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v1 [

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2013

JOINT LOCALIZATION AND CLOCK SYNCHRONIZATION FORWIRELESS SENSOR NETWORKS

Sundeep Prabhakar Chepuri, Geert Leus, and Alle-Jan van der Veen

Faculty of Electrical Engineering, Mathematics and Computer ScienceDelft University of Technology, 2628 CD Delft, The Netherlands

E-mail: s.p.chepuri@tudelft.nl, g.j.t.leus@tudelft.nl, a.j.vanderveen@tudelft.nl.

ABSTRACT

A fully-asynchronous network with one target sensor and afew anchors (nodes with known locations) is considered. Lo-calization and synchronization are traditionally treatedas twoseparate problems. In this paper, localization and synchro-nization is studied under a unified framework. We presenta new model in which time-stamps obtained either via two-way communication between the nodes or with a broadcastbased protocol can be used in a simple estimator based onleast-squares (LS) to jointly estimate the position of the tar-get node as well as all the unknown clock-skews and clock-offsets. The Cramer-Rao lower bound (CRLB) is derived forthe considered problem and is used as a benchmark to analyzethe performance of the proposed estimator.

Index Terms— Clock synchronization, clock-skew, clock-offset, localization, wireless sensor networks.

1. INTRODUCTION

Localization and clock synchronization are two key compo-nents of any self-organizing location-aware wireless sensornetwork (WSN). A WSN enables distributed information pro-cessing tasks like data sampling, information fusion, and othertime-based tasks [1]. Every node in the network has an au-tonomous clock. These individual clocks in a WSN drift fromeach other due to imperfections in the oscillator, aging andother environmental variations. It is essential to calibrate theseimperfections from time to time to achieve a network-widetime coherence. A plethora of algorithms for clock synchro-nization can be found in [2]. For the data to be meaning-ful, the location where the data is acquired is often required.Computing the location of the nodes is commonly calledlo-calization, and is a well-studied topic [3].

Even though localization and clock synchronization aretightly coupled, traditionally they are treated as two sepa-rate problems. Recently, for an anchorless and a fully asyn-chronous network, a global least-squares (GLS) estimator basedon a two-way time-stamp exchange protocol for joint clocksynchronization and ranging has been proposed in [4]. Ex-ploiting the broadcast nature of the wireless medium, an asym-

metrical time-stamping and passive listening (ATPL) protocolwas proposed in [5] for joint clock synchronization and rang-ing. Subsequently, the estimated pairwise distances can beused as an input to the least-squares (LS) based range-squaredmethod for localization. Joint estimation of the position andthe clock parameters of a sensor based on the two-way time-stamp exchange protocol has been considered in [6], wherean synchronous network is considered where only the sensornode suffers from clock-skews and clock-offsets and the an-chors are assumed to be synchronized. In [7], localizationof the sensor node in a fully-asynchronous network has beenstudied, where the main focus is on localization but also cer-tain approximations of the clock parameters are required tosolve the problem.

In this paper, we again consider fully-asynchronous net-work consisting of one sensor and a few anchors, and inves-tigate localization and clock synchronization under a unifiedframework. We propose a linear data model for joint localiza-tion and clock synchronization. The data model is generic inthe sense that it can be used with either the two-way rangingprotocol or the ATPL protocol. This is used in an estima-tor based on LS to jointly estimate the position of the sensornode and all the unknown clock-skews and clock-offsets. TheCramer-Rao Lower Bound (CRLB) is derived for the consid-ered problem and is used as a benchmark to analyze the per-formance of the proposed estimators.Notation:Upper (lower) bold face letters are used for matrices (columnvectors);(·)T denotes transposition;⊙ (⊘) refers to element-wise matrix or vector product (division);(.)⊙2 denotes theelement-wise matrix or vector squaring;bdiag(.) a block di-agonal matrix with the matrices in its argument on the maindiagonal;1N (0N ) denotes theN × 1 vector of ones (zeros);IN is an identity matrix of sizeN ; E(.) denotes the expecta-tion operation;⊗ is the Kronecker product.

