Robust Relative Location Estimation in Wireless Sensor ... - IEEE Xplore

IEEE TRANSACTIONS ON MOBILE COMPUTING,

VOL. 11,

NO. 6,

JUNE 2012

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Robust Relative Location Estimation in Wireless Sensor Networks with Inexact Position Problems Wei-Yu Chiu, Member, IEEE, Bor-Sen Chen, Fellow, IEEE, and Chang-Yi Yang Abstract—In this paper, the relative location estimation problem, a prominent issue faced by several applications in wireless sensor networks (WSNs), is considered. Sensors are classified into two categories: location-aware and location-unaware sensors. To estimate the positions of location-unaware sensors, exact positions are often assumed for location-aware sensors. However, in practice, such precise data may not be available. Therefore, determining the positions of location-unaware sensors in the presence of inexact positions of location-aware sensors is the primary focus of this study. A robust min-max optimization method is proposed for the relative location estimation problem by minimizing the worst-case estimation error. The corresponding optimization problem is originally nonconvex, but after it is transformed into a convex semidefinite program (SDP), it can be solved by existing numerical techniques. In the presence of inexact positions of location-aware sensors, the robustness of the proposed approach is validated by simulations under different WSN topologies. Modified maximum-likelihood (ML) estimation and second-order cone programming (SOCP) relaxation methods have been used for localization in comparison with the proposed approach. Index Terms—Relative location estimation, inexact position problem, wireless sensor networks (WSNs), maximum-likelihood (ML) estimation, second-order cone program (SOCP), semidefinite program (SDP).

Ç 1

INTRODUCTION

A

CCURATE relative location estimation (or self-configuration) is a key requirement of various applications in WSNs [1], [2], [3]. In the literature on location estimation in WSNs, location-aware nodes are also referred to as anchors, beacons, landmarks, or reference devices, whereas location-unaware nodes are simply termed sensors, or blindfolded devices [4], [5], [6], [7], [8]. The self-configuration process can be divided into three stages [7]. The first stage relates to obtaining distances or angles between neighboring sensor nodes. In this stage, either of three popular techniques could be applied, namely time (difference) of arrival (T(D)OA), angle of arrival (AOA), and received signal strength (RSS). The RSS method is adopted in this study due to its simplicity and low cost. The reader can refer to [5], [9], [10], [11], [12] for more details about RSS-based localization. The second stage involves estimating the positions of location-unaware nodes according to the measurements obtained in the first stage. The third stage is optional, and involves refining the location estimation [13], [14]. In this study, we mainly focus on the second stage.

. W.-Y. Chiu is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544. E-mail: [email protected]. . B.-S. Chen is with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu City, Taiwan 30013, R.O.C. E-mail: [email protected]. . C.-Y. Yang is with the Department of Computer Science and Information Engineering, Penghu University of Science and Technology, Penghu, Taiwan 88046, R.O.C. E-mail: [email protected]. Manuscript received 24 Oct. 2009; revised 14 Apr. 2011; accepted 6 May 2011; published online 26 May 2011. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TMC-2009-10-0454. Digital Object Identifier no. 10.1109/TMC.2011.111. 1536-1233/12/$31.00 ß 2012 IEEE

Some research works on relative location estimation problems assume that the exact positions of location-aware nodes are known. However, in reality, such position information has errors [15] and the estimation performance will deteriorate if these errors are neglected. The reasons for these errors can include inaccurate range measurements, algorithm artifacts, or a combination of both. For related studies on location errors for location-aware nodes, the reader can refer to [15], [16], [17] for geographical routing problems, and [18], [19], [20] for target localization problems. For addressing a relative location estimation problem, convex optimization methods have been used in previous literature. For example, Doherty et al. first studied convex position estimation in WSNs [21], and several convex approaches have since then been developed. Biswas et al. [6] proposed a semidefinite programming (SDP) relaxation method. In this approach, the refinement after positioning was also addressed. Tseng [8] studied the second-order cone programming (SOCP) relaxation technique; however, SOCP relaxation causes the estimated locations to lie in the convex hull of the location-aware sensor nodes [20]. In other words, it fails if there are a large number of locationunaware sensors positioned outside the convex hull. Apart from the convex positioning methods, many studies employ conventional maximum likelihood (ML) estimation methods for relative location estimation in WSNs. For instance, Chang and Liao [22] proposed a probability-based ML estimation approach if some prior information is available. Moses et al. [2] used ML estimation for AOA and TOA measurements, whereas Patwari et al. [4], [5] employed the same estimation method for TOA and RSS measurements and studied the Cramer-Rao lower bound (CRLB). The performance of ML estimation for Published by the IEEE CS, CASS, ComSoc, IES, & SPS

