Research ArticleResearch on Algorithm of Three-Dimensional Wireless SensorNetworks Node Localization

Jiang Minlan Luo Jingyuan and Zou Xiaokang

The Department of Electronic Information Technology Zhejiang Normal University Jinhua 321004 China

Correspondence should be addressed to Jiang Minlan xx99zjnucn

Received 29 December 2015 Revised 24 April 2016 Accepted 10 May 2016

Academic Editor Stephane Evoy

Copyright copy 2016 Jiang Minlan 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 three-dimensional wireless sensor networks node localization algorithm based on multidimensional scalinganchor nodes which is used to realize the absolute positioning of unknown nodes by using the distance between the anchor nodesand the nodes The core of the proposed localization algorithm is a kind of repeated optimization method based on anchor nodeswhich is derived from STRESS formula The algorithm employs the Tunneling Method to solve the local minimum problem inrepeated optimization which improves the accuracy of the optimization results The simulation results validate the effectiveness ofthe algorithm Random distribution of three-dimensional wireless sensor network nodes can be accurately positioned The resultssatisfy the high precision and stability requirements in three-dimensional space node location

1 Introduction

Node localization technology is one of the key technolo-gies of wireless sensor network (WSN) Node localizationapproaches in WSN locate unknown nodes through thelocation of anchor nodes by an algorithm or a rule Thetwo-dimensional (2D) WSN node positioning technologyhas been mature but research in three-dimensional (3D)nodes localization has not been rich In practical applicationthe space distribution of wireless sensor is often 3D ratherthan 2D due to the shape of ground and the environmentSo localization in 2D plane is not enough and research onlocalization algorithms in the 3D space is necessary With theincrease of dimension the computational complexity and theerror increase greatly Even though the error of node positionis so little it will be apparent after the 3D calculation Inaddition it is necessary to combine theWSN system positionwith node position to realize accurate positioning of externalnodes Therefore the node localization algorithm requireshigher precision [1ndash3]

In recent years many new ideas and solutions have beenproposed to solve the problem of WSN node localizationin 3D The theory that originated in psychometrics multidi-mensional scaling (MDS) positioning technology proposedby Shang et al provides a new method for node localization

In the existing MDS algorithms MDS-MAP MDS-MAP(P)and MDS-MAP(PR) algorithm are the most influentialMDS-MAP is a centralized location algorithm based onMDS First MDS-MAP uses multidimensional scaling tech-nique to build a local map of a series of neighboring nodesin 3D and to achieve global network positioning throughcoordinate conversion MDS-MAP algorithm makes full useof MDS advantages and avoids the error caused by theshortest path distance in place of the real distance As aresult its positioning accuracy has been improved to a cer-tain extent but its biggest disadvantage requires centralizedcalculation which is not suitable for large-scale uneven dis-tribution network MDS-MAP(P) and MDS-MAP(PR) arethe improvement based onMDS-MAPdistributed algorithmThrough dividing the whole network into many subnetsand using MDS-MAP to compute each subnet to get localcoordinate system MDSMAP(P) and MDS-MAP(PR) thenrealize the entire network position by transform and fusionof local coordinates The complexity of algorithm will greatlyincrease since the transform and fusion It will introduce thetransmission error and lead to the accumulation of errorThe bigger the network is the more apparent the error willbe And the computational complexity will be a geometricgrowth [4 5]

Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 2745109 9 pageshttpdxdoiorg10115520162745109

2 Journal of Sensors

The improved positioning algorithm mentioned in thispaper uses anchor nodes calibration technology of MDSto directly get the absolute coordinates of unknown nodeswhich requires neither integration of local coordinate systemnor conversion from the relative coordinates to the absolutecoordinates As a consequence it avoids the cumulative errorsproduced in this process At the same time taking intoaccount the MDS optimization of local minimum problemthis paper employs Tunneling Method algorithm to solve therepeated optimization algorithm of values in local minimumproblem The simulation result demonstrates that the MDSnode localization algorithm based on Tunneling Method(TMDS) achieves high positioning accuracy and high stabil-ity

The rest of the paper is organized as follows In Sec-tion 2 the node localization algorithm based on MDS isdescribed in detail Section 3 introduces node localizationalgorithm based on MDS Section 4 reports the simulationscenarios and analysis results Finally a conclusion is given inSection 5

2 Node Localization Algorithm Based on MDS

21 NodeCoordinates CalculationBased on theMDS Assumethat WSN in 3D space is made up of 119899 unknown nodes and119898 anchor nodes 119883 = [119909

119894119886]119899times3

represents the unknown nodecoordinate matrix 119862 = [119888

119896119886]119898times3

represents the anchor nodescoordinate matrix 120575

119894119895represents the measuring distance

between unknownnodes 119894 and 119895 120591119894119896represents themeasuring

distance between unknown node 119894 and anchor nodes 119896 Thedistance can be expressed by the minimum hop betweentwo nodes and also can be expressed by the shortest pathdistance between two nodes or any other distance such asthe Euclidean distance To guarantee the accuracy this papermeasures the distance between one-hop neighbor nodes bythe Euclidean method The problem of node localizationis using repeated optimization algorithm to calculate thecoordinates of unknown nodesmatrix119883 under the conditionthat 120575

119894119895and 120591

119894119896distance and the anchor nodes coordinate

matrix 119862 have been known [6 7]Matrix 119883

119899times3represents the coordinates of the 119899 nodes in

3D space For any two nodes 119897 and ℎ their Euclidean distancecan be shown as follows

119889119897ℎ

2(119883) = 119889

119897ℎ

2=

3

sum

119886=1

(119909119897119886minus 119909ℎ119886)2

=

3

sum

119886=1

(119909119897119886

2+ 119909ℎ119886

2minus 2119909119897119886119909ℎ119886)

(1)

For three nodes in 3D space their corresponding matrixis as follows

119883 =[

[

[

11990911

11990912

11990913

11990921

11990922

11990923

11990931

11990932

11990933

]

]

]

(2)

1198632(119883) represents the square of the distance matrix and it can

be represented as follows

1198632(119883) =

[

[

[

[

0 11988912

211988913

2

11988921

20 11988923

2

11988931

211988932

20

]

]

]

]

=

3

sum

119886=1

[

[

[

[

1199091119886

21199091119886

21199091119886

2

1199092119886

21199092119886

21199092119886

2

1199093119886

21199093119886

21199093119886

2

]

]

]

]

+

3

sum

119886=1

[

[

[

[

1199091119886

21199092119886

21199093119886

2

1199091119886

21199092119886

21199093119886

2

1199091119886

21199092119886

21199093119886

2

]

]

]

]

minus 2

3

sum

119886=1

[

[

[

11990911198861199091119886

11990911198861199092119886

11990911198861199093119886

11990921198861199091119886

11990921198861199092119886

11990921198861199093119886

11990931198861199091119886

11990931198861199092119886

11990931198861199093119886

]

]

]

=

[

[

[

[

[

[

[

[

[

[

3

sum

119886=1

1199091119886

2

3

sum

119886=1

1199092119886

2

3

sum

119886=1

1199093119886

2

]

]

]

]

]

]

]

]

]

]

[1 1 1]

+[

[

[

1

1

1

]

]

]

[

3

sum

119886=1

1199091119886

2

3

sum

119886=1

1199092119886

2

3

sum

119886=1

1199093119886

2]

minus 2

3

sum

119886=1

119883119886119883119886

119879= 119862119890119879+ 119890119862119879minus 2

3

sum

119886=1

119883119886119883119886

119879

(3)

where119883119886is the 119886 column of matrix119883119862 is the column vector

which is composed of sum3

119886=1119883119894119886119883119894119886

119879 Vector 119890 is a columnmatrix and its elements are 1

22 The Repeated Optimization Algorithm Based on theAnchor Node Repeated optimization algorithm aims at find-ing coordinate matrix 119883 representing the unknown nodesrsquolocation The algorithm makes nodes distance calculated bycoordinates be closer to the real measuring distance Whenthe difference is smaller coordinate values in a matrix areclose to the real coordinates of nodes Using the repeatedoptimization method find the coordinate matrix119883 The firststep is to build a function about matrix 119883 as the objectivefunctionThen find the minimization inequality of the objec-tive function Finally obtain the repeat optimization formulaWhen the formula reaches its minimum value 119883 is thecoordinate for unknown nodes Regard the difference squaresumbetween computing distance and themeasuring distanceas target function of repeated optimization algorithm inMDStechnology [8ndash10]

Journal of Sensors 3

According to the Euclidean Metric the distance betweenthe unknown nodes 119894 and 119895 can be obtained by

119889119894119895 (

119883) = (

119873

sum

119886=1

(119909119894119886

minus 119909119895119886)

2

)

12

(4)

According to matrix 119883 the distance between the unknownnode 119894 and the anchor node 119896 can be obtained by

119889119894119896 (

119883 119862) = (

119873

sum

119886=1

(119909119894119886

minus 119888119896119886)2)

12

(5)

In order to get the coordinate matrix 119883 we usually useKruskalrsquos STRESS formula to set the objective function asfollows

120590 (119883) =

119899

sum

119894=1

119899

sum

119895=119894+1

120596119894119895[120575119894119895minus 119889119894119895 (

119883)]

2

+

119898

sum

119894=1

119898

sum

119896=1

120592119894119896[120591119894119896minus 119889119894119896 (

119883 119862)]2

= 1205901 (

119883) + 1205902 (

119883)

(6)

where the weights 120596119894119895and 120592

119894119896decide their corresponding

sumrsquos influence on objective function This paper applies thecalculation method of weight proposed by the literature [6]and weights can be calculated by

120596119894119895= exp(minus

120575119894119895

2

1205781198941198952)

120592119894119896

= exp(minus

120591119894119896

2

1205781198941198962)

(7)

Otherwise both of them are zero 120578119894119895in (7) represents the

biggest distance between unknown node 119894 or 119895 and othernodes 120578

119894119896represents the biggest distance between unknown

node 119894 and other nodes or anchor node 119896 For a given matrix119883 the value of corresponding objective function can beobtained according to formula (3) Our goal is to find 119883whichmakes the objective functionminimum Namely min-imize the objective function and find the optimal solution119883

of objective functionThere are many ways to solve the local minimum prob-

lem of repeated optimization method in minimizing theobjective function for example Dimensionality Reductionmultivalue random testing method Distance Smoothingand Tunneling Method Among those methods multivaluerandom testing method may attain good optimal result butit has great randomicity and poor stability DimensionalityReduction is complex and large in the amount of calculationin the 3D space environment Distance Smoothing is simplein calculation and does well in positioning in 2D spaceenvironment But its global search is poor in the 3D spaceenvironment [8] On the contrary the repeated optimizationmethod based on tunneling can almost find the globaloptimal value due to cycle search method that first searches

in the horizontal direction and then searches in the verticaldirection and then searches in the horizontal direction againTherefore Tunneling Method used in this paper solves thelocal minimum problem of repeated optimization algorithmAnd the MDS 3D node localization algorithm based onTunneling Method is proposed

3 MDS Location Algorithm Based onthe Tunneling Method

31 Tunneling Method Algorithm Tunneling Method is anefficient certain global optimization method which was firstproposed byMontalco and Levy It builds tunneling functionin the current local minimum and then gets smaller localminimum through minimizing tunneling function to escapefrom the current local minima and then cyclic calculationuntil the global minimum is found Tunneling functionprovides amethod that uses local optimization to solve globaloptimization problem It only requires its target functionto be continuously differentiable The tunneling functions(auxiliary functions) have the same form and the process ofseeking error 120576 is easy [8]

