Localization fr om Mere Connectivity Shang Univ ersity of Missour iColumbia Columbia MO shangymissour i PDF document - DocSlides

Localization fr om Mere Connectivity Shang Univ ersity of Missour iColumbia Columbia MO  shangymissour i PDF document - DocSlides

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edu Wheeler Ruml ing Zhang Mar kus romherz alo Alto Research Center alo Alto CA 94304 uml yzhang fromherz parccom ABSTRA CT It is often useful to kno the geographic ositions of no des in comm unications net ork but adding GPS receiv ers or other soph ID: 23154

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Localization fr om Mere Connectivity Shang Univ ersity of Missour i–Columbia Columbia, MO 65211 shangy@missour i.edu Wheeler Ruml, ing Zhang, Mar kus romherz alo Alto Research Center alo Alto CA 94304 uml, yzhang, fromherz @parc.com ABSTRA CT It is often useful to kno the geographic ositions of no des in comm unications net ork, but adding GPS receiv ers or other sophisticated sensors to ev ery no de can exp ensiv e. presen an algorithm that uses connectivit information| who is within comm unications range of whom|to deriv the lo cations of the no des in the net ork. The metho can tak adv an tage of additional information, suc as estimated dis- tances et een neigh ors or kno wn ositions for certain an- hor no des, if it is ailable. The algorithm is based on ul- tidimensional scaling, data analysis tec hnique that tak es time for net ork of no des. Through sim ulation studies, demonstrate that the algorithm is more robust to measuremen error than previous prop osals, esp ecially when no des are ositioned relativ ely uniformly throughout the plane. urthermore, it can ac hiev comparable results using man few er anc hor no des than previous metho ds, and ev en yields relativ co ordinates when no anc hor no des are ailable. Categories and Subject Descriptors C.2.3 Computer Systems Organization ]: Net ork Op- erations General erms Algorithms, erformance eyw ords osition estimation, no de lo calization, ultilateration, ul- tidimensional scaling, ad-ho net orks, sensor net orks 1. INTR ODUCTION Large-scale net orks with undreds and ev en thousands of ery small, battery-p ered and wirelessly connected sensor and actuator no des are ecoming realit [5]. or example, future sensor net orks will in olv ery large um er of 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. MobiHoc’03, June 1–3, 2003, Annapolis, Maryland, USA. Cop yright 2003 CM 1-58113-684-6/03/0006 ... 5.00. densely deplo ed sensor no des er ph ysical space. In par- ticular, the no des are ypically highly resource-constrained (pro cessor, memory and er), ha limited comm unica- tion range, are prone to failure, and are put together in ad-ho net orks. Imagine net ork of sensors sprinkled across large build- ing or an area suc as forest. ypical tasks for suc net- orks are to send message to no de at giv en lo cation (without kno wing whic no de or no des are there, or ho to get there), to retriev sensor data (e.g., sound or tem- erature lev els) from no des in giv en region, and to nd no des with sensor data in giv en range. Most of these tasks require kno wing the ositions of the no des, or at least relativ ositions among them. With net ork of thou- sands of no des, it is unlik ely that the osition of eac no de has een pre-determined. No des could equipp ed with global ositioning system (GPS) to pro vide them with ab- solute osition, but this is curren tly costly solution. In this pap er, presen metho for computing the o- sitions of no des giv en only basic information that is lik ely to already ailable, namely whic no des are within com- unications range of whic others. The metho d, MDS- MAP has three steps. Starting with the giv en net ork connectivit information, rst use an all-pairs shortest- paths algorithm to roughly estimate the distance et een eac ossible pair of no des. Then use ultidimensional scaling (MDS), tec hnique from mathematical psyc hology to deriv no de lo cations that t those estimated distances. Finally normalize the resulting co ordinates to tak in to accoun an no des whose ositions are kno wn. As will demonstrate, this simple tec hnique often out- erforms existing metho ds. urthermore, it requires only connectivit information to pro duce meaningful result. If the distances et een neigh oring no des can estimated, that information can easily incorp orated in to the pair- wise shortest-path computation during the rst step of the algorithm. MDS yields co ordinates that pro vide the est t to the estimated pairwise distances, but whic lie at an arbitrary rotation and translation. If the co ordinates of an no des are kno wn, they can used to deriv the ane transformation of the MDS co ordinates that allo ws the est matc to the kno wn ositions. Only three suc ‘anc hor no des are necessary to pro vide absolute ositions for all the no des in the net ork. The next section of the pap er describ es MDS-MAP in more detail. will then pro vide an erview of previous prop os- als efore presen ting our empirical ev aluation. Our presen- tation fo cuses on cen tralized ersion of the algorithm, al- 201