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location estimation is highly accurate if the corresponding likelihood optimization problem can be solved, for example, by grid search. However, the solving process often takes a great deal of time. Although other iterative search techniques may be applied for the likelihood optimization problem, the accuracy usually depends heavily on the starting point of the solving process. Therefore, a different formulation of the relative location estimation problem that does not rely on a good starting point and can be solved by existing numerical techniques is needed. In this study, the location errors of location-aware sensors are assumed to be bounded [20], [23] with an unknown distribution. A robust relative location estimation method is proposed to deal with the inexact position problem. Using the distance measurements obtained from the RSS technique, a min-max optimization method is proposed to minimize the worst estimation error (the maximum error). Therefore, it is the worst-case scenario design. Owing to the nonconvexity of the proposed optimization formulation, it is then transformed into a convex optimization formulation, which is solvable by existing numerical techniques [24]. The aims of this paper are stated as follows: Unlike existing studies that focus on a Gaussian model for the inexact position problem [25], [26], this paper examines the problem from the worst-case perspective. In other words, exact knowledge of the location error distribution is not needed in the proposed approach. To the best of our knowledge, the worst-case scenario for the relative location estimation in WSNs has not been fully investigated. Furthermore, the estimation performance is conventionally evaluated for a particular deployment of sensors, with a theoretical lower bound. Conversely, in this paper, for a fair comparison of different approaches, the performance has been evaluated by the average of random deployments, which means that different topologies of sensors have been considered. Although only the RSS measurement technique is employed, the proposed approach is very flexible in the sense that other techniques such as TOA and TDOA can be easily integrated into the estimation scheme. In contrast to the ML estimators for localization, whose accuracy depends heavily on the starting point for the solving process, the proposed method is relatively robust. In comparison to the SOCP methods, which result in a poor performance if location-unaware nodes lie outside the convex hull of the location-aware nodes, a more accurate performance is achieved by the proposed method, as evidenced by our simulation results. Convex positioning techniques have been applied to target localization problems in the presence of location errors of locationaware sensors [20]. This is the first study (that we are aware of) that has considered convex optimization methods for relative location estimation with the inexact position problem. The rest of this paper is organized as follows: Related and future works are examined in Section 2. Section 3 describes the notations used and the scheme concerned. Motivated by [4], [6], and [20], a robust relative location estimation scheme is proposed in Section 4. Section 5 presents numerical results for the proposed approach and other location estimation methods. Finally, some concluding remarks are given in Section 6.

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RELATED AND FUTURE WORKS

In the context of the relative location estimation problem, a hybrid Cramer-Rao bound (HCRB) was examined for passive localization with uncertain anchor positions in [25], [26]. However, the presented bounds in [25], [26] may not necessarily be tight in certain circumstances [27]. It is noted that all bounds for the estimation error depend on the distribution of measurements and hence could vary over the different distributions. Conventionally, an estimation problem involves a comparison between the proposed method and the theoretical Cramer-Rao bound (CRB) [28]. The bound can be derived for a particular distribution of measurements. For most cases, a Gaussian model is adopted because of its simplicity, thus resulting in a closed-form bound. However, since the distribution of location errors is assumed to be bounded with unknown statistical properties in this study, the CRB cannot be derived. Even if a particular distribution is assumed, the proposed min-max optimization method cannot be proven to be unbiased and hence, the comparison between the proposed estimator and the theoretical bound could be misleading. Therefore, the CRB analysis is not included in this paper. See [4], [6], [7], [20], [29], [30] for more related works. In [7], [29], and [30], mobile sensor localization was considered. The inclusion of mobile sensors could be a future research direction. In [4], ML estimators were used for relative location estimation and will be compared to the proposed approach in the simulation. Besides, the distance measurement model in this study is based on [4]. In [6], SDP localization methods were proposed. Some SDP relaxation and problem formulation techniques are used in this paper. In [20], a target localization problem was dealt with. The consideration of anchor location errors motivates our study of the inexact position problem. Lastly, it should be mentioned that the non-line-of-sight (NLOS) measurement propagation is a very important issue that still needs to be addressed, as most existing studies on relative location estimation problems in WSNs only focus on the line-of-sight (LOS) environment, including this study. This issue is mostly considered in the mobile location estimation problem [31], [32], [33], [34]. For those studies on NLOS mitigation in WSNs, such as [35], [36], exact positions of location-aware sensors are assumed. For WSNs, the NLOS problem is difficult to address because each range estimate could be LOS or NLOS, and cannot be easily identified. The WSN relative location estimation in the presence of both the inexact position and the NLOS problems could be an interesting research topic in the future. Remark 1. For other WSN localization studies, the reader can refer to [13], [14], [37], [38], [39], [40], [41], [42]. A fundamental upper bound for relative location estimation accuracy using TOA was derived in [37]. A parking management system for relative location estimation of vehicles was examined in [40]. In [41], a segmentationaided and density-aware hop-count localization algorithm was proposed. Vision-based bearing measurements were employed for cooperative localization in [42]. Localization problems were handled by successive

CHIU ET AL.: ROBUST RELATIVE LOCATION ESTIMATION IN WIRELESS SENSOR NETWORKS WITH INEXACT POSITION PROBLEMS

refinement in [13] and [14]. Distributed location algorithms with constrained sensor resources were discussed in [38]. A GPS-free positioning scheme was studied in [39]. More approaches to obtain the topology of a WSN can be found in [43], [44], [45], [46], [47], [48].

3

NOTATION AND PROBLEM FORMULATION

Suppose that there are N location-unaware sensors and M location-aware sensors. For n ¼ 1; 2; . . . ; N, xn 2 R2 represent the unknown positions of location-unaware sensors to be estimated. For m ¼ 1; 2; . . . ; M, let z m 2 R2 denote the true positions of location-aware sensors. In practice, the known positions of location-aware sensors are corrupted with location errors [19], [20]. The relationship between the true position z m and the inexact position ^zm can be defined by zm ¼ ^zm þ m ; for m ¼ 1; 2; . . . ; M;

ð1Þ

km k :

ð2Þ

where

In (2), k k represents the euclidean norm. For example, if qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ¼ ðmx my ÞT 2 R2 , then km k ¼ 2mx þ 2my . Equation (1) indicates that the true position z m , which is unavailable, can be obtained by perturbing the inexact position ^z m , which is available, with the bounded error . The existence of the location error m for m ¼ 1; 2; . . . ; M in (1) is called the “inexact position problem” in this study. By using the RSS technique for range estimation, the relative distance measurement d^nm between sensors can be obtained as [4], [5] d^nm ¼ d0 10

p0 ^ pnm 10np

nm

¼ dnm 1010np ;

