IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 7, JULY 2010 647

Mobile Beacon-Based 3D-Localization withMultidimensional Scaling in Large Sensor Networks

Eunchan Kim, Student Member, IEEE, Sangho Lee, Chungsan Kim, and Kiseon Kim, Senior Member, IEEE

Abstract—Localization is essential in wireless sensor networksto handle the reporting of events from sensor nodes. For 3-D applications, we propose a mobile beacon-based localizationusing classical multidimensional scaling (MBL-MDS) by takingfull advantage of MDS with connectivity and measurements.To further improve location performance, MBL-MDS adopts aselection rule to choose useful reference points, and a decisionrule to prevent a failure case due to reference points placed onthe same plane. Simulation results show improved performanceof MBL-MDS in terms of location accuracy and computationcomplexity.

Index Terms—Sensor networks, localization, mobile beacon,multidimensional scaling.

I. INTRODUCTION

RECENTLY, research on localization has been activelyinvestigated in large-scale wireless sensor networks, with

many schemes introduced in various simulation environments.Most of them target locating nodes in 2-D space, e.g., proba-bilistic method for relative location estimation [1]. In practice,sensor nodes are deployed in 3-D space. Locating nodes in 3-D space does not simply reflect just the addition of one extradimension to the locating problem, and thus it is necessary todevelop 3-D localization [2].

One effective approach is range-based location using mobilebeacons because each mobile beacon can directly provide itscurrent position to sensor nodes and sensor nodes can measuredistances to the mobile beacon [3], [4]. Zhang et al. [3]encourages a sensor node to improve location accuracy withan unscented Kalman filter (UKF). While their scheme givesa converged node position under continuous measurements, itoften fails in the improvement due to discontinuous measure-ments when a mobile node flies out of the listenable range of asensor node. Hu et al. [4] uses a least-square method for initialstate of the same UKF of Zhang’s scheme but a least-squaremethod is sensitive to deployment of reference positions. As agood approach for discontinuous measurements, multidimen-sional scaling (MDS) has been widely used to locate nodeswith static beacons in 2-D space [5]. In 3-D localization withmobile beacons, MDS is still appropriate because all requireddistances are obtainable, but there is scarcely any effort to useMDS. Besides, there are open problems that MDS requires

Manuscript received March 30, 2010. The associate editor coordinating thereview of this letter and approving it for publication was C.-F. Chiasserini.

The authors are with the Gwangju Institute of Science and Tech-nology, Gwangju 500-712, Korea (e-mail: {tokec, faith, only2442,kskim}@gist.ac.kr).

This work was supported by the Korea National Research Foundation GrantK20901000004-09E0100-00410, and partially by the World Class University(WCU) program at GIST (Project No. R31-2008-000-10026-0).

Digital Object Identifier 10.1109/LCOMM.2010.07.100513

expensive computation for large number of reference positionsand relocation on the coordinates of reference positions.

In this letter, we propose a mobile beacon-based localizationusing classical multidimensional scaling (MBL-MDS). MBL-MDS applies MDS to 3-D localization using a mobile beaconand adopts two rules to improve location performance byhandling the problems: 1) a selection rule to choose sufficientsets of reference positions among all received ones, and 2)a decision rule to determine which of the two candidates isin the correct node position, in the case that given referencepositions are placed on the same plane.

II. PRINCIPLES FOR THE PROPOSED SCHEME

This section describes multidimensional scaling (MDS) andlinear transformation to locate a sensor node 𝒔 that has the 𝑀number of beacon points as P = [𝒙1,𝒙2, . . . ,𝒙𝑀 ].

A. Classical Multidimensional Scaling

For points X = [𝒔,𝒙1, . . . ,𝒙𝑀 ], distances between all pairsof points are given as [D]𝑖𝑗 = 𝑑𝑖𝑗 = ∣∣𝒙𝑖 − 𝒙𝑗 ∣∣ for 𝑖, 𝑗 =0, 1, ⋅ ⋅ ⋅ ,𝑀 , where 𝒙0 = 𝒔. Then, MDS computes relativepoints X′ = [𝒔′,𝒙′

1, . . . ,𝒙′𝑀 ] from D as follows [5]:

For given distance matrix D, symmetric matrix 𝒟2 has thesquare of the distance for each component, such that [𝒟2]𝑖𝑗 =

