairs that represen the distance from no de to no de min is the robustness threshold, computed from the measuremen noise as describ ed in Section 3. 1: for all pairs j; ij in Me as do 2: for all pairs in Me as do 3: Remo j; from Me as 4: for all pairs in Me as do 5: for all pairs m; lm in Me as do 6: if then 7: con tin ue 8: Retriev ik from Me as 9: Retriev il from Me as 10: if IsR obust lj min AND IsR obust ij ik min AND IsR obust ij il lj min AND IsR obust ik il min then 11: Add i; j; to Quads 12: Remo from Me as 2.2 Computing Inter -Cluster ransf ormations In Phase I, the transformations et een co ordinate sys- tems of connected clusters are computed from the nished cluster lo calizations. This transformation is computed nding the rotation, translation, and ossible reection that bring the no des of the lo cal co ordinate systems in to est coincidence [8]. After Phase is complete for the clusters, the ositions of eac no de in eac lo cal co ordi- nate system are shared. As long as there are at least three non-collinear no des in common et een the lo caliza- tions, the transformation can computed. By testing if these three no des form robust triangle, sim ultaneously

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Algorithm Computes osition estimates for the cluster cen- tered at no de This algorithm do es breadth-rst searc in to eac disconnected subgraph of the erlap graph created from Quads and nds the most complete lo calization ossible. the end of this algorithm, cs best is set con taining pairs j; where is the estimate for the x-y osition of no de An neigh- ors of not presen in cs best ere not lo calizable. 1: cs best := 2: for eac disconnected subgraph of the erlap graph do 3: cs := 4: Cho ose quad from the erlap graph. 5: := (0 0) osition of the origin no de 6: := ab 0) First neigh or sets -axis 7: := ab ac bc ab ac 8: := ac ; ac Lo calize the second neigh- or relativ to the rst 9: Add a; ), b; ), and c; to cs 10: for eac ertex visited in breadth-rst searc in to the erlap graph do 11: if the curren quadrilateral con tains no de that has not een lo calized et then 12: Let the ositions of the three previously lo calized no des. 13: := Trila tera te aj bj cj 14: Add j; to cs 15: if length( cs length( cs best then 16: cs best := cs guaran tee non-collinearit and the same resistance to ip am biguities as Phase of the algorithm. 3. AN AL YSIS 3.1 Pr oof of Rob ustness In order for Algorithm to pro duce correct graph re- alization, ust ensure that our use of robust quads pre- en ts oth ip and ex am biguities. Since distance mea- suremen ts ma ha arbitrary noise cannot guaran tee correct realization in all cases instead can only predict the probabilit of ha ving no ips based on our denition of robustness. It is dicult to quan tify this probabilit for an en tire graph, so instead fo cus on the probabilit of an individual error. That is, dene an \error" as the real- ization of single robust quad with one ertex ipp ed or exed from its correct lo cation. By deriving the orst-case probabilit of error, will pro our rst theorem: Theorem 1. or normal ly-distribute distanc me asur e- ment noise with standar deviation we an onstruct obustness test such that the worst-c ase pr ob ability of err or is ounde d. First, pro that the use of robust quadrilaterals rules out the ossibilit of ex am biguities as seen in Figure 3b. This kind of ex am biguit ccurs only when rigid graph ecomes non-rigid the remo al of single edge [7]. If the graph is suc that no single edge remo al will mak it non- rigid, the graph is dundantly rigid and no ex am biguities are ossible. The robust quad has six edges. By remo ving an edge, are left with 5-edged graph, whic ust rigid according to the follo wing theorem [10]: C PSfrag replacemen ts BC BC AB CD Figure 7: diagram of quadrilateral for deriving the orst-case probabilit of ip error. ertex is eing trilaterated from the kno wn ositions of er- tices A, B, and D. Its distance to ertex is used to disam biguate et een the ossible lo cations and testing whic of CD and is closer to the measured distance et een and D. Theorem (Laman’s Theorem) et gr aph have exactly dges wher is the numb er of vertic es. is generic al ly rigid in if and only if every sub gr aph with vertic es has or fewer dges. Our robust quad with its missing edge has ertices and edges, satisfying the condition in Laman’s Theorem. Since ev ery 3-v ertex subgraph has or few er edges and ev ery 2- ertex subgraph has or few er edges, the 5-edged quad is rigid. Th us, the 6-edged robust quad is redundan tly rigid. Therefore, ex am biguities are imp ossible for graph con- structed of robust quads. Unlik ex am biguities, ips cannot ruled out based on the graph structure alone. Since distance measuremen ts are noisy they ma cause ertex to incorrectly ipp ed in computed realization. Th us, deriv the orst-case probabilit of realizing ipp ed ertex. Figure depicts the scenario in whic ertex could ecome incorrectly ipp ed. In this example, ertex is eing trilaterated with resp ect to the kno wn ositions of ertices A, B, and D. emp orarily ignoring ertex D, can pinp oin to ossible lo ca- tions: and the in tersection oin ts of circles cen- tered at and B, of radius and BC disam biguate this ossible ip, use the kno wn distance to ertex as follo ws. compute distances CD and Whic hev er distance is closer to the measured distance CD will deter- mine whether or is selected during trilateration. The probabilit of an incorrect ip is equal to the proba- bilit that the measured distance CD will closer to the incorrect distance than to the correct distance CD Note that the problem has an in trinsic symmetry: namely disam biguating the osition of based on is equiv alen to disam biguating based on C. Assuming the random mea- suremen noise is zero-mean, there ust measuremen error of magnitude CD for an incorrect ip to realized. can deriv this alue from the graph in Fig- ure 7. or simplicit constrain the gure to left-righ symmetric, although the probabilit of error will only de- crease breaking this symmetry In this problem, tak the alues of AB and as giv en. will later elimi-

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nate maximizing the error with resp ect to it. First, compute the alues of CD and as: CD AB sin sin(2 and AB sin(2 sin sin sin Com bin- ing these yields er CD (3) AB sin sin sin sin sin(2 (4) Since are in terested in the orst-case probabilit of error, minimize er with resp ect to taking the partial deriv ativ of er and setting it equal to zero. nd that er is minimized when This can substituted in to Equation and the resulting equation simplied to yield er AB sin (5) Th us, if the true distance is and the measured distance is random ariable then the orst-case probabilit of error is er ). If measuremen noise is zero- mean Gaussian with standard deviation the orst-case probabilit of error is er er (6) where ( connotes the in tegration of the unit normal prob- abilit densit function from to innit This equation tells us that for arbitrary measuremen noise with standard deviation can ho ose threshold min for the robust- ness test. Only those triangles for whic sin min where is the shortest side and is the smallest angle, will treated as robust. By ho osing min to some constan ultiple of ound the probabilit of error. This pro es Theorem 1. or the sim ulation results presen ted in this pap er, min as hosen to or Gaussian noise, this ounds the probabilit of error for giv en robust quadrilateral to less than 1%. Ho ev er, for the ypical case, the probabilit is signican tly less than 1%, th us osing minimal threat to the stabilit of the lo calization algorithm. 3.2 Computational Complexity It is imp ortan that an distributed lo calization algorithm scalable to large net orks. In this section discuss the computational and comm unications eciency of the al- gorithm presen ted in Section 2. In general, nding re- alization of graph is NP-hard [20]. are able to do it in olynomial time ecause our algorithm purp osefully oids no des that ma ha osition am biguities (i.e., ips or exes) at the cost of failing to nd all ossible realiza- tions. It is these am biguities whic cause the general case to blo up com binatorially Our algorithm gro ws linearly with resp ect to the um er of no des when there are (1) neigh ors er no de. urthermore, since this computation is distributed across the net ork, eac no de erforms (1) computation. If the no de degree is not constan t, eac no de’s computation aries with the third er of the um er of neigh ors. Algorithm 1, whic nds the set of robust quadrilaterals in lo cal cluster, has orst-case run time where is the maxim um no de degree. It can implemen ted with run time using etter data structures. In practice, the algorithm is uc more ecien ecause eac neigh or is generally not connected to ev ery other neigh or. In this algorithm, simply en umerate the robust quadrilaterals in the cluster, th us the orst-cast um er of robust quadrilat- erals is whic is (1) for graph of ounded degree. Algorithm 2, whic solv es for osition estimates for one cluster, has run time where is the um er of robust quadrilaterals. In the orst case, Finding the in ter-cluster transformations for one cluster has run time ). are nding transformations, eac of whic ma tak time to compute ecause the regis- tration problem tak es linear time in the um er of erlap- ping ertices. Again, for graph of ounded degree, these computations tak (1) time. The only stage of the algorithm that en tails comm uni- cation erhead is the initial step where eac no de shares its measured distances with its neigh ors. If assume that non-o erlapping clusters do not share the same hannel (due to range limitations), the comm unications erhead is ecause measuremen ts are eing shared. In prac- tice, this is implemen ted eac no de sending one pac et of constan size for distance measuremen and one pac et of size to share other measuremen ts. 4. EXPERIMENT AL RESUL TS In order to measure the eectiv eness of our algorithm on real sensor net orks, implemen ted it on-b oard func- tioning sensor net ork. The net ork is constructed of Cric k- ets, hardw are platform dev elop ed and supplied MIT [16]. Cric ets are hardw are-compatible with the Mica2 Motes de- elop ed at Berk eley with the addition of an Ultrasonic trans- mitter and receiv er on eac device. This hardw are enables the sensor no des to measure in ter-no de ranges using the time dierence of arriv al (TDoA) et een Ultrasonic and RF sig- nals. Although the Cric ets can ac hiev ranging precision of around cm on the lab enc h, in practice, the ranging error can as large as cm due to o-axis alignmen of the sending and receiving transducers. 4.1 Ev aluation Criteria One criteria whic ev aluate the erformance of the algorithm is ho the computed lo calization diers from kno wn ground truth. This error is expressed as =1 (7) where is the um er of no des, and comp ose the lo calized osition of no de and and comp ose the true osition of no de This metric is simply the mean-square error in Euclidean 2D space. It is useful to compare to the mean-square error in the ra distance measuremen ts, since the error mo del of the measuremen ts determines the minim um ac hiev able of an ideal lo calization algorithm [18]. The mean-square error of the distance measuremen ts is =1 (8) where is the um er of in ter-no de distances, is the measured alue of distance and is the true alue of distance Another useful metric is the prop ortion of no des success- fully lo calized the algorithm. Let the um er of