ð3Þ

where p^nm ¼ pnm þ nm ; pnm ¼ p0 10np log10

dnm : d0

ð4Þ

In (3), p0 represents the received power at a reference distance d0 , np is the path loss exponent [49], dnm is the euclidean distance between two sensors, p^nm is the measured power, and nm models the fading channel, which is a zero-mean Gaussian with variance 2p . Due to the ranging limitation, pairwise estimation d^nm may not be available. Let d~ be the threshold for a possible connection. For n ¼ 1; 2; . . . ; N, define INðnÞ ¼ IN1 ðnÞ [ IN2 ðnÞ

ð5Þ

as the index set that indicates neighbors of the nth locationunaware sensor, where ~ xn x n0 k dg; IN1 ðnÞ ¼ fn0 : n < n0 N; kx ~ IN2 ðnÞ ¼ fm : 1 m M; kx xn z m k dg:

ð6Þ

Therefore, d^nm is available if and only if m 2 INðnÞ. Note that IN1 ðnÞ represents the connection between locationunaware sensors, whereas IN2 ðnÞ represents the connection between location-unaware and location-aware sensors.

937

Based on (6), the relative distance dnm in (3) can be formulated as kx xn z m k; if m comes from IN2 ðnÞ dnm ¼ ð7Þ kx xn x m k; if m comes from IN1 ðnÞ: The relative location estimation problem can be described as follows. Given the measured relative distances d^nm in (3) for n ¼ 1; 2; . . . ; N and m 2 INðnÞ, and the inexact positions of location-aware sensors ^zm 2 R2 in (1) for m ¼ 1; 2; . . . ; M, it is desirable to estimate the positions of location-unaware sensors x n 2 R2 for n ¼ 1; 2; . . . ; N.

4

ROBUST RELATIVE LOCATION ESTIMATION FORMULATION

In this section, the relative location estimation problem is formulated as a minimization problem that minimizes the distance errors among location-unaware sensors, which is relevant to IN1 ðnÞ, and the distance errors between locationunaware and location-aware sensors, which is relevant to IN2 ðnÞ. To facilitate understanding, the proposed robust scheme is divided into two parts. In the first part, only the set IN1 ðnÞ is considered. A semidefinite program relaxation approach is employed to form a convex optimization problem for robust relative location estimation; in the second part, only the set IN2 ðnÞ is considered. Due to inexact knowledge of the positions of location-aware sensors, the worst-case error is considered to be minimized and a min-max optimization approach is proposed to treat the robust relative location estimation problem. By employing the “S-procedure” [24], the min-max problem will then be transformed into a SDP problem to simplify the design procedure. Finally, a robust relative location estimation scheme will be developed by combining the two parts into a convex optimization scheme. In the first part, based on the information IN1 ðnÞ, the unknown positions of location-unaware sensors can be obtained by solving min

N 1 X

X

n¼1

n0 2IN1 ðnÞ

X

kjx xn x n 0 k2 d^2nn0 j:

ð8Þ

It should be noted that since n0 comes from IN1 ðnÞ as shown in (8), d^nn0 relates to kx xn xn 0 k according to (7). Let e n denote the nth unit vector in RN . For convenience, we define enn 0 ¼ e n en0 and X ¼ ðx x1 x 2 . . . xN Þ 2 R2N . The optimization problem in (8) is equivalent to T N 1 X X enn 0 Y XT e nn 0 2 ^ dnn0 min 0 ð9Þ X 0 X I2 n¼1 n0 2IN1 ðnÞ s:t: Y ¼ XT X: However, (9) is not convex. The SDP relaxation method [6] is then employed to relax Y ¼ XT X into Y XT X, which is equivalent to Y XT 0 ð10Þ X I2 (see Schur complement in [24]).

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Utilizing (9) and (10), (8) can be transformed into the following SDP: T N 1 X X e nn0 Y XT enn 0 2 ^ dnn0 min 0 X;Y 0 X I2 n¼1 n0 2IN1 ðnÞ ð11Þ Y XT s:t: 0: X I2 In this part, the relative location estimation is done by solving the above constrained optimization problem based on the information IN1 ðnÞ. In the second part, consider IN2 ðnÞ, the information between location-unaware and location-aware sensors. A min-max optimization approach is proposed for the location estimation of location-unaware sensors in the following: min X

N X n¼1

X

max

km k

ðkx xn z m k d^nm Þ2

m2IN2 ðnÞ

ð12Þ

s:t: z m ¼ ^z m þ m ; m 2 IN2 ðnÞ: It should be noted that since m comes from IN2 ðnÞ as shown in (12), d^nm relates to kx xn zm k according to (7). Equation (12) is the worst-case design, with as the error bound of km k in (2). In this part, based on the information IN2 ðnÞ, the relative location estimation is done by minimizing the estimation error in the presence of location errors of location-aware sensors from the worstcase perspective. By applying the Taylor expansion, the term kx xn z m k in (12) can be expanded as xn ^zm k kx xn z m k ¼ kx Let nm ¼

Tm ðx xn ^ zm Þ kx xn ^ zm k .