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though will briey men tion ho the computation can distributed. will examine the erformance of MDS-MAP on net orks of 100 to 200 no des, with no de lo cations either hosen randomly or according to rough grid or hexagon la out. consider ariet of no de densities (no des er comm unications radius) and ariet of ranging errors when using the estimated distances et een neigh ors. will see that MDS-MAP reco ers more accurate maps of no de lo cations while using uc less information. 2. LOCALIZA TION USING MDS-MAP consider the no de lo calization problem under dif- feren scenarios. In the rst, only pro ximit (or connectiv- it y) information is ailable. Eac no de only kno ws what no des are nearb presumably means of some lo cal com- unication hannel suc as radio or sound, but not ho far these neigh ors are or in what direction they lie. In the second scenario, the pro ximit information is enhanced kno wing the distances, erhaps with limited accuracy et een neigh oring no des. In oth cases, the net ork is represen ted as an undirected graph with ertices and edges The ertices corre- sp ond to the no des, of whic there exist sp ecial no des with kno wn ositions, whic will call anc hors. or the pro ximit y-only case, the edges in the graph corresp ond to the connectivit information. or the case with kno wn distances to neigh ors, the edges are asso ciated with alues corresp onding to the estimated distances. assume that all the no des eing considered in the ositioning problem form connected graph, i.e., there is path et een ev ery pair of no des. There are ossible outputs when solving the lo cal- ization problem. One is relativ map and the other is an absolute map. The task of nding relativ map is to nd an em edding of the no des in to either o- or three-dimensional space that results in the same neigh or relationships as the underlying net ork. Suc relativ map can pro vide cor- rect and useful information ev en though it do es not neces- sarily include accurate absolute co ordinates for eac no de. Relativ information ma all that is obtainable in situa- tions in whic erful sensors or exp ensiv infrastructure cannot installed, or when there are not enough anc hors presen to uniquely determine the absolute ositions of the no des. urthermore, some applications only require relativ ositions of no des, suc as in some direction-based routing algorithms. Sometimes, ho ev er, an absolute map is re- quired. The task of nding an absolute map is to determine the absolute geographic co ordinates of all the no des. This is needed in applications suc as geographic routing and target disco ering and trac king. As will sho elo w, our metho can oten tially gener- ate oth results, dep ending on the um er of anc hor no des. The metho rst generates relativ map of the net ork and then transforms it to absolute ositions if sucien an- hors are ailable. Before describ the details of our metho d, rst in tro duce ultidimensional scaling (MDS), with fo cus on classical MDS, whic is used to generate the relativ map. 2.1 Multidimensional Scaling (MDS) Imagine small cloud of colored eads susp ended in mid- air. haracterize the arrangemen t, one could measure the straigh t-line distance et een eac pair of eads. If the cloud ere shattered and the eads fell to the o or, one could imagine trying to recreate the arrangemen based on the recorded in terp oin distances. One ould try to deter- mine lo cation for eac ead suc that the distances in the new arrangemen matc hed the desired distances. This recre- ation pro cess is exactly the problem that ultidimensional scaling (MDS) solv es. In tuitiv ely it is clear that while the distances will more than enough to determine co ordinates, the result of MDS will an arbitrarily rotated and ipp ed ersion of the true original la out ecause the in terp oin distances mak no reference to an absolute co- ordinates. MDS has its origins in psyc hometrics and psyc hoph ysics. It can seen as set of data analysis tec hniques that dis- pla the structure of distance-lik data as geometrical pic- ture [1]. MDS starts with one or more distance matrices (or similarit matrices) that are presumed to ha een deriv ed from oin ts in ultidimensional space. It is usually used to nd placemen of the oin ts in lo w-dimensional space, usually o- or three-dimensional, where the distances e- een oin ts resem ble the original similarities. MDS is of- ten used as part of exploratory data analysis or informa- tion visualization. By visualizing ob jects as oin ts in lo w-dimensional space, the complexit in the original data matrix can often reduced while preserving the essen tial information. MDS is closely related to principal comp onen analysis, and is also related to factor analysis and cluster analysis. There are man yp es of MDS tec hniques. They can classied according to whether the similarit data is qualita- tiv (nonmetric MDS) or quan titativ (metric MDS). They can also classied according to the um er of similar- it matrices and the nature of the MDS mo del. Classical MDS uses one matrix. Replicated MDS uses sev eral matri- ces, represen ting distances measuremen ts tak en from sev eral sub jects or under dieren conditions. eigh ted MDS uses distance mo del whic assigns dieren eigh to eac dimension. Finally there is distinction et een deter- ministic and probabilistic MDS. In deterministic MDS, eac ob ject is represen ted as single oin in ultidimensional space, whereas in probabilistic MDS eac ob ject is repre- sen ted as probabilit distribution er the en tire space. fo cus on classical metric MDS in this pap er. Classical metric MDS is the simplest case of MDS: the data is quan ti- tativ and the pro ximities of ob jects are treated as distances in Euclidean space [15]. The goal of metric MDS is to nd conguration of oin ts in ultidimensional space suc that the in ter-p oin distances are related to the pro vided pro ximities some transformation (e.g., linear transfor- mation). If the pro ximit data ere measured without error in Euclidean space, then classical metric MDS ould ex- actly recreate the conguration of oin ts. In practice, the tec hnique tolerates error gracefully due to the erdeter- mined nature of the solution. Because classical metric MDS has closed-form solution, it can erformed ecien tly on large matrices. Let ij refer to the pro ximit measure et een ob jects and The Euclidean distance et een oin ts im and in an 202
