Research Article
A clustering approach for error beacon filtering in underwater wireless sensor networks
International Journal of Distributed Sensor Networks 2016, Vol. 12(12) Ó The Author(s) 2016 DOI: 10.1177/1550147716681793 ijdsn.sagepub.com
Linfeng Liu1,2, Jingli Du2 and Dongyue Guo2
Abstract Underwater wireless sensor networks are the enabling technology for the aquatic environmental monitoring and exploring and have attracted much attention recently. Due to the highly hostile and unpredictable underwater environments, some beacon nodes tend to move or be damaged. Therefore, the unknown nodes will be positioned with larger error, which abases the value of data collected by sensor nodes. In order to solve the beacon error problem, this article proposes an error beacon filtering algorithm based on K-means clustering. First, the coordinate of each beacon is calculated through an improved trilateration method, and then the beacon with the maximum positioning error is filtered out via the K-means clustering algorithm. The remaining beacons repeat the above processes until the distance error of each beacon does not exceed a preset threshold. The analysis of simulation results indicates that the error beacons can be accurately found and filter out through our proposed error beacon filtering algorithm (based on K-means clustering), and thus the localization accuracy is enhanced. Besides, error beacon filtering algorithm also has a provable low complexity. Keywords Underwater wireless sensor networks, error beacon filtering, localization algorithm, K-means clustering
Date received: 6 September 2016; accepted: 8 November 2016 Academic Editor: Miguel Ardid
Introduction Underwater acoustic networks (underwater wireless sensor networks (UWSNs)) consist of abundant lowcost sensor nodes tied to underwater vehicles, and the nodes are deployed to monitor the underwater environment collaboratively over the interest area.1 In order to explore the underwater world, UWSNs have attracted wide attention, and many specific applications have emerged, such as environmental monitoring, natural disaster prevention, and distributed tactical surveillance, where the node localization is always very significant.2 If each sensor node cannot provide its accurate coordinate, the data collected by sensor nodes may give wrong interpretations for the physical events.3 However, the underwater environment is more complex than the terrestrial environment, and the underwater
characteristics bring several new challenges as follows. (1) The nodes with limited batteries are more prone to be exhausted, so they should be recharged timely. Unfortunately, it is very hard to access underwater nodes.4 (2) Radio wave is not feasible underwater, because it requires a large antenna and a high transmission power, and thus the acoustic communication
1
Laboratory of Computer Network and Information Integration, Ministry of Education, Southeast University, Nanjing, China 2 School of Computer Science & Technology, Nanjing University of Posts and Telecommunications, Nanjing, China Corresponding author: Linfeng Liu, School of Computer Science & Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China. Email:
[email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (http://www.uk.sagepub.com/aboutus/ openaccess.htm).
2
International Journal of Distributed Sensor Networks the error ones, and thus the error beacon nodes can be filtered out according to the mutual localization results among all beacon nodes. This work is the extension of our early work,10 the main differences between the two papers are as follows: (1) the algorithm is given more illustrations, (2) the theoretical analysis of algorithm has been improved and extended, and (3) more simulations have been done and supplied.
Related works Figure 1. A typical architecture of UWSNs.