2. NETWORK MODEL

We consider a fully-asynchronous network withM anchorsand one sensor (node 0) as shown in Fig. 1. We assume one

A∈R3K×11

︷ ︸︸ ︷

t01 1K −t10 −1K 0K 0K 0K 0K e10 0K 0K

t02 1K 0K 0K −t20 −1K 0K 0K 0K e02 0K

t03 1K 0K 0K 0K 0K −t30 −1K 0K 0K e03

θ∈ R11×1

︷ ︸︸ ︷

c0

c1

c2

c3

τ 0

=

n∈R9×1

︷ ︸︸ ︷

n01

n02

n03

.

(5)

A∈R3K×9

︷ ︸︸ ︷

t01 1K −t10 −1K 0K 0K e10 0K 0K

t02 1K 0K 0K −t20 −1K 0K e02 0K

t03 1K 0K 0K 0K 0K 0K 0K e03

θ∈R9×1

︷ ︸︸ ︷

c0

c1

c2

τ 0

= −

t∈R3K×1

︷ ︸︸ ︷

0K 0K

0K 0K

−t30 −1K

c3 +

n∈R9×1

︷ ︸︸ ︷

n01

n02

n03

.

(6)

node 3

Sensor

Anchor

node 0

node 2

node 1(c0?,x0?)

(c1?,x1)

(c2?,x2)

(c3?,x3)

node M

(cM = [1, 0]T ,xM )

Fig. 1: An illustration of the network model, together withthe know and unknown parameters. Light shaded lines referto the passive listening links.

of the nodes has a relatively stable clock oscillator and is usedas a clock reference. All the other nodes suffer from clock-skews and clock-offsets. The network model considered hereis the same as the model considered in [5].

All the nodes are distributed over anl-dimensional space,with l = 2 (plane) orl = 3 (3-D space). Let the vectorxi ∈ R

l×1 denote the coordinates of theith node. The co-ordinates of all the anchors are collected in a matrixX =[x1,x2, . . . ,xM ] ∈ R

l×M . The unknown coordinates of thesensor are collected inx0. The distance between theith nodeand thejth node is denoted by

dij = dji = ‖xi − xj‖2 =√

‖xi‖2 − 2xTi xj + ‖xj‖2

(1)Let ti be the local time at theith node andt be the refer-

ence time. We then assume that the relation between the localtime and the reference time can be given by a first-order affineclock model [4],

ti = ωit+ φi ⇔ t = αiti + βi (2)

whereωi ∈ R+ is the clock-skew,φi ∈ R is the clock-offset,αi = ω−1

i andβi = −ω−1i φi are the synchronization parame-

ters of theith node. Without loss of generality, we use anchorM as absolute time reference, i.e.,[ωM , φM ] = [1, 0]. Theunknown synchronization parameters are collected inα =[α0, α1, . . . , αM−1]

T andβ = [β0, β1, . . . , βM−1]T . The un-

known clock-skews and clock-offsets are respectively givenby

ω = 1M ⊘α and φ = −β ⊘α. (3)

Nodes in the network can communicate with each othereither via a two-way communication protocol [4] or an ATPLprotocol [5] as illustrated in Fig. 1.

3. PROBLEM FORMULATION

In this paper, we focus on the two-way communication proto-col for deriving the data model. The transmission and recep-tion time-stamps are recorded independently at local time co-ordinates both during the forward and the reverse links. Thetime-stamp recorded at theith node when thekth iterationmessage departs to thejth node is denoted byT (k)

ij , and uponarrival of the corresponding message, thejth node records thetime-stampT (k)

ji . For the sake of generality, we do not put anyconstraints on the sequence of forward links and reverse linksor the number of time-stamps recorded [4,5].

