An Improved Localization Algorithm for Wireless Sensor Network ...
JOURNAL OF NETWORKS, VOL. 7, NO. 6, JUNE 2012
991
An Improved Localization Algorithm for Wireless Sensor Network Based on the Selection of Benchmark Anchor Node Xiaohui Chen* College of Computer and Information Technology, China Three Gorges University, Yichang, China Email: [email protected]
Jinpeng Chen, Jing He, Bangjun Lei College of Computer and Information Technology, China Three Gorges University, Yichang, China Email: [email protected]
Abstract—To improve the localization precision of nodes in Wireless sensor network (WSN), this paper analyses the source of the localization error in the least square localization algorithm (LSL), and it can be concluded that the minimum cumulative relative distance error of the benchmark anchor node directly influence the localization precision. Based on the analyses, this paper proposes an improved least square algorithm, which is the minimum cumulative related distance error in LSL (LSL-MCR) algorithm. In the improved algorithm, we give a new principle to choose the benchmark anchor node. The simulation results indicate that the proposed algorithm can improve the localization precision of the unknown nodes. Index Terms—WSN, localization precision, least square method, benchmark anchor node
I. INTRODUCTION The Wireless sensor network (WSN) is an emerging technology, and it is widely applied in many fields. The WSN consists of a collection of wireless networked lowpower sensor devices, with each node integrating an embedded microprocessor, radio, and a limited amount of storage [1]. It comprehensively uses the microelectronics technology, embedded computation technique, wireless communication technology, distributed information processing technology. Its aim is to perceive, collect and process the information of the perceived objects by the distributed sensors in the region, and sends the information to the observers. With the recent advances in WSN research, applications employing WSN have become quite popular [2]. Today, WSN is widely used in industrial automation, traffic control, environmental monitoring, and military reconnaissance and so on. To Manuscript received September 28, 2011; revised November 1, 2011; accepted November 15, 2011. This researcher is supported by the Research and Development Project of Science and Technology of Yichang (Grant No.A2011-30214); the National Natural Science Foundation of China (Grant No. 60972162); * Corresponding author: Xiaohui Chen
© 2012 ACADEMY PUBLISHER doi:10.4304/jnw.7.6.991-996
make the information obtained from the sensor nodes effectively, we should also precisely know the location of sensor nodes. Therefore, the basis and key technology is the precise localization of the nodes in WSN, and the information which lacks precise localization is meaningless. The localization of nodes in WSN can be achieved by the perceived information and information processing of nodes. To achieve more accurate localization, great deals of localization algorithms have already been developed in WSN. Savarese proposed two localization algorithms: cycle accuracy-Cooperative ranging [3] and Two-Phase localization [4] that can decrease the influence of distance error to localization; Avvides [5] proposed n-hop multilateration primitive localization algorithm, and he used Kalman filtering technique to calculate the accurate coordinates circularly, it reduced the accumulation error; The existent localization refinement algorithms are based on the circulatory method [6]-[10]. At present, the refinement methods that already existed can reduce localization errors in different degree, but there are still many deficiencies: 1) The existent refinement methods focus on the improvement of algorithms, and obtain high precision with the method of circulation, but its data operations are excessive, and it increases the energy of the nodes. 2) The measures of the refinement algorithm are multiple, and the information exchange between the nodes is frequent, so it increases the cost of the networks’ energy. The least square localization algorithm (LSL) is simple and has small calculating amount, so it is beneficial to save the energy of the nodes in WSN. In consideration of these, it draws the researchers’ extensive attention [11][15]. The localization precision in LSL is mainly affected by the distance measurement error, and the distance measurement error in RSSI is mainly due to the influence of multipath propagation and reflection, so the distance measurement error in RSSI is large. The weighted least square localization algorithm can decrease the influence of localization error with the method that decreases the
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impact of the anchor nodes that have bigger errors [12][14]. The weighted matrix is the main factor to affect the localization precision in the weighted least square localization algorithm. Based on the above issues, this paper aims at the influence of the anchor nodes in the least square algorithm, and then optimizes the selection of the benchmark anchor node. So the goal to improve the localization precision can be achieved. The main work of this paper includes some aspects as follows: 1) Analyses the source of the localization error in the least square localization algorithm (LSL). 2) Analyses the influence of localization precision caused by the relationship between the distance measurement error of each anchor node and the distance measurement error of the benchmark anchor node, and then proposes the selection principle of the benchmark anchor node. 3) Designs and emulates the improved LSL algorithms. 4) Summarizes the performance of the improved LSL algorithms. II. THE MODEL OF NODES’ LOCALIZATION The trilateration is a classical deterministic localization method. We adopt the trilateration to calculate coordinate of the unknown node. It respectively uses the three anchor nodes as the center of three circles, and similarly uses the distance between unknown nodes and anchor nodes as the radius of three circles, so the three circles can be intersect at one point. If we have the known points A(x1, y1), B(x2, y2), C(x3, y3) and the unknown point D(x, y) (which position we are looking for) is d1, d2, d3 from the these points respectively. According to the neighboring nodes’ localization information and the distance of the nodes, the unknown node coordinates can be calculated. The model of the localization figure is shown in Fig.1.
A (x1,y1)
C (x3,y3)
d1
d2 B (x2,y2)
Figure 1. The model of Nodes’ localization.