Tunneling Method consists of a series of circulationsEvery circulation includes two steps local minimizing stepand tunneling step

Step 1 (local minimizing step) Starting from an initialpoint it applies local minimization algorithm such as quasi-Newton method gradient method or the conjugate gradientmethod to obtain the first local minimum value point 119909

1

lowast ofthe objective function 119891(119909) [8]

Step 2 (tunneling step) First define the tunneling functionin 1199091

lowast as follows

119879 (119909 1199091

lowast) =

119891 (119909) minus 119891 (1199091

lowast)

[(119909 minus 1199091lowast)119879(119909 minus 119909

1lowast)]

120582 (8)

where (119909 minus 1199091

lowast)119879(119909 minus 119909

1

lowast) is called the pole of tunneling

function and 120582 is the strength of (119909 minus 1199091

lowast)119879(119909 minus 119909

1

lowast) Then it

seeks a new point such that119879(119909 1199091

lowast) le 0 namely it finds out

1199092

lowast= 1199091

lowast such that 119891(1199092

lowast) le 119891(119909

1

lowast) 1199092

lowast will be the originalpoint of next circulation until it gets a better minimumpoint

The key of tunneling function algorithm is how to findthe point which meets 119879(119909 119909

1

lowast) le 0 Namely we apply the

minimizing tunneling function to find a local minimum 1199092

lowastSeveral discussions on it are as follows

(i) If 1199092

lowast= 1199091

lowast increase 120582 to make 1199091

lowast no longer thelocalminimumpoint of119879(119909 119909

1

lowast)That aim is to avoid

getting 1199091

lowast again by minimizing 119879(119909 1199091

lowast) for falling

into endless loop

4 Journal of Sensors

Table 1 The elements of the tunneling function 1205911(119883) and their purpose

Purpose Element(1) Zero point if STRESS is equal to local minimum STRESS 120591 (119883) = 120590 (119883) minus 120590 (119883

lowast)

(2) Zero points are the lowest tunneling function values 120591 (119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

(3) Avoid a zero point at119883lowast by erecting a pole 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

119875 (119883)

(4) Avoid a zero point at irrelevant transformations of 119883lowast 119875 (119883) =1003817100381710038171003817119863 (119883lowast) minus 119863 (119883)

1003817100381710038171003817

2

119908

(5) Ensure sufficiently strong pole (use pole strength parameter120582 0 lt 120582 lt 1) 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

119875 (119883)

(6) Avoid attraction to the horizon 120591 (119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

(1 +

1

119875 (119883)

)

(7) Extend working range of the pole (use width parameter 120596120596 gt 1) 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

(1 +

120596

119875 (119883)

)

(8) Have multiple poles to avoid different119883119896

1205911(119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

119903

prod

119896=1

(1 +

120596

119875119896(119883)

)

1119903

(ii) If 1199092

lowast= 1199091

lowast and 119891(1199092

lowast) gt 119891(119909

1

lowast) the tunneling

function can be constructed as follows

119879 (119909 1199091

lowast)

=

119891 (119909) minus 119891 (1199091

lowast)

[(119909 minus 1199092lowast)119879(119909 minus 119909

2lowast)]

1205821

[(119909 minus 1199091lowast)119879(119909 minus 119909

1lowast)]

120582

(9)

where 1205821is the strength of (119909 minus 119909

2

lowast)119879(119909 minus 119909

2

lowast)

Selecting the appropriate value of 1205821 the aim is to

make 1199092

lowast no longer the local minimum point of119879(119909 119909

1

lowast) and avoid getting 119909

2

lowast again by minimizing119879(119909 119909

1

lowast)

(iii) If 1199092

lowast= 1199091

lowast and 119891(1199092

lowast) lt 119891(119909

1

lowast) 1199092

lowast is the secondoriginal point and then start next circulation

Among the three situations the former two may appearmany times Every time tunneling function needs to berebuilt tominimize the function until the third situation turnsup and begins the next circulationThis circulation will neverstop until the tunneling function cannot find a smaller localminimum in a certain period of time The last local minimawill be regarded as the global minimum

32 MDS Positioning Algorithm Based on Tunneling MethodIn order to find another configurationwith the same STRESSthe tunneling function must have several characteristicsSome of these characteristics are met by the tunnelingfunction originally defined by Groenen and Heiser [9] Thetunneling function is defined as follows

120591 (119883) = 120590 (119883) minus 120590 (119883lowast) (10)

where 119883lowast is the local minimum point of 120590(119883) The elements

of the tunneling function 1205911(119883) and their purpose are given in

Table 1 The final tunneling function 1205911(119883) can be expressed

as follows

1205911 (

119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582 prod119903

119896=1(120596 + 119875

119896 (119883))1119903

prod119903

119896=1119875119896 (

119883)1119903

(11)

where 119875119896(119883) is

119875119896 (

119883) =1003817100381710038171003817119863119896(119883lowast) minus 119863119896 (

119883)1003817100381710038171003817

2

119908

= sum

119894lt119895

120596119894119895(119889119894119895(119883lowast) minus 119889119894119895 (

119883))

2

(12)

where 119903 represents the number of119883119896rsquos available

1205911(119883) mentioned above meets all conditions of the tun-

neling function The tunneling function is shown as formula(11) which can be treated as the ratio of two functions basedon119883 Assume that 119875(119883) gt 0 120591

1(119883) can be written as

1205911 (

119883) =

119872 (119883)

119875 (119883)

(13)

where119872(119883) and119873(119883) are respectively as follows

119872(119883) = 119873 (119883) (120596 + 119875 (119883)) (14)

119873(119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582 (15)

1205911(119883) can get a certain pole through (14) and (15) Assume

that we can find 119884 which makes formula (16) right

1205911 (

119883) =

119872 (119883)

119875 (119883)

le

119872 (119884)

119875 (119884)

= 1205911 (

119884) (16)

Let us multiply both sides with 119875(119883) Then we can obtain

119872(119883) minus

119872(119884)

119875 (119884)

119875 (119883) le 0 (17)

or

119865 (119902119883) = 119872 (119883) minus 119902119875 (119883) le 0 (18)

We find 119883 that meets 119865(119902119883) lt 0 and then 1205911(119883) lt

1205911(119884) is right As a result the iterative optimization algorithm

will be used to optimize 119865(119902119883) Namely it minimizesoptimization 120591

1(119883)

We carry out iterative minimization operations onradic1205911(119883) since 120591

1(119883) have the same zero and stagnation point

Journal of Sensors 5Ite

ratio

ns

86

42

Pole width2

6

10

Pole strength

200

600

1000

(a)

Itera

tions

86

42

Pole width2

6

10

Pole strength

100

200

(b)

Figure 1 Strength parameters and width parameters configuration effect chart

and radic1205911(119883) is easier to build its optimization framework

For every optimization step and optimization function usingformulas (13) (16) (17) and (18) 119865(119902119883) can be got bystep-by-step operation According to formula (18) optimize119865(119902119883) or optimize

119872(119883) = radic119873 (119883)radic(

119903

prod

119896=1

(119875119896 (

119883) + 120596)1119903

)

minusradic(

119903

prod

119896=1

119875119896 (

119883)1119903

)

(19)

Specific optimization steps are as follows

Step 1 119873(119883) will be optimized by optimizing the root ofpositive function |120590(119883) minus 120590(119883

lowast)| Assume that there is 119883

lowast

which meets 120590(119883) gt 120590(119883lowast) namely

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816= 120590 (119883) minus 120590 (119883

lowast) (20)

We verify the update data for each iteration If the assumptionis wrong then there is 120590(119883) le 120590(119883

lowast) and the tunneling step

will be stopped Finally119883lowast can be found and optimization of

119873(119883) can be accomplished

Step 2 Optimize the output of 119875119896(119883) through 119889

119894119895(119883119896) to be

instead of 120575119894119895in formulas (4) (5) (6) and (7) and then realize

the optimization of119903

prod

119896=1

(119875119896 (

119883) + 120596)1119903

(21)

Step 3 Optimize the output of 119875119896(119883) and minusradic119875

119896(119883) to realize

the optimization of

minusradic(

119903

prod

119896=1

119875119896 (

119883)1119903

) (22)

The optimization step of 119875119896(119883) is the same as Step 2

According to Cauchy-Schwarzrsquos inequality the optimizationfor 119889119894119895(119883) and minus119889

119894119895(119883) is to realize the optimization for

minusradic119875119896(119883) The second optimization methods will be used

to optimize 119889119894119895(119883) [10] Then minus119889

119894119895(119883) will be optimized by

Holder inequality

119865(119902119883) will be optimized by the steps mentioned aboveand 119883 will be got which satisfies 119865(119902119883) lt 0 and makesformula (16) right Such cycle can get global minimum andrealize global optimization

33 Parameters of TunnelingMethod TheTunnelingMethodneeds a pole strength parameter and a pole width param-eter Figure 1 is the three-dimensional effect graph ofthe STRESS function under different strength parame-ters and width parameters The strength parameter 1120582 is115 12 13 110 whose number adds up to 18 Thewidth parameters 120596 are 05 1 2 3 4 5 6 7 8 whose numberadds up to 9 The number of iterations is 1000 We can drawa conclusion from Figure 1 that when strength parameterand width parameter are larger more iterations are neededto get the same result But when the strength parametersand width parameters are too small the result is not whatwe want Heiser carried out a lot of experiments and finallyconcluded that when the strength parameters are 13 andwidth parameter is 1198994 in Tunneling Method the result ofiteration is the best where 119899 is the number of iterationelements [8]

4 Simulation Results and Analysis

The TMDS 3D node localization algorithm simulation isoperated on Matlab 2009(a) (64 bits) to research its per-formance Analyze TMDS from three aspects the nodecommunication radius the number of nodes and the pro-portion of anchor node The simulation environment is

6 Journal of Sensors

0 2 4 6 8 10

0

5

10

0

2

4

6

8

10

y-axisx-axis

z-a

xis

Anchor node Unknown node

Figure 2 The distribution map in 3D for unknown node andanchor

described as follows the size of cube geometric model is10m times 10m times 10m and the nodes are placed in this model(the node communication radius is the same and all thenodes are stationary) The model is shown in Figure 2 Setthe simulation environments as follows when the iterationcalculation rule is the same node communication radius isalso the same Repeated optimization algorithm iterative timeis 60 The iterative computation error threshold is 00001Node localization error threshold 119890119903119903119900119903 119901 is 005mThe errorin distancemeasurement is taken as 15When the unknownnode location error meets 119890119903119903119900119903 le 119890119903119903119900119903 119901 the node will beregarded as the accurate localization of nodes

41 Node Communication Radius Influence on the Perfor-mance of the Algorithm Figure 3 shows the impact of nodecommunication radius on the performance of TMDS algo-rithmThe simulation environment is as follows the numberof unknown nodes 119873 is 50 which randomly distribute in a10m lowast 10m lowast 10m cube modelThe number of anchor nodes119872 is 8 which distribute in the 8 vertex positions of the cubemodel Change the node communication radius to analyze itsimpact on the performance of the algorithm