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Y-P osition (cm) 50 100 150 200 −20 20 40 60 80 100 120 140 160 180 200 Ground truth Localized positions X-P osition (cm) Figure 8: comparison of no de ositions as lo cal- ized our algorithm to the true ositions of the no des on ph ysical Cric et cluster. ositions are computed Phase of the algorithm, cluster lo cal- ization. The exp erimen in olv ed 16 no des, one of whic could not lo calize; th us only 15 are sho wn. no des successfully lo calized in the cluster cen tered at no de and the total um er of no des in this cluster. In one cluster, the prop ortion of no des lo calized is =k or the en tire net ork, dene the cluster suc ess ate as (9) This metric tells us the erage ercen tage of no des that ere lo calizable er cluster. Our nal metric con eys the prop ortion of no des in the entir net ork that could lo calized in to single co ordi- nate system. Since some clusters will ha transformations et een them and others will not, the net ork ma split in to separate subgraphs, eac of whic is lo calized with resp ect to all its no des, but is not rigidly lo calized with resp ect to the other subgraphs. call these subgraphs for ests Nat- urally it is desirable for there to only single forest that con tains ev ery no de in the net ork. Th us, another use- ful metric is the lar gest for est size whic is the um er of no des in the largest forest. This metric can expressed as ercen tage dividing the total um er of no des in the net ork. 4.2 Accuracy Study: Hard war Deployment Figure sho ws the results of the rst exp erimen t. In this exp erimen t, 16 cric ets ere placed in pseudo-random, 2-dimensional arrangemen t. Ground truth as measured man ually with the aid of grid on the surface. The small circles depict the ositions of eac no de as computed the lo calization algorithm running on-b oard the cric et in the ottom-most osition of the gure. The ositions sho wn are for Phase of the algorithm, where ositions are trilaterated using robust quadrilaterals. No least-squares optimization as erformed. The true osition of eac no de is sho wn with an \x." line et een the oin ts sho ws the amoun of ositioning error. The error metrics for the exp erimen sho wn in Figure Y-P osition (cm) −100 100 200 300 400 −150 −100 −50 50 100 150 200 250 300 Ground truth Localized positions X-P osition (cm) Figure 9: The lo calized ositions of 40 Cric ets in ph ysical net ork. The \holes" in the net ork are where no des could not lo calize, and th us only 38 are sho wn. The co ordinate transformations et een eac cluster ere computed and used to render the lo calized ositions in the single co ordi- nate system seen here. Ground truth ositions are erlaid, with lines sho wing the amoun of error for eac no de. The dotted line depicts the exten of one cluster. are as follo ws: metric alue 5.18 cm 7.02 cm 15 16 94 15 16 94 The fact that (the lo calization error) is only sligh tly larger than (the measuremen error) tells us that our algorithm erformed ell relativ to the qualit of the distance mea- suremen ts ailable. In addition, all no des but one ere suc- cessfully lo calized, indicating that the algorithm pro vided go lo calization co erage of the cluster. second exp erimen t, the results of whic are sho wn in Figure 9, demonstrates oth Phase and Phase of the algorithm. Once again, for simplicit the optional least- squares relaxation phase is omitted. total of 40 no des ere placed in meter region. The RF and ultrasound ranges of eac Cric et ere arbitrarily restricted so that only 12 neigh ors ere rangeable from eac no de. Then, cluster lo calization as erformed separately on v no des, dividing the net ork in to v clusters. The range of one cluster is sho wn dotted line in the gure. Phase of the algorithm, running on eac of the v clusters, lo calized its no des in lo cal co ordinate system. ransformations e- een eac pair of co ordinate systems with at least three no des in common ere computed Phase of the algo- rithm. Figure sho ws the lo calized ositions of eac no de as small circles, erlaid with the ground truth. The lo calized ositions of three no des are used to bring the en tire net ork in to registration with the global co ordinate system used ground truth. The error metrics for the Figure exp erimen are:

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Figure 10: The oce o orplan used for sensor net- ork sim ulation. Dark lines are the alls of the building and ligh t-colored lines represen the graph edges et een no des. Eac edge represen ts dis- tance measuremen that no de can erform. Mea- suremen ts cannot tak en through alls. metric alue 4.38 cm 6.82 cm 97 38 40 95 As in the rst exp erimen t, these results sho that lo caliza- tion error as not uc greater than measuremen error. 4.3 Scalability Study: Simulated Deployment ha tested the three phases of our algorithm on ariet of sim ulated net orks in order to ev aluate its scal- abilit ey ond the ph ysical exp erimen ts erformed in Sec- tion 4.2. In this pap er, sim ulate an en vironmen based on an actual o orplan of an oce building, sho wn in Fig- ure 10. placed 183 no des uniformly and randomly in the o-dimensional region, but connectivit is only ailable et een no des that are within the maxim um ranging dis- tance and not obstructed alls. The o orplan has three ro oms and one hallw and is appro ximately square with eac side 10 long. When ev aluating the algorithm’s erformance, are in- terested in ho oth no de degree and measuremen noise aect the results. No de degree as aried hanging the maxim um ranging distance. also consider three dieren degrees of measuremen noise: 1. Zero noise, where all measuremen ts are exact. Sim ula- tions without noise giv an upp er ound on ho uc lo calization is ossible for net ork. Without noise, an unlo calizable no des ust due to disconnection or non-rigidit in the graph structure. 2. Noise with cm, similar to that of Cric et device in ideal circumstances. 3. Noise with 10 cm. This gure is to designed to sim ulate sensor net orks with more imprecise ranging capabilit Figure 11 sho ws the sim ulation results for the building en- vironmen t. Eac data oin on the plots represen single run of the sim ulation, whic lo calizes as man no des as os- sible. As one ould exp ect, the abilit of the algorithm to lo calize go es do wn as the measuremen noise increases. In- terestingly the algorithm is nearly as eectiv with cm noise as with zero noise. With more noise, the algo- rithm is still eectiv e, but the requiremen ts for no de degree are higher. Note that the largest forest size rarely ob- tained 100% ev en with high no de degree due to obstruction alls. In practical deplo ymen t, no des ould ha to strategically placed around do orw ys to ac hiev 100% forest size. 4.4 Err or Pr opagation Cluster-based lo calization algorithms generally suer from or error propagation haracteristics ecause they ha no absolute reference oin ts as constrain ts. sho that our approac h, using robust quads, signican tly reduces the amoun of error propagated er approac hes based on basic trilateration. Figure 12a sho ws lo calization results of our algorithm after Phase and on sim ulated net ork of 100 no des. No des ere randomly placed within the square region, eac with maxim um ranging distance of 350 cm. Distance measure- men ts ere corrupted Gaussian noise with cm. In order to compare to ground truth, pic three no des as \anc hors". These no des are used solely for transforming et een the separate co ordinate systems of the algorithm and ground truth, and are not used the algorithm at run-time. The anc hor no des are closely-spaced so that er- rors can accum ulate to ards the edges of the net ork. In con trast, Figure 12b sho ws lo calization results for the same net ork, but with an algorithm that uses trilateration alone and do es not hec for quad robustness. The arious error metrics for three sim ulation runs are as follo ws. Eac as run with dieren amoun of measure- men noise, The error metrics for the sim ulation without robust quads are also sho wn: metric Our algorithm w/o robust quads 1.0 cm 3.0 cm 5.0 cm 5.0 cm 4.43 cm 14.39 cm 16.22 cm 54.87 cm 0.91 0.85 0.79 0.95 0.93 0.87 0.75 0.99 Sho wn in: Figure 12a Figure 12b This comparison demonstrates that robust quads signi- can tly reduce error propagation. 4.5 Localization of Mobile Nodes An adv an tage of our algorithm is that it handles no de mobilit ell ecause eac cluster lo calization can re- computed quic kly Ev en on lo er device, the cluster lo calization phase can tak less than one second for 15{20 neigh ors. Th us, as no des mo e, Phase can simply re- eated to eep up. urthermore, excluding mobile no des from the transformation computation in Phase I, it do es not need to rep eated. There are practical issues with time dierence of arriv al (TDoA) distance estimation that complicate the handling of no de mobilit Sp ecically since eac no de-to-no de dis- tance is estimated at dieren momen in time, mo ving no de will corrupt the self-consistency of the distance mea- suremen ts. Suc discrepancies will urt the erformance of

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(a) Plots of cluster success ra te, versus node degree or the building envir onment 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) (b) Plots of lar gest orest size, versus node degree or the building envir onment 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Largest forest size (p ercen t) 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Largest forest size (p ercen t) 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Largest forest size (p ercen t) Figure 11: (a) The cluster success rate ersus the erage no de degree for three dieren lev els of mea- suremen noise in the building en vironmen of Figure 10. Eac data oin sho ws the alue of these quan tities for single sim ulation run. mo ving erage of the data oin ts is erlaid on eac plot. (b) The size of the largest forest ersus erage no de degree for three dieren lev els of measuremen noise. the lo calization algorithm and oten tially cause large errors. In our exp erimen ts solv this problem measuring the distance to mo ving no des using only ultrasound pulses gen- erated the mo ving no de itself. This all other no des will sense the same ph ysical pulse and generate measure- men estimate for the same momen in time. These mea- suremen ts are then shared within eac cluster. Mobilit also complicates the handling of noise, since out- liers and noisy measuremen ts can misconstrued as ob- serv ed motion. address this issue rst feeding the ra measuremen ts in to er-edge Kalman Filter with state ariables: the no de-to-no de distance and the rate of hange of the distance. The lter sim ultaneously smo oths noisy measuremen ts and eliminates outliers rejecting measure- men ts with noise inconsisten with the lter state. The l- ters for static distances are tuned with single pro cess noise that limits mobilit and the lters for mobile distances are tuned to allo mobilit Algorithm is then run on the out- puts of the lters instead of the ra measuremen ts. These er-link lters use uc less state and computation than Kalman Filter in olving all no des sim ultaneously (e.g. [22]). nal issue in lo calization with mobilit is that trilatera- tion can inaccurate when trac king mo ving device, since it do es not generally use all distance constrain ts ailable. Using large um er of constrain ts is imp ortan for mobile lo calization, whic has more noise than static lo calization. Th us, ha found it imp ortan to use least-squares op- timization, emplo ying all distance constrain ts for osition renemen t, after computing an initial estimate in Phase I. Figure 13 sho ws our exp erimen tal results lo calizing mo- bile no de. Six stationary no des ere deplo ed in roughly circular conguration, as sho wn the small circles in the gure. no de as attac hed to an autonomous rob ot placed in the cen ter of the stationary no des. Once activ ated, the rob ot randomly tra ersed rectangular space. The lo caliza- tion as computed the sensor net ork as logged er time and man ually sync hronized with calibrated video camera. The video as ost-pro cessed to obtain the ground truth rob ot path with sub-cen timeter accuracy This path as then compared to the path computed the lo calization al- gorithm. The lo calization algorithm computed osition estimate for the rob ot roughly once er second for min-