The time-of-flight for a line-of-sight (LOS) transmissionbetween theith node and thejth node can be defined asτij =ν−1dij , whereν denotes the speed of a wave in a medium.Using (2),τij can be written in terms of the local clock coor-dinates as

τij = (αjT(k)ji + βj)− (αiT

(k)ij + βi) + n

(k)ij

(4)

wheren(k)ij denotes the aggregate measurement error on the

time-stamps.In all there areK time-stamps recorded at each node and

the time-stamps recorded at theith node are collected intij =

[T(1)ij , T

(2)ij , . . . , T

(K)ij ]T ∈ R

K×1. The direction (forward or

reverse) of thekth link is denoted bye(k)ij , wheree(k)ij = 1 for

transmission from nodei to nodej ande(k)ij = −1 for trans-mission from nodej to nodei. The direction information iscollected in a vectoreij = [e

(1)ij , e

(2)ij , . . . , e

(K)ij ]T ∈ R

K×1.

The error vector is denoted bynij = [n(1)ij , n

(2)ij , . . . , n

(K)ij ]T ∈

RK×1.

For the sake of exposition, we consider the example of anetwork withM = 3 anchors and one sensor (node 0) witha of two-way communication protocol between each of thesensor-anchor pairs. Let the clock parameters correspondingto the ith node be collected in a vectorci = [αi, βi] andτ 0 = [τ0,1, τ0,2, . . . , τ0,M ]T ∈ R

M×1. The pairwise dis-tances of the sensor to each anchor will bed0 = ντ 0. Wecan now write (4) for all theK time-stamps collected in amatrix-vector form shown in (5) on top of this page. Mov-ing the known columns corresponding toc3 = [1, 0]T (clockreference) to one side, we can re-write (5) shown in (6) ontop of this page. The generalization of (6) for anyM > 2 is

straightforward.In case we adopt the broadcast based ATPL protocol, the

matrixA will have additional rows corresponding to the pas-sive listening links [5]. The detailed derivation of the lineardata model for the ATPL protocol can be found in [5].

The generalized linear model for either the two-way com-munication or ATPL protocol can be written as

Aθ = t+ n (7)

whereA ∈ RKM×3M , θ ∈ R

3M×1, t ∈ RKM×1, andn ∈

RKM×1 with the structures detailed in (6) for the two-way

communication protocol or in [5] for the ATPL protocol.The aim of this work is to estimate the positionx0 of

the target node along with all the unknown clock parameters.The position of the target nodex0 can be computed using therange estimates obtained by solving (7). This is presented asa two-step approach in the next section, where in the first stepwe estimate the unknown clock parameters and the pairwisedistances of the sensor to each anchor, and use this estimatedpairwise distances to compute the position of the target nodein the second step. Alternatively, we can formulate a singleestimation problem to jointly compute the position of the tar-get node as well as all the unknown clock parameters and thisis the main contribution of this paper.

4. TWO-STEP APPROACH

In the two-step approach, we first solve for all the unknownclock parameters and the pairwise distances of the sensor toeach anchor and then use the range estimates in a LS estimatorto compute the location.

4.1. Joint synchronization and ranging: step I

For K ≥ 3, matrixA is tall and left-invertible. Hence, theunknown parameters inθ can be estimated using LS, i.e.,

θ = (ATA)−1AT t. (8)

Subsequently, the clock-skewsω, clock-offsetsφ can be ob-tained using the relation in (3), and the pairwise distancesofthe sensor to each anchor using the relationd0 = ντ 0.

4.2. Localization from estimated pairwise distances: stepII

Pairwise distances form a major input to any localization scheme.Using the pairwise distance estimates obtained in (8), the co-ordinates of the sensor node can be estimated using range-squared localization algorithms.