We can draw a conclusion form Figure 3(a) that theproportion of accurate positioning nodes will increase withthe increase of node communication radius When the nodecommunication radius increases to 8m the number ofaccurate positioning nodes reaches 50 And with the contin-uous increase of the node communication radius accuratepositioning of unknown nodes rate will be maintained at1 We can draw from Figure 3(b) that the average error ofaccurate positioning of node localization gradually decreaseswith the increase of node communication radius Whennode communication radius increases from 4m to 8m theaverage error decreases fast When node communicationradius changes from 8m to 12m the error is stable

When the communication radius is small network con-nectivity is low and the number of unknown nodes that cancommunicate with the node is reduced and a small node

communication radius (119877 = 8m) will get higher accuratepositioning node and low node localization error rate in theTMDS test The positioning result will be stable when theradius is increased

42 Influence of Network Nodes on the Performance of TMDSFigure 4 shows that when the node communication radius119877 is 8 the network node influences the TMDS algorithmAccording to the analysis of 41 when the communicationradius is 8m it is easier to analyze other factors influence ofnodes on the performance of the algorithm So in this sectionthe simulation environment is set as follows The number ofanchor nodes 119872 is 8 which are respectively distributed onthe 8 vertex positions of the cubemodel Nodes have the sameradius (119877 = 8m) of communication Change the proportionof anchor nodes by changing the number of unknown nodesand then analyze the influence of network nodes on theperformance of the algorithm

We can draw a conclusion from the curve of Figure 4that with the number of nodes increasing from 38 to 78 thatis the numbers of unknown nodes increase from 30 to 70accurate positioning of unknownnodes rate has been 1Whenthe network nodes number is 83 that is the unknown nodesnumber is 75 accurate localization rate begins to declineThe results show that when TMDS algorithm works in anenvironment of few anchor nodes and many network nodesthe location accuracy of unknownnodes can bemaintained at1 So when the TMDS algorithmworks in environment of fewanchor nodes the unknown nodes can be located accuratelyand the result is significant

43 Influence of the Proportion of Anchor Nodes on thePerformance of TMDS Figure 5 shows the proportion ofanchor nodes influence on the performance of algorithmThe simulation environment is as follows the number ofunknown nodes 119873 is 100 which are randomly distributed inthe space model Eight anchor nodes are distributed in the 8vertex positions of the cubemodel and others are distributedrandomly in the space model All the nodes have the samecommunication radius (119877 = 8m) Change the number of theanchor nodes to analyze the proportion of anchor impact onthe performance of the algorithm

We can draw form Figure 5(a) that the number ofaccurate positioning nodes will increase with the increaseof the proportion of anchor nodes When the proportion ofanchor nodes is 01 the number of accurate positioning nodesreaches 100 And the proportion of anchor nodes changesfrom 015 to 05 the rate of accurate positioning of unknownnodeswill bemaintained at 100Namelywhen the proportionof anchor nodes is greater than or equal to 015 TMDSalgorithm can locate the unknown nodes accurately

We can draw from Figure 5(b) that the unknown nodelocalization error gradually decreases with the increase of theproportion of anchor nodes When the proportion of anchornodes changes from 01 to 016 the unknown nodes locationerror decreases fast When the proportion of anchor nodeschanges from 016 to 05 the average location error fluctuatesa little But the positioning error is within the threshold rangeThe result of accurate positioning of unknownnodes is stable

Journal of Sensors 7

3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

50

Node communication radius

Num

ber o

f acc

urat

e pos

ition

ing

node

s

(a)

3 4 5 6 7 8 9 10 11 1200

01

02

03

04

05

06

07

08

Node communication radius

Aver

age e

rror

(m)

(b)

Figure 3 Node communication radius influence on the performance of TMDS

35 40 45 50 55 60 65 70 75 80 8520

25

30

35

40

45

50

55

60

65

70

Num

ber o

f acc

urat

e pos

ition

ing

node

s

Numbers of network nodes(a)

35 40 45 50 55 60 65 70 75 80 85000

002

004

006

008

010

012

Aver

age e

rror

(m)

Numbers of network nodes(b)

Figure 4 Influence of network nodes on the performance of TMDS

44 The Comparison between MDS Algorithm and TMDSAlgorithm Tables 2 and 3 represent MDS algorithm andTMDS algorithm for 10 times the continuous operationunder the same simulation environment respectively Thecontent of the table includes the average positioning errorof unknown nodes number of accurate positioning nodesnode location accuracy rate and the program running timeresults data The same simulation environment settings areas follows the number of anchor nodes 119872 is 8 which arerespectively distributed on the 8 vertex positions on a cubemodel all nodes have the same radius of communication(119877 = 8m)

From the accuracy of positioning algorithm TMDSalgorithm can achieve 9 times accurate positioning of all the

unknownnode in 10 times continuous operation whose loca-tion accuracy rate was 1 only 1 time accurately positioningnode number is 49 and its accurate localization rate is 098We can draw from Tables 2 and 3 that TMDS algorithmcan basically realize accurate positioning of the node MDSalgorithm can achieve 4 times the accurate positioning of allthe unknown node in 10 times the continuous operation andhave 2 times the low positioning rate respectively 008 and01

From the running time of the algorithm TMDS andMDS can complete the algorithm in a relatively short periodof time Time of MDS algorithm that completes the basicalgorithm within 13 ssim14 s is relatively stable TMDS takesa long time because of iterative computation of repeated

8 Journal of Sensors

005 010 015 020 025 030 035 040 045 05040

50

60

70

80

90

100N

umbe

r of a

ccur

ate p

ositi

onin

g no

des

Proportion of anchor nodes

(a)

005 010 015 020 025 030 035 040 045 050000

001

002

003

004

005

006

007

008

Aver

age e

rror

(m)

Proportion of anchor nodes

(b)

Figure 5 Influence of the proportion of anchor nodes on the performance of TMDS

Table 2 MDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 02881 00238 00031 00144 00360 00274 00162 02670 00152 00226Number of accurate positioning nodes 4 47 50 50 33 45 50 5 50 48Node location accuracy rate 008 094 1 1 066 09 1 01 1 096Program running time (s) 0142 0140 0137 0136 0131 0132 0137 0133 0137 0138

Table 3 TMDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 00101 00147 00079 00104 00197 00065 00090 00064 00070 00054Number of accurate positioning nodes 50 50 50 50 49 50 50 50 50 50Node location accuracy rate 1 1 1 1 098 1 1 1 1 1Program running time (s) 15754 34418 17085 33918 36783 33358 36259 36219 36137 36363

optimization The selection of the first local minimum valuefor repeating optimization of the algorithm of iterativecalculation is random and the global optimal iterations timeis different The distribution of 50 unknown nodes in modelspace is random So the time to complete the positioning ofall nodes will be different To sum up the Tunneling Methodis applied to solve the problem of local minimum value ofrepeated optimization algorithm and achieves good results inthe positioning of WSN

5 Conclusion

This paper uses the Tunneling Method technology in theWSN 3D node localization and proposes a new 3D wirelesssensor network node positioningmethod based onTunnelingMethod algorithm Innovation points of the TMDS localiza-tion algorithm are as follows

(i) TMDS algorithm is a kind of WSN distributednodes localization algorithm based on anchor node

multidimensional scaling technique The locationalgorithm directly calculates the absolute location ofthe unknown nodes by Euclidean distance betweenanchor nodes andnodes without coordinate transfor-mation and network fusion operation So it avoids thecumulative error

(ii) The core of TMDS is a kind of repeated optimizationmethod based on the new anchor nodes The methodis deduced from STRESS algorithm using TunnelingMethod to solve the local minimum problem ofrepeated optimization method to improve the accu-racy of the optimization results

(iii) The TMDS algorithm is an improved algorithm basedon MDS which makes use of the difference betweenthe nodes The calculation results are stable and thestability of the location effect is also better

The tunneling function algorithm is a certain globaloptimization method It can search the global minimum

Journal of Sensors 9

point in the minimization problem search and avoid fallinginto local minimum to get a smaller minimum point Repeatthis process to obtain the global optimal solution It improvesthe convergence speed of the algorithm and enhances theability of the global optimization The algorithm provides anewmethod in 3DWSN node positioningThe experimentalresult shows that the application of the repeated optimiza-tion algorithm and global optimization theory in 3D nodelocalization algorithm in WSN improves the rate of accuratepositioning of unknown nodes reduces the unknown nodelocation error and ensures that the nodeswhich are randomlydistributed in a three-dimensional wireless sensor networkcan be positioned accurately and stably In addition there areseveral limitations in the proposed method The tunnelingfunction requires its target function to be continuouslydifferentiable and the search time and computation willincrease when WSNs have large scale and high node density

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundations of China (Grant no 51305407)

References

[1] M Demirbas X Lu and P Singla ldquoAn in-network queryingframework for wireless sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 20 no 8 pp 1202ndash12152009

[2] V K Chaurasiya N Jain and G C Nandi ldquoA novel distanceestimation approach for 3D localization in wireless sensornetwork using multi dimensional scalingrdquo Information Fusionvol 15 no 1 pp 5ndash18 2014

[3] J-K Lee Y Kim J-H Lee and S-C Kim ldquoAn efficient three-dimensional localization scheme using trilateration in wirelesssensor networksrdquo IEEE Communications Letters vol 18 no 9pp 1591ndash1594 2014

[4] M Shon M Jo and H Choo ldquoAn interactive cluster-basedMDS localization scheme for multimedia information in wire-less sensor networksrdquo Computer Communications vol 35 no15 pp 1921ndash1929 2012

[5] H-B Chen D-Q Wang F Yuan and R Xu ldquoA MDS-basedlocalization algorithm for underwater wireless sensor networkrdquoinProceedings of theMTSIEEE SanDiegoConference AnOceanin Common (OCEANS rsquo13) San Diego Calif USA September2013

[6] 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

[7] E Kim S Lee C Kim and K Kim ldquoMobile beacon-based3D-localization with multidimensional scaling in large sensornetworksrdquo IEEE Communications Letters vol 14 no 7 pp 647ndash649 2010

[8] P J Groenen W J Heiser and J J Meulman ldquoGlobal opti-mization in least-squares multidimensional scaling by distancesmoothingrdquo Journal of Classification vol 16 no 2 pp 225ndash2541999

[9] P J F Groenen and W J Heiser ldquoThe tunneling method forglobal optimization in multidimensional scalingrdquo Psychome-trika vol 61 no 3 pp 529ndash550 1996

[10] J De Leeuw and W Heiser ldquoMultidimensional scaling withrestrictions on the configurationrdquo in Multivariate Analysis pp501ndash522 North-Holland Publishing 1980

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SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

2 Journal of Sensors

The improved positioning algorithm mentioned in thispaper uses anchor nodes calibration technology of MDSto directly get the absolute coordinates of unknown nodeswhich requires neither integration of local coordinate systemnor conversion from the relative coordinates to the absolutecoordinates As a consequence it avoids the cumulative errorsproduced in this process At the same time taking intoaccount the MDS optimization of local minimum problemthis paper employs Tunneling Method algorithm to solve therepeated optimization algorithm of values in local minimumproblem The simulation result demonstrates that the MDSnode localization algorithm based on Tunneling Method(TMDS) achieves high positioning accuracy and high stabil-ity

The rest of the paper is organized as follows In Sec-tion 2 the node localization algorithm based on MDS isdescribed in detail Section 3 introduces node localizationalgorithm based on MDS Section 4 reports the simulationscenarios and analysis results Finally a conclusion is given inSection 5