becomes the typical physical layer technology in UWSNs. Nevertheless, the acoustic channel is characterized by its limited bandwidth, high bit error rate, path loss, motion-induced Doppler shift, and so on.5 (3) Underwater sensor nodes are liable to move or be damaged6,7 due to the water current caused by external forces such as earthquake, tide, wind velocity, underwater creature touch, or strong electromagnetic interference, which will lead to a dynamic network topology.8 All above unique characteristics are possible to result in the damage or inaccurate localization of beacon nodes. Generally, a typical architecture for threedimensional (3D) UWSNs is shown in Figure 1, where there are three types of nodes: surface buoys, beacon nodes, and ordinary sensor nodes.9 The surface buoys can get the coordinates from their equipped global positioning system (GPS). The beacon nodes are submerged underwater, and thus GPS is not feasible for beacon nodes. They should communicate with the surface buoys to obtain their X-Y plane coordinates (the Z-axis coordinates can be estimated via the pressure sensors). Besides, the beacon nodes also help ordinary sensor nodes do localizations. However, the localizations by beacons are usually unavailable due to the beacon damage (such as the hit from water current and the touch of underwater creatures) and signal interference in the underwater environment. Hence, some beacon nodes probably provide the inaccurate coordinate information for the localizations of ordinary nodes, and these beacon nodes are referred to as the error beacons. To this end, this work proposes an error beacon filtering algorithm (EBFA), which can effectively improve the localization accuracy through filtering out the error beacons. Several error beacon nodes which cannot provide accurate references for ordinary nodes should be filtered out to avoid the aggravation of localization error. However, it is very difficult to find the error beacon nodes because we usually have no pre-knowledge about
Localization schemes have been extensively investigated in wireless sensor networks or UWSNs, and these schemes can be divided into two groups: anchor-based schemes and anchor-free schemes.11 In the anchorbased schemes, the beacon nodes get their coordinates in advance through carrying GPS receiver or even they are artificially pre-configured. The beacon nodes broadcast periodically their coordinate information. Subsequently, the ordinary nodes estimate their coordinates by calculating the distances or angles to the nearest beacon nodes, especially, some measurement techniques such as received signal strength indicator (RSSI), time of arrival (TOA), and time difference of arrival (TDOA) are usually utilized in the process. Zhang et al.12 proposed a multi-anchor nodes collaborative localization (MANCL) algorithm. First, the well-localized nodes within one hop are prone to become the reference nodes if the ordinary nodes cannot receive four beacon signals, and the selection criterion is related to the energy, trust value, and distance. Then, an improved Euclidean distance estimation method is adopted to localize the ordinary nodes. Finally, the remaining un-localized ordinary nodes complete their localizations with the help of two-hop anchor nodes. The anchor-free schemes13 determine the ordinary nodes’ coordinates through exploiting the connectivity or distance information among nodes, and thus the assistances of beacon nodes are unnecessary. The anchor-free schemes are especially suitable for the networks where nodes are hardly deployed, such as the battlefield environment or special warfare environment. Generally, the network protocol without beacon nodes is more complex than that with beacon nodes. In addition, the anchor-based schemes can further be classified into the static beacon node localization and mobile beacon node localization. In Cheng et al.,14 an underwater positioning scheme (UPS) is proposed, where ordinary nodes record the receiving time of beacon messages, and then the time difference is transformed into the range distance after receiving four beacon messages. Finally, the ordinary nodes apply the trilateration method to estimate their own coordinates. UPS reduces the communication overhead and does
Liu et al. not require the time synchronization, so the cost of UPS is relatively low. In Rahman et al.,15 with the help of a mobile beacon node, Cayley–Menger is used to determine the node coordinates. The distance between nodes is measured through combining the radio and acoustic signals which are free from the phenomenon of multi-path fading. In Zhang and Liang,16 the distance between nodes is calculated by a new ranging method named round-trip time of flight (RTOF), and then the ordinary nodes complete localizations using an improved particle swarm optimization (PSO) algorithm, which adds a Gaussian decreasing inertia weight and a kind of competition mechanism. This scheme can improve the localization accuracy and localization efficiency with less beacon nodes. Nonetheless, the above literatures do not take into account the mobility of underwater nodes, which are only applicable in static underwater networks. The mobility issue in node localization has also been reviewed. Ojha and Misra17 used spatially correlated mobility pattern of UWSNs to estimate the node coordinates. In the initial stage of localization, there are only three beacon nodes. If an ordinary node cannot get enough information, it will assume that the node moves according to some specific rules, and the future coordinates can be easily predicted. When the ordinary nodes can communicate with at least three beacon nodes at the original and predicted positions, and then the coordinates of ordinary nodes can be determined. The outstanding advantage of this algorithm is that it is energy efficient as a result of the ‘‘silent localization.’’ In Zhu et al.,18 a localization scheme based on mobility prediction for UWSNs is introduced. The localization process is divided into two parts: the beacon nodes utilize the modified covariance algorithm to estimate their prediction models to reduce the position error, while the ordinary nodes choose the well-localized reference nodes to get their positions and speed by a node-selection strategy. The algorithm increases the localization coverage and decreases the localization error compared with scalable localization scheme with mobility prediction (SLMP) algorithm.19 But the prediction gives a poor accuracy especially when the underwater environment is hostile. A multihop location (MLA) in UWSNs is also proposed in Zhu et al.,20 where the routing nodes are introduced to solve the problem of isolated nodes. First, the shortest paths from beacon nodes to ordinary nodes are found through a greedy approach. Subsequently, the shortest paths are fitted into a straight distance using the cosine method. Finally, the trilateration is repeatedly performed to localize the ordinary nodes. This algorithm has much higher localization accuracy than determined maximum likelihood (DML) algorithm.21 Some researchers also take notice of the measurement errors in localization process. Liu et al.22
3 combined the time synchronization and the node localization, which corrects the bias in the range estimation and improves the propagation delay in estimation when the stratification effect of underwater medium is considered. In addition, in order to further increase the localization accuracy, an advanced tracking algorithm interacting multiple model (IMM) is employed to handle the mobile case. Wu and Li23 proposed an improved underwater acoustic network localization algorithm, which considers the measurement error caused by the sound velocity distortion and signal refraction. It uses an improved linear difference method to correct the measurement offset, which improves the localization accuracy. Simultaneously, a strategy similar to the greedy algorithm reduces the redundancy of the calculation results. However, none of these works take the issue of error beacons into consideration. If beacon nodes move, its coordinate information will become obsolete or even wrong. Therefore, the ordinary nodes will be positioned more inaccurately under the assistance of these error beacons. To deal with the error beacon problem, this article proposes the EBFA based on K-means clustering. The coordinate of each beacon is calculated by an improved trilateration, and then the error beacons are filtered out by the K-means clustering algorithm.24
EBFA based on K-means clustering Suppose that plenty of sensors nodes are deployed in a 3D underwater space D2IR3. A small part of beacon nodes provides error reference coordinates. Let BN denote the beacon nodes set, where BN = {b1, b2, ., bn}. The number of the beacon nodes is n. Suppose that each node can transmit and receive messages with enough power and obtain the distance between nodes through RSSI.
Algorithm description The EBFA based on K-means clustering will calculate the coordinate of each beacon by an improved trilateration, and then the distance differences exceeding a distance threshold are divided into two categories by K-means clustering method. Afterward, the beacon with the maximum positioning error is filtered out. All error beacons will be found until the distance differences are lower than the threshold. The following steps explain EBFA in detail: Step 1. The first beacon bi is selected randomly and five nearest beacons of bi are found. Four of the nearest beacons will be used to position bi, and this process will be repeated C54 times (C54 denotes the number of combinations), which produces C54 coordinate results of bi. In detail, suppose b1, b2, b3, and
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International Journal of Distributed Sensor Networks b4 position bi, then the coordinate of bi is calculated as b , b2 , b3
L b1
Lbb1i , b2 , b3 , b4
=
+
1
+
b , b2 , b3 i
db 1
b , b3 , b4
b , b2 , b4
i b ,b ,b db 1 2 3 i
L b1
i b ,b ,b db 1 2 4 i
+
1
+
b , b2 , b4 i
db 1
L b1
b , b3 , b4
i b ,b ,b db 1 3 4 i
+
1
+
b , b3 , b4 i
db 1
L b2
i b , b3 , b4 i
db 2
1
ð1Þ
b , b3 , b4 i
db 2
where Lbb1i , b2 , b3 = (xbi 1 , b2 , b3 , b4 , ybi 1 , b2 , b3 , b4 , zbi 1 , b2 , b3 , b4 ) represents the coordinate of bi, and dbbi1 , b2 , b3 represents the mean distance from b1, b2, and b3 to bi. Step 2. The distance difference between the estimated coordinate and real one of bi is computed as Figure 2. Example diagram of node localization.