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(a) Our algorithm Y-P osition (cm) 500 1000 1500 500 1000 1500 Localized node position Unlocalizable node Anchor node X-P osition (cm) (b) Algorithm without robust quads Y-P osition (cm) 500 1000 1500 500 1000 1500 Localized node position Unlocalizable node Anchor node X-P osition (cm) Figure 12: (a) Our algorithm’s lo calized ositions for sim ulated net ork compared to ground truth. Lines sho the amoun of error for eac no de’s o- sition. The three no des used to compute the trans- formation to the ground truth’s co ordinate system are sho wn with small circles. The large dotted cir- cle depicts the maxim um ranging distance of no de. (b) Lo calization of the same net ork using basic tri- lateration without hec king for quad robustness. utes. Since discrete computations ere made, eac of these separate lo calizations could compared to ground truth. The mean-square error, computed from these alues is 2.59 cm. Th us, our lo calization algorithm is sho wn to successful at lo calizing net orks with mobile no des. 5. CONCLUSION ha demonstrated an algorithm that successfully lo- calizes no des in sensor net ork with noisy distance mea- suremen ts, using no eacons or anc hors. Sim ulations and exp erimen ts sho ed the relationship et een measuremen (a) Lo calized path of mobile no de Y-P osition (cm) 50 100 150 200 −180 −160 −140 −120 −100 −80 −60 −40 −20 20 Localized path True path X-P osition (cm) (b) Magnitude of lo calization error vs. time Error, Euclidean distance (cm) 20 40 60 80 100 120 140 160 180 200 10 Time (s) (c) Exp erimen tal setup Figure 13: (a) The path of mobile no de computed our lo calization algorithm compared to ground truth er min ute erio d. sensor no de as attac hed to mobile rob ot (an autonomous o or acuum) that randomly co ered rectangular space. Six static no des, depicted as circles, ere used to lo calize this mobile no de er time. Ground truth (dashed) as obtained from calibrated video. (b) The Euclidean distance et een the mobile no de’s lo calized osition and ground truth er time. (c) photo of the exp erimen tal setup.

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noise and abilit of net ork to lo calize itself. As long as the error mo del of the measuremen noise is kno wn, the al- gorithm cop es with this noise refusing to lo calize those no des whic ha am biguous ositions. urthermore, ev en with no noise, eac no de in the net ork ust ha appro xi- mately degree 10 or more efore 100% no de lo calization can attained. As noise increases, so will the connectivit re- quiremen ts. The Cric et platform has mo derate amoun of noise and th us exercises our algorithm’s tolerance for noisy distance measuremen ts. ha also sho wn that the algo- rithm adapts to no de mobilit ltering the underlying measuremen ts. or future ork, are in terested in extending our ph ys- ical exp erimen ts to ev en larger no de deplo ymen ts that also include obstructions. Also, ould lik to use the prin- ciple of robust quads to compute the optimal placemen of additional no des so that partially lo calized graph ecomes fully lo calizable. Finally it ould useful to further re- ne our approac to allo \passiv e" mobile no des to lo calize without transmitting. 6. CKNO WLEDGEMENTS This ork as funded gran from Pro ject Oxygen and supp orted in part National Science oundation Graduate Researc ello wship. Additional funding as pro- vided in part the Institute for Securit ec hnology Stud- ies (ISTS) at Dartmouth College, NSF ard 0225446, ONR ards N00014-01-1-0675, N00014-02-C-0210, and N00014- 03-1-0879, and ARP ASK program ard F30602-00- 2-0585. thank the Cric et pro ject for supplying hardw are and programming assistance and the BMG (Building Mo del Gen- eration) pro ject for o orplans. atric Nic hols wrote the constrained eacon graph generator. are grateful to Erik Demaine for useful discussions and oin ters to literature. 7. REFERENCES [1] Bulusu, N., Heidemann, J., and Estrin, D. GPS-less lo cost outdo or lo calization for ery small devices. IEEE Personal Communic ations Magazine (Octob er 2000), 28{34. [2] Capkun, S., Hamdi, M., and Huba ux, J.-P. GPS-free ositioning in mobile ad-ho net orks. In Pr dings of the 34th Hawaii International Confer enc on System Scienc es (2001). [3] Doher ty, L., Pister, K. S. J., and Gha oui, L. E. Con ex osition estimation in wireless sensor net orks. In Pr c. IEEE INF OCOM (Anc horage, AK, April 2001). [4] Eren, T., Goldenber g, D., Whiteley, W., ang, Y. R., Morse, A. S., Anderson, B. D. O., and Belhumeur, P. N. Rigidit computation, and randomization in net ork lo calization. In Pr c. IEEE INF OCOM (Marc 2004). [5] Estrin, D., Go vind an, R., and Heidemann, J. Em edding the in ternet: in tro duction. Commun. CM 43 (2000), 38{41. [6] Grabo wski, R., and Khosla, P. Lo calization tec hniques for team of small rob ots. In Pr c. IEEE IR OS (Maui, Ha aii, Octob er 2001). [7] Hendrickson, B. Conditions for unique graph realizations. SIAM J. Comput. 21 (1992), 65{84. [8] Horn, B. K. P. Closed form solution of absolute orien tation using unit quaternions. Journal of the Optic al So ciety (April 1987), 629{642. [9] Ji, X., and Zha, H. Sensor ositioning in wireless ad-ho sensor net orks using ultidimensional scaling. In Pr c. IEEE INF OCOM (Marc 2004). [10] Laman, G. On graphs and rigidit of plane sk eletal structures. J. Engine ering Math (1970), 331{340. [11] Na gp al, R., Shr obe, H., and Ba chra ch, J. Organizing global co ordinate system from lo cal information on an ad ho sensor net ork. In Pr c. IPSN (P alo Alto, CA, April 2003), pp. 333{348. [12] Niculescu, D., and Na th, B. based ositioning in ad ho net orks. Kluwer journal of ele ommunic ation Systems (2003), 267{280. [13] Niculescu, D., and Na th, B. Error haracteristics of ad ho ositioning systems (APS). In Pr c. 5th CM MobiHo (T oky o, Ma 2004). [14] tw ari, N., I, A. O. H., Perkins, M., Correal, N. S., and O’Dea, R. J. Relativ lo cation estimation in wireless sensor net orks. IEEE ans. Signal Pr ess. 51 (August 2003), 2137{2148. [15] Priy antha, N. B., Balakrishnan, H., Demaine, E., and Teller, S. Anc hor-free distributed lo calization in sensor net orks. ec h. Rep. 892, MIT Lab. for Comp. Sci., April 2003. [16] Priy antha, N. B., Chakrabor ty, A., and Balakrishnan, H. The Cric et lo cation-supp ort system. In Pr c. 6th CM MobiCom (Boston, MA, August 2000). [17] Sa arese, C., Rabaey, J., and Langendoen, K. Robust ositioning algorithms for distributed ad-ho wireless sensor net orks. In USENIX nnual ch. Conf. (Mon terey CA, June 2002), pp. 317{327. [18] Sa vvides, A., Garber, W., Adlakha, S., Moses, R., and Sriv ast a, M. B. On the error haracteristics of ultihop no de lo calization in ad-ho sensor net orks. In Pr c. IPSN (P alo Alto, CA, April 2003), pp. 317{332. [19] Sa vvides, A., Han, C.-C., and Sriv ast a, M. B. Dynamic ne-grained lo calization in ad-ho net orks of sensors. In Pr c. 7th CM MobiCom (Rome, Italy 2001), pp. 166{179. [20] Saxe, J. B. Em eddabilit of eigh ted graphs in k-space is strongly NP-hard. In Pr c. 17th lerton Conf. Commun. Contr ol Comput. (1979), pp. 480{489. [21] Simic, S. N., and Sastr y, S. Distributed lo calization in wireless ad ho net orks. ec h. Rep. UCB/ERL M02/26, UC Berk eley Decem er 2001. [22] Smith, A., Balakrishnan, H., Gora czk o, M., and Priy antha, N. rac king mo ving devices with the cric et lo cation system. In Pr c. 2nd CM MobiSys (Boston, MA, June 2004), pp. 190{202. [23] Teller, S., Chen, J., and Balakrishnan, H. erv asiv ose-a are applications and infrastructure. IEEE Computer Gr aphics and Applic ations (July/August 2003), 14{18.

mitedu ABSTRA CT This pap er describ es distributed lineartime algorithm for lo calizing sensor net ork no des in the presence of range mea suremen noise and demonstrates the algorithm on ph ysi cal net ork in tro duce the probabilistic notion of obu ID: 23817