2 Node Localization Algorithm Based on MDS

21 NodeCoordinates CalculationBased on theMDS Assumethat WSN in 3D space is made up of 119899 unknown nodes and119898 anchor nodes 119883 = [119909

119894119886]119899times3

represents the unknown nodecoordinate matrix 119862 = [119888

119896119886]119898times3

represents the anchor nodescoordinate matrix 120575

119894119895represents the measuring distance

between unknownnodes 119894 and 119895 120591119894119896represents themeasuring

distance between unknown node 119894 and anchor nodes 119896 Thedistance can be expressed by the minimum hop betweentwo nodes and also can be expressed by the shortest pathdistance between two nodes or any other distance such asthe Euclidean distance To guarantee the accuracy this papermeasures the distance between one-hop neighbor nodes bythe Euclidean method The problem of node localizationis using repeated optimization algorithm to calculate thecoordinates of unknown nodesmatrix119883 under the conditionthat 120575

119894119895and 120591

119894119896distance and the anchor nodes coordinate

matrix 119862 have been known [6 7]Matrix 119883

119899times3represents the coordinates of the 119899 nodes in

3D space For any two nodes 119897 and ℎ their Euclidean distancecan be shown as follows

119889119897ℎ

2(119883) = 119889

119897ℎ

2=

3

sum

119886=1

(119909119897119886minus 119909ℎ119886)2

=

3

sum

119886=1

(119909119897119886

2+ 119909ℎ119886

2minus 2119909119897119886119909ℎ119886)

(1)

For three nodes in 3D space their corresponding matrixis as follows

119883 =[

[

[

11990911

11990912

11990913

11990921

11990922

11990923

11990931

11990932

11990933

]

]

]

(2)

1198632(119883) represents the square of the distance matrix and it can

be represented as follows

1198632(119883) =

[

[

[

[

0 11988912

211988913

2

11988921

20 11988923

2

11988931

211988932

20

]

]

]

]

=

3

sum

119886=1

[

[

[

[

1199091119886

21199091119886

21199091119886

2

1199092119886

21199092119886

21199092119886

2

1199093119886

21199093119886

21199093119886

2

]

]

]

]

+

3

sum

119886=1

[

[

[

[

1199091119886

21199092119886

21199093119886

2

1199091119886

21199092119886

21199093119886

2

1199091119886

21199092119886

21199093119886

2

]

]

]

]

minus 2

3

sum

119886=1

[

[

[

11990911198861199091119886

11990911198861199092119886

11990911198861199093119886

11990921198861199091119886

11990921198861199092119886

11990921198861199093119886

11990931198861199091119886

11990931198861199092119886

11990931198861199093119886

]

]

]

=

[

[

[

[

[

[

[

[

[

[

3

sum

119886=1

1199091119886

2

3

sum

119886=1

1199092119886

2

3

sum

119886=1

1199093119886

2

]

]

]

]

]

]

]

]

]

]

[1 1 1]

+[

[

[

1

1

1

]

]

]

[

3

sum

119886=1

1199091119886

2

3

sum

119886=1

1199092119886

2

3

sum

119886=1

1199093119886

2]

minus 2

3

sum

119886=1

119883119886119883119886

119879= 119862119890119879+ 119890119862119879minus 2

3

sum

119886=1

119883119886119883119886

119879

(3)

where119883119886is the 119886 column of matrix119883119862 is the column vector

which is composed of sum3

119886=1119883119894119886119883119894119886

119879 Vector 119890 is a columnmatrix and its elements are 1

22 The Repeated Optimization Algorithm Based on theAnchor Node Repeated optimization algorithm aims at find-ing coordinate matrix 119883 representing the unknown nodesrsquolocation The algorithm makes nodes distance calculated bycoordinates be closer to the real measuring distance Whenthe difference is smaller coordinate values in a matrix areclose to the real coordinates of nodes Using the repeatedoptimization method find the coordinate matrix119883 The firststep is to build a function about matrix 119883 as the objectivefunctionThen find the minimization inequality of the objec-tive function Finally obtain the repeat optimization formulaWhen the formula reaches its minimum value 119883 is thecoordinate for unknown nodes Regard the difference squaresumbetween computing distance and themeasuring distanceas target function of repeated optimization algorithm inMDStechnology [8ndash10]

Journal of Sensors 3

According to the Euclidean Metric the distance betweenthe unknown nodes 119894 and 119895 can be obtained by

119889119894119895 (

119883) = (

119873

sum

119886=1

(119909119894119886

minus 119909119895119886)

2

)

12

(4)

According to matrix 119883 the distance between the unknownnode 119894 and the anchor node 119896 can be obtained by

119889119894119896 (

119883 119862) = (

119873

sum

119886=1

(119909119894119886

minus 119888119896119886)2)

12

(5)

In order to get the coordinate matrix 119883 we usually useKruskalrsquos STRESS formula to set the objective function asfollows

120590 (119883) =

119899

sum

119894=1

119899

sum

119895=119894+1

120596119894119895[120575119894119895minus 119889119894119895 (

119883)]

2

+

119898

sum

119894=1

119898

sum

119896=1

120592119894119896[120591119894119896minus 119889119894119896 (

119883 119862)]2

= 1205901 (

119883) + 1205902 (

119883)

(6)

where the weights 120596119894119895and 120592

119894119896decide their corresponding

sumrsquos influence on objective function This paper applies thecalculation method of weight proposed by the literature [6]and weights can be calculated by

120596119894119895= exp(minus

120575119894119895

2

1205781198941198952)

120592119894119896

= exp(minus

120591119894119896

2

1205781198941198962)

(7)

Otherwise both of them are zero 120578119894119895in (7) represents the

biggest distance between unknown node 119894 or 119895 and othernodes 120578

119894119896represents the biggest distance between unknown

node 119894 and other nodes or anchor node 119896 For a given matrix119883 the value of corresponding objective function can beobtained according to formula (3) Our goal is to find 119883whichmakes the objective functionminimum Namely min-imize the objective function and find the optimal solution119883

of objective functionThere are many ways to solve the local minimum prob-

lem of repeated optimization method in minimizing theobjective function for example Dimensionality Reductionmultivalue random testing method Distance Smoothingand Tunneling Method Among those methods multivaluerandom testing method may attain good optimal result butit has great randomicity and poor stability DimensionalityReduction is complex and large in the amount of calculationin the 3D space environment Distance Smoothing is simplein calculation and does well in positioning in 2D spaceenvironment But its global search is poor in the 3D spaceenvironment [8] On the contrary the repeated optimizationmethod based on tunneling can almost find the globaloptimal value due to cycle search method that first searches

in the horizontal direction and then searches in the verticaldirection and then searches in the horizontal direction againTherefore Tunneling Method used in this paper solves thelocal minimum problem of repeated optimization algorithmAnd the MDS 3D node localization algorithm based onTunneling Method is proposed

3 MDS Location Algorithm Based onthe Tunneling Method

31 Tunneling Method Algorithm Tunneling Method is anefficient certain global optimization method which was firstproposed byMontalco and Levy It builds tunneling functionin the current local minimum and then gets smaller localminimum through minimizing tunneling function to escapefrom the current local minima and then cyclic calculationuntil the global minimum is found Tunneling functionprovides amethod that uses local optimization to solve globaloptimization problem It only requires its target functionto be continuously differentiable The tunneling functions(auxiliary functions) have the same form and the process ofseeking error 120576 is easy [8]

Tunneling Method consists of a series of circulationsEvery circulation includes two steps local minimizing stepand tunneling step

Step 1 (local minimizing step) Starting from an initialpoint it applies local minimization algorithm such as quasi-Newton method gradient method or the conjugate gradientmethod to obtain the first local minimum value point 119909

1

lowast ofthe objective function 119891(119909) [8]

Step 2 (tunneling step) First define the tunneling functionin 1199091

lowast as follows

119879 (119909 1199091

lowast) =

119891 (119909) minus 119891 (1199091

lowast)

[(119909 minus 1199091lowast)119879(119909 minus 119909

1lowast)]

120582 (8)

where (119909 minus 1199091

lowast)119879(119909 minus 119909

1

lowast) is called the pole of tunneling

function and 120582 is the strength of (119909 minus 1199091

lowast)119879(119909 minus 119909

1

lowast) Then it

seeks a new point such that119879(119909 1199091

lowast) le 0 namely it finds out

1199092

lowast= 1199091

lowast such that 119891(1199092

lowast) le 119891(119909

1

lowast) 1199092

lowast will be the originalpoint of next circulation until it gets a better minimumpoint

The key of tunneling function algorithm is how to findthe point which meets 119879(119909 119909

1

lowast) le 0 Namely we apply the

minimizing tunneling function to find a local minimum 1199092

lowastSeveral discussions on it are as follows

(i) If 1199092

lowast= 1199091

lowast increase 120582 to make 1199091

lowast no longer thelocalminimumpoint of119879(119909 119909

1

lowast)That aim is to avoid

getting 1199091

lowast again by minimizing 119879(119909 1199091

lowast) for falling

into endless loop

4 Journal of Sensors

Table 1 The elements of the tunneling function 1205911(119883) and their purpose

Purpose Element(1) Zero point if STRESS is equal to local minimum STRESS 120591 (119883) = 120590 (119883) minus 120590 (119883

lowast)

(2) Zero points are the lowest tunneling function values 120591 (119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

(3) Avoid a zero point at119883lowast by erecting a pole 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

119875 (119883)

(4) Avoid a zero point at irrelevant transformations of 119883lowast 119875 (119883) =1003817100381710038171003817119863 (119883lowast) minus 119863 (119883)

1003817100381710038171003817

2

119908

(5) Ensure sufficiently strong pole (use pole strength parameter120582 0 lt 120582 lt 1) 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

119875 (119883)

(6) Avoid attraction to the horizon 120591 (119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

(1 +

1

119875 (119883)

)

(7) Extend working range of the pole (use width parameter 120596120596 gt 1) 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

(1 +

120596

119875 (119883)

)

(8) Have multiple poles to avoid different119883119896

1205911(119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

119903

prod

119896=1

(1 +

120596

119875119896(119883)

)

1119903

(ii) If 1199092

lowast= 1199091

lowast and 119891(1199092

lowast) gt 119891(119909

1

lowast) the tunneling

function can be constructed as follows

119879 (119909 1199091

lowast)

=

119891 (119909) minus 119891 (1199091

lowast)

[(119909 minus 1199092lowast)119879(119909 minus 119909

2lowast)]

1205821

[(119909 minus 1199091lowast)119879(119909 minus 119909

1lowast)]

120582

(9)

where 1205821is the strength of (119909 minus 119909

2

lowast)119879(119909 minus 119909

2

lowast)

Selecting the appropriate value of 1205821 the aim is to

make 1199092

lowast no longer the local minimum point of119879(119909 119909

1

lowast) and avoid getting 119909

2

lowast again by minimizing119879(119909 119909

1

lowast)

(iii) If 1199092

lowast= 1199091

lowast and 119891(1199092

lowast) lt 119891(119909

1

lowast) 1199092

lowast is the secondoriginal point and then start next circulation

Among the three situations the former two may appearmany times Every time tunneling function needs to berebuilt tominimize the function until the third situation turnsup and begins the next circulationThis circulation will neverstop until the tunneling function cannot find a smaller localminimum in a certain period of time The last local minimawill be regarded as the global minimum