Tm ðx xn ^zm Þ þ oðkm kÞ: kx xn ^z m k

ð13Þ

ð14Þ

Using (13) and (14), the min-max problem in (12) can be transformed into min X

N X n¼1

max

jnm j

X

ðkx xn ^zm k nm d^nm Þ2 :

X

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max k‘‘n n d^n k2

jnm j

0

B B s:t: ‘ n ¼ B @

1

kx xn ^z ðnn;1Þ k kx xn ^z ðn;2Þ k ... kx xn ^z ðn;jIN2 ðnÞjÞ k

C C C; n ¼ 1; 2; . . . ; N: A

ð15Þ

m2IN2 ðnÞ

To facilitate the ensuing discussion, let us denote IN2 ðnÞ ¼ fðn; 1Þ; ðn; 2Þ; . . . ; ðn; jIN2 ðnÞjÞg, where jIN2 ðnÞj represents the size of IN2 ðnÞ and ðn; mÞ > ðn; m0 Þ if m > m0 . Therefore, ðn; mÞ is strictly increasing in m. For example, if N ¼ 10, M ¼ 5, and IN2 ð1Þ ¼ f1; 3; 5g, then jIN2 ð1Þj ¼ 3, ð1; 1Þ ¼ 1, ð1; 2Þ ¼ 3, and ð1; 3Þ ¼ ð1; jIN2 ð1ÞjÞ ¼ 5. It can be noted that, by the definitions in (6), 1 jIN2 ðnÞj M and 1 ðn; 1Þ ðn; jIN2 ðnÞjÞ M if IN2 ðnÞ 6¼ ;. Denote 0 1 0 ^ 1 nðn;1Þ dnðn;1Þ B nðn;2Þ C B C C; and d^n ¼ B d^nðn;2Þ C: n ¼ B ð16Þ @ A @ A ... ... nðn;jIN2 ðnÞjÞ d^nðn;jIN2 ðnÞjÞ Adopting the notations in (16), (15) can be rewritten as

ð17Þ

Note that, from (16), jnm j ; m 2 IN2 ðnÞ ) kn k

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 jIN2 ðnÞj ¼ n :

ð18Þ

Using (18), (17) can be relaxed into min X

N X n¼1

max k‘‘n n d^n k2

kn kn

1 kx xn ^z ðn;1Þ k ð19Þ B kx xn ^z ðn;2Þ k C B C s:t: ‘ n ¼ B C; n ¼ 1; 2; . . . ; N: @ A ... kx xn ^z ðn;jIN2 ðnÞjÞ k P ‘n n d^n k2 can be transIn (19), minX N n¼1 maxkn kn k‘ formed into min X

N X

0

n ð20Þ

n¼1

s:t: max k‘‘n n d^n k2 n ; n ¼ 1; 2; . . . ; N: kn kn

Unfortunately, (20) is nonconvex. To obtain a convex formulation so that the existing numerical algorithms can be employed to derive a solution, some conditions are transformed or relaxed as shown in the following four steps ðS1ÞðS4Þ. ðS1Þ First step: Letting Y ¼ X T X and Ln ¼ ‘ n ‘ Tn

We have jnm j :

min

VOL. 11,

ð21Þ

with ½L Ln ij as the element in the ith row and jth column of the matrix Ln , it can be readily shown that T en en Y XT : ð22Þ ½L Ln ii ¼ ^z ðn;i ^z ðn;i X I2 n;iÞ n;iÞ Meanwhile, by the Cauchy-Schwartz inequality, we have T en Y ½L Ln ij ^zðn;i X n;iÞ

XT I2

en ^z ðn;j n;jÞ

:

ð23Þ

ðS2Þ Second step: The constraint in (20) is transformed into a matrix inequality constraint, as shown in the following. Theorem 1. The constraint max k‘‘n n d^n k2 n ;

ð24Þ

kn kn

is equivalent to 9n 0 s:t: IjIN2 ðnÞj ð‘‘n d^n ÞT

ð‘‘n d^n Þ sn n

!

n

ð25Þ ; 2

IjIN2 ðnÞj

0

0

n


where ( sn ¼ trace

Ln ‘ Tn

‘n 1

IjIN2 ðnÞj d^Tn

d^n ^ dTn d^n

!) :

Proof. It can be noted that (24) holds true if and only if the statement kn k2 n2 ) k‘‘n n d^n k2 n

ð26Þ

holds true. By using the notations in (16), (26) can be further expressed as the statement in the following matrix form: T IjIN2 ðnÞj 0 n n 0 0 n2 1 1 ! T ð27Þ IjIN2 ðnÞj ð‘‘n d^n Þ n n ) 0: 1 1 sn n ð‘‘n d^n ÞT Finally, employing the S-procedure [24], (27) holds true if and only if (25) holds true, which completes the proof. u t ðS3Þ Third step: To transform (20) into a convex optimization problem, Y ¼ XT X and Ln ¼ ‘ n ‘ Tn in (21) are relaxed into Y XT X and Ln ‘ n ‘Tn , which are equivalent to (see Schur complement in [24]) Ln ‘n Y XT 0; ð28Þ 0 and ‘Tn 1 X I2 respectively. ðS4Þ Fourth step: Incorporating (22), (23), (25), and (28) in steps ðS1ÞðS3Þ into (20), a convex formulation modified from (20) is obtained as min

X;Y ;L Ln ;‘‘n ;n ;n

N X n¼1

n þ c

N jIN 2 ðnÞj X X n¼1

½L Ln ij

j>i

! ‘ n d^n ðn 1ÞIjIN2 ðnÞj 0; s:t: ð‘‘n d^n ÞT n sn n n2 ( !) IjIN2 ðnÞj d^n Ln ‘n sn ¼ trace ; ‘ Tn 1 d^Tn d^Tn d^n !T ! ð29Þ en en Y XT ; ½L Ln ii ¼ ^z ðnn;iiÞ ^z ðnn;iiÞ X I2 !T ! en en Y XT ½L Ln ij ; ^z ðnn;iiÞ ^z ðnn;jjÞ X I2 Ln ‘n Y XT 0; n 0; n 0; 0; ‘ Tn 1 X I2 n ¼ 1; 2; . . . ; N; i; j ¼ 1; 2; . . . ; jIN2 ðnÞj; j > i: Remark 2. For convex relaxation techniques, such as those presented in this section, a penalty function is usually needed so that the original problem can be approximated accurately by the relaxed problem. In (29), the term P PjIN2 ðnÞj c N ½L Ln ij with the positive number c in the j>i n¼1 cost function serves this purpose. The value of c is typically very small, for example, c 2 ½108 ; 104 . Yang