𝑑2𝑖𝑗 . Then, symmetric matrix B, defined as B ≜ X′⊤X′, canbe obtained from 𝒟2 as

B = −1

2J𝒟2J, J = I𝑀+1 − 1

𝑀+11𝑀+11

⊤𝑀+1, (1)

where I𝑀+1 is an (𝑀+1×𝑀+1) identity matrix and 1𝑀+1 =[1, 1, ⋅ ⋅ ⋅ , 1]⊤ is an (𝑀+1× 1) unit vector. Decomposing Busing a singular value decomposition (SVD) gives

B = UVU⊤, (2)

where V = diag(𝜆1, 𝜆2, ⋅ ⋅ ⋅ , 𝜆𝑀+1), 𝜆1 ≥ 𝜆2 ≥ ⋅ ⋅ ⋅ ≥ 𝜆𝑀+1

is a matrix of which diagonals are eigenvalues of B and U =[𝒖1,𝒖2, ⋅ ⋅ ⋅ ,𝒖𝑀+1] is a matrix composed of the eigenvectors𝒖𝑖 for 𝜆𝑖. Thus, the relative points X′ can be obtained as

X′ = B1/2 = UV1/2. (3)

Finally, because the 𝑀+1 points are placed in a 𝑝-dimensional space (𝑀+1 > 𝑝), rank(B) = 𝑝, the relativepoints X′ for X are obtained by selecting the dominanteignevalues and corresponding eigenvectors as

X′ = U𝑝V1/2𝑝 , (4)

where V1/2𝑝 = diag(𝜆

1/21 , 𝜆

1/22 , ⋅ ⋅ ⋅ , 𝜆1/2

𝑝 ) and U𝑝 =[𝒖1,𝒖2, ⋅ ⋅ ⋅ ,𝒖𝑝].

1089-7798/10$25.00 c⃝ 2010 IEEE

648 IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 7, JULY 2010

1x

1r

R

2xKx

ix

s

2r ir Kr

Fig. 1. Gathered beacon points 𝒙𝑖 and measured distances 𝑟𝑖, for 𝑖 =1, 2, ⋅ ⋅ ⋅ , 𝐾 , inside the listenable range 𝑅 of the 𝑛-th sensor node 𝒔.

B. Linear Transformation to Adjust P′ Close to P

Given two matrices P′ = [𝒙′1,𝒙

′2, . . . ,𝒙

′𝑀 ] and P =

[𝒙1,𝒙2, . . . ,𝒙𝑀 ], there are some known solutions for deter-mining the parameters to linearly transform P′ most close toP, such that

arg min𝑐,R,𝒕

𝑀∑𝑖=1

∣∣𝑐R𝒙′𝑖 + 𝒕− 𝒙𝑖∣∣2. (5)

Of these solutions, Arun’s method [6] is used in ourproposed scheme, because: 1) the matrix P′ is based on thedistances between all pairs of points in P; hence, the scalebetween P′ and P does not change, and 2) Arun’s methodprovides low complexity by using the unit scale 𝑐 = 1. InArun’s method, a rotation matrix R and a translation vector 𝒕are given by

R = UV⊤, 𝒕 = 𝝁𝑥 −R𝝁𝑥′ , (6)

where 𝝁𝑥=1𝑀

∑𝑀𝑖=1 𝒙𝑖 and 𝝁𝑥′ = 1

𝑀

∑𝑀𝑖=1 𝒙

′𝑖. Also, U and

V are the SVD results of the centralized matrices as P̄P̄′⊤ =USV⊤, where the centralized matrices are

P̄=P−𝝁𝑥 ⋅1⊤𝑀 , P̄′=P′−𝝁𝑥′ ⋅1⊤

𝑀 . (7)

III. PROPOSED LOCATION SCHEME: MBL-MDS

This section presents a system model and the proposedMBL-MDS with respect to a certain sensor node. The othernodes can locate themselves in the same manner.

A. System Model

As shown in Fig. 1, a mobile beacon flies over sensornodes. While traveling every beacon distance, the mobilebeacon broadcasts a beacon packet that contains its currentposition at the sending point (beacon point). To this end,we assume that the beacon has mobility, self-localizability,and sufficient energy to travel over the sensing area. Everynode has the same limited listenable range 𝑅 to receivebeacon packets. Receiving a beacon packet, a sensor nodeobtains the beacon point 𝒙𝑖 and measures the distance 𝑟𝑖 to𝒙𝑖, 𝑖 = 1, 2, ⋅ ⋅ ⋅ ,𝐾 . Based on the received signal-strengthtechnique, the measurement error 𝜖𝑖 in a measured distance 𝑟𝑖is usually proportional to the actual distance 𝑑𝑖, modeled asthe Gaussian random variable [3]

𝑟𝑖 = 𝑑𝑖 + 𝜖𝑖, 𝜖𝑖 ∼ 𝑁(0, 𝜎2𝑖 ), (8)

with variance of the measurement error 𝜎2𝑖 = (𝛾 ⋅ 𝑑𝑖)2. The

𝛾 is a ratio of standard deviation of the measurement error tothe actual distance.

