A Distributed Anchor Node Selection Algorithm Based on ...

Research ArticleA Distributed Anchor Node Selection Algorithm Based on ErrorAnalysis for Trilateration Localization

Yingsheng Fan 12 Xiaogang Qi 13 Baoguo Yu3 and Lifang Liu4

1 School of Mathematics and Statistics Xidian University Xirsquoan Shaanxi 71007 China2Basic Courses Department Zhejiang Police College Hangzhou Zhejiang 310053 China3State Key Laboratory of Satellite Navigation System and Equipment Technology Shijiazhuang 050000 China4School of Computer Science and Technology Xidian University Xirsquoan Shaanxi 710071 China

Correspondence should be addressed to Xiaogang Qi xgqixidianeducn

Received 19 April 2018 Revised 27 October 2018 Accepted 18 November 2018 Published 5 December 2018

Academic Editor Alessandro Gasparetto

Copyright copy 2018 Yingsheng Fan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a distributed anchor node selection algorithm based on error analysis for trilateration localization (EATL)The influence of distance measurement error on localization accuracy is discussed from two aspects condition number of triangleformed by the three anchor nodes and the relative position between the unknown node and the three anchor nodes Based on theerror analysis three principles for optimizing the selection of anchor nodes are given and then an algorithm for selecting anchornodes on the ring is proposed

1 Introduction

In a distributed sensor network for most applications suchas target tracking environmental monitoring [1] the geo-graphical information of sensor nodes needs to be knownEstimation of node position is a fundamental requirementin distributed sensor networks One possible solution isto install GPS receiver for each sensor node (or similarsystem such as BeiDou Navigation Satellite System) butthis scheme is limited by the characteristics of distributedsensor network itself On the one hand the cost of nodewith GPS system will be two orders of magnitude higherthan that of ordinary node [2] On the other hand accord-ing to the different applications sensor nodes are oftendeployed in the interior city buildings or even forestenvironment and satellite signals are easily affected bymany factors such as multipath interference and occlusion[3] The localization accuracy is poor or even affecting itsusability

Cooperative localization [4 5] is a new idea to realizehigh accuracy positioning in GPS denied environmentsThe basic idea is to use the following information to assistnode localization such as the distance information obtained

from the communication between sensor nodes the relativevelocity information via Doppler shift measurement in thedynamic network In large-scale distributed networks dueto the limited communication capability nodes only caninteract with their neighbors All nodes will form a connectedmultihop network According to the different applicationscenarios coordinates here can be standard coordinatessuch as latitude and longitude or relative coordinate systemOn the one hand arranging a large number of anchornodes is expensive On the other hand in some applicationssuch as battlefield environments anchor nodes only can bedeployed around the network This does not guarantee thatall unknown nodes are adjacent to anchor nodes and obtainenough localization information Consider an application ina two-dimensional network as shown in Figure 1 anchornodes are arranged around the network and unknown nodesare arranged in the network Initially anchor nodes broadcastthe coordinate information Due to the limitation of thecommunication radius only a few unknown nodes canobtain enough localization information to estimate their owncoordinatesTheunknownnodewhich has completed its ownlocation will be an anchor node to assist other unknownnodes to be located Through this kind of information

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 7295702 12 pageshttpsdoiorg10115520187295702

2 Mathematical Problems in Engineering

anchor nodeunknown node

anchor nodeunknown node

positionedunknown node

Figure 1 Cooperative localization

interaction between nodes cooperative localization of thewhole network will be completed

Localization techniques of the distributed sensor networkcan be divided into two categories range-based localiza-tion algorithm and range-free localization algorithm Thispaper only considers range-based localization algorithmThistechnique includes three main categories Trilateration-basedalgorithm [6] Maximum Likelihood-based algorithm [7]and Multidimensional Scaling (MDS)-based algorithm InTrilateration-based algorithm unknown nodes measure thedistances to three neighbor anchor nodes and then use thisinformation to estimate their locations The specific mathe-matical model will be given in the next section When thereare more than three neighbor anchor nodes the unknownnode should choose three to estimate its location MaximumLikelihood-based algorithm is different from Trilateration-based algorithm When the unknown node has more thanthree neighbor anchor nodes it will utilize the informationof all anchor nodes to construct an overdetermined equationand find the least square solution TheMDS algorithm basedon distance measurement is divided into twomain categoriesThe first category is classical MDS [8 9] This algorithmprovides relative coordinate to a seemingly nonconvex local-ization problem using only singular value decompositionWithin the classical MDS framework the complete Euclideandistance matrix is needed but this matrix is often verydifficult to be obtained The second category is to constructa pressure function [10ndash12] and use SMACOF [13] algorithmto minimize the pressure function and get the relative coor-dinate estimate SMACOF algorithm is an iterative solutionwhich will lead to high computational complexity AlthoughTrilateration- based algorithm and Maximum Likelihood-based algorithmneed a certain number of anchor nodes theircomputational complexity is low and they are suitable forcooperative localization of large-scale distributed networks

The main distance measurement techniques of range-based localization algorithm are Received Signal StrengthIndication (RSSI) [14] Time-of-Arrival (TOA) [15] andUltrawideband (UWB) [16] There is also a tradeoff betweendevice cost and range accuracy Using RSSI technique ischeap but the accuracy of measurement may be low usingTOA technique needs to guarantee time synchronization andusing UWB technique can achieve accuracies on the orderof centimeters but at the expense of high device and energycosts