Then we can get the following equations:
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a1 x + b1 y = c1 . a2 x + b2 y = c2
(1)
(2)
Where
ai = 2(xi − x3) ,i = 1,2. bi = 2(yi − y3) 2 2 2 2 2 2 ci = (xi − x3 ) + (yi − y3 ) − (di − d3 ) With the simultaneous equations, we can calculate the coordinate of the unknown node D. b2 c 1 − x = a b − 1 2 a y = 2 c1 − a 1b2 −
b1 c 2 a 2 b1 a1c 2 a 2 b1
III. LEAST-SQUARE LOCALIZATION ALGORITHM(LSL) In the localization of wireless sensor network, the wireless signal transmission will cause attenuation mainly due to the influence of transmission medium, multipath propagation, reflection, non-line of sight, antenna gain, so it will produce error when measure the distance. When the error appears, these three circles will not have one common intersection. Therefore, we cannot get the coordinates of the nodes. In order to solve this problem, the least square localization algorithm (LSL) has been adopted. The LSL algorithm is a mathematical optimization technique, and it is concise and practical for the WSN which attaches importance to energy consumption. The specific steps are shown as follows: The coordinate of the unknown node is (x, y), and there are n anchor nodes within the communication distance of the unknown node. The anchor node coordinates are (x1, y1), (x2, y2), (x3, y3)… (xn, yn) respectively, then we can get the following expressions.
( x1 − x) 2 + ( y1 − y ) 2 = d12 2 2 2 ( x2 − x ) + ( y2 − y ) = d 2 . .... 2 ( x − x) + ( y − y ) 2 = d 2 n n n
d3
( x1 − x) 2 + ( y1 − y ) 2 = d12 2 2 2 ( x2 − x) + ( y2 − y ) = d 2 . 2 2 2 ( x3 − x) + ( y3 − y ) = d 3
Subtracting the third equation from the first and the second equation in (1), then we can get the following expression.
(3)
The other equations subtract the Nth equation respectively, then we can get the following expressions. 2(x1 - xn)x+2(y1 - yn)y= (x12 - xn2) +(y12 - yn2) - (d12 −dn2) 2 2 2 2 2 2 2(x2 - xn)x+2(y2 - yn)y= (x2 - xn ) +(y2 - yn ) -(d2 −dn ) M 2(x - x )x+2(y - y )y= (x 2 - x 2) +(y 2 - y 2) -(d 2 −d 2) n-1 n n-1 n n-1 n n-1 n n-1 n And we can use (4) to instead of it. (4) AX * = B * . Where
JOURNAL OF NETWORKS, VOL. 7, NO. 6, JUNE 2012
x1 − x n x − xn A = 2 × 2 M x n−1 − x n B*
993
y1 − y n y2 − yn . M y n −1 − y n
( (
) ( ) (
) )
(
) (
)
2x1 − xj x + 2y1 − yj y =(x12 − x2j)+(y12 − y2j )−(d12 − d2j )−(e1 − ej) 2x2 − xj x + 2y2 − yj y =(x22 − x2j)+(y22 − y2j )−(d22 − d2j )−(e2 − ej). .... 2 2 2 2 2 2 2xn − xj x + 2yn − yj y =(xn − xj )+(yn − yj)−(dn − dj )−(en − ej) (7) AX = B * + E . So X = ( AT A)−1 AT (B* + E) . (8) = ( AT A) −1 AT B* + ( AT A) −1 AT E Where
( x12 − x n2 ) + ( y12 − y n2 ) − (d 12 − d n2 ) . ( x 22 − x n2 ) + ( y 22 − y n2 ) − (d 22 − d n2 ) = M2 2 2 2 2 2 ( x n - 1 − x n ) + ( y n − 1 − y n ) − (d n − 1 − d n ) x . X * = y
The X* can be solved. X * = ( AT A) −1 AT B * .
(5) So we can calculate the accurate location of the unknown node.
B∗
IV. THE ANALYSIS OF LOCALIZATION PRECISION IN LSL A. Noun Definition In order to achieve better effect, we define some nouns as follows: Reference anchor node: The anchor node which participates in the localization of the unknown node. Benchmark equation: In the (3), each distance equation based on the distance between the anchor node and the unknown node constitutes the localization equations; during the process of reduced-order, each equation subtracts an equation (usually it is the Nth equation), so the Nth equation is called the benchmark equation. Benchmark anchor node: The anchor node that corresponds to the benchmark equation. Cumulative relative distance error: The distance measurement error of other each anchor node subtracts the distance measurement error of the j# anchor node, and then averages the sum of the absolute value of the difference, and it can be described , we call it the cumulative as n
∑
i=1 i≠ j
ei − e
j
/( n − 1 )
relative distance error of j# anchor node. B. The Analysis of Localization Error Caused by the Distance Measurement Error in the LSL Algorithm The RSSI is usually used to measure the distance between the anchor nodes and the unknown node, and it will cause attenuation mainly due to the influence of transmission medium, multipath propagation, reflection and so on, so it will produce distance measurement error in the localization of wireless sensor network. Based on the (3), we introduce the distance error, then (x - x)2 + (y 1 1 2 (x2 - x) + (y2 2 (xn -1 - x) + (yn -1 (xn - x)2 + (yn
2
- y)2 = d1 + e1 2 . - y)2 = d2 + e2 M 2 - y)2 = d n - 1 + en −1
(6)
2
- y)2 = d n + en
The other equations subtract the Jth equation respectively, then
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( x 12 − x 2j ) + ( y 12 − 2 2 2 ( x2 − x j ) + ( y2 − M = ( x 2j − 1 − x 2j ) + ( y 2j − 1 − ( x 2 − x 2 ) + ( y 2 − j j+1 j+1 M (x2 − x2) + ( y2 − n j n
2 2 2 y j ) − ( d j − 1 − d j ) y 2j ) − ( d 2j + 1 − d 2j ) y 2j ) − ( d n2 − d 2j )
y 2j ) − ( d 12 − d 2j ) y 2j ) − ( d 22 − d 2j )
( e j − e1 ) (e − e ) 2 j . M E = ( e j − e j −1 ) ( e j − e j +1 ) M (e − e ) n j
(9)
In order to expound clearly, we suppose the fellows:
F = ( AT A) −1 AT Then X = F (B * + E ) = X
*
+ FE .
