Joint TOA-Based Sensor Synchronization and Localization Using ...

Communications (ICC), 2014 IEEE International Conference on

Joint TOA-Based Sensor Synchronization and Localization Using Semidefinite Programming Reza Monir Vaghefi and R. Michael Buehrer Wireless @ Virginia Tech, Virginia Tech, Blacksburg, VA 24061 E-mail: {vaghefi, buehrer}@vt.edu synchronization techniques is provided in [7]. Recently, it has been shown the joint synchronization and localization provide significant improvement over two-step approaches, since there is a close relationship between them and they can be performed simultaneously [8]. The Cram´erRao Lower Bound (CRLB) and the maximum likelihood (ML) estimator of joint synchronization and localization are studied in [8]. Unfortunately, the ML estimator has a severely nonlinear and nonconvex cost function which does not have a closedform solution. The ML estimator can be solved approximately by iterative methods which require an appropriate initialization [9], [10]. The performance of iterative methods is dependent on the initial point. An iterative method may converge to a local minimum (or a saddle point) resulting in large estimation errors. To deal with this problem, several approaches such as linear estimators [11], [12] and convex relaxation techniques [13]–[16] have been introduced. Although linear estimators have closed-form solutions, their performance is poor when the signal-to-noise ratio (SNR) is moderate to low. Several linear estimators for joint synchronization and localization are derived in [8], [17], [18]. Convex relaxation techniques, such as semidefinite programming (SDP) and second order cone programming, provide another solution for the ML convergence problem. In this technique, the nonlinear and nonconvex ML problem is relaxed into a convex optimization problem. The advantage of an SDP is that the cost function has no local minima or saddle points and thus convergence to the global minimum is guaranteed [19], [20]. The disadvantage of the SDP techniques is that they are sub-optimal and optimal accuracy cannot be expected in all conditions. In this work, asynchronous sensor localization and synchronization are studied. The clock parameters and the locations of the anchor nodes are assumed to be known, while those of the source node are unknown and need to be determined. Therefore, the goal is not only to estimate the location of the source node but also to synchronize it by determining its unknown clock parameters. A joint synchronization and localization framework is considered, as it is expected to provide considerably better accuracy. The system model is introduced and the corresponding ML estimator is formulated. A novel SDP estimator is derived by relaxing the nonconvex and nonlinear ML estimator into a convex one. The CRLB of the system model is also derived as a benchmark for comparison. Computer simulations are conducted to evaluate the performance of the proposed estimator.

Abstract—In this paper, asynchronous sensor localization using time-of-arrival (TOA) measurements is studied. Accurate TOAbased localization requires perfect timing synchronization between the source and anchor nodes. In asynchronous networks, the anchor nodes are assumed to be synchronized, while the clock of the source node needs be synchronized with those of the anchor nodes. Although synchronization and localization are typically performed separately, in this work a joint synchronization and localization framework is considered, as it is expected to provide significant improvement over two-step approaches. The clock parameters (clock offset and skew) of the source node are estimated jointly with its location. The corresponding Cram´er-Rao lower bound (CRLB) and the maximum likelihood (ML) estimator of the system model are derived. The ML estimator is highly nonlinear and nonconvex which must be solved with computationally complex algorithms. Alternatively, a novel semidefinite programming (SDP) estimator is introduced by relaxing the original ML minimization problem into a convex problem. Computer simulations show that the proposed SDP estimator outperforms other previously proposed estimators. Index Terms—Time-of-arrival (TOA), localization, synchronization, semidefinite programming (SDP).