Dbb1i , b2 , b3 , b4 =
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b1 , b2 , b3 , b4 2 2 2 xi + ybi 1 , b2 , b3 , b4 yi + zbi 1 , b2 , b3 , b4 zi xi
ð2Þ
Moreover, each beacon should reserve a variable X[bi] to record the number of found error beacons, and 0 initially. X ½bi Step 3. The distance differences exceeding a threshold will be divided into two categories: accurate and inaccurate, by the K-means clustering method (set K = 2).25 Step 4. The localization results of a beacon are compared. If the beacon is considered inaccurate, then X ½bi + 1. The beacon with the maximum X ½bi positioning error is filtered out and marked as an error beacon. Step 5. Steps 1–4 are repeated until all distance differences are lower than the threshold after removing the found error beacons. The following example describes the EBFA algorithm briefly. Suppose that there are 10 nodes b1, b2, ., b10 and each node requires the localization. As
shown in Figure 2, b2, b3, b4, b5, and b6 position b1, and b5, b6, b7, b8, and b9 position b2. The distance differences exceeding the threshold occur in the localization process of b1 and b2. The categories of beacons are shown in Table 1, where D() bi denotes the distance difference between the estimated coordinate and real one, and ( ) indicates the beacon set for the localization of beacon bi. d is a predefined distance threshold, which is set according to the network environments. The value of X[bi] is given in Table 2. As is shown in Table 2, the value of X[b2] is the maximum, so b2 is marked as an error beacon. The remaining nine beacons repeat the process after removing b2.
Time complexity of EBFA The time complexity of EBFA is mainly contributed by the Step 1 to Step 4. The time complexity of Step 1 is O(n2 ); the time complexity of Step 2 is O(n); the sorting time complexity of Step 3 is O(2nt);O(n), where t is the count of iterations; and the time complexity of Step 4 is O(n). The time complexity of EBFA is O(n2 ), which is acceptable.
Table 1. The diagram of beacons classification. Beacon
Neighboring aided beacons
D() bi
d
Accurate/inaccurate category
b1
b2, b3, b4, b5 b2, b3, b4, b6 b2, b3, b5, b6 b2, b4, b5, b6 b3, b4, b5, b6 b5, b6, b7, b8 b5, b6, b7,b9 b5, b6, b8, b9 b5, b7, b8, b9 b6, b7, b8, b9
11.5 20.5 0.25 0.15 0.6 16.3 22.4 0.34 0.1 0.09
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
Inaccurate category Inaccurate category Dbi d Dbi d Accurate category Inaccurate category Inaccurate category Dbi d Dbi d Dbi d
b2
Liu et al.
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Table 2. The diagram of value of X[bi]. X[b1]
X[b2]
X[b3]
X[b4]
X[b5]
X[b6]
X[b7]
X[b8]
X[b9]
X[b10]
2
4
2
2
3
3
2
1
1
0
Mathematical analysis In general, the deployment of sensor nodes tossed from the air to the ground obeys the normal distribution. Let X-coordinate, Y-coordinate, and Z-coordinate of the ordinary node obey the following distribution: X N (mx , dx ), Y N (my , dy ), and Z N (mz , dz ), respectively. The real coordinate and the estimated coordinate of the beacon node bi are denoted by (xi , yi , zi ) and (xi , yi , zi ), respectively. The real distance and the estimated distance between beacons is di and di , respectively. The impact of the error beacons is analyzed as follows. First, the real coordinate of the localized node is calculated from the following equation set 8 2 2 2 2 > < (x xa ) + (y ya ) + (z za ) = da 2 2 2 2 (x xb ) + (y yb ) + (z zb ) = db > : 2 (x xc )2 + (y yc )2 + (z zc )2 = dc
ð3Þ
0 1 x 1 ðB @yA=A +D Þ ð4Þ z 0 1 2(xa xc ) 2(ya yc ) 2(za zc ) where A = @ 2(xb xc ) 2(yb yc ) 2(zb zc ) A, 2(xa xb ) 2(ya yb ) 2(za zb ) 0 2 1 x a x c 2 + y a 2 y c 2 + za 2 zc 2 and B = @ x b 2 x c 2 + y b 2 y c 2 + zb 2 zc 2 A , x c 2 x b 2 + y a 2 y b 2 + za 2 zb 2 0 2 1 2 dc da B C D = @ dc 2 db 2 A. D indicates that the localization 2 2 db da results are related with the distance. A and B show the localization results are also related with the coordinates of aided beacons.1To simplify the formulations, let 0 a1 b1 c1 A = @ a2 b2 c2 A, then we obtain that a3 b3 c3 0