(10)
Where
X * = ( AT A)−1 AT B* By using the Cauchy-Schwarz inequality, we can get the identification error of X X − X * = FE 2 ≤ F F ⋅ E 2 2
The xi,yi,and di are known, so the matrix A and vector F can be obtained. When the Euclidean norm of error vector E is smaller, the Euclidean norm of vector FE can be smaller. Based on the above conclusion, we can get:
X − X*
2
∝ E
2
The Euclidean norm of vector (X-X*) can be transformed to (11). X*−X
2
x* − x = * y − y
= ( x * − x) 2 + ( y * − y ) 2
.
(11)
2
According to (11), the Euclidean norm of vector (X- X*) is just the localization error of the unknown node. Based on the above proof, a theorem can be concluded: Theorem 1: If the Euclidean norm of E is smaller, the localization error of the unknown node can be smaller. V. THE MINIMUM CUMULATIVE RELATED DISTANCE ERROR IN LSL (LSL-MCR)
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the
E
localization error of each unknown node is the distance between the calculated coordinate and standard coordinate. The simulation is shown by the Fig.2. 2 1.9 1.8 1.7 The localization error
A. The Analysis Based on the Selection of the Benchmark Anchor Node In the traditional LSL algorithm, the selection of the benchmark anchor node is random. How to further select the benchmark anchor node from the anchor nodes reasonably? Some algorithms select the anchor node that has the minimum distance measurement error as the benchmark anchor node. These methods cannot improve the localization precision greatly, even decrease the localization precision. Based on the theorem 1, we know that the Euclidean norm of E has profound effect on the localization error of the unknown node. We attempt to seek the condition that
1.6 1.5 1.4 1.3 1.2 1.1
2
get minimum. We can conclude that the
cumulative relative distance error of the anchor node has great effect on the localization precision of the unknown node. Selecting the benchmark anchor node reasonably is beneficial to reduce the extra errors and improve the localization precision. Based on (9), we can get the following expressions. e1 − e j n 2 E2= M = ∑ (ei − e j ) . i =1 en − e j i≠ j 2
Then E
2 2
=
∑ (e n
i =1 i≠ j
i
− ej
)
2
.
( ) ∑(e − e ) ∝ ∑ e − e 2
E2 ∝ E2 =
n
i =1 i≠ j
2
i
j
n
i =1 i≠ j
i
j
Based on the Theorem 1, we choose the anchor node j# as the benchmark anchor node. Where
j = min j
n
∑
i = 1,i ≠ j
1
2 3 4 5 6 The cumulative relative distance error of the benchmark anchor node
7
Figure 2. The comparison of the error when the reference
anchor node is taken as the benchmark anchor node respectively.
We can conclude from the Fig.2 that the localization error increases with the increase of the cumulative relative distance error of the benchmark anchor node. So theorem 2 can be concluded: Theorem 2: If the anchor node that has smaller cumulative relative distance error is selected as the benchmark anchor node, the localization precision of the unknown node will be higher. Generally, it is difficult to get the exact measurement error, and the measurement error is proportional to the distance in RSSI, that is to say:
X − X*
2
∝ E
2
∝ D
2
.
Where (d j − d 1 ) (d j − d 2 ) M D = ( d j − d j − 1 ) ( d − d ) j +1 j M (d − d ) j n
e i − e j /(n − 1).
B. The simulation analysis based on the selection of the anchor node The simulation analyses the impact on the localization precision caused by the cumulative relative distance error of the benchmark anchor node. Number the reference anchor node from 1#~n# according to the cumulative relative distance error in ascending order, and take the 1#~n# reference anchor node as the benchmark anchor node respectively, and then the LSL algorithm is adopted to calculate the location of the unknown node. The performance of the selection of the anchor node can be evaluated by a series of simulations. To make the analysis more general, 100 unknown nodes are selected to emulate the performance. Specific parameters are showing as follows: 100 unknown nodes are randomly deployed, and there are 7 random anchor nodes around each unknown node. The error is a random distribution and is proportional to the distance. The accurate distance between the unknown node and the anchor node is dc*. Set the measurement error of the distance is dc*×er, in which er is the random noise of [-0.1, 0.1] and its mean value is 0. The © 2012 ACADEMY PUBLISHER
1
So we can choose the anchor node j# which has the minimum cumulative relative distance error as the benchmark anchor node, that is to say
j = min j
n
∑
i = 1,i ≠ j
d i − d j /(n − 1)
C. The Design of the Minimum Cumulative Related Distance Error in LSL (LSL- DMC) Based on the above analysis, we select some nodes that the distance between each anchor node around the unknown node and unknown node is approximate as the corrected anchor nodes. So the distance measurement error between each anchor node around the unknown node and unknown node is approximate. Then we select the corrected anchor node which has the minimum cumulative relative distance error as the benchmark anchor node. That is to say, we select the central point from {e1 , e 2 , L , e n } as the benchmark anchor node,
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and then the LSL algorithm is adopted to improve the localization precision of the unknown node. The specific steps to select the benchmark anchor node from the corrected anchor nodes are shown as follows: 1) Obtain the distance between each anchor node and the unknown node. 2) Take a distance as a number. 3) Other each distance subtracts the distance of an anchor node, and then calculates the summation of the differences. DClusi =
n
∑(d
i =1& i ≠ j
i
− d j ).
(12)
4) Take the minimum i to Dmin, and take the maximum to Dmax, and then take the corresponding anchor node of Dmin as the benchmark anchor node. The steps of LSL-DMC are shown as follows: a) Aiming at the unknown node, obtain the distance between each anchor node around the unknown node and the unknown node, and then form the distance set. c) Selecting the corrected anchor nodes from the anchor nodes, and further select the benchmark anchor node from the corrected anchor nodes. d) The LSL algorithm is adopted to calculate accurate location of the unknown node. VI. THE SIMULATION RESULTS The performance of the proposed localization algorithms can be evaluated by a series of simulations.100 unknown nodes are randomly deployed, and there are 7 random anchor nodes around each unknown node. The error is a random distribution and is proportional to the distance. The accurate distance between the unknown node and the anchor node is dc*. Set the measurement error of the distance is dc*×er, in which er is the random noise of [-0.3, 0.3] and its mean value is 0. The localization error of each unknown node is the distance between the calculated coordinate and standard coordinate. In consideration of the whole situation, we adopt the mean square deviation of the all nodes’ localization error. We compare the estimated locations of a number of nodes with their actual locations. The comparison of the corrected effect in the two algorithms is shown as follows: 8 LSL LSL MCR
7
-
6 5 4 3 2 1 0
0
10
20
30
40
50
60
70
80
90
100
Figure 3. Corrected effect of the two algorithms.