Pa1 a2

a3

d1

d2

d3

Figure 2 The ideal model of Trilateration

It should be noted that even in idealized setups withno obstacles or other external factors relatively small errorfrom noisy sensor measurements can induce much largererrors in node position estimate [17] In Trilateration-basedalgorithms effect of rangemeasurement error on localizationaccuracy is mainly related to the selection of anchor node In[6] an anchor node selection scheme based on the minimumcondition number is proposed to improve the localizationaccuracy However this scheme only considers the influenceof the distribution of three anchor nodes on localization accu-racy Actually the relative position between the unknownnode and the three anchor nodes will also have a great impacton localization accuracy In Trilateration-based algorithmswhen the unknown node completes the estimation of itslocation it will become an anchor node to assist otherunknown nodes to be located In this process the iterativeerror will be produced Accumulation of measurement errorwill increase rapidly if the anchor nodes are not selectedproperly Therefore the selection of anchor nodes will havean important influence on localization accuracy

The main contributions of this paper are as follows(1)The influence of distance measurement error on local-

ization accuracy is discussed from two aspects conditionnumber of triangle formed by the three anchor nodes and therelative position between the unknown node and the threeanchor nodes

(2) Based on error analysis an anchor nodes selectionscheme (EATL) is proposed which can effectively improvethe localization accuracy

The rest of the paper is organized as follows The math-ematical model of Trilateration localization is provided inSection 2 and error analysis is provide in Section 3 Based onthe analysis in Section 3 an anchor nodes selection algorithmis given in Section 4 Simulation analysis is provided inSection 5 while Section 6 concludes the paper

2 Mathematical Model ofTrilateration Localization

We use 119875(119909 119910) to represent the unknown node as shownin Figure 2 three anchor nodes within the communicationradius of 119875 are 1198861(1199091 1199101) 1198862(1199092 1199102) and 1198863(1199093 1199103) respec-tively

Mathematical Problems in Engineering 3

D

Pa1 a2

a3

d1d2

d3

Δd2

d2

Figure 3 Distance measurement error of Trilateration

a

b

c

d

e

f

a1a2

a3

Figure 4 The actual model of Trilateration

The exact Euclidean distances without any noise between119875 and 1198861 119875 and 1198862 119875 and 1198863 are 1198891 1198892 and 1198893 respectivelyThe coordinates of 119875 can be obtained by solving (1)

(119909 minus 1199091)2 + (119910 minus 1199101)2 = 11988912(119909 minus 1199092)2 + (119910 minus 1199102)2 = 11988922(119909 minus 1199093)2 + (119910 minus 1199103)2 = 11988932

(1)

We get (2) by using the third formula minus the firstformula and the third formula minus the second formula in(1)

2 (1199093 minus 1199091) 119909 + 2 (1199103 minus 1199101) 119910 = 11988912 minus 11988932 + 11990932 minus 11990912 + 11991032 minus 119910122 (1199093 minus 1199092) 119909 + 2 (1199103 minus 1199102) 119910 = 11988912 minus 11988932 + 11990932 minus 11990912 + 11991032 minus 11991012

(2)

In practice themeasured distances aremodeled as a noisyversion of the actual node distances For example the actualdistance between anchor node 1198862 and 119875 is 1198892 the measureddistance is 11988921015840 and Δ1198892 denotes the distance measurementerror In this case the three circles intersect to form an area119863 as the shaded area shown in Figure 3 In Figure 4 thethree circles intersect with one another From elementarygeometry we know that the lines 119886119887 119888119889 and 119890119891 will intersectat one point When we solve (2) we get the coordinates of theintersection point actually

Table 1 An example of ill-conditioned equation

(x y) Actual Distance MeasurementDistance

Anchor 1 (815831060) 149220 149215Anchor 2 (836011308) 151560 151500Anchor 3 (869171728) 160724 160761

3 Error Analysis

When the distance measurement error Δ119889 exists the coor-dinates of the unknown node 119875 obtained by solving (2) willalso be inaccurate Even if Δ119889 is small the coordinates of119875 obtained by inappropriate anchor node combination mayalso have great errors See the example in Table 1 (119909 119910) refersto the coordinates of the three anchor nodes the third columnis the actual distance between the unknown node and theanchor node and the fourth column is the measurementdistance with errors

Based on actual distances we can get the exact coordi-nates of 119875 as (790 157) by solving (2) But frommeasurementdistances the solution is (760 402) It can be seen fromTable 1 that the differences between actual distances andmeasurement distances are negligible but the differences ofsolutions especially ordinates are several hundred metersTherefore it is very important to study which factors willaffect the solution of (2) To simplify the analysis we rewrite(2) as follows

119860119883 = 119887 (3)

where 119860 is a coefficient matrix 119883 denotes the coordinates of119875 and 119887 is a column vector We have

119860 = [2 (1199093 minus 1199091) 2 (1199103 minus 1199101)2 (1199093 minus 1199092) 2 (1199103 minus 1199102)] 119883 = [119909119910]

and 119887 = [11988912 minus 11988932 + 11990932 minus 11990912 + 11991032 minus 1199101211988912 minus 11988932 + 11990932 minus 11990912 + 11991032 minus 11991012] (4)

119860 is completely described by the coordinates of the threeanchor nodes and can also be understood as the distributionof the three anchor nodes in the two-dimensional space Theelements in 119887 are completely determined by the coordinates ofthe three anchor nodes and the distances from the unknownnode to the anchor nodes That is the relative position ofthe unknown node and the three anchor nodes will havea decisive influence on 119887 In this paper the error analysiswill be made from the distribution of three anchor nodesand the relative positions of the unknown nodes and thethree anchor nodes In Section 311 we discuss the situationwhen there is only distance measurement error that is weassume that the coordinates of the three anchor nodes 11988611198862 and 1198863 are accurate In Section 312 we will discuss thesituation when both the anchor nodes position error and

