Novel Centroid Localization Algorithm for Three-Dimensional Wireless

Novel Centroid Localization Algorithm for Three-Dimensional Wireless Sensor Networks Hongyang Chen1, Pei Huang2, Marcelo Martins1, Hing Cheung So3, and Kaoru Sezaki1 1Institute of Industrial Science, The University of Tokyo, Tokyo, 153-8505 Japan 2 Department of Computer Science and Engineering, Michigan State University, USA 3Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China sensor networks, which transformed 3D underwater Abstract—In this paper, we propose a new localization localization to 2D. In this paper, a novel localization algorithm algorithm which can be effectively used in three-dimensional (3D) for three-dimensional WSNs, which can obtain high location wireless sensor networks. This scheme needs no additional performance, is developed based on the centroid theorem of hardware support and can be implemented in a distributed way. The proposed method can improve the location accuracy with coordinate-tetrahedron [9][11]. relatively low communication traffic and computing complexity. This paper makes four major contributions to the Simulation results show that the performance of the proposed localization problem in WSNs. First, we present a practical, algorithm is superior to that of the conventional centroid fast and easy-to-use localization scheme with relatively high algorithm. accuracy and low cost for WSNs. Second, we propose the Index Terms—WSNs, Centroid Localization algorithm, centroid theorem of coordinate-tetrahedron in the volume- localization scheme, Volume-coordinate system. coordinate system, which acts as a key component of our estimation approach. Third, the proposed localization I. INTRODUCTION algorithm improves location accuracy than the centroid Energy-efficient wireless sensor networks (WSNs) can be localization algorithm. Fourth, the proposed algorithm can be helpful in many different areas such as military surveillance, effectively applied in 3D WSNs, which is proving to be a medical care, environmental monitoring, public service and promising technique for several application scenarios such as in home automation applications. Design and analysis of WSNs space, under water, or on hilly terrains. The rest of the paper is has been a major research topic in the computer and organized as follows. Section II describes the details of the communication fields [1]. In a significant amount of new algorithm and its derivation. In Section III, simulation applications, for example humidity and temperature results are reported and a comparative study of the localization monitoring, data collected by sensor nodes should include the performance is conducted. Finally, Section IV gives the information of their physical locations. Otherwise, the data concluding remarks. will be irrelevant. Because of the constraints in size, power, and computing capabilities in WSNs, it is challenging to II. ALGORITHM DEVELOPMENT provide satisfactory localization algorithms to meet high accuracy requirement [2]. In this section, we are going to derive a novel node Localization algorithms for WSNs can be divided into two localization method for 3D WSNs. Before proceeding, we categories: (i) range-based and (ii) range-free protocols. review the centroid algorithm. Range-based protocols [3,6,7] employ absolute point-to-point 2.1 Review of Centroid Algorithm distance or angular information to identify the locations Bulusu and Heidemann [2] have proposed the centroid among neighboring nodes. The measurements used in range- localization algorithm, which is a range-free, proximity-based, based localization including angle-of-arrival (AOA), received coarse-grained localization algorithm. The algorithm signal strength indicator (RSSI) [7], time-of-arrival (TOA) [3] implementation contains three core steps. First, all anchors and time-difference-of-arrival (TDOA) [6] schemes. On the send their positions to all sensor nodes within their other hand, we may adopt the range-free approach in which transmission range. Each unknown node listens for a fixed time one can find the positions of non-anchor nodes by calculating period t and collects all the beacon signals it receives from various their distances from the designated but sparse anchor nodes reference points. Second, all unknown sensor nodes calculate with known positions, also known as landmarks [4]. Although their own positions by a centroid determination from all n various localization algorithms have been proposed for two- positions of the anchors in range. dimensional (2D) sensor networks, there are relatively few The centroid localization algorithm, which uses anchor localization schemes for three-dimensional WSNs. Tian et nodes (reference nodes), containing location al.[8] developed a multilateration algorithm which could information (xi , yi ) , to estimate node position. After reliably localize and synchronize underwater sensor networks receiving these beacons, a node estimates its location using the by acoustic ranging. Cheng et al. [10] proposed a novel following centroid formula: underwater localization algorithm for sparse 3D acoustic