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The comparison of the average errors in the two algorithms is shown as table1.
TABLE I. THE COMPARISON OF THE AVERAGE ERRORS IN THE TWO ALGORITHMS Algorithm
LSL
LSL_MCR
Average error
1.6075
1.2319
Based on the simulation results, we can conclude that the localization precision of the most nodes in the LSL_MCR algorithm has greater improvement than the LSL algorithm, and the phenomenon of big errors is eliminated partly. As shown in Fig3 and Table1, it is clear that the LSL-MCR has better localization precision than the LSL algorithm. VII. CONCLUSION This paper analyzes the source of the localization error in the least square localization (LSL) algorithm, and proposes an improved least square localization algorithm. The key novelty of the improved algorithm (LSL-MCR) is the selection of the benchmark anchor node. We have also compared the localization precision of the unknown nodes in the two proposed algorithms. The simulations have been conducted to show the effectiveness of the improved localization algorithm. This paper mainly aims at the simulation in 2d layout environment. For future study, we will do more researches on the 3d physical model of the nodes. ACKNOWLEDGMENT This research is supported by the National Natural Science Foundation of China (Grant No.60972162), 2010.1-2012.12. This research is supported by the Research and Development Project of Science and Technology of Yichang (Grant No.A2011-302-14). REFERENCES [1] I. F. Akyildiz, W. L. Su, Y. Sankarasubramaniam, and E. Cayirci.: A survey on sensor networks. J. IEEE Communications Magazine. vol.40, No. 8, p. 102--114 (Aug. 2002) [2] Onur Ozdemir, Ruixin Niu, Pramod K.Varshney.: Channel Aware Target Localization With Quantized Data in Wireless Sensor Networks. J. IEEE transactions on signal processing, vol.57, no.3 (2009) [3] Savarese C, Rabaey JM., Beutel J.: Localizationing in distributed ad-hoc wireless sensor network. Proceeding of the 2001 IEEE International Conference on Acoustics, Speech, and Signal Salt Lake (2001) [4] Savarese C, Rabaey JM., Langendoen K. Robust.: Localization Algorithms for Distributed Ad Hoc Wireless Sensor Networks. Proceedings of the USENIX Technical Annual Conference, Monterey, USA (2002) [5] Avvides A., Park H., Srivastava MB.: The bits AND flops of the N-hop multilateration primitive for node localization problems. Proceeding of the 1st ACM International
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[14]
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Workshop on Wireless Sensor Networks and Applications, Atlanta (2002) Bergamo P., Mazzini G.: Localization in sensor networks with fading and mobility. Processing of the 13th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, NJ, USA (2002) Guha S., Murty R N., Sirer E G.: A unified node and event localization framework using non-convex constraints. Proceeding of the 6th ACM International Symposium on Mobile Ad Hoc Networking and Computing, IL USA (2005) Cao M., Anderson B.D.O., Morse A.S.: Sensor network localization with imprecise distances. J. Systems and Control Letters. vol.55, pp.887--893 (2006) Srirangarajan S., Tewfik A H., Luo Z.Q.: Distributed sensor network localization with inaccurate anchor positions and noisy distance information. IEEE International Conference on Acoustics, Speech and Signal Processing, HI, USA (2007) Yonghong Guo, Jingwen Wan, Ning Yu, Renjian Feng.: Hop-based Refinement Algorithm for Localization in Wireless Sensor Networks. J. Computer Engineering. No.3, pp. 145-147,151 (2009) Xiaohui Chen, Jing He, Jinpeng Chen.: An improved localization algorithm for wireless sensor network. J. Intelligent Automation and Soft Computing, 2011, Vol.17, 507-517. Ning Yu, Jiangwen Wan, Renjian Feng. Localization refinement algorithms for wireless sensor networks. High Technology Letters[J]. 2008,18(10) Shufeng Wang, Yibin Hou, etal. Error Analysis of Least Squares Method and Optimization for WSN [J].Journal of System Simulation, 2009, 10, Vol.21, No.19. Jiangang Wang, Fubao Wang, Weijun Duan .: Application of Weighted Least Square Estimates on Wireless Sensor Network Node Localization[J].Application Research of Computers.2006, No.9. Xiaohui Chen, Jing He, Bangjun Lei.: An Improved Algorithm of Sensors Localization in Wireless Sensor Network. Proceedings of the 2011 CSEE International Conference, Wuhan, China, 2011, 8.
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Xiaohui Chen was born in Sichuan China in 1967. He received his master’s degree from Huazhong University of Science and Technology (HUST), Wuhan, China, in 1996, He is currently an associate professor in the college of computer and information technology, China Three Gorges University. His research interests include wireless sensor network, data mining, and intelligent control. Jinpeng Chen was born in Hubei China in 1986. He received his bachelor's degree from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2010, He is seeking for his master’s degree in the college of computer and information technology of China Three Gorges University now. His research interests include wireless sensor network, intelligent control, and embedded system. Jing He was born in Hubei province, China in 1987, he received his bachelor's degree from Huazhong University of Science and Technology (HUST) , Wuhan, China,in 2010, and he is seeking for his master’s degree in the college of computer and information technology , China Three Gorges University now. His research fields include wireless sensor networks, embedded system. Bangjun Lei was born in 1973. He got his Ph.D title from TUDelft, the Netherlands in 2003. He is professor at the College of Computer and Information Technology, China Three Gorges University. He is active in low-level image processing, 3-D imaging and computer vision. He enjoys developing practical multimedia applications.