B. Decision Rule for Resolving Ambiguous Reflection

Let P = [𝒙1,𝒙2, . . . ,𝒙𝑀 ] and 𝒓 = [𝑟1, 𝑟2, ⋅ ⋅ ⋅ , 𝑟𝑀 ]⊤

denote 𝑀 beacon points and their corresponding measureddistances, respectively. The beacon points are chosen out oftotal 𝐾 number of beacon points by a selection rule to bementioned in the next subsection. Based on P and 𝒓, a sensornode makes a distance matrix D as

D =

[0 𝒓⊤

𝒓 D𝑃

], (9)

where [D𝑃 ]𝑖𝑗=𝑑𝑖𝑗= ∣∣𝒙𝑖−𝒙𝑗 ∣∣ for 𝑖, 𝑗 = 1, 2, ⋅ ⋅ ⋅ ,𝑀 . UsingMDS, a sensor node computes X′ = [𝒔′,P′] from D, where𝒔′ and P′ are relative positions of the sensor node and theselected beacon points, respectively.

In relocating X′ in the coordinates of beacon points, wehave to consider the ambiguous reflection case when allbeacon points are almost placed on the same plane in Fig.1. In this case, there exist two possible candidates for nodeposition; one is the correct position and the other is a reflectedposition with respect to the coplanar. For the correct position,a decision rule of MBL-MDS first computes both candidatesas

𝒔+ = R+𝒔′ + 𝒕+, 𝒔− = R−𝒔′ + 𝒕− (10)

where R+=UV⊤, and 𝒕+=𝝁𝑥−R+𝝁𝑥′ in (6). The reflectedrotation matrix R− and the corresponding translation vectorare given as

R− = UΣV⊤, 𝒕− = 𝝁𝑥−R−𝝁𝑥′ , (11)

where Σ = diag(1, 1,−1). Then the correct position from thetwo possible candidates is determined as

�̃� =

{𝒔+ if 𝒔+(𝑧) ≤ 𝝁𝑥(𝑧)

𝒔− otherwise(12)

where 𝒔+(𝑧) and 𝝁𝑥(𝑧) are z-values of 𝒔+ and 𝝁𝑥, becausethe sensor node is always located below the beacon points.

C. Locating Node using Selection Rule

MDS and the linear transformation technique result inexponential growth in complexity, such as 𝑂(𝐾3) for the 𝐾number of beacon points due to SVD computation. To reducecomplexity, the selection rule of MBL-MDS collects 𝜅 setscomposed of 𝑀 beacon points (𝑀 ≪ 𝐾), and then MDS andlinear transformation are applied to each set, independently.Hence, the complexity of MBL-MDS is denoted by 𝑂(𝜅 ⋅𝑀3)and linearly increases with 𝜅, 1 ≤ 𝜅 ≤ ⌊𝐾/𝑀⌋, where a realnumber inside ⌊⋅⌋ is mapped to the next smallest integer.

Selecting well-distributed beacon points is also importantto improve location accuracy. To that end, a selection rulechooses 𝑀 beacon points in a set as follows:

1) Compute the mean of the given beacon points 𝝁𝑥 anddividing the x-y plane of beacon points into 4 parts(quadrants) with the center of 𝝁𝑥.

2) Gather 𝑀 beacon points by randomly selecting the⌊𝑀/4⌋-tuple beacon points from each quadrant.

3) Make a matrix P with the selected beacon points andthe corresponding distance vector 𝒓.

KIM et al.: MOBILE BEACON-BASED 3D-LOCALIZATION WITH MULTIDIMENSIONAL SCALING IN LARGE SENSOR NETWORKS 649

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(a) Node deployment (b) Effect of selection and decision rules in MBL-MDS (c) Location performance of MBL-MDS

Hu's scheme [4]Zhang's scheme [3]

MBL-MDS

RMSE TimeRMSE

MDS onlyMDS+Selection

MBL-MDS

Time

Fig. 2. Simulation results: (a) Node deployment, (b) Effect of selection and decision rules in MBL-MDS, and (c) Location performance of MBL-MDS.