4 Mathematical Problems in Engineering

the distance measurement error exist We assume that thedistance measurement error Δ119889 is a random variable whichobeys uniform distribution Δ119889 sim 119880(minus120576 120576)31 The Influence of the Distribution of Anchor Nodes on

Localization Accuracy

311 The Influence of Distance Measurement Error In thispaper we always assume that a matrix norm sdot is 2-normthat is to say sdot 2 = radic120582max(119860119879119860) According to the propertyof matrix norm we have 119860119861 le 119860 sdot 119861

When the distance measurement error exists the erroris mainly reflected on the column vector 119887 in (3) 119887 can berewritten as follows

1198871015840= [[

(1198891 + Δ1198891)2 minus (1198893 + Δ1198893)2 + 11990932 minus 11990912 + 11991032 minus 11991012(1198891 + Δ1198892)2 minus (1198893 + Δ1198893)2 + 11990932 minus 11990912 + 11991032 minus 11991012]] (5)

and thus using 1198871015840 it follows that119860 (119883 + Δ119883) = 119887 + Δ119887 (6)

where Δ119883 is the error of solution and

Δ119887 = 1198871015840 minus 119887= [[

(2Δ1198891 sdot 1198891 minus 2Δ1198893 sdot 1198893) + (Δ11988912 minus Δ11988932)(2Δ1198892 sdot 1198892 minus 2Δ1198893 sdot 1198893) + (Δ11988922 minus Δ11988932)]] (7)

Here we assume that the matrix 119860 is nonsingular and wecan obtain the following

Δ119883 = 119860minus1 (Δ119887) (8)

From (3) and (8) we can get 1119883 le 119860119887 andΔ119883 le 119860minus1 lowast Δ119887 Then

Δ119883119883 le 10038171003817100381710038171003817119860minus110038171003817100381710038171003817 lowast 119860 Δ119887119887 (9)

where Δ119883119883 denotes relative error of solutionΔ119887119887 denotes relative error of column vector 119887 and119860minus1 lowast 119860 denotes the condition number of coefficientmatrix 119860 where

119888119900119899119889 (119860) = 10038171003817100381710038171003817119860minus110038171003817100381710038171003817 lowast 119860 (10)

119860minus1lowast119860 and Δ119887119887will determine the upper boundof Δ119883119883312The Influence ofDistanceMeasurement Error andAnchorNodes Position Error When both the anchor nodes positionerror and the distance measurement error exist the error ismainly reflected in the coefficient matrix 119860 and the columnvector 119887 Equation (6) should be rewritten as follows

(119860 + Δ119860) (119883 + Δ119883) = 119887 + Δ119887 (11)

Here Δ119887 is expressed as follows

Δ119887= [[

(2Δ1198891 sdot 1198891 minus 2Δ1198893 sdot 1198893) + (Δ11988912 minus Δ11988932) + (2Δ1199093 sdot 1199093 minus 2Δ1199091 sdot 1199091) + (Δ11990932 minus Δ11990912) + (2Δ1199103 sdot 1199103 minus 2Δ1199101 sdot 1199101) + (Δ11991032 minus Δ11991012)(2Δ1198892 sdot 1198892 minus 2Δ1198893 sdot 1198893) + (Δ11988922 minus Δ11988932) + (2Δ1199093 sdot 1199093 minus 2Δ1199092 sdot 1199092) + (Δ11990932 minus Δ11990922) + (2Δ1199103 sdot 1199103 minus 2Δ1199102 sdot 1199102) + (Δ11991032 minus Δ11991022)]](12)

We assume that Δ119860 is small and can satisfy 119860minus1 sdotΔ119860 lt 1 If Δ119860 is very large the coordinates error of theanchor node itself is very large Substituting (3) in (11) yields

Δ119883 = 119860minus1 [Δ119887 minus Δ119860 sdot 119883 minus Δ119860 sdot Δ119883] (13)

Computing the norm we have

Δ119883 le 10038171003817100381710038171003817119860minus110038171003817100381710038171003817 (10038171003817100381710038171003817Δ11988710038171003817100381710038171003817 + Δ119860 sdot 119883 + Δ119860 sdot Δ119883) (14)

And then

(1 minus 10038171003817100381710038171003817119860minus110038171003817100381710038171003817 sdot Δ119860) Δ119883le 10038171003817100381710038171003817119860minus110038171003817100381710038171003817 (10038171003817100381710038171003817Δ11988710038171003817100381710038171003817 + Δ119860 sdot 119883) (15)

So we have the following

(1 minus 10038171003817100381710038171003817119860minus110038171003817100381710038171003817 sdot Δ119860) Δ119883119883le 10038171003817100381710038171003817119860minus110038171003817100381710038171003817(119860 sdot

10038171003817100381710038171003817Δ11988710038171003817100381710038171003817119887 + Δ119860) (16)

From inequality (16) we obtain the following

Δ119883119883le 119860 sdot 10038171003817100381710038171003817119860minus1100381710038171003817100381710038171 minus 119860 sdot 1003817100381710038171003817119860minus11003817100381710038171003817 (Δ119860 119860) (

10038171003817100381710038171003817Δ11988710038171003817100381710038171003817119887 + Δ119860119860 )(17)

In (17) Δ119887119887 denotes relative error of column vector119887 and Δ119860119860 denotes relative error of coefficient matrix