x +…+ x y +…+ y M P =⎛ 1 N 1 N ⎞ (1) 1 = λ (xest , yest ) ⎜ , ⎟ on the line M 1M 2 . If , then the volume ⎝ N N ⎠ PM 2 The centroid localization algorithm is simple but the coordinates of P can be calculated as follows: location error is high due to the centroid formula. Besides, the 1 2 conventional centroid localization algorithm only focuses on v + λv v = i i , i = 1,2,3,4 . (4) node self-localization for 2D networks. Practical nodes of i 1+ λ WSNs are generally arranged in 3D scenarios such as in space, 1 2 under water, or on hilly terrains, rather than on pure 2D where vi and vi are the volume coordinates of points M 1 planes, thus 3D position information of nodes is commonly and M 2 , respectively. required. Node self-localization algorithm for 3D WSNs is studied in this paper. ⎡ 1 1 1 1 ⎤ ⎢ ⎥ 2.2 Development of Novel Centroid Algorithm for 3D x x x x = ⎢ 1 2 3 4 ⎥ WSNs Let A . Let K(x,y ,z ) be the ⎢ y1 y2 y3 y4 ⎥ ⎢ ⎥ This subsection describes our proposed 3D positioning z z z z algorithm for WSNs in detail. Wan et al. [11] proposed a ⎣ 1 2 3 4 ⎦ localization algorithm for mobile system based on a linear unknown target node location and (xi , yi , zi ) be the known relationship between the rectangular and the volume location of the ith anchor node. The random 4 anchor coordinates. Our work is inspired by [11]. Based on [11], we nodes , , , will be used to calculate the position of extend their work and propose an improved centroid algorithm A1 A2 A3 A4 for WSNs. Our proposed localization algorithm will not use target node. The anchor nodes A1 , A2 , A3 , A4 can form a 2D centroid theorem, but present the centroid theorem of tetrahedron. Then using Lemma 1 we can get the volume coordinate-tetrahedron in the volume-coordinate system, coordinates of the unknown node K (v , v , v , v ) as which acts as a key component of our estimation approach. 1 2 3 4 Using centroid theorem of coordinate-tetrahedron [9]-[11], the follows proposed positioning algorithm can be used in 3D WSNs and 1 1 1 1 also can improve location accuracy than the conventional 1 x x2 x3 x4 centroid localization algorithm. v = V = 1 KA2 A3 A4 The linear relationship between the rectangular and the 6 y y2 y3 y4 (5) volume coordinates will be reviewed first as shown in Eqs.(5)- z z2 z3 z4 (12), which have been presented in [11]. Then the detailed derivation procedure for the centroid theorem of coordinate- = 1 + + + (A11 A21 x A31 y A41z) tetrahedron in the volume-coordinate system will be described. 6 Lemma 1: Assume that the rectangular coordinate of the 1 1 1 1 vertex A of the tetrahedron is (x , y , z ) (i = 1,2,3,4) . i i i i 1 x1 x x3 x4 v = V = Then its signed volume can be expressed in the form of 2 A1KA3 A4 6 y y y y determinant as follows: 1 3 4 (6) 1 1 1 1 z1 z z3 z4 − − − x 2 x1 y2 y1 z2 z1 1 x x x x 1 (2) 1 V = 1 2 3 4 = x −x y − y z − z = (A + A x + A y + A z) 6 y y y y 6 3 2 3 2 3 2 12 22 32 42 1 2 3 4 − − − 6 x4 x3 y4 y3 z4 z3 z1 z2 z3 z4 1 1 1 1 Lemma 2: The volume of the general tetrahedron is also given 1 x1 x2 x x4 by the determinant v =V = 3 A1A2KA4 6 y y y y 0 1 1 1 1 1 2 4 (7) 2 2 2 z z z z 1 0 r12 r13 r14 1 2 4 1 V 2 = 1 r 2 0 r 2 r 2 (3) 12 23 24 = 1 + + + 288 2 2 2 (A13 A23x A33 y A43z) 1 r13 r23 0 r34 6 2 2 2 1 r14 r24 r34 0 where rij is the range between vertex Ai and Aj . Lemma 3: In the volume-coordinate system, let P be a point