32 MDS Positioning Algorithm Based on Tunneling MethodIn order to find another configurationwith the same STRESSthe tunneling function must have several characteristicsSome of these characteristics are met by the tunnelingfunction originally defined by Groenen and Heiser [9] Thetunneling function is defined as follows

120591 (119883) = 120590 (119883) minus 120590 (119883lowast) (10)

where 119883lowast is the local minimum point of 120590(119883) The elements

of the tunneling function 1205911(119883) and their purpose are given in

Table 1 The final tunneling function 1205911(119883) can be expressed

as follows

1205911 (

119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582 prod119903

119896=1(120596 + 119875

119896 (119883))1119903

prod119903

119896=1119875119896 (

119883)1119903

(11)

where 119875119896(119883) is

119875119896 (

119883) =1003817100381710038171003817119863119896(119883lowast) minus 119863119896 (

119883)1003817100381710038171003817

2

119908

= sum

119894lt119895

120596119894119895(119889119894119895(119883lowast) minus 119889119894119895 (

119883))

2

(12)

where 119903 represents the number of119883119896rsquos available

1205911(119883) mentioned above meets all conditions of the tun-

neling function The tunneling function is shown as formula(11) which can be treated as the ratio of two functions basedon119883 Assume that 119875(119883) gt 0 120591

1(119883) can be written as

1205911 (

119883) =

119872 (119883)

119875 (119883)

(13)

where119872(119883) and119873(119883) are respectively as follows

119872(119883) = 119873 (119883) (120596 + 119875 (119883)) (14)

119873(119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582 (15)

1205911(119883) can get a certain pole through (14) and (15) Assume

that we can find 119884 which makes formula (16) right

1205911 (

119883) =

119872 (119883)

119875 (119883)

le

119872 (119884)

119875 (119884)

= 1205911 (

119884) (16)

Let us multiply both sides with 119875(119883) Then we can obtain

119872(119883) minus

119872(119884)

119875 (119884)

119875 (119883) le 0 (17)

or

119865 (119902119883) = 119872 (119883) minus 119902119875 (119883) le 0 (18)

We find 119883 that meets 119865(119902119883) lt 0 and then 1205911(119883) lt

1205911(119884) is right As a result the iterative optimization algorithm

will be used to optimize 119865(119902119883) Namely it minimizesoptimization 120591

1(119883)

We carry out iterative minimization operations onradic1205911(119883) since 120591

1(119883) have the same zero and stagnation point

Journal of Sensors 5Ite

ratio

ns

86

42

Pole width2

6

10

Pole strength

200

600

1000

(a)

Itera

tions

86

42

Pole width2

6

10

Pole strength

100

200

(b)

Figure 1 Strength parameters and width parameters configuration effect chart

and radic1205911(119883) is easier to build its optimization framework

For every optimization step and optimization function usingformulas (13) (16) (17) and (18) 119865(119902119883) can be got bystep-by-step operation According to formula (18) optimize119865(119902119883) or optimize

119872(119883) = radic119873 (119883)radic(

119903

prod

119896=1

(119875119896 (

119883) + 120596)1119903

)

minusradic(

119903

prod

119896=1

119875119896 (

119883)1119903

)

(19)

Specific optimization steps are as follows

Step 1 119873(119883) will be optimized by optimizing the root ofpositive function |120590(119883) minus 120590(119883

lowast)| Assume that there is 119883

lowast

which meets 120590(119883) gt 120590(119883lowast) namely

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816= 120590 (119883) minus 120590 (119883

lowast) (20)

We verify the update data for each iteration If the assumptionis wrong then there is 120590(119883) le 120590(119883

lowast) and the tunneling step

will be stopped Finally119883lowast can be found and optimization of

119873(119883) can be accomplished

Step 2 Optimize the output of 119875119896(119883) through 119889

119894119895(119883119896) to be

instead of 120575119894119895in formulas (4) (5) (6) and (7) and then realize

the optimization of119903

prod

119896=1

(119875119896 (

119883) + 120596)1119903

(21)

Step 3 Optimize the output of 119875119896(119883) and minusradic119875

119896(119883) to realize

the optimization of

minusradic(

119903

prod

119896=1

119875119896 (

119883)1119903

) (22)

The optimization step of 119875119896(119883) is the same as Step 2

According to Cauchy-Schwarzrsquos inequality the optimizationfor 119889119894119895(119883) and minus119889

119894119895(119883) is to realize the optimization for

minusradic119875119896(119883) The second optimization methods will be used

to optimize 119889119894119895(119883) [10] Then minus119889

119894119895(119883) will be optimized by

Holder inequality

119865(119902119883) will be optimized by the steps mentioned aboveand 119883 will be got which satisfies 119865(119902119883) lt 0 and makesformula (16) right Such cycle can get global minimum andrealize global optimization

33 Parameters of TunnelingMethod TheTunnelingMethodneeds a pole strength parameter and a pole width param-eter Figure 1 is the three-dimensional effect graph ofthe STRESS function under different strength parame-ters and width parameters The strength parameter 1120582 is115 12 13 110 whose number adds up to 18 Thewidth parameters 120596 are 05 1 2 3 4 5 6 7 8 whose numberadds up to 9 The number of iterations is 1000 We can drawa conclusion from Figure 1 that when strength parameterand width parameter are larger more iterations are neededto get the same result But when the strength parametersand width parameters are too small the result is not whatwe want Heiser carried out a lot of experiments and finallyconcluded that when the strength parameters are 13 andwidth parameter is 1198994 in Tunneling Method the result ofiteration is the best where 119899 is the number of iterationelements [8]

4 Simulation Results and Analysis

The TMDS 3D node localization algorithm simulation isoperated on Matlab 2009(a) (64 bits) to research its per-formance Analyze TMDS from three aspects the nodecommunication radius the number of nodes and the pro-portion of anchor node The simulation environment is

6 Journal of Sensors

0 2 4 6 8 10

0

5

10

0

2

4

6

8

10

y-axisx-axis

z-a

xis

Anchor node Unknown node

Figure 2 The distribution map in 3D for unknown node andanchor

described as follows the size of cube geometric model is10m times 10m times 10m and the nodes are placed in this model(the node communication radius is the same and all thenodes are stationary) The model is shown in Figure 2 Setthe simulation environments as follows when the iterationcalculation rule is the same node communication radius isalso the same Repeated optimization algorithm iterative timeis 60 The iterative computation error threshold is 00001Node localization error threshold 119890119903119903119900119903 119901 is 005mThe errorin distancemeasurement is taken as 15When the unknownnode location error meets 119890119903119903119900119903 le 119890119903119903119900119903 119901 the node will beregarded as the accurate localization of nodes

41 Node Communication Radius Influence on the Perfor-mance of the Algorithm Figure 3 shows the impact of nodecommunication radius on the performance of TMDS algo-rithmThe simulation environment is as follows the numberof unknown nodes 119873 is 50 which randomly distribute in a10m lowast 10m lowast 10m cube modelThe number of anchor nodes119872 is 8 which distribute in the 8 vertex positions of the cubemodel Change the node communication radius to analyze itsimpact on the performance of the algorithm

We can draw a conclusion form Figure 3(a) that theproportion of accurate positioning nodes will increase withthe increase of node communication radius When the nodecommunication radius increases to 8m the number ofaccurate positioning nodes reaches 50 And with the contin-uous increase of the node communication radius accuratepositioning of unknown nodes rate will be maintained at1 We can draw from Figure 3(b) that the average error ofaccurate positioning of node localization gradually decreaseswith the increase of node communication radius Whennode communication radius increases from 4m to 8m theaverage error decreases fast When node communicationradius changes from 8m to 12m the error is stable

When the communication radius is small network con-nectivity is low and the number of unknown nodes that cancommunicate with the node is reduced and a small node

communication radius (119877 = 8m) will get higher accuratepositioning node and low node localization error rate in theTMDS test The positioning result will be stable when theradius is increased

42 Influence of Network Nodes on the Performance of TMDSFigure 4 shows that when the node communication radius119877 is 8 the network node influences the TMDS algorithmAccording to the analysis of 41 when the communicationradius is 8m it is easier to analyze other factors influence ofnodes on the performance of the algorithm So in this sectionthe simulation environment is set as follows The number ofanchor nodes 119872 is 8 which are respectively distributed onthe 8 vertex positions of the cubemodel Nodes have the sameradius (119877 = 8m) of communication Change the proportionof anchor nodes by changing the number of unknown nodesand then analyze the influence of network nodes on theperformance of the algorithm

We can draw a conclusion from the curve of Figure 4that with the number of nodes increasing from 38 to 78 thatis the numbers of unknown nodes increase from 30 to 70accurate positioning of unknownnodes rate has been 1Whenthe network nodes number is 83 that is the unknown nodesnumber is 75 accurate localization rate begins to declineThe results show that when TMDS algorithm works in anenvironment of few anchor nodes and many network nodesthe location accuracy of unknownnodes can bemaintained at1 So when the TMDS algorithmworks in environment of fewanchor nodes the unknown nodes can be located accuratelyand the result is significant

43 Influence of the Proportion of Anchor Nodes on thePerformance of TMDS Figure 5 shows the proportion ofanchor nodes influence on the performance of algorithmThe simulation environment is as follows the number ofunknown nodes 119873 is 100 which are randomly distributed inthe space model Eight anchor nodes are distributed in the 8vertex positions of the cubemodel and others are distributedrandomly in the space model All the nodes have the samecommunication radius (119877 = 8m) Change the number of theanchor nodes to analyze the proportion of anchor impact onthe performance of the algorithm

We can draw form Figure 5(a) that the number ofaccurate positioning nodes will increase with the increaseof the proportion of anchor nodes When the proportion ofanchor nodes is 01 the number of accurate positioning nodesreaches 100 And the proportion of anchor nodes changesfrom 015 to 05 the rate of accurate positioning of unknownnodeswill bemaintained at 100Namelywhen the proportionof anchor nodes is greater than or equal to 015 TMDSalgorithm can locate the unknown nodes accurately

We can draw from Figure 5(b) that the unknown nodelocalization error gradually decreases with the increase of theproportion of anchor nodes When the proportion of anchornodes changes from 01 to 016 the unknown nodes locationerror decreases fast When the proportion of anchor nodeschanges from 016 to 05 the average location error fluctuatesa little But the positioning error is within the threshold rangeThe result of accurate positioning of unknownnodes is stable

Journal of Sensors 7

3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

50

Node communication radius

Num

ber o

f acc

urat

e pos

ition

ing

node

s

(a)

3 4 5 6 7 8 9 10 11 1200

01

02

03

04

05

06

07

08

Node communication radius

Aver

age e

rror

(m)

(b)

Figure 3 Node communication radius influence on the performance of TMDS

35 40 45 50 55 60 65 70 75 80 8520

25

30

35

40

45

50

55

60

65

70

Num

ber o

f acc

urat

e pos

ition

ing

node

s

Numbers of network nodes(a)

35 40 45 50 55 60 65 70 75 80 85000

002

004

006

008

010

012

Aver

age e

rror

(m)

Numbers of network nodes(b)