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et al. [20] confirmed that the performance of the corresponding relaxation is not sensitive to the value of c if c is within the interval ½108 ; 104 . Until now, the robust relative location estimations for location-unaware sensors were performed separately according to IN1 ðnÞ and IN2 ðnÞ. The corresponding convex optimization approaches based on the information of IN1 ðnÞ and IN2 ðnÞ are presented in (11) and (29), respectively. In order to utilize all the information INðnÞ, which is the union of IN1 ðnÞ and IN2 ðnÞ, the proposed relative location ^ P combines (11) and (29) as the following estimation X convex optimization problem: ^ P ¼ argX X þ

min

X;Y ;L Ln ;‘‘n ;n ;n

N1 X

N X n¼1

n þ c

X e nn0 T Y 0 X 0

n¼1 n 2IN1 ðnÞ

N jIN 2 ðnÞj X X n¼1

XT I2

½L Ln ij

j>i

e0nn 0

d^2nn0

! ðn 1ÞIjIN2 ðnÞj ‘ n d^n s:t: 0; ð‘‘n d^n ÞT n sn n n2 ( !) IjIN2 ðnÞj d^n Ln ‘n sn ¼ trace ; ð30Þ ‘ Tn 1 d^Tn d^Tn d^n !T ! en en Y XT ½L Ln ii ¼ ; ^zðn;i ^ z ðn;i X I2 n;iÞ n;iÞ !T ! en en Y XT ½L Ln ij ; ^z ðn;i ^z ðn;j X I2 n;iÞ n;jÞ Ln ‘n Y XT 0; n 0; n 0; 0; ‘ Tn 1 X I2 n ¼ 1; 2; . . . ; N; i; j ¼ 1; 2; . . . ; jIN2 ðnÞj; j > i: In (30), X ¼ ðx x1 x 2 . . . x N Þ 2 R2N represents the posi^ P represents the correspondtion matrix of interest, and X ing estimator of the decision variable X. The remaining variables are Y 2 RNN , Ln 2 RjIN2 ðnÞjjIN2 ðnÞj , ‘ n 2 RjIN2 ðnÞj1 , n 2 R, and n 2 R for n ¼ 1; 2; . . . ; N. It can be noted that all constraints in (30) are expressed by matrix inequalities. The constrained optimization in (30) is termed a SDP problem, which is convex. Hence, the global optimum can be achieved by applying existing numerical techniques, for example, the interior-point methods [24]. Remark 3 (Difference from target localization). Generally speaking, a target localization problem may be regarded as a special case of relative location estimation problems when the number of location-unaware sensor nodes is equal to one. However, the corresponding situations are quite different in two ways. First, collaborative sensor nodes are involved for self-configuration, whereas the concerned target is uncooperative [20]. In such a case, the TOA technique can be employed for relative location estimation, but not for target localization. Instead, the TDOA technique is often used [19] in a target localization problem. Second, for target localization, the position of interest is estimated according to the information between it and the sensors of known locations. However,

940


due to the ranging limitation, some sensors of interest in a relative location estimation problem need to be located by the information between sensors of unknown locations. Therefore, target localization methods cannot be directly applied to relative location estimation problems. In contrast to target localization problems, the proposed formulation in (30) employs not only the information between location-aware and location-unaware sensors, i.e., IN2 ðnÞ, but also the information between location-unaware sensors, i.e., IN1 ðnÞ. The use of IN1 ðnÞ differentiates the relative location estimation problem from a target localization problem. Remark 4. The first constraint in (30) corresponds to the worst-case scenario design, i.e., the maximum error introduced by the inexact positions of location-aware sensors is minimized. This differentiates the proposed approach from existing studies. Instead of considering normal distributions of location errors, the proposed approach utilizes the knowledge of the error bound in (2). In particular, the inexact position problem for relative location estimation in WSNs has been dealt with from the worst-case perspective in this paper. The proposed approach may be applicable to a monitoring system for biological groups. For example, assume that we are interested in the influence of humidity on a certain biological group in a certain area. Sensors with humidity detectors are then deployed to the region of interest (ROI). However, it may not be cost effective to carefully position all sensors such that their corresponding locations are known immediately after the deployment. A basic approach is to intentionally place some sensors as location-aware nodes. The remaining sensors would then be randomly deployed by robots and hence, are regarded as locationunaware nodes. The topology of the deployed sensors could be obtained by using existing localization approaches [4], [6], [8], [21]. Once the topology has been obtained, all location-unaware sensors are then regarded as locationaware nodes with a certain degree of location error, which corresponds to the error bound in this study. The bound can be estimated by using prior simulation results. Extending the aforementioned example, let us now suppose that another factor, which can have a great impact on the biological group, has been found, e.g., temperature. Hence, either the deployed sensors need to be equipped with temperature-sensing functionality or new sensors with this functionality are to be added to the existing topology. The latter is chosen due to the lower costs involved. Once again, the random deployment is adopted. The final localization problem is the inexact position problem considered in this paper and hence, can be dealt with by the proposed approach. This study provides a framework for related problems, which require the localization process more than twice. In such a process, blindfolded nodes will become reference nodes with a certain degree of location error after the estimation. The inexact problem occurs when new sensors are added into the ROI and the reference nodes resulting from the estimation are used as the location-aware sensors for another localization process. It can be noted that, for

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future work, as this study has primarily focused on mathematical formulations, several practical issues are yet to be addressed, such as the update of the value of , the effect of changes in the propagation environment, and the interference. More experiments and tests should also be carried out.