A D 1 1 1 1 Since 2 = 1 , then D’s volume coordinate will DA 4 1 x1 x 2 x 3 x v = V = A Q 4 A 1 A 2 A 3 K 6 y y y y be (0, V/2, V/2,0 ) using Lemma 3. Because 3 = 2 , Q’s 1 2 3 (8) QD z z z z 1 2 3 volume coordinate will be (0, V/3, V/3, V/3 ) using Lemma 1 A M = ( A + A x + A y + A z ) 3. Since 1 = 3 , thus, the M’s volume coordinate will 6 14 24 34 44 MQ be (V/4, V/4, V/4, V/4 ) using Lemma 3. The values of v1 , v2 , v3 and v4 can also be calculated by

(3) using Lemma 2. From (5)-(8), we get A1 ⎡ v ⎤ ⎡ A A A A ⎤ ⎡ 1 ⎤ 1 11 21 31 41 A2 ⎢v ⎥ ⎢ A A A A ⎥ ⎢ x ⎥ ⎢ 2 ⎥ = 1 ⎢ 12 22 32 42 ⎥ ⎢ ⎥ ⎢ v 3 ⎥ 6 ⎢ A13 A 23 A33 A 43 ⎥ ⎢ y ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v A A A A z ⎣ 4 ⎦ ⎣ 14 24 34 44 ⎦ ⎣ ⎦ (9) ⎡ 1 ⎤ A ⎢ ⎥ 3 1 ∗ x = A ⎢ ⎥ A 6 ⎢ y ⎥ 4 ⎢ ⎥ Fig.1. Tetrahedron A1A2A3A4 ⎣ z ⎦ Then using the M’s (the barycenter of tetrahedron ⎡v ⎤ ⎡1⎤ 1 A A A A ) volume coordinate we get ⎢v ⎥ ⎢x⎥ 1 2 3 4 ⎢ 2 ⎥ =VA−1 ⎢ ⎥ (10) ⎡v1 ⎤ ⎡ a1 ⎤ ⎡1 4⎤ ⎡ a1 ⎤ ⎢v3 ⎥ ⎢y⎥ ⎢ ⎥ ⎢ ⎥ ⎢v ⎥ ⎢a ⎥ ⎢1 4⎥ ⎢a ⎥ v z = 1 ⎢ 2 ⎥ − ⎢ 2 ⎥ = ⎢ ⎥ − ⎢ 2 ⎥ ⎣ 4 ⎦ ⎣ ⎦ hc1 . (13) V ⎢v3 ⎥ ⎢a3 ⎥ ⎢1 4⎥ ⎢a3 ⎥ ⎡a1 b1 c1 d1 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣v4 ⎦ ⎣a 4 ⎦ ⎣1 4⎦ ⎣a 4 ⎦ −1 a2 b2 c2 d2 Let A = ⎢ ⎥ From (11) and (13), the estimated position for unknown ⎢ ⎥ a3 b3 c3 d3 target nodes can be calculated using ⎢ ⎥ a b c d = T −1 T ⎣ 4 4 4 4 ⎦ Zc1 (Gc Gc ) Gc hc1 (14) Then (10) can be converted into To summarize, our proposed localization algorithm is = hc Gc Zc (11) summarized as follows: = T 1. All anchors send their positions to all sensor nodes Where Zc [x,y,z] within their transmission range. Each unknown node v a b c d collects all the beacon signals it receives from various ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡ 1 1 1 ⎤ reference points. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 v a b c d 2. Each unknown node randomly selects four anchor = ⎢ 2 ⎥ − ⎢ 2 ⎥ , = ⎢ 2 2 2 ⎥ hc G c nodes in range to form a series of tetrahedrons. V ⎢v3 ⎥ ⎢a3 ⎥ ⎢ b 3 c 3 d 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 3. Apply the proposed centroid theorem of coordinate- ⎣v4 ⎦ ⎣a4 ⎦ ⎣b 4 c 4 d 4 ⎦ tetrahedron in the volume-coordinate system described Using least square (LS) algorithm, we get in this subsection to calculate the barycenter of each − tetrahedron, then we use the average coordinates of = T 1 T (12) Z c (Gc Gc ) Gc hc these barycenters as the final estimated position of ith unknown node, = . As shown in Fig. 1, D is the midpoint of A2 A4 , Q is the i 1,2,M barycenter of triangle A2 A3 A4 , and M is the centroid of III. SIMULATION RESULTS tetrahedron . The volume coordinates for , A1 A2 A3 A4 A1 In this section, simulation results are presented and analyzed. The performance evaluation focuses on the position A2 , A3 , A4 will be (V,0,0,0 ) , (0, V,0,0 ) , (0,0, V,0 ) , estimation accuracy of the proposed algorithm. Since the , respectively. (0,0,0, V ) conventional centroid localization algorithm can only obtain 2D localization performance on 2D scenario, we simply