Mathematical Problems in Engineering 5

100

90

80

70

60

50

40

30

20

10

01009080706050403020100

a1a2

a3

a3

a3

Figure 5 The distribution of three anchor nodes

119860 When Δ119860 is small inequality (17) is similar to (9) andthe upper bound of Δ119883119883 is determined by the conditionnumber of coefficient matrix 119860 and Δ119887119887313 The Influence of 119888119900119899119889(119860) on Localization AccuracyFrom (9) and (17) the upper bound of Δ119883119883 is depen-dent on 119860minus1 lowast 119860 In (1) we use the third formula minusthe first and the second formula respectively and then get thecoefficient matrix

1198601198861 = [2 (1199091 minus 1199092) 2 (1199101 minus 1199102)2 (1199091 minus 1199093) 2 (1199101 minus 1199103)] (18)

Similarly we can also use the first formula minus the secondand the third formula respectively Then we will get

1198601198862 = [2 (1199092 minus 1199091) 2 (1199102 minus 1199101)2 (1199092 minus 1199093) 2 (1199102 minus 1199103)]and 1198601198863 = [2 (1199093 minus 1199091) 2 (1199103 minus 1199101)2 (1199093 minus 1199092) 2 (1199103 minus 1199102)]

(19)

119888119900119899119889(1198601198861)+119888119900119899119889(1198601198862)+119888119900119899119889(1198601198863) indicates the collineardegree of three anchor nodes From the theory of ill-conditioned matrix the greater the condition number of119860 isthe more sensitive the linear equation with coefficient matrix119860 is to Δ119860 and Δ119887 When three anchor nodes constitute anequilateral triangle the sum of the condition numbers willobtain a minimal value 51963

But in most cases it is difficult to find three anchor nodesthat just form an equilateral triangle In order to illustrate theinfluence of the condition number on localization accuracywe do the experiment as follows As shown in Figure 5 600unknownnodes are randomly deployed in a two-dimensionalarea of size 100lowast100The coordinates of anchor nodes 1198861 and1198862 are (25 25) and (75 25) respectively The x-coordinate of1198863 is 50 and the y-coordinate starts from 27 sampling every05 meters sampling 100 times and observing the influenceof the condition number of triangles formed by the threeanchor nodes on localization accuracy We assume that each

20

18

16

14

12

10

8

6

4

2

0

Δ

5 10 15 20 25 30 35 40 45 50 55 60 6570

Condition number

Figure 6 Δ1015840 for different condition number

unknown node can communicate directly with any anchornode and distance measurement error Δ119889 obeys the uniformdistribution 119880(minus2 2)

The MAE of the location estimates is given by

Δ 119894 = radic(119909119894 minus 119909119894)2 + (119910119894 minus 119910119894)2Δ = sum119873119894=1 Δ 119894119873 (20)

where 119873 is the number of unknown nodes (119909119894 119910119894) arethe exact coordinates of unknown node 119894 and (119909119894 119910119894) are theestimation coordinates Δ 119894 denotes the distance between theestimated position and the real position of unknown node 119894Δ denotes MAE

To give a dimensionless form we define Δ 1198941015840 as followsΔ 1198941015840 = Δ 119894(13)sum31 10038161003816100381610038161003816Δ11988911989411989510038161003816100381610038161003816 Δ1015840 = sum119873119894=1 Δ 1198941015840119873

(21)

where Δ119889119894119895 denotes the distance measurement error betweenunknownnode 119894 and anchor node 119895The relationship betweenΔ1015840 and the condition number is shown in Figure 6

As shown in Figure 6 Δ1015840 has an increased trend withthe increase of condition number but it is not a completelymonotonically increasing relationship This shows that Δ1015840is not completely determined by condition number of thetriangle formed by the anchor nodes When the conditionnumber is less than 18 Δ1015840 is less than 36 In the processof computer simulation we find that the condition numberof the triangle formed by the three anchor nodes has thefollowing function relationship with the degree of innerangles (1205721 1205722 1205723) of the triangle

119888119900119899119889 (1198601198861) + 119888119900119899119889 (1198601198862) + 119888119900119899119889 (1198601198863)= cot(12057212 ) + cot(12057222 ) + cot (12057232 )

(22)

6 Mathematical Problems in Engineering

y

a1(00) a2(20) a2(200)

x

a3(11)

a3(1010)

Figure 7 The condition numbers of two isosceles right-angletriangles

From this equation when the condition number is 18 theminimum interior angle is 13∘

When the shape of a triangle is determined its conditionnumber is also determined As shown in Figure 7 thecondition number of isosceles right-angle triangle formed by1198861 1198862 and 1198863 is 62361 and the condition number of isoscelesright-angle triangle formed by 11988611015840 11988621015840 and 11988631015840 is also 62361

Although condition numbers of the two triangles are thesame their influences on error Δ119883 are different

From

1198601198861 = [2 (1199091 minus 1199092) 2 (1199101 minus 1199102)2 (1199091 minus 1199093) 2 (1199101 minus 1199103)] (23)

we obtain the following

1198601198861minus1= 12 (1199093 minus 1199091) (1199103 minus 1199101) minus 2 (1199093 minus 1199092) (1199103 minus 1199102)sdot [ (1199103 minus 1199102) minus (1199103 minus 1199101)minus (1199093 minus 1199092) (1199093 minus 1199091) ]

(24)

Substituting 1198861(0 0) 1198862(2 0) and 1198863(1 1) in (24) yields1198601198861minus1 = [ 02500 minus0250002500 02500 ] but when we use 11988610158401(0 0) 11988610158402(20 0)and 11988610158403(10 10) it follows that (11986010158401198861 )minus1 = [ 00250 minus0025000250 00250 ] =(110)1198601198861minus1 In the triangle formed by three anchor nodesthe larger the shortest edge the smaller the elements in 119860minus1