5

SIMULATION RESULTS

Numerical simulations were performed in this section to verify the robustness of the proposed approach in (30). For comparison with our method, the maximum likelihood estimation modified from [4] and the second-order cone programming relaxation method modified from [8] are employed to position the coordinates of locationunaware sensors and are presented in Section 5.1. Section 5.2 describes the figure of merit for all simulations. To illustrate the pros and cons of different estimation methods, a scenario with different parameters is considered in Section 5.3.

5.1 Other Relative Location Estimation Methods Modified ML estimation [4] and SOCP methods [8] were used in the simulation. Originally, these were only designed in the absence of the inexact position problem. To enable comparison with the proposed method, these relative location estimation methods are modified in this paper by replacing the true location zm with the inexact location ^zm for m ¼ 1; 2; . . . ; M. The corresponding estimators are presented as follows. The ML estimation modified from [4] is obtained as ^ ML ¼ argX min X X

þ

N 1 X

X

n¼1 m2IN1 ðnÞ

N X X n¼1 m2IN2 ðnÞ

ln kxx

d^2nm ln kxx ^ 2 n zm k

d^2nm xm k 2 n x

2

2 ð31Þ

;

which can be dealt with by nonlinear least squares algorithms [50]. The main drawback of using the ML method in (31) is the requirement of a starting point for the solving process. The optimality of the obtained solution often depends on the choice of starting points. In the simulation, two starting points ðXÞ, true positions of location-unaware sensors, and ðCÞ, the centroid of the neighboring location-aware sensors, were used. For example, if a location-unaware sensor x n connects with locationaware sensors z1 , z 2 , and z 3 , then the nth column of the starting point ðXÞ is set by ^z 1 þ ^z2 þ ^z 3 : 3 If x n does not connect with any location-aware sensors, then the nth column of the starting point ðXÞ is set by the centroid of the region monitored. To distinguish the resulting estimates obtained by solving (31), let us denote ^ MLðCÞ as the use of the starting points ðXÞ and ^ MLðXÞ and X X ðCÞ, respectively. It can be noted that the starting point ðXÞ is an ideal choice and not achievable in practice, whereas ðCÞ is a natural choice and realistic. The SOCP method modified from [4] is obtained as


^ SOCP ¼ argX min X

X;ynm

þ

N 1 X

X

jynm d^2nm j

n¼1 m2IN1 ðnÞ

N X X

jynm d^2nm j

ð32Þ

n¼1 m2IN2 ðnÞ

xn x m k2 ; n ¼ 1; 2; . . . ; N 1; m 2 IN1 ðnÞ s:t: ynm kx xn ^z m k2 ; n ¼ 1; 2; . . . ; N; m 2 IN2 ðnÞ; ynm kx which is convex and can be solved by interior-point methods [24]. The physical meaning of (32) can be readily understood by replacing all inequality constraints with equality constraints. The relaxation of feasible solutions, i.e., using inequality constraints rather than equality constraints, transforms the problem into a convex SOCP, which is solvable with existing solvers. However, it is known that this relaxation can result in a “convex-hull” problem [20]. In particular, the performance can severely degrade if location-unaware sensors mostly lie outside the convex hull of ^z m , m ¼ 1; 2; . . . ; M. Therefore, a good deployment of sensors is needed for a good performance of SOCP method, for example, location-aware nodes positioned on the edges or vertices of the region monitored. It is constructive to examine the iteration complexity for producing results of the relative location estimation approaches mentioned previously. It should be noted that, for one particular optimization problem, the iteration complexity can vary with different algorithms used. In the ensuing discussion, for simplicity, only one particular algorithm is considered for each optimization problem. Hopefully, the reader can have an idea of the order of magnitude. For the ML estimator in (31), the Levenberg-Marquardt method in [51] is concerned. Let us denote the objective function of (31) by . The corresponding iteration complexity is Oð 2 Þ, where satisfies kr k . For the SOCP and the proposed estimators in (32) and (30), primal-dual algorithms in [52] and [53] are considered, respectively. The main computation effort for solving the SOCP in (32) comes from the number of second-order cones, denoted by K. pThe iteration complexity can be expressed as ffiffiffiffi Oð K log 01 Þ, where 0 represents the reduced duality gap [52]. Referring to (32), we have K¼

N1 X

jIN1 ðkÞj þ

k¼1

N X

jIN2 ðkÞj:

k¼1

For the proposed estimator in (30), the dimension of the variables, denoted by K0 , mainly contributes to the iteration 3 complexity, which is OðK0 2 log 01 Þ [53]. In order to use the algorithms in [53], the optimization problem in (30) is transformed into a standard form by introducing extra variables to eliminate the absolute function in the objective space and to replace the inequality constraints with equality constraints. By doing so, we have the following estimation of K0 : K0 > 4N þ 2

N1 X k¼1

jIN1 ðkÞj þ

N X k¼1

jIN2 ðkÞj:

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5.2 Figure of Merit All simulations have been carried out in MATLAB and K ¼ 50 Monte Carlo (MC) runs were performed. Solvers, such as SeDuMi [54] and the MATLAB function “lsqnonlin,” are readily available and hence, are used to produce results of the mentioned estimators. The root mean square error (RMSE) of an estimator is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u K u1 X t ^ ðkÞ X ðkÞ k2 kX K k¼1 ð33Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N K u 1 X X ðkÞ

ðkÞ 2 ðkÞ ðkÞ 2 ¼t x^n xn ; þ y^n yn K n¼1 k¼1 where ðkÞ ðkÞ ðkÞ

X ðkÞ ¼ x 1 x 2 . . . x N ðkÞ

¼

x1

ðkÞ

y1

ðkÞ

x2

ðkÞ

y2

ðkÞ

. . . xN ...

