Research Letter Distributed Source Localization Based on TOA Measurements in Wireless Sensor Networks

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1 Researh Letters in Eletronis Volume 2009, Artile ID , 4 pages doi: /2009/ Researh Letter Distributed Soure Loalization Based on TOA Measurements in Wireless Sensor Networks Wanzhi Qiu and Efstratios Skafidas National ICT Australia, Department of Eletrial and Eletroni, The University of Melbourne, Parkville, Vitoria 3010, Australia Correspondene should be addressed to Wanzhi Qiu, wanzhiqiu128@yahoo.om Reeived 30 Otober 2008; Aepted 30 Deember 2008 Reommended by Muhammad Taher Abuelma atti We study the problem of soure loalization in multihop wireless sensor networks. A fully distributed algorithm based on sensor measurements of time of arrivals (TOAs) is proposed. In ontrast to entralized methods where all TOA measurements are transmitted via ertain routes to a entral loation (the sink) for proessing, the proposed method distributes the proessing among the relay nodes on the routes to the sink. Fusion strategies are proposed so that the raw and intermediate data are progressively proessed, and only the refined results are further relayed. As a result, the proposed sheme has improved flexibility and salability sine it does not impose any speial requirements on the sink node. The proposed distributed strategy also has the potential to save energy and bandwidth due to redued radio transmissions. Copyright 2009 W. Qiu and E. Skafidas. This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. 1. Introdution In entralized wireless sensor network (WSN) soure loalization, the transmission of all the raw measurements to a entral point is required and the proessing is performed in a entralized manner. These shemes have the potential to ahieve optimum solutions due to the availability of the full data set at the proessing point. However, the entralized systems typially have low energy effiieny and low salability due to exessive radio transmissions and dependeny on a entral proessing node. Distributed methods based on lustering have been proposed [1]. In [2], the intermediate result is yled through the network on speially designed routes, and eah node makes a small adjustment to the reeived estimate based on deepest desent on a ost funtion, then passes updated estimate to one of its neighbours. In this paper, we propose a fully distributed soure loalization algorithm based on TOA measurements in multihop wireless sensor networks. The proposed sheme distributes the loalization proess among the relay nodes along the routes to the (arbitury) node where the loalization estimate is needed. Whenever determined feasible, the intermediate estimate is produed or refined by inorporating TOAs and intermediate estimates available at the loal node. Data fusion strategies are proposed so that the raw and intermediate data are progressively proessed and only the refined results are further relayed. 2. The Data Model and Centralized Proessing As in [3] and illustrated in Figure 1, forasignalemittedby the soure at time T (unknown), the TOA measurement at node i (i = 1,..., N, N 3) is modelled by t i = T + R i + ε i, i = 1,..., N, (1) where ε i denotes measurement noise and the propagation speed. The noise samples are assumed to be unorrelated with zero mean and variane σ 2. No assumption is made about the probability of density funtion of the noise. The distane between the soure and a node loated at (x i, y i )is given by R i = (x x i ) 2 +(y y i ) 2. We assume that a nominal soure position (x n, y n )is available. This nominal position, whih is lose to the true