978-1-4244-2108-4/08/$25.00 © 2008 IEEE extend the conventional Centroid localization algorithm to the between two sensor nodes. three-dimensional space via adding z-coordinate. We consider an experiment region in a 3D cubic region 1.2 with a size of 100 m x 100 m x 100 m. As can be seen from 1.1 Centroid Algorithm Proposed Algorithm [4], centroid location is robust under the effect of the 1 irregularity of the radio pattern. The reason is that the 0.9 centroid algorithm does not depend on hop-count and hopsize that the effect of degree of irregularity（DOI） is abated by 0.8 the aggregation of beaconed information. So similar to 0.7

[2][5], we also assume the sensor nodes have the same 0.6 maximum radio range R, which is used for normalization only. 0.5 Firstly, we deploy 100 sensor nodes randomly on the three- dimensional plane for the centroid localization algorithm. The location error•0.4 ratio of radio range• number of sensor nodes and the radio range of sensor nodes 0.3 will be varied then. The simulation results are averaged over 0.2 100 network instances. The simulation results are shown in 30 40 50 60 70 the radio range of sensor nodes Figs.2-3. To begin with, we investigate the location errors resulted Fig.3. Location error with 20 anchor nodes from the conventional centroid localization algorithm and the proposed algorithm. As observed from Fig. 2, the proposed IV. CONCLUSION algorithm has smaller location error in general with respect to We present a new localization method that improves the different number of anchor nodes in the WSNs. Herein, the basic centroid localization algorithm significantly. It is shown number of unknown nodes is fixed to 100. Generally, the in the simulation results that the proposed algorithm can location error decreases as the number of anchor nodes improve location accuracy than the conventional centroid increases. For the same number of anchor nodes and in the localization algorithm. The performance of our proposed same WSN environment, the position error in the proposed scheme has identified that it has potential application advantage for 3D WSNs.. scheme is smaller than that in the conventional centroid localization algorithm. For example, with 20 anchor nodes, our proposed algorithm has an average error of approximately REFERENCES 0.54R, while the conventional centroid localization algorithm [1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “A has an average error about 0.7R. survey on sensor networks,” IEEE Communications Magazine, vol. 40, pp.102-113, Aug. 2002. 0.95 [2] N. Bulusu, J. Heidemann, and D. Estrin, “GPS-less low cost outdoor localization for very small devices”, IEEE Personal 0.9 CommunicationsMagazine, 7(5):28–34, October 2000. [3] L. Girod and D. Estrin, “Robust range estimation using acoustic and 0.85 multimodal sensing,” in Proc. IEEE International Conference on Intelligent Robots and Systems, vol 3, pp. 1312-1320, 2001. 0.8 [4] T. He, C. Huang, B. M. Blum, J. A. Stankovic, and T. Abdelzaher.

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