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dimensional space is ij =1 ik When the geometrical mo del ts the pro ximit data er- fectly the Euclidean distances are related to the pro ximities transformation ij ij ). In classical metric MDS, linear transformation mo del is assumed, i.e., ij bp ij The distances are determined so that they are as close to the pro ximities as ossible. There are ariet of ys to dene \close". common one is least-squares denition, whic is used classical metric MDS. In this case, dene where is linear transformation of the pro ximities, and is matrix of errors (residuals). Since is function of the co ordinates the goal of classical metric MDS is to calculate the suc that the sum of squares of is minimized, sub ject to suitable normalization of In classical metric MDS, is shifted to the cen ter and co ordinates can computed from the double cen tered through singular alue decomp osition (SVD). or an matrix for oin ts and dimensions of eac oin t, it can sho wn that ij =1 ij =1 ij =1 =1 ij =1 ik (1) The double cen tered matrix on the left hand side (call it is symmetric and ositiv semidenite. erforming sin- gular alue decomp osition on giv es us AV The co ordinate matrix ecomes Retaining the rst largest eigen alues and eigen ectors leads to solution in lo er dimension. This implies that the summation er in Eq. (1) runs from to instead of This is the est lo w-rank appro ximation in the least- squares sense. or example, for 2-D net ork, tak the rst largest eigen alues and eigen ectors to construct the est 2-D appro ximation. or 3-D net ork, tak the rst largest eigen alues and eigen ectors to construct the est 3-D appro ximation. In nonmetric (ordinal) MDS (rst dev elop ed Shep- ard [14]), the goal is to establish monotonic relationship et een in ter-p oin distances and the desired distances. In- stead of trying to directly matc the giv en distances, one is satised if the distances et een the oin ts in the solution fall in the same rank ed order as the corresp onding distances in the input matrix. The adv an tage of nonmetric MDS is that no assumptions need to made ab out the underly- ing transformation function. The only assumption is that the data is measured at the ordinal lev el. Just as classical MDS, nonmetric MDS can also applied to the lo calization problem. By adopting more exible mo del, the eects of few highly incorrect measuremen ts migh more easily tolerated. 2.2 The MDSMAP Algorithm Based on classical MDS, our metho d, called MDS-MAP consists of three steps: 1. Compute shortest paths et een all pairs of no des in the region of consideration. The shortest path dis- tances are used to construct the distance matrix for MDS. 2. Apply classical MDS to the distance matrix, retaining the rst (or 3) largest eigen alues and eigen ectors to construct 2-D (or 3-D) relativ map. 3. Giv en sucien anc hor no des (3 or more for 2-D, or more for 3-D), transform the relativ map to an abso- lute map based on the absolute ositions of anc hors. In step 1, rst assign distances to the edges in the connectivit graph. When the distance of pair of neigh- or no des is kno wn, the alue of the corresp onding edge is the measured distance. When only ha connectivit in- formation, simple appro ximation is to assign alue to all edges. Then classical all-pairs shortest-path algorithm, suc as Dijkstra’s or Flo yd’s algorithm, can applied. The time complexit is ), where is the um er of no des. In step 2, classical MDS is applied directly to the dis- tance matrix. The core of classical MDS is singular alue decomp osition, whic has complexit of ). The result of MDS is relativ map that giv es lo cation for eac no de. Although these lo cations ma accurate relativ to one an- other, the en tire map will arbitrarily rotated and ipp ed relativ to the true no de ositions. In step 3, the relativ map is transformed through linear transformations, whic include scaling, rotation, and reec- tion. The goal is to minimize the sum of squares of the errors et een the true ositions of the anc hors and their transformed ositions in the MDS map. Computing the transformation parameters tak es time, where is the um er of anc hors. Applying the transformation to the whole relativ map tak es time. 3. RELA TED ORK No de lo calization has een topic of activ researc in recen ears. detailed surv ey of the area is pro vided High to er and Borriello [6]. Ho ev er, few approac hes for lo cating no des in an ad-ho net ork are describ ed. Most systems use some kind of range or distance information and man of them rely on erful eacon no des with extreme capabilities, suc as radio or laser ranging devices. Dohert y’s [4] con ex constrain satisfaction approac for- ulates the lo calization problem as feasibilit problem with radial constrain ts. No des whic can hear eac other are constrained to lie within certain distance of eac other. This con ex constrain problem is in turn solv ed e- cien semi-denite programming (an in terior oin metho d) to nd globally optimal solution. or the case with di- rectional comm unication, the metho form ulates the lo cal- ization problem as linear programming problem, whic is solv ed an in terior oin metho d. The metho requires cen tralized computation. or the tec hnique to ork ell, it needs anc hor no des to placed on the outer oundary preferably at the corners. Only in this conguration are the constrain ts tigh enough to yield useful conguration. When all anc hors are lo cated in the in terior of the net ork, the osition estimation of outer no des can easily collapse to ard the cen ter, whic leads to large estimation errors. or example, with 10% anc hors, the error of unkno wns is on the order of the radio range. With anc hors in 200-no de 203