Using the data in Table 1 we will get 119860 = [ 106686 133518663182 83877 ]and 119888119900119899119889(119860) = 17096

The large condition number indicates that the matrix isill-conditioned and very sensitive to disturbances Further wecan obtain 119860minus1 = [ 08946 minus14241minus70733 113787 ] The large elements in 119860minus1will lead to a large error of location estimate

32 Influence of the Relative Positions of the Unknown Nodesand Anchor Nodes on Localization Accuracy In this sectionwe only discuss the situation in Section 311 that is thecoordinates of the anchor nodes are precise and only distancemeasurement error exists According to the expression of Δ119887in Section 312 when Δ119860 is small the elements in Δ119887 are

p

a3

a1

a1 a2

a3

a3

Figure 8 Two isosceles right-angle triangles

mainly affected by the relative position of unknown node andthe three anchor nodes This is similar to the case of Δ119887

When estimating the coordinates of 119901 there are twogroups of anchor nodes 1198861 1198862 1198863 and 11988611015840 11988621015840 11988631015840 as shown inFigure 8The triangle 119886110158401198862101584011988631015840 is the expansion of the triangle119886111988621198863 We assume that line 1198861101584011988621015840 is 119873 times as long as line11988611198862

In the case of fixed distance measurement error Δ119889 fortriangle 119886111988621198863 we haveΔ119887 = [[

(2Δ1198891 sdot 1198891 minus 2Δ1198893 sdot 1198893) + (Δ11988912 minus Δ11988932)(2Δ1198892 sdot 1198892 minus 2Δ1198893 sdot 1198893) + (Δ11988922 minus Δ11988932)]]and Δ119883= 119860minus1sdot [[(2Δ1198891 sdot 1198891 minus 2Δ1198893 sdot 1198893) + (Δ11988912 minus Δ11988932)(2Δ1198892 sdot 1198892 minus 2Δ1198893 sdot 1198893) + (Δ11988922 minus Δ11988932)]]

(25)

For triangle 119886110158401198862101584011988631015840 we haveΔ1198871015840= [[

119873 sdot (2Δ1198891 sdot 1198891 minus 2Δ1198893 sdot 1198893) + (Δ11988912 minus Δ11988932)119873 sdot (2Δ1198892 sdot 1198892 minus 2Δ1198893 sdot 1198893) + (Δ11988922 minus Δ11988932)]]and Δ1198831015840= 119860minus1sdot [[[(2Δ1198891 sdot 1198891 minus 2Δ1198893 sdot 1198893) + 1119873 (Δ11988912 minus Δ11988932)(2Δ1198892 sdot 1198892 minus 2Δ1198893 sdot 1198893) + 1119873 (Δ11988922 minus Δ11988932)

]]]

(26)

As Δ119889 sim 119880(minus120576 120576) the expectation and variance of Δ119889are 119864(Δ119889) = 0 and 119863(Δ119889) = 12057623 The expectation of theelements in Δ119883 and Δ1198831015840 is 0 To compare the variance of theelements in Δ119883 and Δ1198831015840 we just need to compare 119863(Δ11988912 minusΔ11988932) and 119863((1119873)(Δ11988912 minus Δ11988932)) As 119863(Δ11988912 minus Δ11988932) =(845)1205764 and 119863((1119873)(Δ11988912 minus Δ11988932)) = (8(45 sdot 1198732))1205764 thevariance of the elements in Δ1198831015840 is smaller than Δ119883 we selecttriangle 119886110158401198862101584011988631015840 to estimate the coordinates of 119901

Mathematical Problems in Engineering 7

In fact when the positions of the three anchor nodes aredetermined (that is matrix 119860minus1 is determined) the elementsin Δ119887 are mainly determined by the relative positions of theunknown nodes and the three anchor nodes In geometry theFermat point of a triangle also called the Torricelli point orFermatndashTorricelli point is a point such that the total distancefrom the three vertices of the triangle to the point is theminimum possible 1198891 1198892 and 1198893 must be satisfied 1198891 + 1198892 +1198893 ge 119889min 119889min is the minimum total distance

Since the value of Δ11988912 minus Δ11988932 is small we assume

Δ1198871198863 = [2Δ1198891 sdot 1198891 minus 2Δ1198893 sdot 11988932Δ1198892 sdot 1198892 minus 2Δ1198893 sdot 1198893] (27)

and in the same way we have

Δ1198871198861 = [2Δ1198892 sdot 1198892 minus 2Δ1198891 sdot 11988912Δ1198893 sdot 1198893 minus 2Δ1198891 sdot 1198891]and Δ1198871198862 = [2Δ1198891 sdot 1198891 minus 2Δ1198892 sdot 11988922Δ1198893 sdot 1198893 minus 2Δ1198892 sdot 1198892]

(28)

From Δ1198871198863 we have Δ1198871198863222 = (Δ1198891 sdot 1198891 minus Δ1198893 sdot 1198893)2 +(Δ1198892 sdot 1198892minusΔ1198893 sdot 1198893)2 In the same way we have Δ1198871198861222 andΔ1198871198862222Then we obtain the following

119864(10038171003817100381710038171003817100381710038171003817Δ119887119886121003817100381710038171003817100381710038171003817100381722) + 119864(10038171003817100381710038171003817100381710038171003817Δ11988711988622

1003817100381710038171003817100381710038171003817100381722) + 119864(10038171003817100381710038171003817100381710038171003817Δ11988711988632

1003817100381710038171003817100381710038171003817100381722)

= 431205762 (11988912 + 11988922 + 11988932)(29)