Figure 4 Influence of network nodes on the performance of TMDS

44 The Comparison between MDS Algorithm and TMDSAlgorithm Tables 2 and 3 represent MDS algorithm andTMDS algorithm for 10 times the continuous operationunder the same simulation environment respectively Thecontent of the table includes the average positioning errorof unknown nodes number of accurate positioning nodesnode location accuracy rate and the program running timeresults data The same simulation environment settings areas follows the number of anchor nodes 119872 is 8 which arerespectively distributed on the 8 vertex positions on a cubemodel all nodes have the same radius of communication(119877 = 8m)

From the accuracy of positioning algorithm TMDSalgorithm can achieve 9 times accurate positioning of all the

unknownnode in 10 times continuous operation whose loca-tion accuracy rate was 1 only 1 time accurately positioningnode number is 49 and its accurate localization rate is 098We can draw from Tables 2 and 3 that TMDS algorithmcan basically realize accurate positioning of the node MDSalgorithm can achieve 4 times the accurate positioning of allthe unknown node in 10 times the continuous operation andhave 2 times the low positioning rate respectively 008 and01

From the running time of the algorithm TMDS andMDS can complete the algorithm in a relatively short periodof time Time of MDS algorithm that completes the basicalgorithm within 13 ssim14 s is relatively stable TMDS takesa long time because of iterative computation of repeated

8 Journal of Sensors

005 010 015 020 025 030 035 040 045 05040

50

60

70

80

90

100N

umbe

r of a

ccur

ate p

ositi

onin

g no

des

Proportion of anchor nodes

(a)

005 010 015 020 025 030 035 040 045 050000

001

002

003

004

005

006

007

008

Aver

age e

rror

(m)

Proportion of anchor nodes

(b)

Figure 5 Influence of the proportion of anchor nodes on the performance of TMDS

Table 2 MDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 02881 00238 00031 00144 00360 00274 00162 02670 00152 00226Number of accurate positioning nodes 4 47 50 50 33 45 50 5 50 48Node location accuracy rate 008 094 1 1 066 09 1 01 1 096Program running time (s) 0142 0140 0137 0136 0131 0132 0137 0133 0137 0138

Table 3 TMDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 00101 00147 00079 00104 00197 00065 00090 00064 00070 00054Number of accurate positioning nodes 50 50 50 50 49 50 50 50 50 50Node location accuracy rate 1 1 1 1 098 1 1 1 1 1Program running time (s) 15754 34418 17085 33918 36783 33358 36259 36219 36137 36363

optimization The selection of the first local minimum valuefor repeating optimization of the algorithm of iterativecalculation is random and the global optimal iterations timeis different The distribution of 50 unknown nodes in modelspace is random So the time to complete the positioning ofall nodes will be different To sum up the Tunneling Methodis applied to solve the problem of local minimum value ofrepeated optimization algorithm and achieves good results inthe positioning of WSN

5 Conclusion

This paper uses the Tunneling Method technology in theWSN 3D node localization and proposes a new 3D wirelesssensor network node positioningmethod based onTunnelingMethod algorithm Innovation points of the TMDS localiza-tion algorithm are as follows

(i) TMDS algorithm is a kind of WSN distributednodes localization algorithm based on anchor node

multidimensional scaling technique The locationalgorithm directly calculates the absolute location ofthe unknown nodes by Euclidean distance betweenanchor nodes andnodes without coordinate transfor-mation and network fusion operation So it avoids thecumulative error

(ii) The core of TMDS is a kind of repeated optimizationmethod based on the new anchor nodes The methodis deduced from STRESS algorithm using TunnelingMethod to solve the local minimum problem ofrepeated optimization method to improve the accu-racy of the optimization results

(iii) The TMDS algorithm is an improved algorithm basedon MDS which makes use of the difference betweenthe nodes The calculation results are stable and thestability of the location effect is also better

The tunneling function algorithm is a certain globaloptimization method It can search the global minimum

Journal of Sensors 9

point in the minimization problem search and avoid fallinginto local minimum to get a smaller minimum point Repeatthis process to obtain the global optimal solution It improvesthe convergence speed of the algorithm and enhances theability of the global optimization The algorithm provides anewmethod in 3DWSN node positioningThe experimentalresult shows that the application of the repeated optimiza-tion algorithm and global optimization theory in 3D nodelocalization algorithm in WSN improves the rate of accuratepositioning of unknown nodes reduces the unknown nodelocation error and ensures that the nodeswhich are randomlydistributed in a three-dimensional wireless sensor networkcan be positioned accurately and stably In addition there areseveral limitations in the proposed method The tunnelingfunction requires its target function to be continuouslydifferentiable and the search time and computation willincrease when WSNs have large scale and high node density

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundations of China (Grant no 51305407)

References

[1] M Demirbas X Lu and P Singla ldquoAn in-network queryingframework for wireless sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 20 no 8 pp 1202ndash12152009

[2] V K Chaurasiya N Jain and G C Nandi ldquoA novel distanceestimation approach for 3D localization in wireless sensornetwork using multi dimensional scalingrdquo Information Fusionvol 15 no 1 pp 5ndash18 2014

[3] J-K Lee Y Kim J-H Lee and S-C Kim ldquoAn efficient three-dimensional localization scheme using trilateration in wirelesssensor networksrdquo IEEE Communications Letters vol 18 no 9pp 1591ndash1594 2014

[4] M Shon M Jo and H Choo ldquoAn interactive cluster-basedMDS localization scheme for multimedia information in wire-less sensor networksrdquo Computer Communications vol 35 no15 pp 1921ndash1929 2012

[5] H-B Chen D-Q Wang F Yuan and R Xu ldquoA MDS-basedlocalization algorithm for underwater wireless sensor networkrdquoinProceedings of theMTSIEEE SanDiegoConference AnOceanin Common (OCEANS rsquo13) San Diego Calif USA September2013

[6] 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

[7] E Kim S Lee C Kim and K Kim ldquoMobile beacon-based3D-localization with multidimensional scaling in large sensornetworksrdquo IEEE Communications Letters vol 14 no 7 pp 647ndash649 2010

[8] P J Groenen W J Heiser and J J Meulman ldquoGlobal opti-mization in least-squares multidimensional scaling by distancesmoothingrdquo Journal of Classification vol 16 no 2 pp 225ndash2541999

[9] P J F Groenen and W J Heiser ldquoThe tunneling method forglobal optimization in multidimensional scalingrdquo Psychome-trika vol 61 no 3 pp 529ndash550 1996

[10] J De Leeuw and W Heiser ldquoMultidimensional scaling withrestrictions on the configurationrdquo in Multivariate Analysis pp501ndash522 North-Holland Publishing 1980

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

Journal of Sensors 3

According to the Euclidean Metric the distance betweenthe unknown nodes 119894 and 119895 can be obtained by

119889119894119895 (

119883) = (

119873

sum

119886=1

(119909119894119886

minus 119909119895119886)

2

)

12

(4)

According to matrix 119883 the distance between the unknownnode 119894 and the anchor node 119896 can be obtained by

119889119894119896 (

119883 119862) = (

119873

sum

119886=1

(119909119894119886

minus 119888119896119886)2)

12

(5)

In order to get the coordinate matrix 119883 we usually useKruskalrsquos STRESS formula to set the objective function asfollows

120590 (119883) =

119899

sum

119894=1

119899

sum

119895=119894+1

120596119894119895[120575119894119895minus 119889119894119895 (

119883)]

2

+

119898

sum

119894=1

119898

sum

119896=1

120592119894119896[120591119894119896minus 119889119894119896 (

119883 119862)]2

= 1205901 (

119883) + 1205902 (

119883)

(6)

where the weights 120596119894119895and 120592

119894119896decide their corresponding

sumrsquos influence on objective function This paper applies thecalculation method of weight proposed by the literature [6]and weights can be calculated by

120596119894119895= exp(minus

120575119894119895

2

1205781198941198952)

120592119894119896

= exp(minus

120591119894119896

2

1205781198941198962)

(7)

Otherwise both of them are zero 120578119894119895in (7) represents the

biggest distance between unknown node 119894 or 119895 and othernodes 120578

119894119896represents the biggest distance between unknown

node 119894 and other nodes or anchor node 119896 For a given matrix119883 the value of corresponding objective function can beobtained according to formula (3) Our goal is to find 119883whichmakes the objective functionminimum Namely min-imize the objective function and find the optimal solution119883

of objective functionThere are many ways to solve the local minimum prob-

lem of repeated optimization method in minimizing theobjective function for example Dimensionality Reductionmultivalue random testing method Distance Smoothingand Tunneling Method Among those methods multivaluerandom testing method may attain good optimal result butit has great randomicity and poor stability DimensionalityReduction is complex and large in the amount of calculationin the 3D space environment Distance Smoothing is simplein calculation and does well in positioning in 2D spaceenvironment But its global search is poor in the 3D spaceenvironment [8] On the contrary the repeated optimizationmethod based on tunneling can almost find the globaloptimal value due to cycle search method that first searches

in the horizontal direction and then searches in the verticaldirection and then searches in the horizontal direction againTherefore Tunneling Method used in this paper solves thelocal minimum problem of repeated optimization algorithmAnd the MDS 3D node localization algorithm based onTunneling Method is proposed

3 MDS Location Algorithm Based onthe Tunneling Method

31 Tunneling Method Algorithm Tunneling Method is anefficient certain global optimization method which was firstproposed byMontalco and Levy It builds tunneling functionin the current local minimum and then gets smaller localminimum through minimizing tunneling function to escapefrom the current local minima and then cyclic calculationuntil the global minimum is found Tunneling functionprovides amethod that uses local optimization to solve globaloptimization problem It only requires its target functionto be continuously differentiable The tunneling functions(auxiliary functions) have the same form and the process ofseeking error 120576 is easy [8]

Tunneling Method consists of a series of circulationsEvery circulation includes two steps local minimizing stepand tunneling step

Step 1 (local minimizing step) Starting from an initialpoint it applies local minimization algorithm such as quasi-Newton method gradient method or the conjugate gradientmethod to obtain the first local minimum value point 119909

1

lowast ofthe objective function 119891(119909) [8]

Step 2 (tunneling step) First define the tunneling functionin 1199091

lowast as follows

119879 (119909 1199091

lowast) =

119891 (119909) minus 119891 (1199091

lowast)

[(119909 minus 1199091lowast)119879(119909 minus 119909

1lowast)]

120582 (8)

where (119909 minus 1199091

lowast)119879(119909 minus 119909

1

lowast) is called the pole of tunneling

function and 120582 is the strength of (119909 minus 1199091

lowast)119879(119909 minus 119909

1

lowast) Then it

seeks a new point such that119879(119909 1199091

lowast) le 0 namely it finds out

1199092

lowast= 1199091

lowast such that 119891(1199092

lowast) le 119891(119909

1

lowast) 1199092

lowast will be the originalpoint of next circulation until it gets a better minimumpoint

The key of tunneling function algorithm is how to findthe point which meets 119879(119909 119909

1

lowast) le 0 Namely we apply the

minimizing tunneling function to find a local minimum 1199092

lowastSeveral discussions on it are as follows

(i) If 1199092

lowast= 1199091

lowast increase 120582 to make 1199091

lowast no longer thelocalminimumpoint of119879(119909 119909

1

lowast)That aim is to avoid

getting 1199091

lowast again by minimizing 119879(119909 1199091

lowast) for falling

into endless loop

4 Journal of Sensors

Table 1 The elements of the tunneling function 1205911(119883) and their purpose

Purpose Element(1) Zero point if STRESS is equal to local minimum STRESS 120591 (119883) = 120590 (119883) minus 120590 (119883

lowast)