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random net ork, the error of unkno wns is more than wice the radio range. Most lo calization metho ds for ad-ho net orks require more information than just connectivit and use more w- erful eacon no des. The ad-ho lo calization tec hniques used in mobile rob ots usually fall in to this category [7, 10]. Mo- bile rob ots use additional dometric measuremen ts for esti- mating the initial rob ot ositions, whic are not ailable in sensor net orks. Man existing lo calization tec hniques for net orks use dis- tance or angle measuremen ts from xed set of reference oin ts or anc hor no des and apply ultilateration or trian- gulation tec hniques to nd co ordinates of unkno wn no des [8, 11]. The distance estimates can obtained from receiv ed signal strength (RSSI) or time-of-arriv al (T oA) measure- men ts. Due to non uniform signal propagation en vironmen ts, RSSI metho ds are not ery reliable and accurate. oA meth- ds ha etter accuracy but ma require additional hard- are at the sensor no des to receiv signal that has smaller propagation sp eed than radio, suc as ultrasound [12]. Em- phasis has een put on algorithms that can executed in distributed fashion on the sensor no des without cen tralized computation, comm unication, or information propagation. The \D V-based" approac Niculescu and Nath [9] is dis- tributed. The \D V-hop" metho ac hiev es an lo cation error of ab out 45% of the radio range for net orks with 100 no des, anc hors, and erage connectivit 7.6. It starts with the anc hor no des. The anc hors o their lo cation to all no des in the net ork. Eac unkno wn no de erforms triangula- tion to three or more anc hors to estimate its wn osition. The metho orks ell in dense and regular top ologies. or sparse and irregular net orks, the accuracy degrades to the radio range. The \D V-distance" metho uses distance e- een neigh oring no des and reduces the lo cation error ab out half of that of \D V-hop". Sa arese et al. prop ose another distributed metho [11]. The metho consists of phases: start-up and rene- men t. or the start-up phase, they use Hop-TERRAIN, an algorithm similar to V-hop. Hop-TERRAIN is run once at the eginning to generate rough initial estimate of the no des lo cations. Again, it needs at least anc hor no des to start. Then the renemen algorithm is run iterativ ely to impro and rene the osition estimates. The algorithm is concerned only with no des within one-hop neigh orho and uses least-squares triangulation metho to determine no de’s osition based on its neigh ors ositions and dis- tances to them. The approac can deliv er lo calization accu- racy within one third of the comm unication range. When the um er of anc hor no des is high, the collab o- rativ ultilateration approac Sa vvides et al. can used [13]. The metho estimates no de lo cations using anc hor lo cations that are sev eral hops and distance measuremen ts to neigh oring no des. global nonlinear op- timization problem is solv ed. The metho has three main phases: 1) formation of collab orativ subtree, whic only includes no des that can uniquely determined, 2) com- putation of initial estimates with resp ect to anc hor no des, 3) osition renemen minimizing the residuals et een the measured distances et een the no des and the distances computed using the no de lo cation estimates. They presen oth cen tralized computation mo del and distributed ap- pro ximation of the cen tralized mo del. The metho orks ell when the fraction of anc hor no des is high. The GPS-less system Bulusu et al. [3] emplo ys grid of eacon no des with kno wn ositions. Eac unkno wn no de sets its osition to the cen troid of the eacons near the un- kno wn. The osition accuracy is ab out one-third of the separation distance et een eacons, so the metho needs high eacon densit to ork ell. Almost all the existing metho ds need some kind of anc hor or eacon no des to start with. Our metho do es not ha this limitation. It builds relativ map of the no des ev en without anc hor no des. With three or more anc hor no des, the relativ map can transformed and absolute co ordi- nates of the no des are computed. Our metho orks ell in situations with lo ratios of anc hor no des and erforms ev en etter on regular net orks. limitation of the cur- ren implemen tation is that it is cen tralized. There are ar- ious ys to apply this metho in decen tralized or dis- tributed fashion. or example, the metho can applied to sub-net orks to obtained regional relativ maps, whic are patc hed together to form an erall map of the net ork. 4. EXPERIMENT AL RESUL TS In our exp erimen ts, ran MDS-MAP on arious top olo- gies of net orks in Matlab. The no des are placed (a) ran- domly with uniform distribution within square area, (b) on square grid with some placemen errors, or (c) on hexagonal grid with some placemen errors. In square grid, assuming is the unit length, no des are ypically placed in an nr nr square. mo del placemen errors for the grid la out as Gaussian noises. With placemen er- ror random alue dra wing from normal distribution (0 1) is added to the no de’s original grid osition. The placemen error in hexagonal grid is dened similarly The anc hor no des are selected randomly The