To get the minimum value of (29) the problem can betransformed into the following optimization problem

min 11988912 + 11988922 + 11988932119904119905 1198891 + 1198892 + 1198893 minus 119889min ge 01198891 ge 01198892 ge 01198893 ge 0(30)

The objective function of (30) is a convex function andthe inequality constraints are all linear functions so the K-Tpoint of (30) must be the global optimal solution The K-Tpoint can be obtained as 1198891 = 1198892 = 1198893 = 119889min3

According to the above analysis the distances betweenthe three anchor nodes and the unknown node should be assimilar as possible

4 Anchor Node Selection Algorithm Based onError Analysis

41 Design Principle The algorithm presented in this paper(EATL) abides by the following three principles

p

Rr

Figure 9 Selection of anchor nodes

(1)Theminimum internal angle of the triangle formed bythe three anchor nodes should be larger than 13∘

(2) The shortest edge of the triangle formed by the threeanchor nodes should be as long as possible

(3)The distances between the three anchor nodes and theunknown node should be as similar as possible

Based on the above principles we can select the anchornodes on the ring centered on the unknown node As shownin Figure 9 119901 is the unknown node nodes marked in red areanchor nodes 119877 is communication radius and 119903 is the innerradius of the ring

We only select anchor nodes on the ring shown inFigure 9 On the one hand it reduces the complexity ofthe algorithm In Trilateration algorithms if unknown nodeselects the optimal combination among all neighbor anchornodes 1198623119873 calculations will be performed where 119873 is theaverage number of neighbor anchor nodes If the unknownnode only selects anchor nodes on the ring then 119873 will bereduced to1198731015840 = 119873sdot(1198772minus1199032)1198772 On the other hand it ensuresthat the distances between the unknown node and the threeanchor nodes are as similar as possible which also satisfiesprinciple (3)

The six anchor nodes marked with black circle areavailable for 119901 in Figure 9 Thus 6 anchor nodes will have11986236 = 20 combinations After narrowing down the selectionrange of anchor nodes we will select 3 among these 6 anchornodes according to principle (1) and principle (2)

For principle (1) the minimum internal angle of thetriangle formed by the three anchor nodes should be largerthan 13∘ This principle is to reduce the collinearity of thethree anchor nodes In order to guarantee principle (2) weset the shortest side length threshold 119905ℎ119903119890119904ℎ119900119897119889 119886 as shownin Table 2 Among all combinations satisfying the thresholdwe select the maximum 119889min(119897) as shown in Table 2

It should be noted that the inner radiuses 119903 and119905ℎ119903119890119904ℎ119900119897119889 119886 are given in Section 5 through simulation exper-iments where 119903 = 06119877 119905ℎ119903119890119904ℎ119900119897119889 119886 = 0811987742 Symbol Description The main parameters and variablesused in the algorithm are shown in Table 2

8 Mathematical Problems in Engineering

Table 2 Explanation of parameters and variables

Notation Description1198991199001198891198901(119894) Unknown node 119894119877 Communication radius119903 ring The ring shown in Figure 91198991199001198891198901(119894)119899119890119894119892ℎ119887119900119903 Neighbor anchor nodes set of unknown node 119894 on 119903 ring recording distance and location information of anchor nodes119873(119894) The number of neighbor anchor nodes of unknown node 1198941198991199001198891198901(119894)1198911198971198861198921 Localization flag of unknown node 119894 The initial value is 0 update to 1 after localization1198991199001198891198901(119894)1198911198971198861198922 Record the time when the unknown node 119894 is located120579min(119897) Minimum internal angle of the 119897th combination of anchor node 119897 = 1 sdot sdot sdot 1198623119873(119894)119889min(119897) The shortest edge of the 119897th combination of anchor node 119897 = 1 sdot sdot sdot 1198623119873(119894)119905ℎ119903119890119904ℎ119900119897119889 119886 The shortest edge threshold119905 Waiting time threshold

Start set r and

k = 1

Update node1(i)neighbor

End

Yes

NoIf N(i) ge 3

combination of anchor nodes

If exist combinations with

to solve the coordinates of node1(i)Update node1(i)flag2 = k and

node1(i)flag1 = 1

Yes

No

threshold_a

k = k + 1 k = k + 1

Calculate ＧＣＨ and dＧＣＨ for every

ＧＣＨ ge13∘ and dＧＣＨ ge tℎresℎold_a

Select the combination with maximum dＧＣＨ

Figure 10 Algorithm flowchart

43 Algorithm Procedure In the beginning initial anchornodes broadcast their location information and unknownnodes collect the information of neighbor anchor nodes andmeasure the distance

The unknown node starts executing the anchor nodesselection algorithm The algorithm flowchart is shown

in Figure 10 After completing localization procedure theunknown node updates 1198991199001198891198901(119894)1198911198971198861198921 = 1 and becomesan anchor node Then it will broadcast its own locationinformation If the unknown node fails to complete the local-ization procedure for example no anchor node informationis collected or the anchor node information does not satisfy

Mathematical Problems in Engineering 9

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

unknown nodeanchor node

(a)

unknown nodeanchor node

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

(b)

Figure 11 Two kinds of anchor nodes arrangement (a) Around the square area (b) In the square area

requirement of the flowchart then it will enter the waitingstate and wait for receiving enough information to completethe localization procedure

In a relatively sparse network some unknown nodes maywait for a long time to find the anchor node informationwhich can satisfy the requirement of the flowchartThereforeaccording to different applications a waiting time threshold 119905maybe setWhen reaching thewaiting time threshold and thelocalization procedure is still not completed the conditions inthe flowchart can be properly relaxed For example the valueof 119903 can be reduced Of course this may reduce localizationaccuracy