I. I NTRODUCTION Wireless sensor networks (WSNs) have been widely used for many military, civilian, and commercial applications. Localization and synchronization are the main components of every WSN. Sensor location information must be known to make its data meaningful [1]. Moreover, synchronization plays an important part in many operations of WSNs such as power management, data fusion, and spectrum allocation [2]. The lack of synchronization among nodes in a WSN is mainly due to their different clock parameters (clock offset and skew). Since sensors have different hardware implementations, each sensor has its own specific clock. Clock synchronization in WSNs is typically performed by transferring a series of time stamp messages among nodes. The two most common clock synchronization techniques are one-way message dissemination and two-way message exchange [2]. On the other hand, localization using time-of-arrival (TOA) method is highly dependent on synchronization. A small difference between the clocks of nodes may lead to a significant localization error. Typically in an asynchronous network, the clocks of the nodes are first synchronized [3], [4] and localization is then performed [5], [6]. However, this approach can lead to a poor synchronization performance and causes error propagation through the network which affects the localization performance dramatically. An overview on the different clock 1

Throughout the paper, the following notation is used. Lowercase and uppercase letters denote scalar values. Bold uppercase and bold lowercase letters denote matrices and vectors, respectively. IM and 0M denote the M × M identity and the M × M zero matrices, respectively. 1M represents the M × 1 vector of all ones. For arbitrary symmetric matrices of equal size, A and B, A  B means that A − B is positive semidefinite.

Anchor Node i Clock

Local Clock Time

Ideal Clock

II. S YSTEM M ODEL

Anchor Node i Clock

...

The internal clocks of the nodes are assumed to be imperfect which makes the internal time drift away from the reference time. The relationship between the internal time of a node and the reference time is modeled as [2], [8], [18] ti = ω i t + θ i

Source Node Clock

Source Node Clock

Real Time

(1)

Fig. 1. The two-way message exchange between the source and anchor node with different clock skews and clock offsets.

where ti and t are the internal time of the ith node and the reference time, respectively. ωi is the clock skew and θi represents the clock offset. Each node has a unique clock, i.e., the clock skew and offset are different for each node. A network with M anchor nodes and one source node is considered. Note that we consider the general case of multiple source nodes in future work. Let x = [x, y]T ∈ R2 and ωx , θx be the unknown coordinates and the clock parameters of the source node, respectively. Let yi = [xi , yi ]T ∈ R2 , i = 1, . . . , M and ωi , θi be the known coordinates and clock parameters of the ith anchor node, respectively. The clock parameters and location of the source nodes are jointly estimated from a series of synchronization messages transferred among the nodes. There are two common clock synchronization techniques in WSNs: one-way message dissemination and twoway message exchanges. In the former, either the source node (forward link) or the anchor nodes (backward link) transmits the synchronization messages. In the latter, both source and anchor nodes transmit the synchronization messages as shown in Fig. 1. Several rounds of messages are usually transferred between the nodes to achieve higher accuracy. At the mth round, the source node transmits the forward signal at the time stamp Tim and the anchor node i receives the signal at the time stamp Rim . The anchor node i then sends back a signal at T¯im and the source node k captures the backward ¯ im . The time stamps Tim and R ¯ im are reported signal at R based on the internal clock of the source node, while Rim and T¯im are reported based on the internal clock of anchor node i. The measured time stamps at the receivers are modeled as ωi ωi Rim = Tim + ωi (ti + nim ) − θx + θi ωx ωx ωx ¯ im = ωx T¯im + ωx (ti + n R ¯ im ) − θi + θx (2) ωi ωi

where c is the propagation velocity. The terms nim and n ¯ im represent measurement errors which are modeled as independent and identically distributed (i.i.d.) Gaussian random variables with variance σi2 and σ ¯i2 , respectively. The variances of the measurements do not vary with time. However, they are dependent on the received SNR and typically modeled as [21] σi2 = ξi dγi σ ¯i2 = ξx dγi

(4)

where ξi and ξx define the relationship between the noise variance and the true distance whose values are dependent on the propagation environment and hardware implementations. γ is the path-loss exponent which typically varies between 2 (free space) and 4 (harsh environments) [21]. In most studies, the variances of the measurements are assumed to be the same for all links. However, in this work, each link has a ¯ im } unique variance. All four time stamps {Tim , Rim , T¯im , R are available to the anchor nodes which are used to estimate the location and clock parameters of the source node. We assume that during L rounds of message exchanges, the clock parameters and location of the source node do not change. To further simplify (2), the L rounds of message exchanges can be averaged ωi ωi Ri = Ti + ωi (ti + ni ) − θx + θi ωx ωx ωx ¯ i = ωx T¯i + ωx (ti + n R ¯i) − θi + θx (5) ωi ωi where Ti =

where ti is the propagation time between the source and the ith anchor node 1 ti = di cq 1 = (x − yi )T (x − yi ) (3) c