data oin ts represen erages er 30 trials in net orks con taining 100 to 200 no des. In the connectivit y-only cases, eac no de only kno ws the iden tities of no des in its neigh orho but not the distance to them. In the kno wn-distance cases, eac no de kno ws the distances to its neigh or no des. The distance information is mo deled as the true distance blurred with Gaussian noise. Assume the true distance is and range error is then the measured distance is random alue dra wing from nor- mal distribution (1 (0 )). The connectivit (a erage um er of neigh ors) is con trolled sp ecifying radio range compare with previous results in [4, 11], the errors of osition estimates are normalized to (i.e., 50% osition er- ror means half of the range of the radio). do not consider mo dels of non-uniform radio propagation or widely arying ranging errors. Both mo deling these phenomena and sim u- lating their eects are ery imp ortan directions for future ork. 4.1 Random Placement In this set of exp erimen ts, 200 no des are placed randomly in 10 10 square. Figure sho ws an example using radio range of 1.5 whic leads to an erage connectivit of 12.1. In the graph, oin ts represen no des and edges rep- resen the connections et een neigh ors who can hear eac other. Figure sho ws the result of MDS-MAP based on the connectivit information of the net ork. Figure 2(a) sho ws the in termediate result of classical MDS on the connectivit matrix. It can seen that the cen ter of the map is the ori- gin (0,0), and it has dieren orien tation than the original 204
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Figure 1: 200 no des randomly placed in 10 10 square. The radio range is 1.5 yielding an erage connectivit of 12.1. net ork in Figure 1. Figure 2(b) sho ws the nal solution of MDS-MAP where the MDS result is transformed based on random anc hor no des, denoted the stars (*) in the net- ork. The circles represen the true lo cations of the no des, and the solid lines represen the errors of the estimated o- sition from the true osition. The longer the line, the larger the error. The erage estimation error in this example is ab out 0.46 When the distances et een neigh ors are kno wn, ev en with limited accuracy the result of MDS-MAP can sig- nican tly impro ed. Figure sho ws the result of MDS-MAP kno wing the distance of neigh ors with 5% range error. Fig- ure 3(a) sho ws the map constructed MDS. Again, it has dieren scale and orien tation than the ones in Figures and 2(a). Figure 3(b) sho ws the nal estimation of MDS-MAP based on the same anc hor no des. It has an erage esti- mation error of 0.24 uc etter than the previous result using connectivit only Figure sho ws the erage erformance of MDS-MAP as function of connectivit and um er of anc hors. Figure 4(a) sho ws results of MDS-MAP based on pro ximit information only The radio ranges are 1.25 1.5 1.75 and resp ectiv ely whic lead to erage connectivit lev els 5.9, 8.9, 12.2, 16.2, and 20.7, resp ectiv ely 3, 4, 6, and 10 random anc hors are used. osition estimates MDS-MAP ha an erage error under 100% in scenarios with just anc hor no des and an erage connectivit lev el of 8.9 or greater. When the connectivit lev el is 12.2 or greater, the errors with just anc hors is quite go d, close to or etter than 50%. On the other hand, when the connectivit is lo w, e.g., 5.9, the errors can large. These results are uc etter than the ones obtained the con ex optimization approac in [4] when the um er of anc hor no des is lo w. or example, with to 10 anc hors in 200-no de random net ork, the con ex optimization approac has an erage estimation error of more than wice the radio range when the radio range is 1.25 and ab e. In con trast, our metho has an error from ab out 80% do wn to 40% as the radio range go es from 1.25 to −4 −2 −4 −3 −2 −1 (a) In termediate MDS result 10 (b) Final osition estimation MDS-MAP Figure 2: Lo cation estimation for the random net- ork using connectivit information only Figure 4(b) sho ws results of MDS-MAP using distance measuremen et een neigh ors with 5% range error. Kno w- ing the distances et een neigh ors leads to uc etter so- lutions when the connectivit in the net ork is high. When the connectivit lev el is 12.2 or greater, the errors are ab out half of those MDS-MAP using pro ximit information only On the other hand, when the connectivit is lo w, e.g., 5.9, kno wing the lo cal distance do es not help uc and the er- rors are still large. These results impro on the results of Hop-TERRAIN [11], esp ecially when the um er of an- hors is small. or example, with 2% anc hors and con- nectivit lev el ab out 12, MDS-MAP has an erage error of ab out 50% whereas Hop-TERRAIN has an error of ab out 90% After the renemen phase in [11], the error is reduced to ab out 18% MDS-MAP should compared to Hop-TERRAIN since it can follo ed renemen phase lik the ones in [11] and [13]. By starting from the et- ter initial estimates generated MDS-MAP renemen phase should nd ev en etter results. Our preliminary ex- erimen ts along this en ue ha een encouraging. 205