ðkÞ

! 2 R2N

yN

represents the true positions of location-unaware sensors at ^ ðkÞ represents the corresponding the kth MC run and X location estimation. For example, if the ML method using ^ ðkÞ stands for ðCÞ as the starting point is considered, then X ^ the kth estimates resulting from the use of XMLðCÞ in (31). It can be noted that the value of RMSE defined in (33) often increases upon increasing N, the number of locationunaware sensors. The RMSE is not normalized by dividing it by N because the whole X is regarded as the unknown parameter to be estimated. ^ ð1Þ Apart from the RMSE, a “boxplot” [55] of fkX ð1Þ ð2Þ ð2Þ ðKÞ ðKÞ ^ ^ X k; kX X k; . . . ; kX X kg was employed as well. The function “boxplot” in MATLAB transforms the input data into a box with several marks. The central mark stands for the median, and the edge marks stand for the 25th and 75th percentiles. The reader can refer to MATLAB “help” for a detailed description of the “boxplot” function. Generally speaking, a boxplot diagram illustrates the distribution of the input data. In our case, the boxplot illustrates the distribution of the location estimation error. For an estimator to be robust, it is desirable to have the corresponding box with a shorter length and a lower median mark.

5.3 Scenario ^p To compare the performance of the proposed estimator X ^ ^ ^ with XMLðCÞ , XMLðXÞ , and XSOCP , a scenario with different parameters has been considered. Suppose that there are 10 location-unaware sensors (i.e., N ¼ 10) to be estimated. All sensors were randomly deployed in a normalized 1 1 m2 area. In particular, for each MC run (for each k in (33)), the deployment ðkÞ ðkÞ ðkÞ ðkÞ

X z 1 z2 z M ¼ randð2; N þ MÞ has been used, where randð2; N þ MÞ represents a 2 ðN þ MÞ matrix with all elements randomly chosen from ½0; 1. The inexact positions ^zm ’s have been generated according to (1) with m randomly chosen from the ball

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TABLE 1 Typical Values of Parameters for Scenario

fm 2 R2 : km k g;

ð34Þ

for m ¼ 1; 2; . . . ; M. Equation (34) can be readily realized by setting m ¼ ðr cos r sin ÞT ;

2.

where r ¼ randð1; 1Þ and ¼ 2 randð1; 1Þ. The typical values of the parameters for the scenario are listed in Table 1. Performance of the proposed method is compared with other methods by adjusting these in turn and the simulation results are discussed as follows: 1.

Number of location-aware sensors M: The estimation error will generally be reduced if the number of location-aware sensors is increased. Fig. 1a illustrates the RMSEs of different relative location estimation methods versus the number of locationaware sensors. As can be observed in Fig. 1a, the ^ MLðXÞ with X as the starting point estimator X ^ MLðCÞ achieves a high level of accuracy, whereas X with a starting point selected by the centroid of the neighboring location-aware sensors has the worst performance of the presented methods. With respect ^ SOCP , the possible probto the performance of X ability of all the estimation locations lying in the convex hull of the location-aware sensor nodes increases if the number of location-aware sensors ^P becomes greater. Overall, the proposed estimator X is relatively robust. Consider the boxplot in Fig. 1b.

3.

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The y-axis denotes the estimation error, whereas the x-axis denotes different estimators as detailed in the caption. An estimator is more robust in the boxplot if the corresponding box has a shorter length, i.e., a narrower error distribution, and a lower central mark (median mark), i.e., a lower average estimation error. As can be observed, the error distribution of ^ SOCP is the widest among the estimators. Note that X ^ ML depends heavily on the the performance of X ^ MLðXÞ can achieve a high starting point. Although X level of accuracy, it is impractical due to the use of the unknown parameter X. With the exception of ^ P has a superior performance in terms of a ^ MLðXÞ , X X lower value of RMSE and a narrower error distribution with lower values of estimation error. ~ When d~ becomes Threshold for a possible connection d: large, it implies that the probability of the connection between two sensors would increase and that the estimation algorithm would obtain more information. Consequently, the performance would improve. The simulation results obtained are illustrated in ^ P has the best ^ MLðXÞ , X Fig. 2. With the exception of X performance among the presented estimators in terms of a lower value of RMSE and a narrower error distribution with lower values of estimation error. It should be noted that although the sensing ~ varying from 0.3 to 0.7 m, seems relatively range d, large with respect to the normalized 1 1 m2 region, it is still possible that a large proportion of locationunaware nodes do not connect with the locationaware nodes due to the random deployment. This phenomenon is related to the “k-coverage problem,” which is often discussed in studies relating to the relationship between the sensing range and the coverage. The reader can refer to [56], [57], [58], [59], [60] for related works. Variance of measured power error p : From (3) and (4), we know that the variance of the relative distance measurement d^nm is exponentially proportional to p .

Fig. 1. Numerical simulation results of different methods with various number of the location-aware sensors (ROI ¼ 1 1m2 , sensor range ¼ 0:5m, ^ SOCP denotes the SOCP method, X ^ MLðRÞ denotes the ML method with a ^ P denotes the proposed method, X N ¼ 10). (a) is “RMSE versus M” plot. X ^ MLðXÞ denotes the ML method with the starting point X. (b) is starting point selected by the centroid of the neighboring location-aware sensors, and X ^P , X ^ SOCP , X ^ MLðCÞ , and X ^ MLðXÞ , respectively. All of the ^ ðkÞ Xk. In x-axis label, “P,” “S,” “C,” and “X” stand for X the boxplot of estimation errors kX following figures have the same notations, except for the x-axis in (a).