b2 c3 c2 b3 1 = A = 1 @ c2 a3 a2 c3 A A A a b b a 2 3
2 3
c1 b3 b1 c3 a1 c 3 c 1 a3 b1 a3 a1 b3
0
1 0 1 0 1 x Dx x 1 ðB 1 ðB @ Dy A = @ y A @ y A = A + DÞ A +D Þ Dz z z 1 ðD D Þ =A ð5Þ
Moreover, the localization error is expressed by the scalar ERf ERf = Dx2 + Dy2 + Dz2
where a, b, and c are the aided beacons. (xa , ya , za ) denotes the real coordinate of the beacon a, and da is the real distance from a to the coordinate (x, y, z) (the coordinate of the localized node). Thus, the real coordinate of the localized nodes is expressed as
= a1 (b2 c3 c2 b3 ) a2 (b1 c3 c1 b3 ) + a3 (b1 c2 where jAj c1 b2 ). The error from trilateration algorithm executions should be taken into account. Set di = di + jid (i = a, b, c), hence the measured 0 1coordinate of the localized x 1 (B + D). Therefore, nodes is calculated as @ y A = A z the localization error of trilateration algorithm is
ð6Þ
Formula (6) transforms the localization error into a scalar, and then the error can be analyzed from each axis. Let xi = xi + jix , yi = yi + jiy , and zi = zi + jiz (i = a, b, c). Thus, the measured of loca0 coordinate 1 x0 lized nodes is expressed as @ y0 A = A1 ðB + DÞ. z0 Therefore, the localization error is written as 0
1 0 01 0 1 Dx0 x x 1 ðB @ Dy0 A = @ y0 A @ y A = A1 ðB + DÞ A +D Þ 0 0 z Dz z ð7Þ
Then the localization error expressed by the scalar ERs is rewritten as ERs = Dx02 + Dy02 + Dz02 .The sign of ERs ERf are discussed from Case I and Case II. Case I. If jix = jiy = jiz = 0, one gets ERs ERf = 0 easily; Case II. If jix 6¼ 0, jiy 6¼ 0, and jiz = 0, then 0
2(jcx jax ) + @ 2(jcx jbx ) A=A 2(jbx jax )
e1 e2 e3
1 0 0 A, 0
1 jcd 2 2dc jcd + 2da jad jad 2 b1 c 2 c 1 b2 +G=D + @ j 2 2dc j + 2db j j 2 A, D=D cd cd bd bd a2 c 1 a1 c 2 A 2 2 j 2d j + 2d j j b a bd bd ad ad a1 b2 a2 b1 and 1
0
1 0 2(jcy jay ) 0 m1 2(jcy jby ) 0 A = @ m2 2(jby jay ) 0 m3
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International Journal of Distributed Sensor Networks
B=B+F =B+ 0 1 jax 2 2xa jax + 2xc jcx jcx 2 + 2ya jay + 2yc jcy jcy 2 B 2 C @ jbx 2xb jbx + 2xc jcx jcx 2 + 2yb jby + 2yc jcy jcy 2 A: 2 2 2 jax 2xa jax + 2xb jbx jbx + 2yb jby + 2yc jcy jcy
B is transformed into the sum of the real beacon value and the beacon error F. D is transformed into the B and the distance error sum of the real distance value D and F + G G. In order to facilitate the analysis, B + D 0 1 n1 = @ n2 A and are jointly analyzed. Define B + D n3 0 1 f1 + g1 F + G = @ f2 + g2 A, so Formula (7) can be rewritten f3 + g3 as 1 Dx0 1 ðB @ Dy0 A = A1 ðB + DÞ A +D Þ = A1 0 Dz 1 ðB +D + F + GÞ A +D Þ ðB
Table 3. Simulation parameters. Parameter Description
Value
jDj N n K rate
100 m3100 m3100 m 300 60 2 20%
RCmax d Count
Deployment space Number of nodes Number of beacon nodes Number of categories Proportion of error beacon nodes in beacon nodes Maximum communication range Distance threshold Number of execution times
10 m 5m 100
0
ð8Þ
Furthermore, ERs ERf can be expressed as (Dx02 Dx2 ) + (Dy02 Dy2 ) + (Dz02 Dz2 ). To observe the sign of ERs ERf , (Dx0 )2 (Dx)2 , (Dy0 )2 (Dy)2 , and (Dz0 )2 (Dz)2 are verified respectively in the Appendix 1. Without loss of generality, the error from Y-axis and Z-axis is ignored temporarily, and thus there are g2 = g3 = 0, n2 = n3 = 0, and f2 = f3 = 0, then there is Dx02 Dx2 .0, Dy02 Dy2 .0, and Dz02 Dz2 .0, the derivation and proof of which are also given in the Appendix 1. Therefore, ERs ERf 0. That is, the localization error with error beacons is higher than that without error beacons which have been filtered out.