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−5 −5 −4 −3 −2 −1 (a) In termediate MDS result (b) Final osition estimation MDS-MAP Figure 3: Estimation for the random net ork using distances et een neigh ors with 5% range error. Figure sho ws the um er of no des participating in the lo cation estimation in MDS-MAP Recall that the largest connected subnet ork is extracted for pro cessing. When the connectivit lev el is lo w, suc as 5.9, ab out 7% of the no des are not connected to the main subnet ork, and hence their ositions are not estimated. Among the no des that are part of the main subnet ork, man of them ha only or connections to their neigh ors. The lac of sucien information to determine the osition of no de causes large errors in MDS-MAP solutions. When the erage connec- tivit exceeds 12.2, the net ork tends to fully connected and all no des can lo calized. Sensitivit to range errors has een ma jor concern for lo calization algorithms. Figure sho ws the eects of increas- ing range errors on the estimation errors. The radio ranges are 1.5 and whic lead to connectivit lev els 12.2 and 20.7. random anc hors are used. The range errors ary from to 50%. Estimation error increases steadily as the range error increases. The errors with larger connectivit (20.7) are more than 10% lo er than those with smaller 10 12 14 16 18 20 22 50 100 150 200 250 Connectivity Error (%R) 3 anchors 4 anchors 6 anchors 10 anchors (a) Using pro ximit information only 10 12 14 16 18 20 22 50 100 150 200 250 Connectivity Error (%R) 3 anchors 4 anchors 6 anchors 10 anchors (b) Using distances et een neigh ors (5% range error). Figure 4: Av erage osition error of MDS-MAP on random net orks of 200 no des. connectivit (12.2) in most cases. The estimation error go es up faster after the range error is er 30%. 4.2 Grid Placement In this set of exp erimen ts, assume that the sensor no des are deplo ed according to some regular structures suc as square or hexagonal grid. Actually no des are placed in the neigh orho of the ertices due to random placemen error. 100 no des are placed on 10 10 grid, with unit edge distance Our results sho that MDS- MAP obtains uc etter results on the grid la out than on the random la out for the same connectivit lev el. Figure sho ws an example of the regular grid with 10% placemen error. The radio range is whic leads to connectivit 5.06. In the graph, oin ts again represen sensor no des and edges represen the connections et een neigh ors. Figure sho ws the result of MDS-MAP based on the connectivit of the net ork. Figure 8(a) sho ws the in termediate result of classical MDS on the connectivit ma- trix. It can seen that the cen ter of the map is the origin (0,0), and it has dieren orien tation than the original net ork in Figure 7. Figure 8(b) sho ws the nal solution of MDS-MAP where the MDS result is transformed based on random anc hor no des (* no des). The circles represen the 206
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10 12 14 16 18 20 22 93 94 95 96 97 98 99 100 Connectivity # of located nodes (%) Figure 5: raction of no des participating in the lo- calization pro cess. This fraction is indep enden of the um er or osition of anc hor no des. 10 15 20 25 30 35 40 45 50 10 20 30 40 50 60 70 80 4 anchers, connectivity: 12.242 and 20.7283 Range error Error (%R) R = 1.5r R = 2 r Figure 6: The eect of range error on the estimation error. true lo cations of sensor no des, and the solid lines represen the errors et een the estimated and true ositions. The longer the line, the larger the error. When the distance measuremen et een neigh ors is ail- able, the result of MDS-MAP is signican tly impro ed. Fig- ure sho ws the result of MDS-MAP kno wing the distance of neigh ors with 5% range error. Figure 9(a) sho ws the map constructed MDS. Again, it has dieren scale and ori- en tation than the ones in Figures and 8(a). Figure 9(b) sho ws the nal estimation of MDS-MAP based on the same anc hor no des. The estimation errors are no ery small. Figure 10 sho ws the erage erformance of MDS-MAP us- ing pro ximit information as function of connectivit and placemen errors, giv en or random anc hors resp ectiv ely The radio ranges are 1.5 2.5 and resp ectiv ely or dieren placemen errors, the same radio range leads to dieren connectivit lev els. With anc hors, osition esti- mates MDS-MAP ha an erage error under 50% for the placemen errors up to 50% in scenarios with an erage connectivit lev el of or greater. With anc hors, osition 10 10 Figure 7: 100 no des placed on 10 10 grid with 10% random placemen error. errors MDS-MAP is reduced to 35% and elo for the same cases. Figure 11 sho ws results of MDS-MAP using distance mea- suremen et een neigh ors with range errors from 5% to 50%. Kno wing the distances et een neigh ors leads to uc etter solutions. When the range error is elo 20%, the estimation errors are elo 25%. Ha ving more anc hors (5 vs. 3) impro es erformance, esp ecially for the case with large range errors. In addition to the exp erimen ts with 10 10 grids, also tried similar exp erimen ts with grids of other sizes, suc as grids (64 no des) and 20 20 grids (400 no des). The erage osition errors obtained MDS-MAP on the dieren size grids are ery similar giv en the same um er of anc hor no des. Similar results are obtained for the hexagonal grid la y- out. MDS-MAP ac hiev es sligh tly etter erformance than for the square grid la out due to the increased regularit of the distances et een neigh oring no des. Figure 12 sho ws an example of the regular grid with 10% placemen error. The radio range is whic leads to connectivit 5.32. Figure 13 sho ws the result of MDS-MAP based on the connectivit of the net ork. Figure 13(a) sho ws the in ter- mediate result of classical MDS on the connectivit matrix. Figure 13(b) sho ws the nal solution of MDS-MAP where the MDS result is transformed based on random anc hor no des (* no des). When using the distance measuremen et een neigh ors, the result of MDS-MAP is again uc etter. Figure 14 sho ws the result of MDS-MAP kno wing the distance of neigh ors with 5% range error. Figure 14 sho ws the map constructed MDS and the nal estima- tion of MDS-MAP based on the same anc hor no des. The estimation errors are no ery small. Figure 15 sho ws the erage erformance of MDS-MAP us- ing pro ximit information as function of connectivit and placemen errors, giv en or random anc hors resp ectiv ely The radio ranges are 1.5 2.5 and 3.5 resp ectiv ely or dieren placemen errors, the same radio range leads to dieren connectivit lev els. With anc hors, osition es- timates MDS-MAP ha an erage error under 30% 207