943

Fig. 2. Numerical simulation results of different methods with various value of d~ (ROI ¼ 1 1m2 ; M ¼ 4; N ¼ 10).

4.

5.

The RMSE becomes large if the variance of d^nm is increased. The trend can be found in Fig. 3. Estimator ^ P has the best performance among the realizable X estimation methods. Error bound : The large bounding error implies that the positions of the location-aware sensors are more inaccurate. Therefore, the RMSE is proportional to . The simulation results for different values of are illustrated in Fig. 4. It should be noted that is the worst possible error for all location-aware nodes. The proposed approach uses the knowledge of and hence, it is a worst-case design. Similarly, ^ P as the relative location estimator results in using X the smallest estimation error, excluding the imprac^ MLðXÞ . tical estimator X Path loss exponent np : The variance of d^nm is inversely exponentially proportional to np , according to (3). This implies that the RMSE will decrease if the value of np increases. The simulation results are illustrated ^ P still has the best performance in Fig. 5. Estimator X among the feasible estimation methods.

From (30) to (32), it is evident that the proposed method has more variables and constraints than the other methods. This means that the proposed method would take more time to derive the relative location estimation. However, from the above simulations, it can be observed that the ^ P outperforms X ^ MLðCÞ and X ^ SOCP . proposed method X Moreover, unlike the ML and SOCP methods, a good starting point for the solving process and a good deployment of sensors are not required in the proposed scheme. Therefore, to minimize the worst estimation error, the proposed min-max method has better performance without exact location knowledge of location-aware sensors. It also has greater potential for robust relative location estimation in WSNs with location errors in the future.

6

CONCLUSIONS

The inexact position problem of location-aware sensors in WSNs has been considered. A robust relative location estimation method based on a min-max optimization method is proposed in the presence of location errors. To

Fig. 3. Numerical simulation results of different methods with various value of p (ROI ¼ 1 1m2 , sensor range ¼ 0:5m, M ¼ 4, N ¼ 10).

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Fig. 4. Numerical simulation results of different methods with various value of (ROI ¼ 1 1m2 , sensor range ¼ 0:5m, M ¼ 4, N ¼ 10).

Fig. 5. Numerical simulation results of different methods with various value of np (ROI ¼ 1 1m2 , sensor range ¼ 0:5m, M ¼ 4, N ¼ 10).

the best of our knowledge, the existing approaches for relative location estimation in WSNs only consider the Gaussian distribution model for the inexact positions of location-aware sensors. In this study, exact knowledge of the location error distribution is not needed because the problem is examined from a worst-case perspective, which has not been fully investigated in the literature. Our simulations have evidenced that the proposed method outperforms the modified ML and SOCP methods. Rather than focusing on some particular topologies of WSNs, the random deployment of sensor nodes has been considered. The influence of different environmental parameters on the localization performance has been examined. There are three benefits from the proposed approach. First, the relative location estimation problem is formulated as a convex optimization problem, which can be solved by existing numerical techniques. The feasibility of the proposed approach is then validated. Second, instead of knowing the distribution of the location error, only an error bound is required. In this way, the proposed approach can be applied to a more general situation. Third, unlike the conventional ML estimation and SOCP methods, a good

starting point for the solving process and a good deployment of sensors are not needed in our robust scheme. Finally, it should be noted that this study aims to provide a framework for the inexact position problem with a feasible solution. Many mathematical formulations and models have been investigated as well. Any future work should include detailed experiments in practical scenarios.

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NO. 6,

JUNE 2012

Bor-Sen Chen received the BS degree from the Tatung Institute of Technology, Taipei, Taiwan, the MS degree from National Central University, Chungli, Taiwan, and the PhD degree from the University of Southern California, Los Angeles, in 1970, 1973, and 1982, respectively. He was a lecturer, associate professor, and professor at the Tatung Institute of Technology from 1973 to 1987. He is currently the Tsing Hua chair professor of electrical engineering and computer science at the National Tsing Hua University, Hsinchu, Taiwan. His current research interests are in control engineering, signal processing, and systems biology. He has received the Distinguished Research Award from the National Science Council of Taiwan four times. He is a research fellow of the National Science Council of Taiwan and holds the excellent scholar chair in engineering. He also received the Automatic Control Medal from the Automatic Control Society of Taiwan in 2001. He was an associate editor of the IEEE Transactions on Fuzzy Systems from 2001 to 2006 and is an editor of the Asian Journal of Control. He is a member of the editorial advisory board of Fuzzy Sets and Systems and the International Journal of Control, Automation and Systems. He was the editor-in-chief of the International Journal of Fuzzy Systems from 2005 to 2008. He is now the editor-in-chief of the International Journal of Systems and Synthetic Biology and a member of the editorial board of BMC Systems Biology. He is a fellow of the IEEE. Chang-Yi Yang received the BS and MS degrees in control engineering from National Chiao Tung University, Taiwan, R.O.C., in 1983 and 1988, respectively. He received the PhD degree in electrical engineering from the National Tsing Hua University in 1992. From 1985 to 1986, he was an assistant researcher at the Mechanical Industry Research Laboratories of the Industrial Technology Research Institute, Taiwan. From 1992 to 1996, he worked on wide-area networking with Southern Information Systems, Inc., Taiwan. He joined the Department of Electronic Engineering at Vanung University, Jhongli, Taiwan, from 1996 to 2004. Since 2004, he has been with the Department of Computer Science and Information Engineering at the National Penghu University of Science and Technology, Makung City, Taiwan, where he is currently an associate professor. His research interests include communication theory and its applications to wireless networks.

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