Simulations EBFA is evaluated by observing the performance variation when adopting different model parameters (such as the number of beacon nodes and the number of error beacons) and by comparing EBFA with other algorithms. Table 3 shows the values of the parameters. The accurate discovery of the error beacons is extremely critical to the localization accuracies of ordinary nodes. This simulation measures the number of found error beacons with different number of beacon nodes. As shown in Figure 3, three plots (the proportion is assigned as 20%, 30%, and 40%, respectively) are observed. The plot with a larger rate is higher than the other plots because there are more error beacon nodes to be found. Besides, the number of found error beacons also grows with the increase in the total number of beacon nodes and the proportion of error beacon nodes. When the number of real error beacons is fixed, more beacons will bring
Figure 3. Number of found error beacons versus rates of error beacons.
more accurate judgments for the error beacons, and thus more error beacons can be found. However, when there are excessive error beacons, the number of found error beacons is much different from the number of real error beacons. When the proportion is set 20% and the number of beacons reaches 60, the number of found error beacons (about 13) is approximately equal to the number of real error beacons (60 3 20% = 12), which indicates that EBFA can effectively detect almost all error beacons especially when the number of beacon nodes and the proportion of error beacons are not very large. In Figure 4, the variance metric denotes the stability of the number P of found errorP beacons. The variance is count count 2 (a½i ( expressed as i=1 j = 1 a½j=count)) =count, where count is the number of execution times, and a[i] is the number of found error beacons at the ith execution. The plot with a larger rate has a sharper fluctuation than the others, which is attributed to the fact that the variable number of the found error beacons becomes larger when there are more error beacons. Moreover, more error beacons give rise to a larger localization error of ordinary nodes as well. Note that some of the error beacons are still not being detected, especially
Liu et al.
Figure 4. Variance versus number of beacons.
Figure 5. Found error beacons versus distance threshold.
Figure 6. Variance versus different thresholds.
when the number of beacon nodes becomes larger, which is attributed to the random deployment of beacons and the original localization error.
7 In EBFA, the error beacons are filtered out according to a distance threshold, that is, the beacons with larger localization error are excluded. Therefore, the value of threshold has a significant influence on the performance of EBFA. Both Figures 5 and 6 illustrate the impacts of the threshold when rate is set 20%. In Figure 5, it can be found that when the threshold is set 10, the number of found error beacons is approximately equal to the number of real error beacons because the proper setting of threshold helps EBFA to find the error beacon nodes accurately. Nevertheless, when the threshold is too small, some accurate beacons are also mistakenly labeled as the error beacons. Moreover, when the threshold is too large, most of the error beacons are not found because EBFA cannot differentiate the accurate beacons and error beacons by a large threshold. Consequently, the proper setting of the threshold is important to the EBFA performance. As depicted in Figure 6, the threshold has an obvious impact on the variance number of found error nodes as well. In general, with the increase in the number of beacon nodes, the plots of variance continue to rise up. When the threshold is 5, the variance number is larger than the others, this is because the number of found error beacons is larger, and thus the deployment of error beacons becomes more random accordingly. Figure 7 compares the number of found beacons of EBFA, Centroid, and Trilateration. Apparently, EBFA overcomes Centroid and Trilateration absolutely. In Centroid, almost all nodes are labeled invalid. This is because Centroid has a stricter requirement for the node distribution and it assumes that the nodes obey the uniform distribution, which does not tally with the random deployment in our simulations. Therefore, it is hard to find error beacons exactly. In Trilateration, all nodes involved in localization will be marked as error beacons provided that this localization result is wrong. Hence, some of the error beacons cannot be found. In EBFA, most of the error beacons can be found in an iterative way, and probabilities of falsely marking the beacon nodes are very small, but the found error beacons are usually a bit more than the real ones. The reason is that some accurate beacons served for the localizations of the neighboring error beacons are also prone to be regarded as the error beacons. As shown in Figure 8, the localization error denotes the coordinate derivations of ordinary nodes from the beacon localizations. The results indicate that EBFA can achieve the lowest localization error after effectively filtering most of the error beacons. Moreover, the scalability of EBFA is also better than those of the other two algorithms, this is because the proportion of the number of found error beacons stays the same approximately, as shown in Figure 3. In summary, when the proper distance threshold is set, EBFA can accurately filter out most of the error
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International Journal of Distributed Sensor Networks
(a)
(b)
Figure 7. Comparisons of the number of found beacons and found error beacons: (a) number of found beacons and (b) number of found error beacons.