4) Estimate a node position �̃�𝑖 based on P and 𝒓, which iscomputed with aforementioned MDS and linear trans-formation.

5) Remove P from the given beacon points.

Repeating steps 1) to 5) for 𝜅 times, MBL-MDS obtains aset 𝑊 = {�̃�𝑖 : 𝑖 = 1, 2, . . . , 𝜅}. Then, the final node position�̂� is determined as the median of �̃�𝑗 such that

�̂� = arg min�̃�𝑗∈𝑊

∑𝑖

∣�̃�𝑖 − �̃�𝑗 ∣, (13)

because the median is much better than the average when thesize of 𝑊 is not large enough and there are some estimatedpositions with large errors.

IV. PERFORMANCE EVALUATION

MBL-MDS was evaluated in the complex 3-D terrain inFig. 2(a); 1000 × 1000 in size and altitude of −300 to 300.The linear track of a mobile beacon starts from [0, 0, 300]⊤

to [𝑥, 𝑦, 𝑧]⊤, where 𝑥, 𝑦, and 𝑧 are randomly selected withuniform distribution in the ranges of 0 ≤ 𝑥, 𝑦 ≤ 1000 and300 ≤ 𝑧 ≤ 𝑧max. After arriving, the mobile beacon heads toanother randomly selected destination point. Beacon pointsare arranged at every 50 distance along the linear track. Thenumber of sensor nodes is 200, the listenable range is 600, and𝛾 in (8) is given as 10% of the actual distances. The numberof selected beacon points in a set is fixed to four (𝑀 = 4).Simulation results were obtained using MATLAB v7.1, 2.0GHz CPU, and 1 Gbytes memory.

Fig. 2(b) presents the effect of selection and decision rulesof MBL-MDS on location performance. All beacon pointsare placed on the same plane, 𝑧max =300. The performanceis shown in terms of location accuracy with dashed linesand average CPU time with dotted lines. MBL-MDS usesboth rules, MDS+Selection uses the only selection rule, andMDS only uses none of two rules. MDS is applied to allcases. Decision rule of MBL-MDS shows the improvementin location accuracy by comparing MBL-MDS to the others;it reduces RMSE to 40% of the others with 80 beaconpoints. Comparing RMSE of MDS only and MDS+Selection,MDS+Selection is superior to MDS only in small beaconnumber due to relatively well-distributed beacon points byits selection rule. Conversely, MDS only is superior in largebeacon number because MDS only has well-distributed beacon

points and improved location accuracy due to many beacon

points in computing MDS. Selection rule of MBL-MDS andMDS+Selection depicts linear growth in computation com-plexity.

Fig. 2(c) presents the location performance of MBL-MDScompared to the conventional location schemes mentionedin Introduction under 𝑧max = 400. This figure shows thatMBL-MDS outperforms the others with respect to RMSE oflocation errors, e.g., MBL-MDS reduces RMSE to about 26%of Zhang’s scheme [3] and 24% of Hu’s schemes [4], with 20beacon points. Also, MBL-MDS takes lower average time tolocate a node than others, about 1.5 times faster than Zhang’sscheme and 2.5 times faster than Hu’s scheme. Note that theUKF mechanism in Zhang’s and Hu’s schemes should run forevery beacon point, while MDS and linear transformation inMBL-MDS run once every 𝑀 beacon points.

V. CONCLUSIONS

In this letter, we proposed a mobile beacon-based local-ization with multidimensional scaling (MBL-MDS) for 3-Dapplications of large-scale sensor networks. To further improvelocation performance in applying MDS, MBL-MDS adoptsa selection rule to choose sufficient sets of beacon points,and a decision rule to resolve ambiguous reflection for anode position. Simulation results confirmed the improvementof MBL-MDS, e.g., MBL-MDS reduces RMSE to 26% ofZhang’s scheme [3] with 1.5 times faster time.

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[3] L. Zhang, Q. Cheng, Y. Wang, and S. Zeadally, “A novel distributedsensor positioning system using the dual of target tracking,” IEEE Trans.Computers, vol. 57, no. 2, pp. 246–260, Feb. 2008.

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[6] K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-squares fitting of two3-D point sets,” IEEE Trans. Pattern Analysis and Machine Intelligence,vol. PAMI-9, no. 5, pp. 698–700, Sep. 1987.