2 2 Researh Letters in Eletronis Soure (x, y) Nominal position (x n, y n ) δy y R ni Node i R i α i δx Figure 1: Soure loalization geometry. soure position (x, y), ould have been obtained from the previous measurements. Suh a situation is typial when the soure is being traked [3]. Hene, to estimate the soure position we require an estimation of [(x x n )(y y n )] T = [δx δy] T,whereT denotes matrix transpose. Using a first-order Taylor expansion of R i about (x n, y n ) and defining R ni = (x n x i ) 2 +(y n y i ) 2,wehaveR i R ni +(x n x i )R ni δx +(y n y i )R ni δy. Therefore, (1) anbe expressed as τ i T + os α i x δx + sin α i δy + ε i, (2) where os α i = (x n x i )R ni, sinα i = (y n y i )R ni,, and τ i = t i R ni /. By olleting all TOA measurements at one point, the entralized proessing sheme formulates the model x = Hθ + w, where x = [ τ 1 τ 2 τ N ] T, θ = [ T δx δy ] T, w = [ ε 1 ε 2 ε N ] T,and 1 os α 1 1 os α 2 H =.. 1 os α N sin α 1 sin α 2. (3). sin α N Based on this linear model, the unknowns T, δx,andδy an be solved by the best linear unbiased estimator (BLUE) θ = [H T H] 1 H T x, and the estimation varianes are given by the diagonal items of the ovariane matrix C θ = σ 2 [H T H] 1, that is, [σ 2 T, σ 2 δx, σ 2 δy ] = [C θ(1, 1), C θ (2, 2), C θ (3, 3)]. Note that, in order to perform the estimation, the sink, where the proessing is performed, has to know the loations of all partiipating nodes as well as their TOA measurements. This ould ause severe ommuniation/energy problems in situations when, for example, nodes are moving and therefore being required to onstantly report their loations to the sink. 3. The Proposed Distributed Proessing As with the entralized proessing, we assume that the sink is the point where the final estimate is needed. However, in the proposed sheme, the sink is just an ordinary node whih does not have additional energy or omputing apaity and does not perform any additional work as ompared to other nodes. Only the nodes whih have a TOA measurement or have reeived any measurements or intermediate estimates (IEs) from the up-stream nodes partiipate in the proess. The priniple is, for eah partiipating node, to proess the available measurements and/or IEs whenever possible and to send the result to the next-hop node (i.e., the down-stream node). The fusion result from the sink node is the final estimate. Note that the proposed sheme does not require any speial arrangements on other funtionalities of the network suh as routing; TOA measurements and IEs are fused naturally while travelling on any established routes to the sink. Different fusion rules have to be used depending on the information available at a node. Basially, three or more TOA measurements an be fused into one IE where a TOA measurement is defined as the loation of the node and its measured τ i, that is, TOA (x i, y i, τ i ).TheIEisdefined as the estimates of the unknowns and their orresponding varianes, that is, IE = ( δx, δy, δ T, σ 2 δx, σ 2 δy, σ 2 T). An IE an be further fused with one measurement, two measurements, or one or more IEs. Fusion is performed at a loal node after reeiving all TOAs and/or IEs from up-stream nodes. When an IE is obtained, all follow-on transmission on this partiular route will involve one IE only. Figure 2 depits a typial fusion senario where the arrows represent data transmissions, and retangular bloks indiate performed fusion funtions whih are desribed in detail below Fuse TOAs(). This funtion fuses N ( 3) TOA measurements into one IE. The formulas used here are exatly the same as in Setion Fuse IEs(). When N (>1) IEs are available, they an be ombined aording to the priniple of BLUE to yield an updated IE. That is, for (δxn, δyn, δ Tn, σδxn 2, σ δyn 2, σ Tn), 2 where n = 1,..., N, the updated IE is given by δx new = σδx 2 Nn=1 new δxn/σδxn 2, δy new = σ δy 2 Nn=1 new δyn/σδyn 2, and Tn new = σt 2 Nn=1 new Tn/σTn, 2 where σδx 2 new = ( N n=1 1/σ 2 δxn ) 1, σ 2 δy new = ( N n=1 1/σ 2 δyn ) 1,andσ 2 T new = ( N n=1 1/σ 2 Tn) 1. As an be seen from the above equations, the BLUE is atually a weighted average of the estimates, giving the more aurate (with smaller varianes) ones larger weights Fuse IE with one TOA(). We now fuse one intermediate estimate IE 0 = ( δx, δy, T, σ 2 δx, σ 2 δy, σ 2 T) with one TOA (x i, y i, τ i )measuredbyaertainnoden i.inotherwords,we refine IE 0 with the new information provided by the TOA measurement. In order to do this, we approximate T by its