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Rob ust Distrib uted Netw ork Localization with Noisy Rang Measurements Da vid Moore John Leonard Daniela Rus Seth eller MIT Computer Science and Ar tiﬁcial Intelligence Labor ator The Stata Center 32 assar Street Cambr idge MA 02139 dcm, jleonard, us teller @csail.mit.edu ABSTRA CT This pap er describ es distributed, linear-time algorithm for lo calizing sensor net ork no des in the presence of range mea- suremen noise and demonstrates the algorithm on ph ysi- cal net ork. in tro duce the probabilistic notion of obust quadrilater als as to oid ip am biguities that other- wise corrupt lo calization computations. form ulate the lo calization problem as o-dimensional graph realization problem: giv en planar graph with appro ximately kno wn edge lengths, reco er the Euclidean osition of eac ertex up to global rotation and translation. This form ulation is applicable to the lo calization of sensor net orks in whic eac no de can estimate the distance to eac of its neigh ors, but no absolute osition reference suc as GPS or xed an- hor no des is ailable. implemen ted the algorithm on ph ysical sensor net- ork and empirically assessed its accuracy and erformance. Also, in sim ulation, demonstrate that the algorithm scales to large net orks and handles real-w orld deplo ymen geome- tries. Finally sho ho the algorithm supp orts lo caliza- tion of mobile no des. Categories and Subject Descriptors C.2.4 Computer-Comm unications Net orks ]: Distrib- uted Systems; C.3 Sp ecial-Purp ose and Application- Based Systems ]: Real-time and em edded systems General erms Algorithms, Design, Exp erimen tation eyw ords Sensor net orks, lo calization, mobilit lo cation-a areness, trac king, erv asiv computing Permission to mak digital or hard copies of all or part of this ork for personal or classroom use is granted without fee pro vided that copies are not made or distrib uted for proﬁt or commercial adv antage and that copies bear this notice and the full citation on the ﬁrst page. cop otherwise, to republish, to post on serv ers or to redistrib ute to lists, requires prior speciﬁc permission and/or fee. SenSys’04, No ember 3–5, 2004, Baltimore, Maryland, USA. Cop yright 2004 CM 1-58113-879-2/04/0011 ... 5.00. 1. INTR ODUCTION This pap er describ es distributed algorithm for lo calizing no des in sensor net ork in whic no des ha the abilit to estimate distance to nearb no des, but suc measure- men ts are corrupted noise. Lo calization is an essen tial to ol for the dev elopmen of lo w-cost sensor net orks for use in lo cation-a are applications and ubiquitous net orking [5, 23]. Distributed computation and robustness in the pres- ence of measuremen noise are ey ingredien ts for practi- cal lo calization algorithm that will giv reliable results er large scale net ork. form ulate this as the follo wing o-dimensional graph realization problem: giv en planar graph with edges of kno wn length, reco er the Euclidean osition of eac ertex up to global rotation and transla- tion. This is dicult problem for sev eral reasons. First, there is often insucien data to compute unique osition assignmen for all no des. Second, distance measuremen ts are noisy comp ounding the eects of insucien data and creating additional uncertain Another problem is lac of absolute reference oin ts or anc hor no des whic could pro vide starting oin for lo calization. Finally it is di- cult to devise algorithms that scale linearly with the size of the net ork, esp ecially if data ust broadcast through the limited comm unications capacit of wireless hannel. presen distributed lo calization algorithm that gets around these diculties lo calizing no des in regions con- structed from obust quadrilater als term that formally dene in Section 2. Lo calization based on robust quads attempts to prev en incorrect realizations of ip am bigui- ties that ould otherwise corrupt lo calization computations. urthermore, sho that the criteria for quadrilateral ro- bustness can adjusted to cop with arbitrary amoun ts of measuremen noise in the system. The dra wbac of our approac is that under conditions of lo no de connectivit or high measuremen noise, the algorithm ma unable to lo calize useful um er of no des. Ho ev er, for man applications, missing lo calization information for kno wn set of no des is preferen tial to incorrect information for an unkno wn set. general result of our sim ulations is that ev en as noise go es to zero, no des in large net orks ust ha degree 10 or more on erage to ac hiev 100% lo calization. high lev el, our net ork lo calization algorithm orks use the term anchor no de to refer to no de that has prior kno wledge of its absolute osition, either man ual initialization or an outside reference suc as GPS. This yp of no de is also called ac on

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PSfrag replacemen ts Figure 1: An example run of our algorithm to es- timate the relativ ositions of no de A’s neigh ors. No des ABCD form obust quad ecause their real- ization is unam biguous ev en in the presence of noise. select as the origin of lo cal co ordinate system and ho ose ositions for B, C, and that satisfy the six distance constrain ts. In the next step, no de is lo calized relativ to the kno wn ositions of ABD us- ing trilateration. This lo calization is unam biguous ecause ABDE also forms robust quadrilateral. Con tin uing, the same pro cedure is used to lo calize no de whic is part of the robust quad ADFE. as follo ws. Eac no de measures distances to neigh oring no des, then shares these measuremen ts with the neigh ors. This \one-hop" information is sucien for eac no de to lo- calize itself and its neigh ors, whic call cluster, in some lo cal co ordinate system. Co ordinate transforms can then computed et een erlapping clusters to stitc them in to global co ordinate system. Suc stitc hing can done in an on-line fashion as messages are routed through the net ork rather than attempting to solv for the full lo calization up fron t. Similar cluster-based approac hes ha een prop osed efore, but often suer from gross lo calization errors due to graph ips that can comp ound er larger distances. Our no el use of robust quadrilaterals ensures that cluster-based lo calization do es not suer from suc errors. Figure depicts an illustrativ run of our algorithm. nd all sets of four no des that are fully connected distance measuremen ts and are \w ell-spaced" suc that ev en in the presence of measuremen noise, their relativ ositions are unam biguous. adopt this quadrilateral as the smallest subgraph that can denitiv ely realized, and dene it as obust quad Additional robust quads can \c hained" to the original quad if they ha no des in common with it. This approac allo ws eac hained quad to lo calize its fourth no de based on the kno wn ositions common to the quads using the standard tec hnique of trilater ation [4, 12]. This use of robust quadrilaterals enables our algorithm to tolerate noise computing unique realizations for graphs that migh otherwise am biguous. Our algorithm has the follo wing haracteristics: 1. It supp orts noisy distance measuremen ts, and is de- signed sp ecically to robust under suc conditions. 2. It is fully distributed, requiring no eacons or anc hors. 3. It lo calizes eac no de correctly with high probabilit or not at all. Th us, rather than pro duce net ork with an incorrect la out, an no des with am biguous lo cations are not used as building blo ks for further lo calization. 4. Cluster-based lo calization supp orts dynamic no de in- sertion and mobilit 1.1 Related ork Eren et al. in [4] pro vide theoretical foundation for net- ork lo calization in terms of graph rigidit theory They sho that net ork has unique lo calization if and only if its underlying graph is generic al ly glob al ly rigid In ad- dition, they sho that certain sub class of globally rigid graphs, trilater ation gr aphs can constructed and lo cal- ized in linear time. tak global rigidit and trilateration graphs one step further with robust quadrilaterals that pro- vide unam biguous lo calizations and tolerate measuremen noise. In [18], Sa vvides et al. deriv the Cram er-Rao lo er ound (CRLB) for net ork lo calization, expressing the exp ected error haracteristics for an ideal algorithm, and compare it to the actual error in an algorithm based on ultilateration. They dra the imp ortan conclusion that error in tro duced the algorithm is just as imp ortan as measuremen error in assessing end-to-end lo calization accuracy In [13] and [12], Niculescu and Nath also apply the CRLB to few gen- eral classes of lo calization algorithms. Their \Euclide an metho is similar to our metho of cluster lo calization in that it dep ends on the trilateration primitiv e. They also state the relev ance of four-no de quadrilaterals. In their case, the quads are constrained with v distance measuremen ts the sixth is computed based on the rst v e. Flip am bi- guities are resolv ed using additional information from neigh- oring no des. Their \D V-c or dinate" propagation metho presen ted in [12] is similar to our metho in that clusters consisting of no de and its one-hop neigh ors are rst lo- calized in lo cal co ordinate systems. Registration is then used to compute the transformations et een neigh oring co or- dinate systems. This idea of lo cal clusters as also prop osed Capkun et al. in [2]. Ho ev er, neither algorithm consid- ers ho measuremen noise can cause incorrect realization of ip am biguit ariet of other researc attempts to solv the lo cal- ization problem using some form of global optimization. Dohert et al. describ ed metho using connectivit con- strain ts and con ex optimization when some um er of ea- con no des are initialized with kno wn ositions [3]. Ji and Zha use ultidimensional scaling (MDS) to erform dis- tributed optimization that is more toleran of anisotropic net ork top ology and complex terrain [9]. Priy an tha et al. eliminate the dep endence on anc hor no des using comm u- nication hops to estimate the net ork’s global la out, then using force-based relaxation to optimize this la out [15]. Other previous ork is based on propagation of lo cation information from kno wn reference no des. Bulusu et al. and Simic et al. prop ose distributed algorithms for lo calization of lo er devices based on connectivit [1, 21]. Other tec hniques use distributed propagation of lo cation informa- tion using ultilateration [11, 19]. Sa arese et al. use o-phase approac using connectivit for initial osition estimates and trilateration for osition renemen [17]. at- ari et al. use one-hop ultilateration from reference no des in ph ysical exp erimen using oth receiv ed signal strength (RSS) and time of arriv al (T oA) [14]. Grab wski and Khosla maximize lik eliho estimator to lo calize small team of rob ots, ac hieving some robustness including motion mo del in their optimization [6]. In this pap er mak sev eral departures from previous researc h. Most imp ortan tly no previous algorithm consid- ers the ossibilit of ip am biguities during trilateration