(2) Zero points are the lowest tunneling function values 120591 (119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

(3) Avoid a zero point at119883lowast by erecting a pole 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

119875 (119883)

(4) Avoid a zero point at irrelevant transformations of 119883lowast 119875 (119883) =1003817100381710038171003817119863 (119883lowast) minus 119863 (119883)

1003817100381710038171003817

2

119908

(5) Ensure sufficiently strong pole (use pole strength parameter120582 0 lt 120582 lt 1) 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

119875 (119883)

(6) Avoid attraction to the horizon 120591 (119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

(1 +

1

119875 (119883)

)

(7) Extend working range of the pole (use width parameter 120596120596 gt 1) 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

(1 +

120596

119875 (119883)

)

(8) Have multiple poles to avoid different119883119896

1205911(119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

119903

prod

119896=1

(1 +

120596

119875119896(119883)

)

1119903

(ii) If 1199092

lowast= 1199091

lowast and 119891(1199092

lowast) gt 119891(119909

1

lowast) the tunneling

function can be constructed as follows

119879 (119909 1199091

lowast)

=

119891 (119909) minus 119891 (1199091

lowast)

[(119909 minus 1199092lowast)119879(119909 minus 119909

2lowast)]

1205821

[(119909 minus 1199091lowast)119879(119909 minus 119909

1lowast)]

120582

(9)

where 1205821is the strength of (119909 minus 119909

2

lowast)119879(119909 minus 119909

2

lowast)

Selecting the appropriate value of 1205821 the aim is to

make 1199092

lowast no longer the local minimum point of119879(119909 119909

1

lowast) and avoid getting 119909

2

lowast again by minimizing119879(119909 119909

1

lowast)

(iii) If 1199092

lowast= 1199091

lowast and 119891(1199092

lowast) lt 119891(119909

1

lowast) 1199092

lowast is the secondoriginal point and then start next circulation

Among the three situations the former two may appearmany times Every time tunneling function needs to berebuilt tominimize the function until the third situation turnsup and begins the next circulationThis circulation will neverstop until the tunneling function cannot find a smaller localminimum in a certain period of time The last local minimawill be regarded as the global minimum

32 MDS Positioning Algorithm Based on Tunneling MethodIn order to find another configurationwith the same STRESSthe tunneling function must have several characteristicsSome of these characteristics are met by the tunnelingfunction originally defined by Groenen and Heiser [9] Thetunneling function is defined as follows

120591 (119883) = 120590 (119883) minus 120590 (119883lowast) (10)

where 119883lowast is the local minimum point of 120590(119883) The elements

of the tunneling function 1205911(119883) and their purpose are given in

Table 1 The final tunneling function 1205911(119883) can be expressed

as follows

1205911 (

119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582 prod119903

119896=1(120596 + 119875

119896 (119883))1119903

prod119903

119896=1119875119896 (

119883)1119903

(11)

where 119875119896(119883) is

119875119896 (

119883) =1003817100381710038171003817119863119896(119883lowast) minus 119863119896 (

119883)1003817100381710038171003817

2

119908

= sum

119894lt119895

120596119894119895(119889119894119895(119883lowast) minus 119889119894119895 (

119883))

2

(12)

where 119903 represents the number of119883119896rsquos available

1205911(119883) mentioned above meets all conditions of the tun-

neling function The tunneling function is shown as formula(11) which can be treated as the ratio of two functions basedon119883 Assume that 119875(119883) gt 0 120591

1(119883) can be written as

1205911 (

119883) =

119872 (119883)

119875 (119883)

(13)

where119872(119883) and119873(119883) are respectively as follows

119872(119883) = 119873 (119883) (120596 + 119875 (119883)) (14)

119873(119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582 (15)

1205911(119883) can get a certain pole through (14) and (15) Assume

that we can find 119884 which makes formula (16) right

1205911 (

119883) =

119872 (119883)

119875 (119883)

le

119872 (119884)

119875 (119884)

= 1205911 (

119884) (16)

Let us multiply both sides with 119875(119883) Then we can obtain

119872(119883) minus

119872(119884)

119875 (119884)

119875 (119883) le 0 (17)

or

119865 (119902119883) = 119872 (119883) minus 119902119875 (119883) le 0 (18)

We find 119883 that meets 119865(119902119883) lt 0 and then 1205911(119883) lt

1205911(119884) is right As a result the iterative optimization algorithm

will be used to optimize 119865(119902119883) Namely it minimizesoptimization 120591

1(119883)

We carry out iterative minimization operations onradic1205911(119883) since 120591

1(119883) have the same zero and stagnation point

Journal of Sensors 5Ite

ratio

ns

86

42

Pole width2

6

10

Pole strength

200

600

1000

(a)

Itera

tions

86

42

Pole width2

6

10

Pole strength

100

200

(b)

Figure 1 Strength parameters and width parameters configuration effect chart

and radic1205911(119883) is easier to build its optimization framework

For every optimization step and optimization function usingformulas (13) (16) (17) and (18) 119865(119902119883) can be got bystep-by-step operation According to formula (18) optimize119865(119902119883) or optimize

119872(119883) = radic119873 (119883)radic(

119903

prod

119896=1

(119875119896 (

119883) + 120596)1119903

)

minusradic(

119903

prod

119896=1

119875119896 (

119883)1119903

)

(19)

Specific optimization steps are as follows

Step 1 119873(119883) will be optimized by optimizing the root ofpositive function |120590(119883) minus 120590(119883

lowast)| Assume that there is 119883

lowast

which meets 120590(119883) gt 120590(119883lowast) namely

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816= 120590 (119883) minus 120590 (119883

lowast) (20)

We verify the update data for each iteration If the assumptionis wrong then there is 120590(119883) le 120590(119883

lowast) and the tunneling step

will be stopped Finally119883lowast can be found and optimization of

119873(119883) can be accomplished

Step 2 Optimize the output of 119875119896(119883) through 119889

119894119895(119883119896) to be

instead of 120575119894119895in formulas (4) (5) (6) and (7) and then realize

the optimization of119903

prod

119896=1

(119875119896 (

119883) + 120596)1119903

(21)

Step 3 Optimize the output of 119875119896(119883) and minusradic119875

119896(119883) to realize

the optimization of

minusradic(

119903

prod

119896=1

119875119896 (

119883)1119903

) (22)

The optimization step of 119875119896(119883) is the same as Step 2

According to Cauchy-Schwarzrsquos inequality the optimizationfor 119889119894119895(119883) and minus119889

119894119895(119883) is to realize the optimization for

minusradic119875119896(119883) The second optimization methods will be used

to optimize 119889119894119895(119883) [10] Then minus119889

119894119895(119883) will be optimized by

Holder inequality

119865(119902119883) will be optimized by the steps mentioned aboveand 119883 will be got which satisfies 119865(119902119883) lt 0 and makesformula (16) right Such cycle can get global minimum andrealize global optimization

33 Parameters of TunnelingMethod TheTunnelingMethodneeds a pole strength parameter and a pole width param-eter Figure 1 is the three-dimensional effect graph ofthe STRESS function under different strength parame-ters and width parameters The strength parameter 1120582 is115 12 13 110 whose number adds up to 18 Thewidth parameters 120596 are 05 1 2 3 4 5 6 7 8 whose numberadds up to 9 The number of iterations is 1000 We can drawa conclusion from Figure 1 that when strength parameterand width parameter are larger more iterations are neededto get the same result But when the strength parametersand width parameters are too small the result is not whatwe want Heiser carried out a lot of experiments and finallyconcluded that when the strength parameters are 13 andwidth parameter is 1198994 in Tunneling Method the result ofiteration is the best where 119899 is the number of iterationelements [8]

4 Simulation Results and Analysis

The TMDS 3D node localization algorithm simulation isoperated on Matlab 2009(a) (64 bits) to research its per-formance Analyze TMDS from three aspects the nodecommunication radius the number of nodes and the pro-portion of anchor node The simulation environment is

6 Journal of Sensors

0 2 4 6 8 10

0

5

10

0

2

4

6

8

10

y-axisx-axis

z-a

xis

Anchor node Unknown node

Figure 2 The distribution map in 3D for unknown node andanchor

described as follows the size of cube geometric model is10m times 10m times 10m and the nodes are placed in this model(the node communication radius is the same and all thenodes are stationary) The model is shown in Figure 2 Setthe simulation environments as follows when the iterationcalculation rule is the same node communication radius isalso the same Repeated optimization algorithm iterative timeis 60 The iterative computation error threshold is 00001Node localization error threshold 119890119903119903119900119903 119901 is 005mThe errorin distancemeasurement is taken as 15When the unknownnode location error meets 119890119903119903119900119903 le 119890119903119903119900119903 119901 the node will beregarded as the accurate localization of nodes

41 Node Communication Radius Influence on the Perfor-mance of the Algorithm Figure 3 shows the impact of nodecommunication radius on the performance of TMDS algo-rithmThe simulation environment is as follows the numberof unknown nodes 119873 is 50 which randomly distribute in a10m lowast 10m lowast 10m cube modelThe number of anchor nodes119872 is 8 which distribute in the 8 vertex positions of the cubemodel Change the node communication radius to analyze itsimpact on the performance of the algorithm

We can draw a conclusion form Figure 3(a) that theproportion of accurate positioning nodes will increase withthe increase of node communication radius When the nodecommunication radius increases to 8m the number ofaccurate positioning nodes reaches 50 And with the contin-uous increase of the node communication radius accuratepositioning of unknown nodes rate will be maintained at1 We can draw from Figure 3(b) that the average error ofaccurate positioning of node localization gradually decreaseswith the increase of node communication radius Whennode communication radius increases from 4m to 8m theaverage error decreases fast When node communicationradius changes from 8m to 12m the error is stable

When the communication radius is small network con-nectivity is low and the number of unknown nodes that cancommunicate with the node is reduced and a small node

communication radius (119877 = 8m) will get higher accuratepositioning node and low node localization error rate in theTMDS test The positioning result will be stable when theradius is increased

42 Influence of Network Nodes on the Performance of TMDSFigure 4 shows that when the node communication radius119877 is 8 the network node influences the TMDS algorithmAccording to the analysis of 41 when the communicationradius is 8m it is easier to analyze other factors influence ofnodes on the performance of the algorithm So in this sectionthe simulation environment is set as follows The number ofanchor nodes 119872 is 8 which are respectively distributed onthe 8 vertex positions of the cubemodel Nodes have the sameradius (119877 = 8m) of communication Change the proportionof anchor nodes by changing the number of unknown nodesand then analyze the influence of network nodes on theperformance of the algorithm

We can draw a conclusion from the curve of Figure 4that with the number of nodes increasing from 38 to 78 thatis the numbers of unknown nodes increase from 30 to 70accurate positioning of unknownnodes rate has been 1Whenthe network nodes number is 83 that is the unknown nodesnumber is 75 accurate localization rate begins to declineThe results show that when TMDS algorithm works in anenvironment of few anchor nodes and many network nodesthe location accuracy of unknownnodes can bemaintained at1 So when the TMDS algorithmworks in environment of fewanchor nodes the unknown nodes can be located accuratelyand the result is significant