L 1 X Tim , L m=1

L 1 X ¯ T¯i = Tim , L m=1

Ri =

L 1 X Rim L m=1

L X ¯i = 1 ¯ im R R L m=1

(6)

and ni and n ¯ i are i.i.d. Gaussian random variables with variance σi2 /L and σ ¯i2 /L, respectively. Note that collecting more rounds of measurement does not change the model but 2

type of convex optimization problem, since its cost function and constraints (which might be nonlinear) are convex. To formulate an SDP estimator for the system model, the cost function of the ML estimator needs to first be formulated as a linear function. Then, the constraints are relaxed into a series of positive semidefinite matrices. The expressions in (7) can be written in vector form as

decreases the effect of the measurement noises. The more rounds of measurements are collected, the lower the variance of the noise is. III. M AXIMUM L IKELIHOOD E STIMATOR The ML estimator has several attractive properties. The ML estimator is asymptotically efficient which means that it can reach the CRLB when the number of measurements tends to infinity [22, Ch. 7]. The CRLB of the system model is derived in Appendix A. Applying the ML estimation directly to the model in (5) is slightly difficult, since an unknown parameter appears in the variance of the noise measurements. To avoid this, a transformation can be applied to the model in (5). According to the invariance property, the ML estimator of the transformed parameters can be determined by applying the transformation to the ML estimator of the original parameters [22, Ch. 7]. Introducing two new variables, the model in (5) can be alternatively written as

n = Rβ − t − βx T1M + αx 1M − α ¯ − t + βx R1 ¯ M − αx 1M + α ¯ = Tβ n where t = [t1 , . . . , tM ]T and T = diag{T1 , . . . , TM }, ¯ = diag{T¯1 , . . . , T¯M }, T

R = diag{R1 , . . . , RM } ¯ = diag{R ¯ 1, . . . , R ¯M } R

n = [n1 , . . . , nM ]T ,

¯ = [¯ n n1 , . . . , n ¯ M ]T

β = [β1 , . . . , βM ]T ,

α = [α1 , . . . , αM ]T .

Defining a new variable h = [tT , βx , αx ]T , (9) is written as

ni = Ri βi − ti − Ti βx + αx − αi ¯ i βx − ti − T¯i βi − αx + αi n ¯i = R

n = b − Ah ¯ − Ah ¯ ¯=b n

(7)

where βi = 1/ωi , αi = θi /ωi , βx = 1/ωx, and αx = θx /ωx . The ML estimator is obtained by maximizing the likelihood function [22, Ch. 7]. Let ϕ = [xT , βx , αx ]T be the vector of unknown parameters to be estimated. Since the distributions of ni and n ¯ i are Gaussian, the ML estimator of the model in (7) is simply obtained by the following minimization problem ˆ = arg min ϕ x,βx ,αx

+

M X

σi−2 (Ri βi − ti − Ti βx + αx − αi ) ¯ i βx − ti − T¯i βi − αx + αi σ ¯i−2 R

 A = IM  ¯ = IM A

i=1

 T1M −1M ,  ¯ M 1M , −R1

b = Rβ − α ¯ = Tβ ¯ + α. b

By using (10), the cost function of the ML estimator in (8) can be alternatively written as ¯n = ¯ T Q¯ nT Qn + n T ¯ nn ¯ T} = Tr{Qnn + Q¯

2


2

(10)

where

i=1

M X

(9)

¯ Ah ¯ T} = ¯ Ah ¯ − b)( ¯ − b) Tr{Q(Ah − b)(Ah − b)T + Q(

. (8)