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(a) Ground truth (b) Alternate realization 3.0 3.0 er 37 er 34 Figure 2: Example graph sho wing (a) true ertex ositions and (b) an alternate realization of the graph in whic in ter-v ertex distances, depicted as lines, are preserv ed almost exactly The error met- ric er is sho wn elo eac realization, with in ter- ertex distances generated from Gaussian distri- bution with mean of the true distance and 35 Th us, in this example, an incorrect realization of the graph ts the constrain ts etter than the ground truth, sho casing wh net ork lo calization is di- cult. due to measuremen noise. Although the requiremen of global rigidit as means to oid ips has een ell estab- lished [4], the eects of measuremen noise on global rigidit are not ell understo d. Our notion of robust quadrilaterals minimizes the probabilit of realizing ip am biguit incor- rectly due to measuremen noise. Error propagation during trilateration is deriv ed in [13], but the oten tial for signi- can error due to ips is not considered. Secondly lik [2] and [12], do not require anc hor no des, enabling lo caliza- tion of net orks without absolute osition information. This haracteristic is imp ortan for lo calization in homogeneous ad-ho net orks, where an no de ma ecome mobile. ur- thermore, man ual eacon initialization can error-prone or imp ossible, for example, in sensor net ork deplo ed mobile rob ot. 1.2 Challenges of netw ork localization The diculties inheren in lo calization can easily demon- strated with an example. Consider the follo wing metric that haracterizes the error for giv en lo calization, er =1 (1) where is the um er of distance measuremen ts, is eac distance computed from the lo calized ositions, and is eac measured distance. Without ground truth, er tells us ho ell computed lo calization ts the constrain ts. Fig- ure sho ws wh minimizing this error metric is insucien for lo calization. In this example, ossible realizations sho wn for the net ork ha similar alues for er but the ground truth actually has higher error than an incorrect realization. This demonstrates the need for an algorithm that appropriately handles no des with am biguous ositions refusing to assign osition to an no de that has more than one ossible lo cus for its osition. no formally address wh an algorithm based primarily on umerical op- timization of the distance constrain ts fails. In graph theory the problem of nding Euclidean osi- tions for the ertices of graph is kno wn as the gr aph e- alization pr oblem Saxe sho ed that nding realization (a) Flip am biguit (b) Discon tin uous ex am biguit PSfrag replacemen ts Figure 3: (a) Flip am biguit ertex can re- ected across the line connecting and with no hange in the distance constrain ts. (b) Discon tin- uous ex am biguit If edge AD is remo ed, then reinserted, the graph can ex in the direction of the arro w, taking on dieren conguration but ex- actly preserving all distance constrain ts. is strongly NP-hard for the o-dimensional case or higher [20]. Ho ev er, kno wing the length of eac graph edge do es not guaran tee unique realization, ecause deformations can exist in the graph structure that preserv edge lengths but hange ertex ositions. Rigidit theory distinguishes et een non-rigid and rigid graphs. Non-rigid graphs can con tin uously deformed to pro duce an innite um er of dieren realizations, while rigid graphs cannot. Ho ev er, in rigid graphs, there are yp es of disc ontinuous defor- mations that can prev en realization from eing unique [7]: 1. Flip ambiguities (Figure 3a) ccur for graph in dimensional space when the ositions of all neigh ors of some ertex span 1)-dimensional subspace. In this case, the neigh ors create mirror through whic the ertex can reected. 2. Disc ontinuous ex ambiguities (Figure 3b) ccur when the remo al of one edge will allo part of the graph to exed to dieren conguration and the remo ed edge reinserted with the same length. This yp of deformation is distinct from ontinuous ex ambigui- ties whic are presen only in non-rigid graphs. In the remainder of this pap er, use \ex am biguit y" to mean the discon tin uous yp e. Graph theory suggests ys of testing if giv en graph has unique realization determining whether or not ip or ex am biguities are presen in sp ecic graph. Ho ev er, neither these sp cic tests nor the gener al principles of gr aph the or etic rigidity apply to the gr aph alization pr oblem when distanc me asur ements ar noisy Since realizations of the graph will rarely satisfy the distance constrain ts exactly al- ternativ realizations can exist that satisfy the constrain ts as ell as or etter than the correct realization, ev en when the graph is rigid in the graph theoretic sense. In this situ- ation, assuming can mo del the measuremen noise as random pro cess, it is desirable to lo calize only those ertices that ha small pr ob ability of eing sub ject to ip or ex am biguit Our algorithm uses robust quadrilaterals as building blo for lo calization, adding an additional constrain e- ond graph rigidit This constrain ermits lo calization of only those no des whic ha high lik eliho of unam bigu- ous realization. presen the algorithm itself in the next section, then justify it deriving the orst-case error lik e- liho in Section 3.

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2. APPR CH or simplicit describ o-dimensional lo calization. Ho ev er, our algorithm extends straigh tforw ardly to three dimensions. dene no de’s neighb ors to those no des that ha direct bidirectional comm unications and ranging capabilit to it. Dep ending on the yp of ranging mec ha- nism used the net ork, these conditions ma alw ys satised together. cluster is no de and its set of neigh- ors. The algorithm can brok en do wn in to three main phases. The rst phase lo calizes clusters in to lo cal co ordinate sys- tems. The optional second phase renes the lo calization of the clusters. The third phase computes co ordinate transfor- mations et een these lo cal co ordinate systems. When all three phases are complete, an lo cal co ordinate system can reconciled in to unique global co ordinate system. Alter- nativ ely the transformation et een an connected pair of clusters can computed on-line haining the individual cluster transformations as messages are passed through the net ork. The three phases of the algorithm are as follo ws: Phase I. Cluster Localiza tion Eac no de ecomes the cen ter of cluster and estimates the relativ lo cation of its neigh ors whic can unam biguously lo calized. call this pro cess cluster lo alization or eac clus- ter, iden tify all robust quadrilaterals and nd the largest subgraph comp osed solely of erlapping ro- bust quads. This subgraph is also trilateration graph as in [4]; our restriction to robust quads pro vides an additional constrain that minimizes the probabilit of realizing ip am biguit osition estimates within the cluster can then computed incremen tally follo wing the hain of quadrilaterals and trilaterating along the as in Figure 1. Phase I. Cluster Optimiza tion (optional) Rene the osition estimates for eac cluster using umerical op- timization suc as spring relaxation or Newton-Raphson with the full set of measured distance constrain ts. This phase reduces and redistributes an accum ulated er- ror that results from the incremen tal approac used in the rst phase. It can omitted when maxim um e- ciency is desired. Note that this optimization imp oses no comm unications erhead since it is erformed er cluster and not the net ork as whole. Phase I. Cluster Transf orma tion Compute transfor- mations et een the lo cal co ordinate systems of neigh- oring clusters nding the set of no des in com- mon et een clusters and solving for the rotation, translation, and ossible reection that est aligns the clusters. This cluster-based approac has the adv an tage that eac no de has lo cal co ordinate system with itself as the ori- gin. The algorithm is easily distributed ecause clusters are lo calized using only distance measuremen ts to immediate neigh ors and et een neigh ors. urthermore, if one no de in the net ork mo es, only the (1) clusters con taining that no de ust up date their osition information. The follo wing sections describ the phases of the algorithm in more detail. 2.1 Cluster Localization The goal of cluster lo calization is to compute the osi- tion of cluster’s no des in lo cal co ordinate system up to (a) (b) PSfrag replacemen ts Figure 4: (a) The robust four-v ertex quadrilateral. The haracteristic features of this subgraph are that eac ertex is connected to ev ery other distance measuremen and that kno wing the lo cations of an three ertices is sucien to compute the lo cation of the fourth using trilateration. (b) Decomp osition of the robust quadrilateral in to four triangles. global rotation and ossible reection. An no des that are not part of the largest subgraph of robust quads in the cluster will not lo calized. Ho ev er, after Phases I-I are complete, the ositions of man of these unlo calized no des can computed using more error prone metho ds that do not rely on robust quads. do not use suc metho ds in this phase since inaccurate osition estimates will com- ounded later phases of the algorithm. Our cluster-based lo calization strategy is similar to that prop osed in [2] except that our use of robust quads sp ecically oids ip am bigu- ities. Quadrilaterals are relev an to lo calization ecause they are the smallest ossible subgraph that can unam bigu- ously lo calized in isolation. Consider the no de subgraph in Figure 4, fully-connected distance measuremen ts. Assuming no three no des are collinear, these distance con- strain ts giv the quadrilateral the follo wing prop erties: 1. The relativ ositions of the four no des are unique up to global rotation, translation, and reection. In graph theory terms, the quadrilateral is glob al ly rigid 2. An globally rigid quadrilaterals sharing three er- tices form 5-v ertex subgraph that is also globally rigid. By induction, an um er of quadrilaterals hained in this manner form globally rigid graph. Despite these useful prop erties of the quadrilateral, global rigidit is not sucien to guaran tee unique graph realization when distance measuremen ts are noisy Th us, further restrict our quadrilateral to obust as follo ws. The quadrilateral sho wn in Figure 4a can decomp osed in to four triangles: ABC, ABD, A CD, and BCD, as sho wn in Figure 4b. If the smallest angle is near zero, there is risk that measuremen error, sa in edge AD, will cause ertex to reected er this sliv er of triangle as sho wn in Figure 5. Accordingly our algorithm iden ties only those triangles with sucien tly large minim um angle as robust. Sp ecically ho ose threshold min based on the measuremen noise and iden tify those triangles that satisfy sin min (2) where is the length of the shortest side and is the small- est angle, as robust. This equation ounds the orst-case probabilit of ip error for eac triangle. See Section for full deriv ation. dene obust triangle to triangle that satises Equation 2. urthermore, dene obust