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An Improved Localization Algorithm for Wireless Sensor Network Based on the Selection of Benchmark Anchor Node Xiaohui Chen* College of Computer and Information Technology, China Three Gorges University, Yichang, China Email: [email protected]
Jinpeng Chen, Jing He, Bangjun Lei College of Computer and Information Technology, China Three Gorges University, Yichang, China Email: [email protected]
Abstract—To improve the localization precision of nodes in Wireless sensor network (WSN), this paper analyses the source of the localization error in the least square localization algorithm (LSL), and it can be concluded that the minimum cumulative relative distance error of the benchmark anchor node directly influence the localization precision. Based on the analyses, this paper proposes an improved least square algorithm, which is the minimum cumulative related distance error in LSL (LSL-MCR) algorithm. In the improved algorithm, we give a new principle to choose the benchmark anchor node. The simulation results indicate that the proposed algorithm can improve the localization precision of the unknown nodes. Index Terms—WSN, localization precision, least square method, benchmark anchor node
I. INTRODUCTION The Wireless sensor network (WSN) is an emerging technology, and it is widely applied in many fields. The WSN consists of a collection of wireless networked lowpower sensor devices, with each node integrating an embedded microprocessor, radio, and a limited amount of storage [1]. It comprehensively uses the microelectronics technology, embedded computation technique, wireless communication technology, distributed information processing technology. Its aim is to perceive, collect and process the information of the perceived objects by the distributed sensors in the region, and sends the information to the observers. With the recent advances in WSN research, applications employing WSN have become quite popular [2]. Today, WSN is widely used in industrial automation, traffic control, environmental monitoring, and military reconnaissance and so on. To Manuscript received September 28, 2011; revised November 1, 2011; accepted November 15, 2011. This researcher is supported by the Research and Development Project of Science and Technology of Yichang (Grant No.A2011-30214); the National Natural Science Foundation of China (Grant No. 60972162); * Corresponding author: Xiaohui Chen
© 2012 ACADEMY PUBLISHER doi:10.4304/jnw.7.6.991-996
make the information obtained from the sensor nodes effectively, we should also precisely know the location of sensor nodes. Therefore, the basis and key technology is the precise localization of the nodes in WSN, and the information which lacks precise localization is meaningless. The localization of nodes in WSN can be achieved by the perceived information and information processing of nodes. To achieve more accurate localization, great deals of localization algorithms have already been developed in WSN. Savarese proposed two localization algorithms: cycle accuracy-Cooperative ranging [3] and Two-Phase localization [4] that can decrease the influence of distance error to localization; Avvides [5] proposed n-hop multilateration primitive localization algorithm, and he used Kalman filtering technique to calculate the accurate coordinates circularly, it reduced the accumulation error; The existent localization refinement algorithms are based on the circulatory method [6]-[10]. At present, the refinement methods that already existed can reduce localization errors in different degree, but there are still many deficiencies: 1) The existent refinement methods focus on the improvement of algorithms, and obtain high precision with the method of circulation, but its data operations are excessive, and it increases the energy of the nodes. 2) The measures of the refinement algorithm are multiple, and the information exchange between the nodes is frequent, so it increases the cost of the networks’ energy. The least square localization algorithm (LSL) is simple and has small calculating amount, so it is beneficial to save the energy of the nodes in WSN. In consideration of these, it draws the researchers’ extensive attention [11][15]. The localization precision in LSL is mainly affected by the distance measurement error, and the distance measurement error in RSSI is mainly due to the influence of multipath propagation and reflection, so the distance measurement error in RSSI is large. The weighted least square localization algorithm can decrease the influence of localization error with the method that decreases the
992
JOURNAL OF NETWORKS, VOL. 7, NO. 6, JUNE 2012
impact of the anchor nodes that have bigger errors [12][14]. The weighted matrix is the main factor to affect the localization precision in the weighted least square localization algorithm. Based on the above issues, this paper aims at the influence of the anchor nodes in the least square algorithm, and then optimizes the selection of the benchmark anchor node. So the goal to improve the localization precision can be achieved. The main work of this paper includes some aspects as follows: 1) Analyses the source of the localization error in the least square localization algorithm (LSL). 2) Analyses the influence of localization precision caused by the relationship between the distance measurement error of each anchor node and the distance measurement error of the benchmark anchor node, and then proposes the selection principle of the benchmark anchor node. 3) Designs and emulates the improved LSL algorithms. 4) Summarizes the performance of the improved LSL algorithms. II. THE MODEL OF NODES’ LOCALIZATION The trilateration is a classical deterministic localization method. We adopt the trilateration to calculate coordinate of the unknown node. It respectively uses the three anchor nodes as the center of three circles, and similarly uses the distance between unknown nodes and anchor nodes as the radius of three circles, so the three circles can be intersect at one point. If we have the known points A(x1, y1), B(x2, y2), C(x3, y3) and the unknown point D(x, y) (which position we are looking for) is d1, d2, d3 from the these points respectively. According to the neighboring nodes’ localization information and the distance of the nodes, the unknown node coordinates can be calculated. The model of the localization figure is shown in Fig.1.
A (x1,y1)
C (x3,y3)
d1
d2 B (x2,y2)
Figure 1. The model of Nodes’ localization.