Let us define a vectorq = [‖x1‖2, ‖x2‖

2, . . . , ‖xM‖2]T ∈R

M×1. Using (1), we can write the pairwise distance of thesensor to each anchor in a vector form as

d0 ⊙ d0 = q− 2XTx0 + ‖x0‖21M

= Xp+ q,(9)

wherep = [xT0 , ‖x0‖

2]T andX = [−2XT ,1M ] ∈ RM×(l+1).

Subsequently, the coordinates of the sensor can be estimated

using LS as follows

p = (XT X)−1XT (d0 ⊙ d0 − q) (10)

providedM ≥ l + 1 such that the matrixX is tall. Theanchor positions can be designed such that the matrixX isleft-invertible.

5. THE JOINT ESTIMATOR

Although clock synchronization and localization problemsaretightly coupled, they have a non-linear relation as can be seenin (9). In case we want to localize the sensor and also estimateall the unknown clock parameters in one linear step, we haveto linearize the relation between the clock parameters and theposition.

In order to do such a joint localization and synchroniza-tion, we exploit the linear relation between the range-squaredmeasurements and the coordinates of the sensor. Instead ofsquaring the estimated pairwise distances in the second stepas in Section 4.2, we linearize the problem by squaring thelinear model in (7), and this is the main contribution of thispaper. A unified framework for localization and synchroniza-tion is essential for applications in WSNs such as joint track-ing of the position and the clock-parameters, for e.g., using astandard Kalman filter.

Hadamard squaring the data model in (7) would result ina linear model and is given by

(Aθ)⊙ (Aθ) = (t+ n)⊙ (t+ n) (11)

and can be further simplified to

(AT ◦AT )T (θ ⊗ θ) = (t+ n)⊙ (t+ n) (12)

The linear model in (12) does not have a unique solution asthe system matrix(AT ◦AT )T is aKM×9M2 matrix whichis generally fat and thus not left-invertible. However, this canbe solved with certain approximations of the clock parametersas in [7].

An alternative approach to linearizing the problem is bytaking the Kroneckor product of the measurements, i.e.,

(Aθ)⊗ (Aθ) = (t+ n)⊗ (t+ n). (13)

Using the matrix propertyPC ⊗ QE = (P ⊗ Q)(C ⊗ E),we can further simplify (13) to the following linear model

A︷ ︸︸ ︷

(A⊗A)

θ︷ ︸︸ ︷

(θ ⊗ θ) = (t+ n)⊗ (t+ n)

=

t︷ ︸︸ ︷

t⊗ t+

w︷ ︸︸ ︷

n⊗ n+ t⊗ n+ n⊗ t

(14)

whereA ∈ RK2M2

×9M2

, andw ∈ RK2M2

×1 is the new er-ror vector. If the matrixA is full column-rank, then it followsthat the matrixA is also full column-rank.

We now introduce two new variables to resolve the clockparameters without ambiguity after the squaring operation.For theith node, we define the variables

γi , α2i and δi , αiβi, (15)

and collect parameters corresponding to theith node in thevector ci = [γi, δi]

T . The clock-skewωi ∈ R+ is alwayspositive and the clock-offsetφi ∈ R can be either positive ornegative. As a result, recovering clock-offsets fromβ2

i with-out ambiguities would be difficult. Hence, we make use of thecross-termδi = αiβi to recover the clock-offset. For all thenodes in the network we havec = [cT0 , c

T1 , c

T2 , . . . , c

TM−1]

T .

Let us define a permutation matrixΠ ∈ R9M2

×9M2

that sortsthe entries ofθ, such thatΠθ = [cT , τ⊙2T

0 , zT ]T . Here, theentries of the vectorz ∈ R

Lz×1 with Lz = 9M2 − 3M , con-sist of the nuisance parameters excludingc andτ⊙2

0 from θ

and is of less interest.We can now re-write (14) as follows

AΠT (Scc+ ν−2Sdd⊙20 + Szz) = t+w (16)

whereSc, Sd, andSz are the selection matrices to selectcolumns ofAΠT corresponding toc, d⊙2

0 , andz, respec-tively. Substituting (9) in (16), we get

AΠT (Scc+ ν

−2SdXap+ Szz) = t− AΠT ν−2Sdq+w.