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D PSfrag replacemen ts BD BD CD CD AD AD Figure 5: An example of ip am biguit realized due to measuremen noise. No de is trilaterated from the kno wn ositions of no des A, B, and C. Measured distances BD and CD constrain the o- sition of to the in tersections of the dashed circles. Kno wing AD disam biguates et een these ositions for D, but small error in AD (sho wn as AD selects the wrong lo cation for D. No de Graph Ov erlap Graph Figure 6: The dualit et een cluster ro oted at and graph of robust quads, whic call an over- lap gr aph In the erlap graph, eac robust quadri- lateral is ertex. Edges are presen et een quads whenev er they share three no des. Th us, if all four no de ositions are kno wn for some quad, an neigh oring quad in the erlap graph can use the three common no des to trilaterate the osition of the unkno wn no de. breadth-rst searc in to the erlap graph from some starting quad, trilaterat- ing along the lo calizes the cluster as describ ed Algorithm 2. Note that the erlap graph for cluster can ha distinct, unconnected subgraphs as sho wn in this example. No des that are unique to one subgraph cannot lo calized with resp ect to those of an unconnected subgraph. quadrilater al as fully-connected quadrilateral whose four sub-triangles are robust. ey feature of our algorithm is that use the ro- bust quadrilateral as starting oin t, and lo calize additional no des haining together connected robust quads. When- ev er quads ha three no des in common and the rst quad is fully lo calized, can lo calize the second quad trilaterating from the three kno wn ositions. natural rep- resen tation of the relationship et een robust quads is the overlap gr aph sho wn in Figure 6. Since three ertices in common mak it ossible to lo calize quads relativ to eac other, it is natural to represen the space as graph of robust quads. Lo calization then amoun ts to tra ersing the erlap graph with breadth-rst searc and trilaterating as go, linear time op eration as in [4]. The en tire algorithm for Phase I, cluster lo calization, is as follo ws: 1. Distance measuremen ts from eac one-hop neigh or are broadcast to the origin no de so that it has kno wl- edge of the et een-neigh or distances. 2. The complete set of robust quadrilaterals in the cluster is computed (Algorithm 1) and the erlap graph is generated. 3. osition estimates are computed for as man no des as ossible via breadth-rst searc in the erlap graph (Algorithm 2). the start of the graph searc h, ho ose ositions for the rst three no des to x the ar- bitrary translation, rotation, and reection. place the origin no de at (0 0) to sp ecify the global trans- lation, the rst neigh or on the -axis to sp ecify the global rotation, and the second neigh or in the osi- tiv direction to sp ecify the global reection. The remaining no des are trilaterated as they are encoun- tered. Algorithm Finds the set of robust quadrilaterals that con tain an origin no de Eac quad is stored as 4-tuple of its ertices and is returned in the set Quads assume that distance mea- suremen ts ha already een gathered as follo ws: Me as is set of ordered pairs that represen the distance from no de to no de min is the robustness threshold, computed from the measuremen noise as describ ed in Section 3. 1: for all pairs j; ij in Me as do 2: for all pairs in Me as do 3: Remo j; from Me as 4: for all pairs in Me as do 5: for all pairs m; lm in Me as do 6: if then 7: con tin ue 8: Retriev ik from Me as 9: Retriev il from Me as 10: if IsR obust lj min AND IsR obust ij ik min AND IsR obust ij il lj min AND IsR obust ik il min then 11: Add i; j; to Quads 12: Remo from Me as 2.2 Computing Inter -Cluster ransf ormations In Phase I, the transformations et een co ordinate sys- tems of connected clusters are computed from the nished cluster lo calizations. This transformation is computed nding the rotation, translation, and ossible reection that bring the no des of the lo cal co ordinate systems in to est coincidence [8]. After Phase is complete for the clusters, the ositions of eac no de in eac lo cal co ordi- nate system are shared. As long as there are at least three non-collinear no des in common et een the lo caliza- tions, the transformation can computed. By testing if these three no des form robust triangle, sim ultaneously

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Algorithm Computes osition estimates for the cluster cen- tered at no de This algorithm do es breadth-rst searc in to eac disconnected subgraph of the erlap graph created from Quads and nds the most complete lo calization ossible. the end of this algorithm, cs best is set con taining pairs j; where is the estimate for the x-y osition of no de An neigh- ors of not presen in cs best ere not lo calizable. 1: cs best := 2: for eac disconnected subgraph of the erlap graph do 3: cs := 4: Cho ose quad from the erlap graph. 5: := (0 0) osition of the origin no de 6: := ab 0) First neigh or sets -axis 7: := ab ac bc ab ac 8: := ac ; ac Lo calize the second neigh- or relativ to the rst 9: Add a; ), b; ), and c; to cs 10: for eac ertex visited in breadth-rst searc in to the erlap graph do 11: if the curren quadrilateral con tains no de that has not een lo calized et then 12: Let the ositions of the three previously lo calized no des. 13: := Trila tera te aj bj cj 14: Add j; to cs 15: if length( cs length( cs best then 16: cs best := cs guaran tee non-collinearit and the same resistance to ip am biguities as Phase of the algorithm. 3. AN AL YSIS 3.1 Pr oof of Rob ustness In order for Algorithm to pro duce correct graph re- alization, ust ensure that our use of robust quads pre- en ts oth ip and ex am biguities. Since distance mea- suremen ts ma ha arbitrary noise cannot guaran tee correct realization in all cases instead can only predict the probabilit of ha ving no ips based on our denition of robustness. It is dicult to quan tify this probabilit for an en tire graph, so instead fo cus on the probabilit of an individual error. That is, dene an \error" as the real- ization of single robust quad with one ertex ipp ed or exed from its correct lo cation. By deriving the orst-case probabilit of error, will pro our rst theorem: Theorem 1. or normal ly-distribute distanc me asur e- ment noise with standar deviation we an onstruct obustness test such that the worst-c ase pr ob ability of err or is ounde d. First, pro that the use of robust quadrilaterals rules out the ossibilit of ex am biguities as seen in Figure 3b. This kind of ex am biguit ccurs only when rigid graph ecomes non-rigid the remo al of single edge [7]. If the graph is suc that no single edge remo al will mak it non- rigid, the graph is dundantly rigid and no ex am biguities are ossible. The robust quad has six edges. By remo ving an edge, are left with 5-edged graph, whic ust rigid according to the follo wing theorem [10]: C PSfrag replacemen ts BC BC AB CD Figure 7: diagram of quadrilateral for deriving the orst-case probabilit of ip error. ertex is eing trilaterated from the kno wn ositions of er- tices A, B, and D. Its distance to ertex is used to disam biguate et een the ossible lo cations and testing whic of CD and is closer to the measured distance et een and D. Theorem (Laman’s Theorem) et gr aph have exactly dges wher is the numb er of vertic es. is generic al ly rigid in if and only if every sub gr aph with vertic es has or fewer dges. Our robust quad with its missing edge has ertices and edges, satisfying the condition in Laman’s Theorem. Since ev ery 3-v ertex subgraph has or few er edges and ev ery 2- ertex subgraph has or few er edges, the 5-edged quad is rigid. Th us, the 6-edged robust quad is redundan tly rigid. Therefore, ex am biguities are imp ossible for graph con- structed of robust quads. Unlik ex am biguities, ips cannot ruled out based on the graph structure alone. Since distance measuremen ts are noisy they ma cause ertex to incorrectly ipp ed in computed realization. Th us, deriv the orst-case probabilit of realizing ipp ed ertex. Figure depicts the scenario in whic ertex could ecome incorrectly ipp ed. In this example, ertex is eing trilaterated with resp ect to the kno wn ositions of ertices A, B, and D. emp orarily ignoring ertex D, can pinp oin to ossible lo ca- tions: and the in tersection oin ts of circles cen- tered at and B, of radius and BC disam biguate this ossible ip, use the kno wn distance to ertex as follo ws. compute distances CD and Whic hev er distance is closer to the measured distance CD will deter- mine whether or is selected during trilateration. The probabilit of an incorrect ip is equal to the proba- bilit that the measured distance CD will closer to the incorrect distance than to the correct distance CD Note that the problem has an in trinsic symmetry: namely disam biguating the osition of based on is equiv alen to disam biguating based on C. Assuming the random mea- suremen noise is zero-mean, there ust measuremen error of magnitude CD for an incorrect ip to realized. can deriv this alue from the graph in Fig- ure 7. or simplicit constrain the gure to left-righ symmetric, although the probabilit of error will only de- crease breaking this symmetry In this problem, tak the alues of AB and as giv en. will later elimi-

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nate maximizing the error with resp ect to it. First, compute the alues of CD and as: CD AB sin sin(2 and AB sin(2 sin sin sin Com bin- ing these yields er CD (3) AB sin sin sin sin sin(2 (4) Since are in terested in the orst-case probabilit of error, minimize er with resp ect to taking the partial deriv ativ of er and setting it equal to zero. nd that er is minimized when This can substituted in to Equation and the resulting equation simplied to yield er AB sin (5) Th us, if the true distance is and the measured distance is random ariable then the orst-case probabilit of error is er ). If measuremen noise is zero- mean Gaussian with standard deviation the orst-case probabilit of error is er er (6) where ( connotes the in tegration of the unit normal prob- abilit densit function from to innit This equation tells us that for arbitrary measuremen noise with standard deviation can ho ose threshold min for the robust- ness test. Only those triangles for whic sin min where is the shortest side and is the smallest angle, will treated as robust. By ho osing min to some constan ultiple of ound the probabilit of error. This pro es Theorem 1. or the sim ulation results presen ted in this pap er, min as hosen to or Gaussian noise, this ounds the probabilit of error for giv en robust quadrilateral to less than 1%. Ho ev er, for the ypical case, the probabilit is signican tly less than 1%, th us osing minimal threat to the stabilit of the lo calization algorithm. 3.2 Computational Complexity It is imp ortan that an distributed lo calization algorithm scalable to large net orks. In this section discuss the computational and comm unications eciency of the al- gorithm presen ted in Section 2. In general, nding re- alization of graph is NP-hard [20]. are able to do it in olynomial time ecause our algorithm purp osefully oids no des that ma ha osition am biguities (i.e., ips or exes) at the cost of failing to nd all ossible realiza- tions. It is these am biguities whic cause the general case to blo up com binatorially Our algorithm gro ws linearly with resp ect to the um er of no des when there are (1) neigh ors er no de. urthermore, since this computation is distributed across the net ork, eac no de erforms (1) computation. If the no de degree is not constan t, eac no de’s computation aries with the third er of the um er of neigh ors. Algorithm 1, whic nds the set of robust quadrilaterals in lo cal cluster, has orst-case run time where is the maxim um no de degree. It can implemen ted with run time using etter data structures. In practice, the algorithm is uc more ecien ecause eac neigh or is generally not connected to ev ery other neigh or. In this algorithm, simply en umerate the robust quadrilaterals in the cluster, th us the orst-cast um er of robust quadrilat- erals is whic is (1) for graph of ounded degree. Algorithm 2, whic solv es for osition estimates for one cluster, has run time where is the um er of robust quadrilaterals. In the orst case, Finding the in ter-cluster transformations for one cluster has run time ). are nding transformations, eac of whic ma tak time to compute ecause the regis- tration problem tak es linear time in the um er of erlap- ping ertices. Again, for graph of ounded degree, these computations tak (1) time. The only stage of the algorithm that en tails comm uni- cation erhead is the initial step where eac no de shares its measured distances with its neigh ors. If assume that non-o erlapping clusters do not share the same hannel (due to range limitations), the comm unications erhead is ecause measuremen ts are eing shared. In prac- tice, this is implemen ted eac no de sending one pac et of constan size for distance measuremen and one pac et of size to share other measuremen ts. 4. EXPERIMENT AL RESUL TS In order to measure the eectiv eness of our algorithm on real sensor net orks, implemen ted it on-b oard func- tioning sensor net ork. The net ork is constructed of Cric k- ets, hardw are platform dev elop ed and supplied MIT [16]. Cric ets are hardw are-compatible with the Mica2 Motes de- elop ed at Berk eley with the addition of an Ultrasonic trans- mitter and receiv er on eac device. This hardw are enables the sensor no des to measure in ter-no de ranges using the time dierence of arriv al (TDoA) et een Ultrasonic and RF sig- nals. Although the Cric ets can ac hiev ranging precision of around cm on the lab enc h, in practice, the ranging error can as large as cm due to o-axis alignmen of the sending and receiving transducers. 4.1 Ev aluation Criteria One criteria whic ev aluate the erformance of the algorithm is ho the computed lo calization diers from kno wn ground truth. This error is expressed as =1 (7) where is the um er of no des, and comp ose the lo calized osition of no de and and comp ose the true osition of no de This metric is simply the mean-square error in Euclidean 2D space. It is useful to compare to the mean-square error in the ra distance measuremen ts, since the error mo del of the measuremen ts determines the minim um ac hiev able of an ideal lo calization algorithm [18]. The mean-square error of the distance measuremen ts is =1 (8) where is the um er of in ter-no de distances, is the measured alue of distance and is the true alue of distance Another useful metric is the prop ortion of no des success- fully lo calized the algorithm. Let the um er of