Then we can get the following equations:
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a1 x + b1 y = c1 . a2 x + b2 y = c2
(1)
(2)
Where
ai = 2(xi − x3) ,i = 1,2. bi = 2(yi − y3) 2 2 2 2 2 2 ci = (xi − x3 ) + (yi − y3 ) − (di − d3 ) With the simultaneous equations, we can calculate the coordinate of the unknown node D. b2 c 1 − x = a b − 1 2 a y = 2 c1 − a 1b2 −
b1 c 2 a 2 b1 a1c 2 a 2 b1
III. LEAST-SQUARE LOCALIZATION ALGORITHM(LSL) In the localization of wireless sensor network, the wireless signal transmission will cause attenuation mainly due to the influence of transmission medium, multipath propagation, reflection, non-line of sight, antenna gain, so it will produce error when measure the distance. When the error appears, these three circles will not have one common intersection. Therefore, we cannot get the coordinates of the nodes. In order to solve this problem, the least square localization algorithm (LSL) has been adopted. The LSL algorithm is a mathematical optimization technique, and it is concise and practical for the WSN which attaches importance to energy consumption. The specific steps are shown as follows: The coordinate of the unknown node is (x, y), and there are n anchor nodes within the communication distance of the unknown node. The anchor node coordinates are (x1, y1), (x2, y2), (x3, y3)… (xn, yn) respectively, then we can get the following expressions.
( x1 − x) 2 + ( y1 − y ) 2 = d12 2 2 2 ( x2 − x ) + ( y2 − y ) = d 2 . .... 2 ( x − x) + ( y − y ) 2 = d 2 n n n
d3
( x1 − x) 2 + ( y1 − y ) 2 = d12 2 2 2 ( x2 − x) + ( y2 − y ) = d 2 . 2 2 2 ( x3 − x) + ( y3 − y ) = d 3
Subtracting the third equation from the first and the second equation in (1), then we can get the following expression.
(3)
The other equations subtract the Nth equation respectively, then we can get the following expressions. 2(x1 - xn)x+2(y1 - yn)y= (x12 - xn2) +(y12 - yn2) - (d12 −dn2) 2 2 2 2 2 2 2(x2 - xn)x+2(y2 - yn)y= (x2 - xn ) +(y2 - yn ) -(d2 −dn ) M 2(x - x )x+2(y - y )y= (x 2 - x 2) +(y 2 - y 2) -(d 2 −d 2) n-1 n n-1 n n-1 n n-1 n n-1 n And we can use (4) to instead of it. (4) AX * = B * . Where
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x1 − x n x − xn A = 2 × 2 M x n−1 − x n B*
993
y1 − y n y2 − yn . M y n −1 − y n
( (
) ( ) (
) )
(
) (
)
2x1 − xj x + 2y1 − yj y =(x12 − x2j)+(y12 − y2j )−(d12 − d2j )−(e1 − ej) 2x2 − xj x + 2y2 − yj y =(x22 − x2j)+(y22 − y2j )−(d22 − d2j )−(e2 − ej). .... 2 2 2 2 2 2 2xn − xj x + 2yn − yj y =(xn − xj )+(yn − yj)−(dn − dj )−(en − ej) (7) AX = B * + E . So X = ( AT A)−1 AT (B* + E) . (8) = ( AT A) −1 AT B* + ( AT A) −1 AT E Where
( x12 − x n2 ) + ( y12 − y n2 ) − (d 12 − d n2 ) . ( x 22 − x n2 ) + ( y 22 − y n2 ) − (d 22 − d n2 ) = M2 2 2 2 2 2 ( x n - 1 − x n ) + ( y n − 1 − y n ) − (d n − 1 − d n ) x . X * = y
The X* can be solved. X * = ( AT A) −1 AT B * .
(5) So we can calculate the accurate location of the unknown node.
B∗
IV. THE ANALYSIS OF LOCALIZATION PRECISION IN LSL A. Noun Definition In order to achieve better effect, we define some nouns as follows: Reference anchor node: The anchor node which participates in the localization of the unknown node. Benchmark equation: In the (3), each distance equation based on the distance between the anchor node and the unknown node constitutes the localization equations; during the process of reduced-order, each equation subtracts an equation (usually it is the Nth equation), so the Nth equation is called the benchmark equation. Benchmark anchor node: The anchor node that corresponds to the benchmark equation. Cumulative relative distance error: The distance measurement error of other each anchor node subtracts the distance measurement error of the j# anchor node, and then averages the sum of the absolute value of the difference, and it can be described , we call it the cumulative as n
∑
i=1 i≠ j
ei − e
j
/( n − 1 )
relative distance error of j# anchor node. B. The Analysis of Localization Error Caused by the Distance Measurement Error in the LSL Algorithm The RSSI is usually used to measure the distance between the anchor nodes and the unknown node, and it will cause attenuation mainly due to the influence of transmission medium, multipath propagation, reflection and so on, so it will produce distance measurement error in the localization of wireless sensor network. Based on the (3), we introduce the distance error, then (x - x)2 + (y 1 1 2 (x2 - x) + (y2 2 (xn -1 - x) + (yn -1 (xn - x)2 + (yn
2
- y)2 = d1 + e1 2 . - y)2 = d2 + e2 M 2 - y)2 = d n - 1 + en −1
(6)
2
- y)2 = d n + en
The other equations subtract the Jth equation respectively, then
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( x 12 − x 2j ) + ( y 12 − 2 2 2 ( x2 − x j ) + ( y2 − M = ( x 2j − 1 − x 2j ) + ( y 2j − 1 − ( x 2 − x 2 ) + ( y 2 − j j+1 j+1 M (x2 − x2) + ( y2 − n j n
2 2 2 y j ) − ( d j − 1 − d j ) y 2j ) − ( d 2j + 1 − d 2j ) y 2j ) − ( d n2 − d 2j )
y 2j ) − ( d 12 − d 2j ) y 2j ) − ( d 22 − d 2j )
( e j − e1 ) (e − e ) 2 j . M E = ( e j − e j −1 ) ( e j − e j +1 ) M (e − e ) n j
(9)
In order to expound clearly, we suppose the fellows:
F = ( AT A) −1 AT Then X = F (B * + E ) = X
*
+ FE .
(10)
Where
X * = ( AT A)−1 AT B* By using the Cauchy-Schwarz inequality, we can get the identification error of X X − X * = FE 2 ≤ F F ⋅ E 2 2
The xi,yi,and di are known, so the matrix A and vector F can be obtained. When the Euclidean norm of error vector E is smaller, the Euclidean norm of vector FE can be smaller. Based on the above conclusion, we can get:
X − X*
2
∝ E
2
The Euclidean norm of vector (X-X*) can be transformed to (11). X*−X
2
x* − x = * y − y
= ( x * − x) 2 + ( y * − y ) 2
.