(17)We next collect the unknowns in the vectorψ = [cT ,pT , zT ]T

∈ RL×1 whereL = 2M + l + 1 + Lz and the columns cor-

responding to the unknowns in the matrix

A = [AΠTSc, ν−2AΠTSdX, AΠTSz] ∈ R

K2M2×L,

(18)and the measurements in the vectort = t− ν−2AΠTSdq ∈R

K2M2×1.

The generalized linear model for joint localization andsynchronization is then given by

Aψ = t+w (19)

The unknown parameters inϕ can be estimated using LS, i.e.,

ψ = (AT A)−1AT t. (20)

Hence, the unknown positionx0 is obtained by solving (20)and the unknown clock-skews and clock-offsets can be ob-tained using (15) and (3) without any ambiguities.

Alternatively, a weighted least-squares (WLS) estimatorinstead of (10) taking the estimation error in (8) or a WLS es-timator instead of (20) pre-whitening the noisew is possible.However, this is not further detailed in this paper.

6. CRAMER-RAO LOWER BOUND

We now derive the CRLB for jointly estimating the clock-skewsω, the clock-offsetsφ, and the coordinates of the sen-sor nodex0, i.e., ψ = [ωT ,φT ,xT

0 ]T based on (7). For an

unbiased estimatorϕ it follows from the CRLB theorem thatE(ˆψ ˆψT ) ≥ F−1 whereF is the Fisher information matrix.If the error vectorn is Gaussian distributed with a varianceσ2, thenF can be computed asF = σ−2JTJ, whereJ is theJacobian matrix given by

J =∂(Aθ − t)

∂ϕ= [Jω Jφ Jx0

] ∈ RKM×(2M+l)

(21)with sub-blocks

Jω = −A(Sα − Sβ ⊙ 1KMφT )⊘ (1KMω

T )⊙2,

Jφ = −ASβ ⊘ 1KMωT ,

Jx0= ν−1TSτ 0

D,

(22)

whereSα, Sβ, andSτ0are selection matrices to select the

columns ofA corresponding toα, β, andτ 0, respectively.TheM × l derivative matrixD is defined as

[D]i,j =

[∂d0

∂x0

]

i,j

=[x0]j − [xi]j‖x0 − xi‖2

(23)

7. NUMERICAL EXAMPLE

A network with one target sensor and5 anchors is considered.Both the target node and anchor nodes are deployed randomlywithin a range of100m. The clock-skewsω and clock-offsetsφ are uniformly distributed in the range[1 − 100ppm, 1 +100ppm] and[−1s, 1s], respectively. We use an observationinterval of 100 s during which the clock parameters are as-sumed to be fixed. We useν = 300m/s and recordK = 10time-stamps. The error vector is assumed to be Gaussian dis-tributed with a varianceσ2. The simulations are averagedover1000 independent Monte Carlo experiments.

10−9

10−8

10−7

10−6

10−3

10−2

10−1

100

σ2 [s]

RMSE

( x 0) [

m]

2−step (two−way)Joint estimator (two−way)RCRB (two−way)2−step (ATPL)Joint estimator (ATPL)RCRLB (ATPL)

Fig. 2: RMSE of the estimated sensor coordinates.

In this paper, we analyze the performance of the proposedestimators in terms of the root mean square error (RMSE) ofthe estimated sensor position, clock-skews, and clock-offsetsfor different values ofσ2. We provide the results for a) two-way communication protocol [4] b) ATPL protocol [5].

10−9

10−8

10−7

10−6

10−7

10−6

10−5

10−4

σ2 [s]

RMSE

(ω )

2−step (two−way)joint estimator (two−way)RCRB (two−way)2−step (ATPL)joint estimator (ATPL)RCRLB (ATPL)

Fig. 3: RMSE of the estimated clock-skews.