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Y-P osition (cm) 50 100 150 200 −20 20 40 60 80 100 120 140 160 180 200 Ground truth Localized positions X-P osition (cm) Figure 8: comparison of no de ositions as lo cal- ized our algorithm to the true ositions of the no des on ph ysical Cric et cluster. ositions are computed Phase of the algorithm, cluster lo cal- ization. The exp erimen in olv ed 16 no des, one of whic could not lo calize; th us only 15 are sho wn. no des successfully lo calized in the cluster cen tered at no de and the total um er of no des in this cluster. In one cluster, the prop ortion of no des lo calized is =k or the en tire net ork, dene the cluster suc ess ate as (9) This metric tells us the erage ercen tage of no des that ere lo calizable er cluster. Our nal metric con eys the prop ortion of no des in the entir net ork that could lo calized in to single co ordi- nate system. Since some clusters will ha transformations et een them and others will not, the net ork ma split in to separate subgraphs, eac of whic is lo calized with resp ect to all its no des, but is not rigidly lo calized with resp ect to the other subgraphs. call these subgraphs for ests Nat- urally it is desirable for there to only single forest that con tains ev ery no de in the net ork. Th us, another use- ful metric is the lar gest for est size whic is the um er of no des in the largest forest. This metric can expressed as ercen tage dividing the total um er of no des in the net ork. 4.2 Accuracy Study: Hard war Deployment Figure sho ws the results of the rst exp erimen t. In this exp erimen t, 16 cric ets ere placed in pseudo-random, 2-dimensional arrangemen t. Ground truth as measured man ually with the aid of grid on the surface. The small circles depict the ositions of eac no de as computed the lo calization algorithm running on-b oard the cric et in the ottom-most osition of the gure. The ositions sho wn are for Phase of the algorithm, where ositions are trilaterated using robust quadrilaterals. No least-squares optimization as erformed. The true osition of eac no de is sho wn with an \x." line et een the oin ts sho ws the amoun of ositioning error. The error metrics for the exp erimen sho wn in Figure Y-P osition (cm) −100 100 200 300 400 −150 −100 −50 50 100 150 200 250 300 Ground truth Localized positions X-P osition (cm) Figure 9: The lo calized ositions of 40 Cric ets in ph ysical net ork. The \holes" in the net ork are where no des could not lo calize, and th us only 38 are sho wn. The co ordinate transformations et een eac cluster ere computed and used to render the lo calized ositions in the single co ordi- nate system seen here. Ground truth ositions are erlaid, with lines sho wing the amoun of error for eac no de. The dotted line depicts the exten of one cluster. are as follo ws: metric alue 5.18 cm 7.02 cm 15 16 94 15 16 94 The fact that (the lo calization error) is only sligh tly larger than (the measuremen error) tells us that our algorithm erformed ell relativ to the qualit of the distance mea- suremen ts ailable. In addition, all no des but one ere suc- cessfully lo calized, indicating that the algorithm pro vided go lo calization co erage of the cluster. second exp erimen t, the results of whic are sho wn in Figure 9, demonstrates oth Phase and Phase of the algorithm. Once again, for simplicit the optional least- squares relaxation phase is omitted. total of 40 no des ere placed in meter region. The RF and ultrasound ranges of eac Cric et ere arbitrarily restricted so that only 12 neigh ors ere rangeable from eac no de. Then, cluster lo calization as erformed separately on v no des, dividing the net ork in to v clusters. The range of one cluster is sho wn dotted line in the gure. Phase of the algorithm, running on eac of the v clusters, lo calized its no des in lo cal co ordinate system. ransformations e- een eac pair of co ordinate systems with at least three no des in common ere computed Phase of the algo- rithm. Figure sho ws the lo calized ositions of eac no de as small circles, erlaid with the ground truth. The lo calized ositions of three no des are used to bring the en tire net ork in to registration with the global co ordinate system used ground truth. The error metrics for the Figure exp erimen are:

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Figure 10: The oce o orplan used for sensor net- ork sim ulation. Dark lines are the alls of the building and ligh t-colored lines represen the graph edges et een no des. Eac edge represen ts dis- tance measuremen that no de can erform. Mea- suremen ts cannot tak en through alls. metric alue 4.38 cm 6.82 cm 97 38 40 95 As in the rst exp erimen t, these results sho that lo caliza- tion error as not uc greater than measuremen error. 4.3 Scalability Study: Simulated Deployment ha tested the three phases of our algorithm on ariet of sim ulated net orks in order to ev aluate its scal- abilit ey ond the ph ysical exp erimen ts erformed in Sec- tion 4.2. In this pap er, sim ulate an en vironmen based on an actual o orplan of an oce building, sho wn in Fig- ure 10. placed 183 no des uniformly and randomly in the o-dimensional region, but connectivit is only ailable et een no des that are within the maxim um ranging dis- tance and not obstructed alls. The o orplan has three ro oms and one hallw and is appro ximately square with eac side 10 long. When ev aluating the algorithm’s erformance, are in- terested in ho oth no de degree and measuremen noise aect the results. No de degree as aried hanging the maxim um ranging distance. also consider three dieren degrees of measuremen noise: 1. Zero noise, where all measuremen ts are exact. Sim ula- tions without noise giv an upp er ound on ho uc lo calization is ossible for net ork. Without noise, an unlo calizable no des ust due to disconnection or non-rigidit in the graph structure. 2. Noise with cm, similar to that of Cric et device in ideal circumstances. 3. Noise with 10 cm. This gure is to designed to sim ulate sensor net orks with more imprecise ranging capabilit Figure 11 sho ws the sim ulation results for the building en- vironmen t. Eac data oin on the plots represen single run of the sim ulation, whic lo calizes as man no des as os- sible. As one ould exp ect, the abilit of the algorithm to lo calize go es do wn as the measuremen noise increases. In- terestingly the algorithm is nearly as eectiv with cm noise as with zero noise. With more noise, the algo- rithm is still eectiv e, but the requiremen ts for no de degree are higher. Note that the largest forest size rarely ob- tained 100% ev en with high no de degree due to obstruction alls. In practical deplo ymen t, no des ould ha to strategically placed around do orw ys to ac hiev 100% forest size. 4.4 Err or Pr opagation Cluster-based lo calization algorithms generally suer from or error propagation haracteristics ecause they ha no absolute reference oin ts as constrain ts. sho that our approac h, using robust quads, signican tly reduces the amoun of error propagated er approac hes based on basic trilateration. Figure 12a sho ws lo calization results of our algorithm after Phase and on sim ulated net ork of 100 no des. No des ere randomly placed within the square region, eac with maxim um ranging distance of 350 cm. Distance measure- men ts ere corrupted Gaussian noise with cm. In order to compare to ground truth, pic three no des as \anc hors". These no des are used solely for transforming et een the separate co ordinate systems of the algorithm and ground truth, and are not used the algorithm at run-time. The anc hor no des are closely-spaced so that er- rors can accum ulate to ards the edges of the net ork. In con trast, Figure 12b sho ws lo calization results for the same net ork, but with an algorithm that uses trilateration alone and do es not hec for quad robustness. The arious error metrics for three sim ulation runs are as follo ws. Eac as run with dieren amoun of measure- men noise, The error metrics for the sim ulation without robust quads are also sho wn: metric Our algorithm w/o robust quads 1.0 cm 3.0 cm 5.0 cm 5.0 cm 4.43 cm 14.39 cm 16.22 cm 54.87 cm 0.91 0.85 0.79 0.95 0.93 0.87 0.75 0.99 Sho wn in: Figure 12a Figure 12b This comparison demonstrates that robust quads signi- can tly reduce error propagation. 4.5 Localization of Mobile Nodes An adv an tage of our algorithm is that it handles no de mobilit ell ecause eac cluster lo calization can re- computed quic kly Ev en on lo er device, the cluster lo calization phase can tak less than one second for 15{20 neigh ors. Th us, as no des mo e, Phase can simply re- eated to eep up. urthermore, excluding mobile no des from the transformation computation in Phase I, it do es not need to rep eated. There are practical issues with time dierence of arriv al (TDoA) distance estimation that complicate the handling of no de mobilit Sp ecically since eac no de-to-no de dis- tance is estimated at dieren momen in time, mo ving no de will corrupt the self-consistency of the distance mea- suremen ts. Suc discrepancies will urt the erformance of