Tr{Q(AHAT − 2AhbT + bbT ) ¯T + b ¯b ¯ T )} ¯ AH ¯ A ¯ T − 2Ah ¯ b + Q(

The estimates of the clock parameters would be: ω ˆ x = 1/βˆx and θˆx = α ˆ x /βˆx . In the two-way message exchange technique, both sets of forward and backward messages are available. However, in one-way message dissemination technique, only one of either the forward set or the backward set is available. Therefore, the minimization is only performed over the set which is available. The problem in (8) is highly nonlinear and nonconvex and its closed-form solution is not available. However, it can be approximately solved by iterative numerical techniques [9], [22], [23]. Since the cost function is nonconvex, there is no guarantee that the solver converges to the global minimum. The difficulty in finding the solution of the ML estimator leads us to employ suboptimal estimators, such as SDP and linear least squares (LLS) estimators. Several linear estimators have been previously derived in the literature [8], [17], [18]. A novel SDP estimator is derived in the next section.

−2 −2 ¯ = diag{¯ where Q = diag{σ1−2 , . . . , σM }, Q σ1−2 , . . . , σ ¯M }, T and H = hh . The first M diagonal elements of the matrix H are  T    1 yi I2 x yi [H]ii = 2 , i = 1, 2, . . . , M (11) xT z −1 c −1

where z = xT x. By using the auxiliary variables above, the ML estimator in (8) is rewritten as ¯ AH ¯ A ¯ T− 2Ah ¯ b ¯ T )} minimize Tr{Q(AHAT− 2AhbT )+ Q( x,z,h,H  T    1 yi I2 x yi subject to [H]ii = 2 , i = 1, . . . , M xT z −1 c −1 H = hhT , z = xT x.

(12)

¯b ¯ T are removed from the cost function, The terms bbT and b since they are constant and do not impact the minimization problem. Now, the problem in (12) has a linear cost function. However, there are still two non-affine constraints which make the problem nonconvex. To convert the problem in (12) into a convex SDP problem, the non-affine operations need to

IV. S EMIDEFINITE P ROGRAMMING The proposed SDP estimator is derived in this section. A SDP includes a linear cost function and a series of constraints which are defined by positive semidefinite matrices. SDP is a 3

be relaxed. Using the Schur complement, the matrix H and the variable z are relaxed and written as a linear matrix inequality (LMI)   I2 x T z=x x ⇒ T  03 x z   H h H = hhT ⇒ T  0M+3 . (13) h 1

TABLE I S UMMARY OF THE C ONSIDERED E STIMATORS .

Although the above approximations may affect the performance of the estimation, they enable us to relax the nonconvex ML estimator into a convex estimator. Using the above relaxations, the nonlinear and nonconvex ML estimator can be formulated as

Estimator

Description

SDP ML LLS2 LLS SEP

The proposed SDP algorithm in (14) The ML estimator in (8) A linear estimator in [17] A linear estimator in [8] A two-step approach: sync. in [4], local. in [11]