(11)
2
According to (11), the Euclidean norm of vector (X- X*) is just the localization error of the unknown node. Based on the above proof, a theorem can be concluded: Theorem 1: If the Euclidean norm of E is smaller, the localization error of the unknown node can be smaller. V. THE MINIMUM CUMULATIVE RELATED DISTANCE ERROR IN LSL (LSL-MCR)
994
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the
E
localization error of each unknown node is the distance between the calculated coordinate and standard coordinate. The simulation is shown by the Fig.2. 2 1.9 1.8 1.7 The localization error
A. The Analysis Based on the Selection of the Benchmark Anchor Node In the traditional LSL algorithm, the selection of the benchmark anchor node is random. How to further select the benchmark anchor node from the anchor nodes reasonably? Some algorithms select the anchor node that has the minimum distance measurement error as the benchmark anchor node. These methods cannot improve the localization precision greatly, even decrease the localization precision. Based on the theorem 1, we know that the Euclidean norm of E has profound effect on the localization error of the unknown node. We attempt to seek the condition that
1.6 1.5 1.4 1.3 1.2 1.1
2
get minimum. We can conclude that the
cumulative relative distance error of the anchor node has great effect on the localization precision of the unknown node. Selecting the benchmark anchor node reasonably is beneficial to reduce the extra errors and improve the localization precision. Based on (9), we can get the following expressions. e1 − e j n 2 E2= M = ∑ (ei − e j ) . i =1 en − e j i≠ j 2
Then E
2 2
=
∑ (e n
i =1 i≠ j
i
− ej
)
2
.
( ) ∑(e − e ) ∝ ∑ e − e 2
E2 ∝ E2 =
n
i =1 i≠ j
2
i
j
n
i =1 i≠ j
i
j
Based on the Theorem 1, we choose the anchor node j# as the benchmark anchor node. Where
j = min j
n
∑
i = 1,i ≠ j
1
2 3 4 5 6 The cumulative relative distance error of the benchmark anchor node
7
Figure 2. The comparison of the error when the reference
anchor node is taken as the benchmark anchor node respectively.
We can conclude from the Fig.2 that the localization error increases with the increase of the cumulative relative distance error of the benchmark anchor node. So theorem 2 can be concluded: Theorem 2: If the anchor node that has smaller cumulative relative distance error is selected as the benchmark anchor node, the localization precision of the unknown node will be higher. Generally, it is difficult to get the exact measurement error, and the measurement error is proportional to the distance in RSSI, that is to say:
X − X*
2
∝ E
2
∝ D
2
.
Where (d j − d 1 ) (d j − d 2 ) M D = ( d j − d j − 1 ) ( d − d ) j +1 j M (d − d ) j n
e i − e j /(n − 1).
B. The simulation analysis based on the selection of the anchor node The simulation analyses the impact on the localization precision caused by the cumulative relative distance error of the benchmark anchor node. Number the reference anchor node from 1#~n# according to the cumulative relative distance error in ascending order, and take the 1#~n# reference anchor node as the benchmark anchor node respectively, and then the LSL algorithm is adopted to calculate the location of the unknown node. The performance of the selection of the anchor node can be evaluated by a series of simulations. To make the analysis more general, 100 unknown nodes are selected to emulate the performance. Specific parameters are showing as follows: 100 unknown nodes are randomly deployed, and there are 7 random anchor nodes around each unknown node. The error is a random distribution and is proportional to the distance. The accurate distance between the unknown node and the anchor node is dc*. Set the measurement error of the distance is dc*×er, in which er is the random noise of [-0.1, 0.1] and its mean value is 0. The © 2012 ACADEMY PUBLISHER
1
So we can choose the anchor node j# which has the minimum cumulative relative distance error as the benchmark anchor node, that is to say
j = min j
n
∑
i = 1,i ≠ j
d i − d j /(n − 1)
C. The Design of the Minimum Cumulative Related Distance Error in LSL (LSL- DMC) Based on the above analysis, we select some nodes that the distance between each anchor node around the unknown node and unknown node is approximate as the corrected anchor nodes. So the distance measurement error between each anchor node around the unknown node and unknown node is approximate. Then we select the corrected anchor node which has the minimum cumulative relative distance error as the benchmark anchor node. That is to say, we select the central point from {e1 , e 2 , L , e n } as the benchmark anchor node,
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995
and then the LSL algorithm is adopted to improve the localization precision of the unknown node. The specific steps to select the benchmark anchor node from the corrected anchor nodes are shown as follows: 1) Obtain the distance between each anchor node and the unknown node. 2) Take a distance as a number. 3) Other each distance subtracts the distance of an anchor node, and then calculates the summation of the differences. DClusi =
n
∑(d
i =1& i ≠ j
i
− d j ).
(12)
4) Take the minimum i to Dmin, and take the maximum to Dmax, and then take the corresponding anchor node of Dmin as the benchmark anchor node. The steps of LSL-DMC are shown as follows: a) Aiming at the unknown node, obtain the distance between each anchor node around the unknown node and the unknown node, and then form the distance set. c) Selecting the corrected anchor nodes from the anchor nodes, and further select the benchmark anchor node from the corrected anchor nodes. d) The LSL algorithm is adopted to calculate accurate location of the unknown node. VI. THE SIMULATION RESULTS The performance of the proposed localization algorithms can be evaluated by a series of simulations.100 unknown nodes are randomly deployed, and there are 7 random anchor nodes around each unknown node. The error is a random distribution and is proportional to the distance. The accurate distance between the unknown node and the anchor node is dc*. Set the measurement error of the distance is dc*×er, in which er is the random noise of [-0.3, 0.3] and its mean value is 0. The localization error of each unknown node is the distance between the calculated coordinate and standard coordinate. In consideration of the whole situation, we adopt the mean square deviation of the all nodes’ localization error. We compare the estimated locations of a number of nodes with their actual locations. The comparison of the corrected effect in the two algorithms is shown as follows: 8 LSL LSL MCR
7
-
6 5 4 3 2 1 0
0
10
20
30
40
50
60
70
80
90
100
Figure 3. Corrected effect of the two algorithms.