that the EBFA can accurately find the error beacons with low complexity. This work is based on the assumption that the number of error beacon nodes is fewer compared with the number of all beacon nodes. The probabilities of finding the error beacon nodes will become smaller when the proportion of error beacon nodes becomes larger. Our future work will focus on the issue that how to revise the coordinate of error beacons such that they can still be available for coordinate references. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Figure 8. Localization error comparisons.
Funding beacons, which effectively improve the localization accuracy of ordinary nodes. Moreover, EBFA performs a favorable scalability with the number of beacon nodes. Moreover, there may be some gaps from theory algorithms to practical application, and EBFA should be improved the practicality according to the test feedbacks in actual UWSN systems.
Conclusion This article explores the problem of error beacons filtering. The error beacons will be found in an iterative way based on an improved trilateration and the K-means clustering method, that is, the error beacon nodes are filtered out according to the mutual localization results among all beacon nodes. Simulation results demonstrate
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China under grant nos. 61373139, 61373137, 61300239, and 71301081; Natural Science Foundation of Jiangsu Province under grant nos BK2012833, BK20130877, and BK20141429; China Postdoctoral Science Foundation under grant nos 2014M560379, 2014M551635, and 2015T80484; Scientific and Technological Support Project (Society) of Jiangsu Province under grant no. BE2013666; Scientific and Technological Support Project (Society) of Lianyungang under grant no. SH1306; and Postdoctoral Science Foundation of Jiangsu Province under grant no. 1302085B.
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Appendix 1 The derivations of (Dx0 )2 (Dx)2 , (Dy0 )2 (Dy)2 , and (Dz0 )2 (Dz)2 are given as
10
International Journal of Distributed Sensor Networks 8 > > > > > > > 2 > > Dx0 Dx2 = > > > > > > > > > > > > > > > > > > > > > < Dy0 2 Dy2 = > > > > > > > > > > > > > > > > > > > > > Dz0 2 Dz2 = > > > > > > > > > :