of the considered estimators and their labels is provided in Table I. The performance of the considered estimators is evaluated for the three synchronization techniques: forward, backward, and two-way. The SEP estimator is only applicable for the two-way message exchange, since separate synchronization and localization cannot be performed in the one-way message dissemination techniques (forward or backward) [2]. Moreover, since the SEP estimator performs synchronization and localization separately, at least two rounds of measurements between the source and anchor nodes are required [2]. However, for joint synchronization and localization, one round of measurements is sufficient. A network with eight anchor nodes and a source node distributed uniformly in a square area 10m × 10m is considered. The location of the anchor nodes is fixed and in each experiment the location of the source node is randomly generated. The clock offset and skew of the nodes are drawn from a uniform distribution U[0, 10−1 ] µs and U[0.95, 1.05], respectively. The transmission times of the nodes Tim and T¯im are drawn from a uniform distribution U[5m, 5m + 1] s and U[5m + 3, 5m + 4] s, respectively. The variances of the measurement noises are determined based on (4). The value of the path-loss exponent γ is set to 2 and assumed to be the same for all nodes. The values of ξx and ξi are assumed to be the same for all nodes. For each location of the source node, 100 realizations are performed. The time measurements are obtained based on the model given in (2). In each noise realization, the clock parameters and transmission times are also changed. Moreover, two rounds of measurements are performed between nodes, i.e., L = 2. The cumulative distribution function (CDF) of the localization error for the compared estimators is given in Fig. 2. The estimators are compared under high and low SNR situations. In the high SNR case, the average error on the measurements in the network is about 10% of the true values, while in low SNR, it is about 25%. The upper and lower rows depict the performance of the estimators in high and low SNR, respectively. The ML estimator, as expected, provides the best performance in all cases, since its solver is initialized with the true values and provides a lower bound of error performance. Among the estimators, LLS has the worst performance, mainly due to several approximations. LLS2 performs significantly better than LLS, especially in high SNR. The reason is that, in LLS2, the effect of clock offset is eliminated by using a time difference calculation and linearization is performed over three parameters rather than four parameters. Therefore, there are fewer approximations in LLS2 than LLS which helps the

¯ T )} ¯ AH ¯ A ¯ T− 2Ah ¯ b minimize Tr{Q(AHAT− 2AhbT )+ Q( x,z,h,H  T    1 yi I2 x yi subject to [H]ii = 2 , i = 1, . . . , M xT z −1 c −1     H h I2 x  0M+3 ,  03 (14) hT 1 xT z which is an SDP optimization problem. The solution of (14) can be found with iterative algorithms such as the interior point method [19], [20]. Although an iterative algorithm is employed to solve an SDP problem, the algorithm is selfinitialized and requires no initialization from the user. Unlike the ML estimator, the cost function of the SDP problem in (14) is linear which ensures that there is only one minimum point. The most important feature of SDP problems is that they can be solved very effectively in polynomial time [19]. In the MATLAB simulations, the standard SDP solvers such as SeDuMi [24] are used to solve SDP problems. If the equality holds in the constraints in (13), the solution of the SDP and ML estimators will be the same. V. S IMULATION R ESULTS Computer simulations are conducted to evaluate the performance of the proposed SDP estimator. The ML estimator is solved by the MATLAB routine fminunc. The solver of the ML estimator is initialized with true values in an attempt to avoid local minima and thus represents a bestcase performance. The proposed SDP is implemented by the CVX toolbox [25] using SeDuMi as the solver [24]. Besides the ML estimator described in the previous sections, three other estimators are selected for comparisons. To show the advantages of joint synchronization and localization, a two-step approach is also included in the simulations. In this technique, the synchronization is first performed in the network by using the method described in [4], where the clock parameters of the source node are estimated. The synchronization method in [4] has a closed-form solution and requires no initialization. Then a classic localization estimator assuming perfect synchronization is employed to estimate the location of the source node. A well-known linear estimator in [11], which also has a closedform solution, is selected for localization. Two other linear estimators in [8] and [17] performing joint synchronization and localization are also included for comparison. A summary 4

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Fig. 2. The CDF of localization error of the compared estimators. There are 8 anchor nodes and one source node in the network. In the upper row, the SNR is high and the average error is about 10% of the true values. In the lower row, the SNR is low and the average error is about 25% of the true values.

initialized with the true values. Despite the fact that the SEP estimator can only be employed in the two-way technique, its performance is better than LLS and LLS2 but worse than SDP. Since the SEP performs the synchronization part based on ML estimation, the accuracy of synchronization is satisfactory. However, the localization part is based on a linear estimation which is subject to performance degradation. Another reason for the poor performance of the SEP in comparison with the ML and SDP might be that synchronization and localization are performed separately. Hence, it seems reasonable to use SEP estimator in high SNR, if a closed-form and simple estimator is required. However, having slightly more complexity, the proposed SDP estimator provides better performance in most situations.