5 Performance Evaluation

51 Simulation Setup Our experiments are run on varioustopologies of networks in Matlab R2017a The 200 unknownnodes are placed randomly with a uniform distributionwithin a 1000lowast1000 square area The 119898 anchor nodes areplaced (a) randomly with a uniform distribution around thesquare area as shown in Figure 11(a) or (b) randomly witha uniform distribution within the square area as shown inFigure 11(b) In order to observe the influence of networkaverage connectivity on localization accuracy we changecommunication radius 119877 between 200 and 300 We assumethat Δ119889 sim 119880(minus1119877 1119877) The performance of differentalgorithms is compared using mean absolute error (MAE) ofthe location estimates We also calculate error bar defined bystandard deviation to compare the stability of the algorithmsThe standard deviation is given as follows

120590 = radic 1119873119873sum119894=1

(Δ 119894 minus Δ)2 (31)

The performance of the proposed algorithm (EATL) iscompared with that of the Maximum Likelihood-based (ML)

threshold_a

20

160150

140130

120 150 140 130120 110

100

1816141210

86420

r

MA

E

Figure 12 MAE varies with 119903 and 119905ℎ119903119890119904ℎ119900119897119889 119886localization algorithm and the minimum condition number-based (FMMC) localization algorithm [6] In ML approachthe unknownnode to be located requires aminimumof sevenneighbor anchor nodes When comparing the performanceof the three algorithms we mainly do simulation on thenetwork topology shown in Figure 11(a) For this type ofnetwork topology the anchor nodes are placed around thenetwork the inside unknown nodes require more iterationsto complete location which may result in greater iterationerror In many application scenarios the anchor nodes canonly be randomly placed around the network such as thebattlefield environment It is more practical to simulate thenetwork topology in Figure 11(a)

52 119903 and 119905ℎ119903119890119904ℎ119900119897119889 119886 We set 119898 = 75 119877 = 200 and Δ119889 sim119880(minus2 2) The network topology is shown in Figure 11(a)We change 119903 from 05119877 to 075119877 sampling every 005119877sampling 6 times Similarly 119905ℎ119903119890119904ℎ119900119897119889 119886 is changed from 06119877to 08119877 sampling every 005119877 sampling 5 times Thus Thereare a total of 30 combinations of 119903 and 119905ℎ119903119890119904ℎ119900119897119889 119886 We doexperiment to solve theMAEof every combinationThevalueof MAE varies with 119903 and 119905ℎ119903119890119904ℎ119900119897119889 119886 is shown in Figure 12

10 Mathematical Problems in Engineering

1 2 3 4 5 6Iteration times

0123456789

101112131415161718

EATLMLFMMC

MA

E

Figure 13 MAE for different iteration times

From Figure 12 with the increase of 119903 and 119905ℎ119890119903119890119904ℎ119900119897119889 119886that is the constraints becoming more and more stringentthe MAE is generally on a downward trend For examplewhen 119903 = 05119877 and 119905ℎ119903119890119904ℎ119900119897119889 119886 = 06119877 119872119860119864 = 1947When 119903 = 06119877 and 119905ℎ119903119890119904ℎ119900119897119889 119886 = 08119877 119872119860119864 = 362However as 119903 increases the areawhere unknownnodes selectanchor nodes continuously decreases which lead to a longwaiting time of the unknown nodes to be located The entirenetwork requires more iteration times When 119903 = 07119877 ittakes more than 10 iteration times to complete the entirenetwork localization The increase in the number of iterationsmeans the accumulation of iteration errors From Figure 12it can also be seen that when 119903 increases to 065119877 continuedincrease of 119903 does not significantly reduce the MAE but theentire network localization time increases significantly Afterbalancing the localization time and localization accuracy inthe following simulations if there are no special instructionswe take 119903 = 06119877 and 119905ℎ119903119890119904ℎ119900119897119889 119886 = 0811987753 MAE Performance We set 119898 = 75 and 119877 =200 to observe that MAE varies with iteration times ofthe three algorithms The network topology is shown inFigure 11(a)

As shown in Figure 13 The MAE of all three algorithmstends to increase with the number of iterations which isdue to the accumulation of errors According to the errorbar of each iteration the standard deviation of the proposedalgorithm is smaller than that of ML and FMMC and theproposed algorithm is more stable

Overall ML algorithm requires 4 iteration times tocomplete the localization and the MAE is 507 FMMCalgorithm requires 5 iteration times and the MAE is 412Theproposed algorithm requires 6 iteration times and theMAE is362 Compared with ML and FMMC the MAE of proposedalgorithm decreased by 286 and 121 respectively

54 Impact of Network Connectivity We set 119898 = 75 and thevalues of 119877 are 200 225 250 275 and 300 respectively Thenetwork topology is shown in Figure 11(a) Figure 14(a) shows

that with the increase of the communication radius of nodesthe network connectivity increases FromFigure 14(b) we cansee that under different communication radius theMAE andstandard deviation of the proposed algorithm are lower thanthat of ML algorithm and FMMC algorithm This shows thatthe proposed algorithm has good stability and scalability

55 Impact of Different Number of Anchor Nodes We set119877 = 200 and the values of 119898 are 20 25 30 35 4045 and 50 respectively The network topology is shownin Figure 11(b) From Figure 15 with the increase in thenumber of anchor nodes the MAE and standard deviationare gradually reduced When the number of anchor nodesexceeds 30 the increase in the number of anchor nodes has noobvious effect on improving the localization accuracy of thenetwork When the number of anchor nodes in the networkis less than 20 the process takes too long and the localizationaccuracy is low