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−6 −4 −2 −4 −2 (a) In termediate MDS result 10 10 (b) Final osition estimation MDS-MAP Figure 8: Estimation on the grid net ork using con- nectivit information only for the placemen errors up to 50% when the connectivit lev el is 14 or greater. With anc hors, osition errors MDS-MAP are reduced sligh tly Figure 16 sho ws results of MDS-MAP using distance mea- suremen et een neigh ors with range errors from 5% to 50%. Again, kno wing the distances et een neigh ors leads to uc etter solutions. When the range error is elo 20%, the estimation errors for lo connectivit lev el (5.2) are elo 30%. With anc hors, osition estimates MDS- MAP ha an erage error under 15% with an erage connectivit lev el of 14 or greater. Ha ving more anc hors (5 vs. 3) impro es its erformance, esp ecially for the case with large range errors. In summary MDS-MAP erforms ell when the lev el of connectivit is er for the grid placemen ts and er 12 for the random placemen t. The um er of anc hor no des needed MDS-MAP is ery small. When there is sucien connectivit anc hors for the grid placemen ts and anc hors for the random placemen are usually enough for MDS-MAP to nd solutions with osition errors less than half of the −6 −4 −2 −6 −4 −2 (a) In termediate MDS result 10 10 (b) Final osition estimation MDS-MAP Figure 9: Estimation on the grid net ork using neigh or distances with 5% error. range of radio. MDS-MAP orks ell when the placemen errors are less than quarter of the radio range and when the range errors are less than 20% of the radio range. or oth the random and grid placemen ts, the osition error of MDS-MAP increases prop ortional to the range error. 5. POSSIBLE EXTENSIONS dra wbac of the curren implemen tation of MDS-MAP is that it requires global information of the net ork and cen tralized computation. One to address this issue is to divide the net ork in to sub-net orks and apply MDS- MAP to eac sub-net ork indep enden tly Since our metho do es not require anc hor no des in building relativ map of sub-net ork, the metho can applied to man sub- net orks in parallel. Then adjacen lo cal maps can com- bined aligning with eac other. In another ords, the complete map of the sensor net ork consists of man smaller patc hes. When three or more anc hors are presen in ei- ther sub-net ork or the whole net ork, an absolute map can computed accordingly Although this patc hing ap- 208
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10 12 14 16 18 20 22 20 40 60 80 100 120 140 160 3 anchors, known connectivity Connectivity Error (%R) 0% placement errors 10% placement errors 20% placement errors 50% placement errors 10 12 14 16 18 20 22 20 40 60 80 100 120 140 5 anchors, known connectivity Connectivity Error (%R) 0% placement errors 10% placement errors 20% placement errors 50% placement errors Figure 10: Av erage osition error of MDS-MAP on grid net orks of 100 no des when using only connec- tivit information. proac requires signican computation within eac patc h, one has considerable exibilit in ho osing whic no des er- form the computation. Preliminary exp erimen ts ha een ery encouraging and detailed results will rep orted in future ork. MDS-MAP can also extended applying more ad- anced MDS tec hniques. Instead of classical metric MDS, other MDS tec hniques suc as ordinal MDS and MDS with missing data can applied. This ma useful to handle non-uniform radio propagation and non-uniform ranging er- rors. ha done some limited exp erimen ts with ordinal MDS. Our results sho that ordinal MDS is etter than classical MDS when the connectivit lev el of the sensor net- ork is lo w, and is comparable with classical MDS when the connectivit lev el is high. Another dra wbac of MDS-MAP is that when the um- er of anc hor no des is large, the erformance of MDS-MAP is not as go as previous metho ds suc as the constrain t- based approac [4], \D V-hop" [9], or Hop-TERRAIN [11]. The reason is that the second step of MDS-MAP the applica- tion of classical MDS, is done without using the ositioning information of anc hor no des. The information is only used in step 3, when the erall structure and distance ratios e- een no des ha already een determined. The approac 10 12 14 16 18 20 22 10 20 30 40 50 60 70 80 90 100 3 anchors, known local distance Connectivity Error (%R) 5% range errors 10% range errors 20% range errors 50% range errors 10 12 14 16 18 20 22 10 20 30 40 50 60 5 anchors, known local distance Connectivity Error (%R) 5% range errors 10% range errors 20% range errors 50% range errors Figure 11: Av erage error of MDS-MAP on grid net- orks of 100 no des when using distance information. of building relativ map irresp ectiv of the co ordinations of anc hor no des is double-edged. It orks nicely when there are few or no anc hor no des, but not as ell when there are more anc hor no des. One solution ma to use more ad- anced MDS tec hnique called the anc hor oin metho [2], where co ordinates of anc hor no des are explicitly used in de- termining the scaling. As men tioned ab e, com bining MDS-MAP with other metho ds is another promising en ue. or example, MDS- MAP can used to get go initial estimates of no de o- sitions, whic is follo ed renemen phase lik the ones in [11] or [13]. Due to the go erformance of MDS-MAP comparing to comp eting metho ds on the cases of lo anc hor no de densities, one can exp ect this o-phase approac to generate go results. 6. CONCLUSIONS In this pap er, presen ted new lo calization metho d, MDS-MAP that orks ell with mere connectivit infor- mation. Ho ev er, it can also incorp orate distance informa- tion et een neigh oring no des when it is ailable. The strength of MDS-MAP is that it can used when there are few or no anc hor no des. Previous metho ds often re- quire ell-placed anc hors to ork ell. or example, the constrain t-based approac in [4] orks ell only when the 209