3 Researh Letters in Eletronis 3 TOA1 Fuse IEs( ) Fuse IE with one TOA( ) Fuse TOAs( ) TOA1 TOA2 TOA2 IE1 IE2 IE5 TOA3 Fuse IE with two TOAs( ) IE4 TOA4 IE3 Nodes with own TOA measurements Nodes without own TOA measurements Figure 2: A typial fusion senario. with updated varianes σδx 2 new = (1/σ2 δx +1/σ2 δx ) 1 and σδy 2 new = (1/σ2 δy +1/σ2 δy ) Fuse IE with two TOAs(). Here, we have two TOA measurements (x 1, y 1, τ 1 ) and (x 2, y 2, τ 2 ) along with one IE IE 0 = ( δx, δy, T, σδx 2, σ δy 2, σ T) 2 available. We first use the BLUE priniple to fuse the two TOAs to obtain initial estimates δx and δy. The result is then further fused with IE 0. As in the previous setion, by denoting T T + ε T,for i = 1, 2, we have z i = a i δx+b i δy+ε i +ε T,wherea i = os α i /, b i = sin α i /,andz i = τ i T. By defining x = [z 1 z 2 ] T a1 b1, H = [ a 2 b 2 ], θ = [δx δy] T, and w = [ε 1 + ε T ε 2 + ε T ] T,wehavex = Hθ + w whih immediately leads to the following BLUE [3]: estimated value T and refine the estimates of δx and δy only. By denoting T T + ε T,whereε T is the error in the estimate of T, the TOA measurement (2)an be expressed by τ i T (os α i /)δx + (sin α i /)δy + ε i + ε T whih an be further simplified as where a = os α i /, b = sin α i /,and z = aδx + bδy, (4) z = z o + ε i + ε T, z o = τ i T. (5) When the error terms (ε i and ε T ) are ignored, (4) anbe represented by a straight line in the δx δy plane. The estimate ( δx, δy) is a point on the same plane, and we propose to use a weighted average of it and its projetion onto line (4) ( δx, δy ) as the new estimate. As far as the node n i an see, ( δx, δy ) lies on the irle entred at (x i, y i )with radius r = z o,where is the propagation speed. It an be seen from (5) that the variane of r is given by σr 2 = 2 (σ 2 + σt). 2 Sine it is too omplex to obtain the exat expressions for the varianes of δx and δy, in order to determine their weights in forming the new estimates, we will use σr 2 to approximate their varianes, that is, σδx 2 σ 2 δy σ 2 r. This approximation is in line with the new information the TOA measurement provides on the soure loation. By denoting K = a 2 + b 2, the projetion oordinates an be expressed by ( δx, δy ) = ( a K z o + b2 K δx ab K δy, leading to new estimates δx new = σ 2 δx new δy new = σ 2 δy new ( δx σ 2 δx ( δy σ 2 δy b K z o ab K δx + a2 K δy ), (6) + δx ) σδx 2, δy + σ 2 δy ), (7) [ δx, δy ] T = [ H T C 1 H ] 1 H T C 1 x, (8) with ovariane matrix C θ = [H T C 1 H] 1, where, with E denoting expetation, C = E ( ( ww T) σ 2 + σt 2 σ 2 ) T = σt 2 σ 2 + σt 2. (9) That is, [σ 2 δx, σ 2 δy ] = [C θ(1, 1), C θ (2, 2)]. This fusion funtion is now summarized as follows. Step 1, use (8) toobtain δx and δy.step2,fuse( δx, δy )and( δx, δy) using (7). 4. Simulation In this setion, we ondut simulation to verify the proposed distributed soure loalization sheme. The tree routing protool and network setup proess inluding parent-hild relationship and address assignment algorithms of the ZigBee [4] standard are followed in the simulation. 100 ZigBee nodes are randomly deployed in a square region with size 500 m by 500 m. The root node (the oordinator in ZigBee s terms) is ating as the sink and is loated at the entre of the lower edge of the region. The nominal loation of the sound soure (target) (with speed = 344 m/s) is the entre of the region, and it has an effetive range R = 200 m. When a node is less than R meters away from the target, it an make a TOA measurement. For eah simulation senario, the soure takes a small move in both x and y diretions, that is, (δx, δy) = (0.5 m,0.5 m), and 1000 runs are onduted with independent and identially distributed zero-mean Gaussian noise samples generated in eah run. The estimation auray is measured by the mean square error over all the runs: MSE = (1/1000) 1000 r=1 ((x x r ) 2 +(y ŷ r ) 2 ), where (x r, y r ) is the estimate of the true soure loation (x, y) in the rth run. Figure 3 shows the estimation errors with varying noise variane. It an be seen that the proposed distributed sheme works, and its performane attains that of the entralized best linear unbiased estimator when the noise is redued to a ertain level.

4 4 Researh Letters in Eletronis MSE Distributed Centralized σ 2 Figure 3: MSE of estimates. In summary, the benefit of the proposed sheme is twofold. First, it has improved flexibility and salability over entralized shemes sine it shares the proessing among multiple nodes and does not pose any speial requirements on the sink node. Seond, it has the potential to save energy/bandwidth onsumption of the network due to redued radio transmissions. Referenes [1] W.-P. Chen, J. C. Hou, and L. Sha, Dynami lustering for aousti target traking in wireless sensor networks, in Proeedings of the 11th IEEE International Conferene on Network Protools (ICNP 03), pp , Atlanta, Ga, USA, November [2] M. G. Rabbat and R. D. Nowak, Deentralized soure loalization and traking, in Proeedings of the IEEE International Conferene on Aoustis, Speeh and Signal Proessing (ICASSP 04), vol. 3, pp , Montreal, Canada, May [3] S.M.Kay,Fundamentals of Statistial Signal Proessing: Estimation Theory, Prentie-Hall, Englewood Cliffs, NJ, USA, [4] ZigBee Speifiation Version 1.0, ZigBee Alliane, 2005.

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