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(a) Plots of cluster success ra te, versus node degree or the building envir onment 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) (b) Plots of lar gest orest size, versus node degree or the building envir onment 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Largest forest size (p ercen t) 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Largest forest size (p ercen t) 10 15 20 25 30 10 20 30 40 50 60 70 80 90 100 PSfrag replacemen ts cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Cluster success rate (p ercen t) cm (no noise) cm (Cric et noise) 10 cm Avg. no de degree Avg. no de degree Avg. no de degree Largest forest size (p ercen t) Figure 11: (a) The cluster success rate ersus the erage no de degree for three dieren lev els of mea- suremen noise in the building en vironmen of Figure 10. Eac data oin sho ws the alue of these quan tities for single sim ulation run. mo ving erage of the data oin ts is erlaid on eac plot. (b) The size of the largest forest ersus erage no de degree for three dieren lev els of measuremen noise. the lo calization algorithm and oten tially cause large errors. In our exp erimen ts solv this problem measuring the distance to mo ving no des using only ultrasound pulses gen- erated the mo ving no de itself. This all other no des will sense the same ph ysical pulse and generate measure- men estimate for the same momen in time. These mea- suremen ts are then shared within eac cluster. Mobilit also complicates the handling of noise, since out- liers and noisy measuremen ts can misconstrued as ob- serv ed motion. address this issue rst feeding the ra measuremen ts in to er-edge Kalman Filter with state ariables: the no de-to-no de distance and the rate of hange of the distance. The lter sim ultaneously smo oths noisy measuremen ts and eliminates outliers rejecting measure- men ts with noise inconsisten with the lter state. The l- ters for static distances are tuned with single pro cess noise that limits mobilit and the lters for mobile distances are tuned to allo mobilit Algorithm is then run on the out- puts of the lters instead of the ra measuremen ts. These er-link lters use uc less state and computation than Kalman Filter in olving all no des sim ultaneously (e.g. [22]). nal issue in lo calization with mobilit is that trilatera- tion can inaccurate when trac king mo ving device, since it do es not generally use all distance constrain ts ailable. Using large um er of constrain ts is imp ortan for mobile lo calization, whic has more noise than static lo calization. Th us, ha found it imp ortan to use least-squares op- timization, emplo ying all distance constrain ts for osition renemen t, after computing an initial estimate in Phase I. Figure 13 sho ws our exp erimen tal results lo calizing mo- bile no de. Six stationary no des ere deplo ed in roughly circular conguration, as sho wn the small circles in the gure. no de as attac hed to an autonomous rob ot placed in the cen ter of the stationary no des. Once activ ated, the rob ot randomly tra ersed rectangular space. The lo caliza- tion as computed the sensor net ork as logged er time and man ually sync hronized with calibrated video camera. The video as ost-pro cessed to obtain the ground truth rob ot path with sub-cen timeter accuracy This path as then compared to the path computed the lo calization al- gorithm. The lo calization algorithm computed osition estimate for the rob ot roughly once er second for min-

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(a) Our algorithm Y-P osition (cm) 500 1000 1500 500 1000 1500 Localized node position Unlocalizable node Anchor node X-P osition (cm) (b) Algorithm without robust quads Y-P osition (cm) 500 1000 1500 500 1000 1500 Localized node position Unlocalizable node Anchor node X-P osition (cm) Figure 12: (a) Our algorithm’s lo calized ositions for sim ulated net ork compared to ground truth. Lines sho the amoun of error for eac no de’s o- sition. The three no des used to compute the trans- formation to the ground truth’s co ordinate system are sho wn with small circles. The large dotted cir- cle depicts the maxim um ranging distance of no de. (b) Lo calization of the same net ork using basic tri- lateration without hec king for quad robustness. utes. Since discrete computations ere made, eac of these separate lo calizations could compared to ground truth. The mean-square error, computed from these alues is 2.59 cm. Th us, our lo calization algorithm is sho wn to successful at lo calizing net orks with mobile no des. 5. CONCLUSION ha demonstrated an algorithm that successfully lo- calizes no des in sensor net ork with noisy distance mea- suremen ts, using no eacons or anc hors. Sim ulations and exp erimen ts sho ed the relationship et een measuremen (a) Lo calized path of mobile no de Y-P osition (cm) 50 100 150 200 −180 −160 −140 −120 −100 −80 −60 −40 −20 20 Localized path True path X-P osition (cm) (b) Magnitude of lo calization error vs. time Error, Euclidean distance (cm) 20 40 60 80 100 120 140 160 180 200 10 Time (s) (c) Exp erimen tal setup Figure 13: (a) The path of mobile no de computed our lo calization algorithm compared to ground truth er min ute erio d. sensor no de as attac hed to mobile rob ot (an autonomous o or acuum) that randomly co ered rectangular space. Six static no des, depicted as circles, ere used to lo calize this mobile no de er time. Ground truth (dashed) as obtained from calibrated video. (b) The Euclidean distance et een the mobile no de’s lo calized osition and ground truth er time. (c) photo of the exp erimen tal setup.

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noise and abilit of net ork to lo calize itself. As long as the error mo del of the measuremen noise is kno wn, the al- gorithm cop es with this noise refusing to lo calize those no des whic ha am biguous ositions. urthermore, ev en with no noise, eac no de in the net ork ust ha appro xi- mately degree 10 or more efore 100% no de lo calization can attained. As noise increases, so will the connectivit re- quiremen ts. The Cric et platform has mo derate amoun of noise and th us exercises our algorithm’s tolerance for noisy distance measuremen ts. ha also sho wn that the algo- rithm adapts to no de mobilit ltering the underlying measuremen ts. or future ork, are in terested in extending our ph ys- ical exp erimen ts to ev en larger no de deplo ymen ts that also include obstructions. Also, ould lik to use the prin- ciple of robust quads to compute the optimal placemen of additional no des so that partially lo calized graph ecomes fully lo calizable. Finally it ould useful to further re- ne our approac to allo \passiv e" mobile no des to lo calize without transmitting. 6. CKNO WLEDGEMENTS This ork as funded gran from Pro ject Oxygen and supp orted in part National Science oundation Graduate Researc ello wship. Additional funding as pro- vided in part the Institute for Securit ec hnology Stud- ies (ISTS) at Dartmouth College, NSF ard 0225446, ONR ards N00014-01-1-0675, N00014-02-C-0210, and N00014- 03-1-0879, and ARP ASK program ard F30602-00- 2-0585. thank the Cric et pro ject for supplying hardw are and programming assistance and the BMG (Building Mo del Gen- eration) pro ject for o orplans. atric Nic hols wrote the constrained eacon graph generator. are grateful to Erik Demaine for useful discussions and oin ters to literature. 7. REFERENCES [1] Bulusu, N., Heidemann, J., and Estrin, D. GPS-less lo cost outdo or lo calization for ery small devices. IEEE Personal Communic ations Magazine (Octob er 2000), 28{34. [2] Capkun, S., Hamdi, M., and Huba ux, J.-P. GPS-free ositioning in mobile ad-ho net orks. In Pr dings of the 34th Hawaii International Confer enc on System Scienc es (2001). [3] Doher ty, L., Pister, K. S. J., and Gha oui, L. E. Con ex osition estimation in wireless sensor net orks. In Pr c. IEEE INF OCOM (Anc horage, AK, April 2001). [4] Eren, T., Goldenber g, D., Whiteley, W., ang, Y. R., Morse, A. S., Anderson, B. D. O., and Belhumeur, P. N. Rigidit computation, and randomization in net ork lo calization. In Pr c. IEEE INF OCOM (Marc 2004). [5] Estrin, D., Go vind an, R., and Heidemann, J. Em edding the in ternet: in tro duction. Commun. CM 43 (2000), 38{41. [6] Grabo wski, R., and Khosla, P. Lo calization tec hniques for team of small rob ots. In Pr c. IEEE IR OS (Maui, Ha aii, Octob er 2001). [7] Hendrickson, B. Conditions for unique graph realizations. SIAM J. Comput. 21 (1992), 65{84. [8] Horn, B. K. P. Closed form solution of absolute orien tation using unit quaternions. Journal of the Optic al So ciety (April 1987), 629{642. [9] Ji, X., and Zha, H. Sensor ositioning in wireless ad-ho sensor net orks using ultidimensional scaling. In Pr c. IEEE INF OCOM (Marc 2004). [10] Laman, G. On graphs and rigidit of plane sk eletal structures. J. Engine ering Math (1970), 331{340. [11] Na gp al, R., Shr obe, H., and Ba chra ch, J. Organizing global co ordinate system from lo cal information on an ad ho sensor net ork. In Pr c. IPSN (P alo Alto, CA, April 2003), pp. 333{348. [12] Niculescu, D., and Na th, B. based ositioning in ad ho net orks. Kluwer journal of ele ommunic ation Systems (2003), 267{280. [13] Niculescu, D., and Na th, B. Error haracteristics of ad ho ositioning systems (APS). In Pr c. 5th CM MobiHo (T oky o, Ma 2004). [14] tw ari, N., I, A. O. H., Perkins, M., Correal, N. S., and O’Dea, R. J. Relativ lo cation estimation in wireless sensor net orks. IEEE ans. Signal Pr ess. 51 (August 2003), 2137{2148. [15] Priy antha, N. B., Balakrishnan, H., Demaine, E., and Teller, S. Anc hor-free distributed lo calization in sensor net orks. ec h. Rep. 892, MIT Lab. for Comp. Sci., April 2003. [16] Priy antha, N. B., Chakrabor ty, A., and Balakrishnan, H. The Cric et lo cation-supp ort system. In Pr c. 6th CM MobiCom (Boston, MA, August 2000). [17] Sa arese, C., Rabaey, J., and Langendoen, K. Robust ositioning algorithms for distributed ad-ho wireless sensor net orks. In USENIX nnual ch. Conf. (Mon terey CA, June 2002), pp. 317{327. [18] Sa vvides, A., Garber, W., Adlakha, S., Moses, R., and Sriv ast a, M. B. On the error haracteristics of ultihop no de lo calization in ad-ho sensor net orks. In Pr c. IPSN (P alo Alto, CA, April 2003), pp. 317{332. [19] Sa vvides, A., Han, C.-C., and Sriv ast a, M. B. Dynamic ne-grained lo calization in ad-ho net orks of sensors. In Pr c. 7th CM MobiCom (Rome, Italy 2001), pp. 166{179. [20] Saxe, J. B. Em eddabilit of eigh ted graphs in k-space is strongly NP-hard. In Pr c. 17th lerton Conf. Commun. Contr ol Comput. (1979), pp. 480{489. [21] Simic, S. N., and Sastr y, S. Distributed lo calization in wireless ad ho net orks. ec h. Rep. UCB/ERL M02/26, UC Berk eley Decem er 2001. [22] Smith, A., Balakrishnan, H., Gora czk o, M., and Priy antha, N. rac king mo ving devices with the cric et lo cation system. In Pr c. 2nd CM MobiSys (Boston, MA, June 2004), pp. 190{202. [23] Teller, S., Chen, J., and Balakrishnan, H. erv asiv ose-a are applications and infrastructure. IEEE Computer Gr aphics and Applic ations (July/August 2003), 14{18.

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