performance. At low SNR, the performance of LLS2 is poor, since its approximations are no longer valid in this situation. In high SNR, LLS2 provides satisfactory performance about 90% of the time. In 10% of the scenarios, LLS2 performs very poor (represented by a long tail in its CDF). That is because in LLS2 the problem of localization is transformed form a circular geometry to a hyperbolic one and the latter is highly sensitive to the noise and the location of the source node. Therefore, in some realizations, where the source node is typically located outside convex hull (especially behind the anchor nodes) or the value of the noise is very high, LLS2 performs very poorly. In both high and low SNR, the performance of the proposed SDP is better than LLS and LLS2 in all three synchronization techniques (forward, backward, two-way). Moreover, it performs as good as the ML estimator for the two-way message exchange technique. In the backward and forward techniques, SDP performs slightly worse than the ML estimator. The reason is that in the one-way techniques, four parameters (2D source location, clock skew, and clock offset) should be relaxed which degrades the performance when the relaxation is not tight enough. On the other hand, in the two-way technique, three parameters should be relaxed, as the effect of clock offset in the cost function is automatically eliminated, and the performance of the proposed SDP is less affected by relaxation. As depicted in Fig. 2d, the proposed SDP performs better than the ML estimator about 20% of the time. The poor performance of the ML estimator might be due to its convergence to local minima, even though the algorithm is

VI. C ONCLUSION In this paper, asynchronous TOA-based sensor localization was examined. It was assumed that the locations of the anchor nodes are known and they are perfectly synchronous, while the location and clock parameters of the source nodes are unknown. A joint synchronization and localization framework was proposed, since it was expected to provide better accuracy than two-step approaches. The CRLB and the ML estimator of the system model were derived. A novel SDP estimator was also derived which jointly estimates the location (localization) and the clock parameters (synchronization) of the source nodes. Simulation results showed that the proposed SDP estimator performs as good as the ML estimator in most situations. Unlike the ML estimator, the proposed SDP requires no initialization and does not have convergence problems. 5

A PPENDIX A C RAM E´ R -R AO L OWER B OUND

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The CRLB expresses a lower bound on the variance of any unbiased estimator [22, Ch. 3]. Let ϕ = [xT , ωx , θx ]T be the vector of unknown parameters to be estimated. The CRLB of ϕ is the diagonal elements of the inverse of the Fisher information matrix obtained by [22, Ch. 3]  2  ∂ ln p [F]ij = −E , i, j = 1, 2, 3, 4, (15) ∂ϕi ∂ϕj where p is the probability density function (liklihood function) of the system model in (5) which is written as
¯ i }i=1,...,M ; {x, ωx , θx } = ln p {Ti , Ri , T¯i , R ! M q Y 2 2 2 − ln 2πωi σi 2πωx2 σ ¯i i=1

− −

M 1X

2 2



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¯ 2 Ri T¯i θx θi − ti − − + . ωx ωi ωx ωi

i=1

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Ri Ti θx θi − ti − + − ωi ωx ωx ωi

(16)

The elements of the Fisher information matrix are calculated as follows:   Fxx fxω fxθ T F =  fxω fωω fωθ  (17) T fxθ fωθ fθθ where

Fxx = fxω =

M X i=1 M X

(σi−2 + σ ¯i−2 )ai aT i ai σi−2 µi + σ ¯i−2 λi

i=1

fxθ = fωω =

M X i=1 M X




ωx−1 ai (−σi−2 + σ ¯i−2 ) σi−2 µ2i + σ ¯i−2 λ2i + 2ωx−2

i=1

fωθ = fθθ =

M X i=1 M X

ωx−1 (−σi−2 µi + σ ¯i−2 λi ) ωx−2 σi−2 + σ ¯i−2

i=1

and

 ai = (x − xi )/cdi




(y − yi )/cdi

µi = ωx−2 (θx − Ti ) λi = ω −1 (di /c + T¯i /ωi − θi /ωi ).

T

x

Finally, the CRLB on the variance of the estimators of the deterministic parameters is Var[ϕi ] ≥ [F−1 ]ii ,

i = 1, 2, 3, 4.

(18)

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