88 29 A fc3 ðb2 + e2 Þ c2 ðb3 + e3 Þg j Ajn1 ðb2 c3 c2 b3 Þ 9 > ð n + f + g Þ 1 1 1 > > > > = > > !2 > > >< = < + ðn2 + f2 + g2 Þ A fc1 ðb3 + e3 Þ c3 ðb1 + e1 Þg j Ajn2 ðc1 b3 b1 c3 Þ 1 > > : A j Aj > fc2 ðb1 + e1 Þ c1 ðb2 + e2 Þg j Ajn3 ðb1 c2 c1 b2 Þ ; > > > + ðn3 + f3 + g3 ÞA > > > > ; : 2 fj Ajg1 ðb2 c3 c2 b3 Þ + j Ajg2 ðc1 b3 b1 c3 Þ + j Ajg3 ðb1 c2 c1 b2 Þg 88 92 9 fc2 ða3 + m3 Þ c3 ða2 + m2 Þg j Ajn1 ðc2 a3 c3 a2 Þ >> ð n1 + f 1 + g1 Þ A > > = > < > > !2 > > > = < + ðn2 + f2 + g2 Þ A fc3 ða1 + m1 Þ c1 ða3 + m3 Þg j Ajn2 ðc3 a1 c1 a3 Þ 1 > > ; : A j Aj > > fc1 ða2 + m2 Þ c2 ða1 + m1 Þg j Ajn3 ðc1 a2 c2 a1 Þ + ðn3 + f3 + g3 ÞA > > > > > > ; : 2 fj Ajg1 ðc2 a3 c3 a2 Þ + j Ajg2 ðc3 a1 c1 a3 Þ + j Ajg3 ðc1 a2 c2 a1 Þg 88 92 9 A fða2 + m2 Þðb3 + e3 Þ ðb2 + e2 Þða3 + m3 Þg > > > > ð n + f + g Þ > > 1 1 1 > > > > > > > > > > > > = < > !2 > > > + ðn2 + f2 + g2 Þ A fðb1 + e1 Þða3 + m3 Þ ða1 + m1 Þðb3 + e3 Þg = < 1 > > + ðn3 + f3 + g3 Þ A fða1 + m1 Þðb2 + e2 Þ ða2 + m2 Þðb1 + e1 Þg > A j Aj > > > > >> > > ; > : > > > > A ð a b b a Þ A ð b a a b Þ A ð a b a b Þ j jn j jn j jn > > 1 2 3 2 3 2 1 3 1 3 3 1 2 2 1 > > > > ; : 2 fj Ajg1 ða2 b3 b2 a3 Þ + j Ajg2 ðb1 a3 a1 b3 Þ + j Ajg3 ða1 b2 a2 b1 Þg Furthermore, the sub-formulas can be derived as !2 o 1 fc3 ðb2 + e2 Þ c2 ðb3 + e3 Þg j Ajn1 ðb2 c3 c2 b3 Þg2 fj Ajg1 ðb2 c3 c2 b3 Þg2 ðDx Þ ðDxÞ = fðn1 + f1 + g1 ÞA A j Aj nn o 2 n1 j Aj 2 ðj Ajg1 Þ2 ðb2 c3 c2 b3 Þ2 + ðn1 + f1 + g1 ÞA ðc3 e2 c2 e3 Þ2 ðn1 + f1 + g1 ÞA = !2 n1 j Aj ðn1 + f1 + g1 ÞA ðb2 c3 c2 b3 Þðc3 e2 c2 e3 Þg 1 + 2 ðn1 + f1 + g1 ÞA A j Aj 0 2
2
Usually, 2 j Ajg1 ðn1 f1 g1 Þ, ðn1 + f1 + g1 ÞA n1 j Aj 2 ðj Ajg1 Þ2 = A ðn1 + f1 + g1 Þ2 j Aj2 g1 ðn1 g1 Þ.A ðn1 g1 Þj Ajg2 0 j Ajn1 ðn1 + f1 + g1 Þ. ðn1 + f1 + g1 ÞA + j Aj2 n1 2 g1 2 2A n1 j Aj ðn1 + f1 + g1 ÞA ðb2 c3 c2 b3 Þðc3 e2 c2 e3 Þ 0, which yields Dx02 Dx2 .0. and ðn1 + f1 + g1 ÞA 02 2 Similarly, there is also Dy Dy .0. Besides !2 n o 1 fða2 + m2 Þðb3 + e3 Þ ðb2 + e2 Þða3 + m3 Þg j Ajn1 ða2 b3 b2 a3 Þ 2 fj Ajg1 ðb2 c3 c2 b3 Þg2 ðDz Þ ðDzÞ = ðn1 + f1 + g1 ÞA A j Aj n n o 2 n1 j Aj 2 ðj Ajg1 Þ2 ða2 b3 b2 a3 Þ2 + ðn1 + f1 + g1 ÞA fa2 e3 b2 m3 + m2 ðb3 + e3 Þ e2 ða3 + m3 Þg2 = ðn1 + f1 + g1 ÞA !2 n1 j Aj ðn1 + f1 + g1 ÞA ða2 b3 b2 a3 Þfa2 e3 b2 m3 + m2 ðb3 + e3 Þ e2 ða3 + m3 Þgg 1 + 2 ðn1 + f1 + g1 ÞA A j Aj .0 0 2
2