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The comparison of the average errors in the two algorithms is shown as table1.
TABLE I. THE COMPARISON OF THE AVERAGE ERRORS IN THE TWO ALGORITHMS Algorithm
LSL
LSL_MCR
Average error
1.6075
1.2319
Based on the simulation results, we can conclude that the localization precision of the most nodes in the LSL_MCR algorithm has greater improvement than the LSL algorithm, and the phenomenon of big errors is eliminated partly. As shown in Fig3 and Table1, it is clear that the LSL-MCR has better localization precision than the LSL algorithm. VII. CONCLUSION This paper analyzes the source of the localization error in the least square localization (LSL) algorithm, and proposes an improved least square localization algorithm. The key novelty of the improved algorithm (LSL-MCR) is the selection of the benchmark anchor node. We have also compared the localization precision of the unknown nodes in the two proposed algorithms. The simulations have been conducted to show the effectiveness of the improved localization algorithm. This paper mainly aims at the simulation in 2d layout environment. For future study, we will do more researches on the 3d physical model of the nodes. ACKNOWLEDGMENT This research is supported by the National Natural Science Foundation of China (Grant No.60972162), 2010.1-2012.12. This research is supported by the Research and Development Project of Science and Technology of Yichang (Grant No.A2011-302-14). REFERENCES [1] I. F. Akyildiz, W. L. Su, Y. Sankarasubramaniam, and E. Cayirci.: A survey on sensor networks. J. IEEE Communications Magazine. vol.40, No. 8, p. 102--114 (Aug. 2002) [2] Onur Ozdemir, Ruixin Niu, Pramod K.Varshney.: Channel Aware Target Localization With Quantized Data in Wireless Sensor Networks. J. IEEE transactions on signal processing, vol.57, no.3 (2009) [3] Savarese C, Rabaey JM., Beutel J.: Localizationing in distributed ad-hoc wireless sensor network. Proceeding of the 2001 IEEE International Conference on Acoustics, Speech, and Signal Salt Lake (2001) [4] Savarese C, Rabaey JM., Langendoen K. Robust.: Localization Algorithms for Distributed Ad Hoc Wireless Sensor Networks. Proceedings of the USENIX Technical Annual Conference, Monterey, USA (2002) [5] Avvides A., Park H., Srivastava MB.: The bits AND flops of the N-hop multilateration primitive for node localization problems. Proceeding of the 1st ACM International
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Workshop on Wireless Sensor Networks and Applications, Atlanta (2002) Bergamo P., Mazzini G.: Localization in sensor networks with fading and mobility. Processing of the 13th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, NJ, USA (2002) Guha S., Murty R N., Sirer E G.: A unified node and event localization framework using non-convex constraints. Proceeding of the 6th ACM International Symposium on Mobile Ad Hoc Networking and Computing, IL USA (2005) Cao M., Anderson B.D.O., Morse A.S.: Sensor network localization with imprecise distances. J. Systems and Control Letters. vol.55, pp.887--893 (2006) Srirangarajan S., Tewfik A H., Luo Z.Q.: Distributed sensor network localization with inaccurate anchor positions and noisy distance information. IEEE International Conference on Acoustics, Speech and Signal Processing, HI, USA (2007) Yonghong Guo, Jingwen Wan, Ning Yu, Renjian Feng.: Hop-based Refinement Algorithm for Localization in Wireless Sensor Networks. J. Computer Engineering. No.3, pp. 145-147,151 (2009) Xiaohui Chen, Jing He, Jinpeng Chen.: An improved localization algorithm for wireless sensor network. J. Intelligent Automation and Soft Computing, 2011, Vol.17, 507-517. Ning Yu, Jiangwen Wan, Renjian Feng. Localization refinement algorithms for wireless sensor networks. High Technology Letters[J]. 2008,18(10) Shufeng Wang, Yibin Hou, etal. Error Analysis of Least Squares Method and Optimization for WSN [J].Journal of System Simulation, 2009, 10, Vol.21, No.19. Jiangang Wang, Fubao Wang, Weijun Duan .: Application of Weighted Least Square Estimates on Wireless Sensor Network Node Localization[J].Application Research of Computers.2006, No.9. Xiaohui Chen, Jing He, Bangjun Lei.: An Improved Algorithm of Sensors Localization in Wireless Sensor Network. Proceedings of the 2011 CSEE International Conference, Wuhan, China, 2011, 8.
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Xiaohui Chen was born in Sichuan China in 1967. He received his master’s degree from Huazhong University of Science and Technology (HUST), Wuhan, China, in 1996, He is currently an associate professor in the college of computer and information technology, China Three Gorges University. His research interests include wireless sensor network, data mining, and intelligent control. Jinpeng Chen was born in Hubei China in 1986. He received his bachelor's degree from Huazhong University of Science and Technology (HUST), Wuhan, China, in 2010, He is seeking for his master’s degree in the college of computer and information technology of China Three Gorges University now. His research interests include wireless sensor network, intelligent control, and embedded system. Jing He was born in Hubei province, China in 1987, he received his bachelor's degree from Huazhong University of Science and Technology (HUST) , Wuhan, China,in 2010, and he is seeking for his master’s degree in the college of computer and information technology , China Three Gorges University now. His research fields include wireless sensor networks, embedded system. Bangjun Lei was born in 1973. He got his Ph.D title from TUDelft, the Netherlands in 2003. He is professor at the College of Computer and Information Technology, China Three Gorges University. He is active in low-level image processing, 3-D imaging and computer vision. He enjoys developing practical multimedia applications.