10−9

10−8

10−7

10−6

10−6

10−5

10−4

10−3

10−2

σ2 [s]

RMSE

( φ)

[s]

2−step (two−way)joint estimator (two−way)RCRB (two−way)2−step (ATPL)joint estimator (ATPL)RCRLB (ATPL)

Fig. 4: RMSE of the estimated clock-offsets.

Fig. 2 shows the RMSE of the estimated sensor positioncomputed using a) the two-step approach, i.e., LS estimator(10) with the range estimates of (8) as the input, and b) theproposed joint estimator. The root CRLB (RCRLB) is alsoprovided for both the considered protocols. The ATPL pro-tocol performs better than the two-way communication pro-tocol due to the additional passive listening links [5]. How-ever, both the estimators for localization are inaccurate as thedependencies with the nuisance parameters are not consid-ered, and this can be resolved using a constrained WLS solu-tions [7].

Fig. 3 and Fig. 4 show the RMSE of the estimated clock-skews and clock-offsets. The ATPL protocol again performsbetter than the two-way communication protocol. In addi-tion, both the estimators for clock-skews and clock-offsets areasymptotically efficient, and meet the CRLB.

The location of the target node can be obtained with a two-step approach using the range estimates obtained from jointsynchronization and ranging. Alternatively, we can formulatelocalization and synchronization under a unified frameworkas a single linear problem. However, this results in a largersystem to solve and is computationally less attractive thanthetwo-step approach. The linear model of the joint synchroniza-tion and localization can be used for joint tracking of the clock

parameters and the position which is an important applicationin a WSN.

8. CONCLUSIONS

We have considered a fully-asynchronous network with onesensor and a few anchors. In this paper, we have addresseda problem in which we estimate all the unknown clock pa-rameters as well as the position of the target node. Loca-tion of the node can be estimated with a two-step approachusing the range estimates. To avoid this two-step approach,we have proposed a generic linear data model for joint lo-calization and clock synchronization. An estimator based onLS to jointly estimate the position of the target sensor node,along with all the unknown clock-skews and clock-offsets hasbeen presented. The proposed estimator for clock-skews andclock-offsets is asymptotically efficient and meets the CRLB,however, the position estimates do not asymptotically achievethe CRLB.

9. REFERENCES

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[2] Yik-Chung Wu, Q. Chaudhari, and E. Serpedin, “Clocksynchronization of wireless sensor networks,”IEEE Sig-nal Process. Mag., vol. 28, no. 1, pp. 124 –138, jan. 2011.

[3] F. Gustafsson and F. Gunnarsson, “Mobile positioningusing wireless networks: possibilities and fundamentallimitations based on available wireless network measure-ments,”IEEE Signal Process. Mag., vol. 22, no. 4, pp. 41– 53, july 2005.

[4] R. T. Rajan and A.-J. van der Veen, “Joint ranging andclock synchronization for a wireless network,” inProc.4th IEEE International Workshop on Computational Ad-vances in Multi-Sensor Adaptive Processing (CAMSAP),dec. 2011, pp. 297 –300.

[5] S. P. Chepuri, R. Rajan, G. Leus, and A.-J. van der Veen,“Joint clock synchronization and ranging: Asymmetricaltime-stamping and passive listening,”IEEE Signal Pro-cess. Lett., vol. 20, no. 1, jan 2013.

[6] Jun Zheng and Yik-Chung Wu, “Joint time synchroniza-tion and localization of an unknown node in wireless sen-sor networks,”IEEE Trans. Signal Process., vol. 58, no.3, pp. 1309 –1320, march 2010.

[7] Yiyin Wang, Xiaoli Ma, and G. Leus, “Robust time-basedlocalization for asynchronous networks,”IEEE Trans.Signal Process., vol. 59, no. 9, pp. 4397 –4410, sept.2011.