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10 Figure 12: 100 no des randomly placed on 10 10 hexagonal grid with 10% placemen error. anc hors are placed at the outside corners and edges and the constrain ts are tigh t. It orks orly when the anc hors are inside the net ork, close to the cen ter. The collab ora- tiv ultilateration approac in [13] also requires anc hors throughout the net ork, as ell as relativ ely large um- er of anc hors, to ork ell. Our metho do es not ha this limitation. It builds relativ map of the no des ev en with- out anc hor no des. With three or more anc hor no des, the relativ map can transformed and absolute co ordinates of the sensor no des are computed. Extensiv sim ulations using arious net ork arrangemen ts and dieren lev els of ranging error sho that the metho is eectiv e, and partic- ularly so for situations with few anc hor no des and relativ ely uniform no de distributions. 7. CKNO WLEDGEMENTS This ork as supp orted in part ARP under con- tract F33615-01-C-1904. 8. REFERENCES [1] I. Borg and Gro enen. Mo dern Multidimensional Sc aling, The ory and Applic ations Springer-V erlag, New ork, 1997. [2] A. Buja, D. F. Sw yne, M. Littman, N. Dean, and H. Hofmann. Gvis: In teractiv data visualization with ultidimensional scaling. Journal of Computational and Gr aphic al Statistics page (to app ear), 2001. [3] N. Bulusu, J. Heidemann, and D. Estrin. GPS-less lo w-cost outdo or lo calization for ery small devices. IEEE Personal Communic ations 7(5):28{34, Oct. 2000. [4] L. Dohert L. E. Ghaoui, and K. Pister. Con ex osition estimation in wireless sensor net orks. In Pr c. Info om 2001 Anc horage, AK, April 2001. [5] D. Ganesan, B. Krishnamac hari, A. o, D. Culler, D. Estrin, and S. Wic er. An empirical study of epidemic algorithms in large scale ultihop wireless −5 −4 −3 −2 −1 (a) In termediate MDS result 10 (b) Final osition estimation MDS-MAP Figure 13: Estimation on the hexagonal grid net- ork using connectivit information only net orks. ec hnical rep ort UCLA/CSD-TR-02-0013, UCLA Computer Science Departmen t, 2002. [6] J. High to er and G. Boriello. Lo cation systems for ubiquitous computing. IEEE Computer 34(8):57{66, Aug. 2001. [7] A. Ho ard, M. J. Mataric, and G. S. Sukhatme. Relaxation on mesh: formalism for generalized lo calization. In Pr c. IEEE/RSJ Int’l Conf. on Intel ligent ob ots and Systems (IR OS01) pages 1055{1060, 2001. [8] A. Nasipuri and K. Li. directionalit based lo cation disco ery sc heme for wireless sensor net orks. In 1st CM Int’l Workshop on Wir eless Sensor Networks and Applic ations (WSNA’02) pages 105{111, tlan ta, GA, Sept. 2002. [9] D. Niculescu and B. Nath. Ad-ho ositioning system. In IEEE Glob eCom No v. 2001. [10] S. I. Roumeliotis and G. A. Bek ey Synergetic lo calization for groups of mobile rob ots. In Pr c. 39th IEEE Conf. on De cision and Contr ol Sydney 210
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−5 −4 −3 −2 −1 (a) In termediate MDS result 10 (b) Final osition estimation MDS-MAP Figure 14: Estimation on the hexagonal grid net- ork using distances with 5% range error. Australia, Dec. 2000. [11] C. Sa arese, J. Rabaey and K. Langendo en. Robust ositioning algorithm for distributed ad-ho wireless sensor net orks. In USENIX chnic al nnual Conf. Mon terey CA, June 2002. [12] A. Sa vvides, C. C. Han, and M. Sriv asta a. Dynamic ne-grained lo calization in ad ho net orks of sensors. In CM/IEEE Int’l Conf. on Mobile Computing and Networking (MOBICON) July 2001. [13] A. Sa vvides, H. ark, and M. Sriv asta a. The bits and ops of the n-hop ultilateration primitiv for no de lo calization problems. In 1st CM Int’l Workshop on Wir eless Sensor Networks and Applic ations (WSNA’02) pages 112{121, tlan ta, GA, Sept. 2002. [14] R. N. Shepard. Analysis of pro ximities: Multidimensional scaling with an unkno wn distance function I. Psychometrika 27:125{140, 219{246, 1962. [15] W. S. orgeson. Multidimensional scaling of similarit Psychometrika 30:379{393, 1965. 10 15 20 25 30 35 20 40 60 80 100 120 3 anchors, known connectivity Connectivity Error (%R) 0% placement errors 10% placement errors 20% placement errors 50% placement errors 10 15 20 25 30 35 10 20 30 40 50 60 70 80 90 5 anchors, known connectivity Connectivity Error (%R) 0% placement errors 10% placement errors 20% placement errors 50% placement errors Figure 15: Av erage error on hexagonal grid net- orks when using only connectivit information. 211
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10 15 20 25 30 10 20 30 40 50 60 70 80 3 anchors, known local distance Connectivity Error (%R) 5% range errors 10% range errors 20% range errors 50% range errors 10 15 20 25 30 10 15 20 25 30 35 40 45 5 anchors, known local distance Connectivity Error (%R) 5% range errors 10% range errors 20% range errors 50% range errors Figure 16: Av erage error on hexagonal grid net- orks when using distances et een neigh oring no des. 212

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