6 Conclusion

This paper proposes an anchor selection algorithm based onerror analysis starting from an example of ill-conditionedlinear equation to show that selecting the right anchornodes combination will make a big difference in localizationaccuracy The influence of distance measurement error onlocalization accuracy is discussed from two aspects condi-tion number of triangle formed by the three anchor nodesand the relative position between the unknown node and thethree anchor nodes Then an algorithm of selecting anchornodes on a ring is proposedThe values of 119903 and 119905ℎ119890119903119890119904ℎ119900119897119889 119886are given through simulation experiments Simulation alsoshows that the performance of the proposed algorithm inMAE and standard deviation are better than those of MLalgorithm and FMMC algorithm

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This project is supported by the National Natural ScienceFoundation of China (Grants No 61877067 61572435) Jointfund project the Ministry of Educationmdashthe China Mobile(No MCM20170103) Xirsquoan Science and Technology Inno-vation Project (Grants No 201805029YD7CG13-6) NingboNatural Science Foundation (Grants No 2016A6100352017A610119) The authors would like to thank ProfessorLi Guojun (Basic Courses Department Zhejiang Police

Mathematical Problems in Engineering 11

200 210 220 230 240 250 260 270 280 290 300Communication radius

10

15

20

25

30

35

40

45

50Av

erag

e net

wor

k co

nnec

tivity

(a)

200 225 250 275 300Communication radius

0123456789

101112

EATLMLFMMC

MA

E

(b)

Figure 14 Impact of network connectivity (a) Network connectivity for different communication radius (b) MAE for different networkconnectivity

20 25 30 35 40 45 50The number of anchor node

0123456789

101112

MA

E

Figure 15 Impact of different number of anchor nodes

College) for his valuable comments during the revision of thepaper

References

[1] Z Kashino J Y Kim G Nejat and B Benhabib ldquoSpatiotem-poral adaptive optimization of a static-sensor network via anon-parametric estimation of target location likelihoodrdquo IEEESensors Journal vol 17 no 5 pp 1479ndash1492 2017

[2] F-B Wang L Shi and F-Y Ren ldquoSelf-localization systems andalgorithms for wireless sensor networksrdquo Journal of Software vol 16 no 5 pp 857ndash868 2005

[3] P-H Tseng Z Ding and K-T Feng ldquoCooperative self-navigation in a mixed LOS and NLOS environmentrdquo IEEETransactions on Mobile Computing vol 13 no 2 pp 350ndash3632014

[4] L Dong ldquoCooperative localization and tracking of mobile AdHoc networksrdquo IEEE Transactions on Signal Processing vol 60no 7 pp 3907ndash3913 2012

[5] F Yin C Fritsche D Jin F Gustafsson and A M ZoubirldquoCooperative localization in WSNs using gaussian mixturemodeling Distributed ECM algorithmsrdquo IEEE Transactions onSignal Processing vol 63 no 6 pp 1448ndash1463 2015

[6] X Du L Sun F Xiao and R Wang ldquoLocalization algorithmbased on minimum condition number for wireless sensornetworksrdquo Journal of Electronics (China) vol 30 no 1 pp 25ndash32 2013

[7] I Guvenc C-C Chong and F Watanabe ldquoAnalysis of a linearleast-squares localization technique in LOS andNLOS environ-mentsrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC2007-Spring) pp 1886ndash1890 April 2007

[8] Y Shang W Ruml Y Zhang and M Fromherz ldquoLocalizationfrom connectivity in sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 15 no 11 pp 961ndash9742004

[9] H Jamali-Rad and G Leus ldquoDynamic multidimensional scal-ing for low-complexity mobile network trackingrdquo IEEE Trans-actions on Signal Processing vol 60 no 8 pp 4485ndash4491 2012

[10] N Saeed and H Nam ldquoCluster based multidimensional scalingfor irregular cognitive radio networks localizationrdquo IEEE Trans-actions on Signal Processing vol 64 no 10 pp 2649ndash2659 2016

[11] J A Costa N Patwari and A O Hero III ldquoDistributedweighted-multidimensional scaling for node localization insensor networksrdquo ACM Transactions on Sensor Networks vol2 no 1 pp 39ndash64 2006

[12] S Kumar R Kumar and K Rajawat ldquoCooperative localizationof mobile networks via velocity-assistedmultidimensional scal-ingrdquo IEEE Transactions on Signal Processing vol 64 no 7 pp1744ndash1758 2016

[13] I Borg and P J F Groenen Modern Multidimensional ScalingTheory and Applications Springer New York NY USA 2005

[14] S Tomic M Beko R Dinis and P Montezuma ldquoDistributedalgorithm for target localization in wireless sensor networksusing RSS and AoA measurementsrdquo Pervasive and MobileComputing vol 37 pp 63ndash77 2017

[15] N Patwari A O Hero III M Perkins N S Correal andR J OrsquoDea ldquoRelative location estimation in wireless sensor

12 Mathematical Problems in Engineering

networksrdquo IEEE Transactions on Signal Processing vol 51 no8 pp 2137ndash2148 2003

[16] K Guo Z Qiu W Meng L Xie and R Teo ldquoUltra-widebandbased cooperative relative localization algorithm and experi-ments for multiple unmanned aerial vehicles in GPS deniedenvironmentsrdquo International Journal of Micro Air Vehicles vol9 no 3 pp 169ndash186 2017

[17] A Savvides W L Garber R L Moses and M B SrivastavaldquoAn analysis of error inducing parameters in multihop sensornode localizationrdquo IEEETransactions onMobile Computing vol4 no 6 pp 567ndash577 2005

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