43 Influence of the Proportion of Anchor Nodes on thePerformance of TMDS Figure 5 shows the proportion ofanchor nodes influence on the performance of algorithmThe simulation environment is as follows the number ofunknown nodes 119873 is 100 which are randomly distributed inthe space model Eight anchor nodes are distributed in the 8vertex positions of the cubemodel and others are distributedrandomly in the space model All the nodes have the samecommunication radius (119877 = 8m) Change the number of theanchor nodes to analyze the proportion of anchor impact onthe performance of the algorithm

We can draw form Figure 5(a) that the number ofaccurate positioning nodes will increase with the increaseof the proportion of anchor nodes When the proportion ofanchor nodes is 01 the number of accurate positioning nodesreaches 100 And the proportion of anchor nodes changesfrom 015 to 05 the rate of accurate positioning of unknownnodeswill bemaintained at 100Namelywhen the proportionof anchor nodes is greater than or equal to 015 TMDSalgorithm can locate the unknown nodes accurately

We can draw from Figure 5(b) that the unknown nodelocalization error gradually decreases with the increase of theproportion of anchor nodes When the proportion of anchornodes changes from 01 to 016 the unknown nodes locationerror decreases fast When the proportion of anchor nodeschanges from 016 to 05 the average location error fluctuatesa little But the positioning error is within the threshold rangeThe result of accurate positioning of unknownnodes is stable

Journal of Sensors 7

3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

50

Node communication radius

Num

ber o

f acc

urat

e pos

ition

ing

node

s

(a)

3 4 5 6 7 8 9 10 11 1200

01

02

03

04

05

06

07

08

Node communication radius

Aver

age e

rror

(m)

(b)

Figure 3 Node communication radius influence on the performance of TMDS

35 40 45 50 55 60 65 70 75 80 8520

25

30

35

40

45

50

55

60

65

70

Num

ber o

f acc

urat

e pos

ition

ing

node

s

Numbers of network nodes(a)

35 40 45 50 55 60 65 70 75 80 85000

002

004

006

008

010

012

Aver

age e

rror

(m)

Numbers of network nodes(b)

Figure 4 Influence of network nodes on the performance of TMDS

44 The Comparison between MDS Algorithm and TMDSAlgorithm Tables 2 and 3 represent MDS algorithm andTMDS algorithm for 10 times the continuous operationunder the same simulation environment respectively Thecontent of the table includes the average positioning errorof unknown nodes number of accurate positioning nodesnode location accuracy rate and the program running timeresults data The same simulation environment settings areas follows the number of anchor nodes 119872 is 8 which arerespectively distributed on the 8 vertex positions on a cubemodel all nodes have the same radius of communication(119877 = 8m)

From the accuracy of positioning algorithm TMDSalgorithm can achieve 9 times accurate positioning of all the

unknownnode in 10 times continuous operation whose loca-tion accuracy rate was 1 only 1 time accurately positioningnode number is 49 and its accurate localization rate is 098We can draw from Tables 2 and 3 that TMDS algorithmcan basically realize accurate positioning of the node MDSalgorithm can achieve 4 times the accurate positioning of allthe unknown node in 10 times the continuous operation andhave 2 times the low positioning rate respectively 008 and01

From the running time of the algorithm TMDS andMDS can complete the algorithm in a relatively short periodof time Time of MDS algorithm that completes the basicalgorithm within 13 ssim14 s is relatively stable TMDS takesa long time because of iterative computation of repeated

8 Journal of Sensors

005 010 015 020 025 030 035 040 045 05040

50

60

70

80

90

100N

umbe

r of a

ccur

ate p

ositi

onin

g no

des

Proportion of anchor nodes

(a)

005 010 015 020 025 030 035 040 045 050000

001

002

003

004

005

006

007

008

Aver

age e

rror

(m)

Proportion of anchor nodes

(b)

Figure 5 Influence of the proportion of anchor nodes on the performance of TMDS

Table 2 MDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 02881 00238 00031 00144 00360 00274 00162 02670 00152 00226Number of accurate positioning nodes 4 47 50 50 33 45 50 5 50 48Node location accuracy rate 008 094 1 1 066 09 1 01 1 096Program running time (s) 0142 0140 0137 0136 0131 0132 0137 0133 0137 0138

Table 3 TMDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 00101 00147 00079 00104 00197 00065 00090 00064 00070 00054Number of accurate positioning nodes 50 50 50 50 49 50 50 50 50 50Node location accuracy rate 1 1 1 1 098 1 1 1 1 1Program running time (s) 15754 34418 17085 33918 36783 33358 36259 36219 36137 36363

optimization The selection of the first local minimum valuefor repeating optimization of the algorithm of iterativecalculation is random and the global optimal iterations timeis different The distribution of 50 unknown nodes in modelspace is random So the time to complete the positioning ofall nodes will be different To sum up the Tunneling Methodis applied to solve the problem of local minimum value ofrepeated optimization algorithm and achieves good results inthe positioning of WSN

5 Conclusion

This paper uses the Tunneling Method technology in theWSN 3D node localization and proposes a new 3D wirelesssensor network node positioningmethod based onTunnelingMethod algorithm Innovation points of the TMDS localiza-tion algorithm are as follows

(i) TMDS algorithm is a kind of WSN distributednodes localization algorithm based on anchor node

multidimensional scaling technique The locationalgorithm directly calculates the absolute location ofthe unknown nodes by Euclidean distance betweenanchor nodes andnodes without coordinate transfor-mation and network fusion operation So it avoids thecumulative error

(ii) The core of TMDS is a kind of repeated optimizationmethod based on the new anchor nodes The methodis deduced from STRESS algorithm using TunnelingMethod to solve the local minimum problem ofrepeated optimization method to improve the accu-racy of the optimization results

(iii) The TMDS algorithm is an improved algorithm basedon MDS which makes use of the difference betweenthe nodes The calculation results are stable and thestability of the location effect is also better

The tunneling function algorithm is a certain globaloptimization method It can search the global minimum

Journal of Sensors 9

point in the minimization problem search and avoid fallinginto local minimum to get a smaller minimum point Repeatthis process to obtain the global optimal solution It improvesthe convergence speed of the algorithm and enhances theability of the global optimization The algorithm provides anewmethod in 3DWSN node positioningThe experimentalresult shows that the application of the repeated optimiza-tion algorithm and global optimization theory in 3D nodelocalization algorithm in WSN improves the rate of accuratepositioning of unknown nodes reduces the unknown nodelocation error and ensures that the nodeswhich are randomlydistributed in a three-dimensional wireless sensor networkcan be positioned accurately and stably In addition there areseveral limitations in the proposed method The tunnelingfunction requires its target function to be continuouslydifferentiable and the search time and computation willincrease when WSNs have large scale and high node density

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundations of China (Grant no 51305407)

References

[1] M Demirbas X Lu and P Singla ldquoAn in-network queryingframework for wireless sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 20 no 8 pp 1202ndash12152009

[2] V K Chaurasiya N Jain and G C Nandi ldquoA novel distanceestimation approach for 3D localization in wireless sensornetwork using multi dimensional scalingrdquo Information Fusionvol 15 no 1 pp 5ndash18 2014

[3] J-K Lee Y Kim J-H Lee and S-C Kim ldquoAn efficient three-dimensional localization scheme using trilateration in wirelesssensor networksrdquo IEEE Communications Letters vol 18 no 9pp 1591ndash1594 2014

[4] M Shon M Jo and H Choo ldquoAn interactive cluster-basedMDS localization scheme for multimedia information in wire-less sensor networksrdquo Computer Communications vol 35 no15 pp 1921ndash1929 2012

[5] H-B Chen D-Q Wang F Yuan and R Xu ldquoA MDS-basedlocalization algorithm for underwater wireless sensor networkrdquoinProceedings of theMTSIEEE SanDiegoConference AnOceanin Common (OCEANS rsquo13) San Diego Calif USA September2013

[6] 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

[7] E Kim S Lee C Kim and K Kim ldquoMobile beacon-based3D-localization with multidimensional scaling in large sensornetworksrdquo IEEE Communications Letters vol 14 no 7 pp 647ndash649 2010

[8] P J Groenen W J Heiser and J J Meulman ldquoGlobal opti-mization in least-squares multidimensional scaling by distancesmoothingrdquo Journal of Classification vol 16 no 2 pp 225ndash2541999

[9] P J F Groenen and W J Heiser ldquoThe tunneling method forglobal optimization in multidimensional scalingrdquo Psychome-trika vol 61 no 3 pp 529ndash550 1996

[10] J De Leeuw and W Heiser ldquoMultidimensional scaling withrestrictions on the configurationrdquo in Multivariate Analysis pp501ndash522 North-Holland Publishing 1980

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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4 Journal of Sensors

Table 1 The elements of the tunneling function 1205911(119883) and their purpose

Purpose Element(1) Zero point if STRESS is equal to local minimum STRESS 120591 (119883) = 120590 (119883) minus 120590 (119883

lowast)

(2) Zero points are the lowest tunneling function values 120591 (119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

(3) Avoid a zero point at119883lowast by erecting a pole 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

119875 (119883)

(4) Avoid a zero point at irrelevant transformations of 119883lowast 119875 (119883) =1003817100381710038171003817119863 (119883lowast) minus 119863 (119883)

1003817100381710038171003817

2

119908

(5) Ensure sufficiently strong pole (use pole strength parameter120582 0 lt 120582 lt 1) 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

119875 (119883)

(6) Avoid attraction to the horizon 120591 (119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

(1 +

1

119875 (119883)

)

(7) Extend working range of the pole (use width parameter 120596120596 gt 1) 120591 (119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

(1 +

120596

119875 (119883)

)

(8) Have multiple poles to avoid different119883119896

1205911(119883) =

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582

119903

prod

119896=1

(1 +

120596

119875119896(119883)

)

1119903

(ii) If 1199092

lowast= 1199091

lowast and 119891(1199092

lowast) gt 119891(119909

1

lowast) the tunneling

function can be constructed as follows

119879 (119909 1199091

lowast)

=

119891 (119909) minus 119891 (1199091

lowast)

[(119909 minus 1199092lowast)119879(119909 minus 119909

2lowast)]

1205821

[(119909 minus 1199091lowast)119879(119909 minus 119909

1lowast)]

120582

(9)

where 1205821is the strength of (119909 minus 119909

2

lowast)119879(119909 minus 119909

2

lowast)

Selecting the appropriate value of 1205821 the aim is to

make 1199092

lowast no longer the local minimum point of119879(119909 119909

1

lowast) and avoid getting 119909

2

lowast again by minimizing119879(119909 119909

1

lowast)

(iii) If 1199092

lowast= 1199091

lowast and 119891(1199092

lowast) lt 119891(119909

1

lowast) 1199092

lowast is the secondoriginal point and then start next circulation

Among the three situations the former two may appearmany times Every time tunneling function needs to berebuilt tominimize the function until the third situation turnsup and begins the next circulationThis circulation will neverstop until the tunneling function cannot find a smaller localminimum in a certain period of time The last local minimawill be regarded as the global minimum

32 MDS Positioning Algorithm Based on Tunneling MethodIn order to find another configurationwith the same STRESSthe tunneling function must have several characteristicsSome of these characteristics are met by the tunnelingfunction originally defined by Groenen and Heiser [9] Thetunneling function is defined as follows

120591 (119883) = 120590 (119883) minus 120590 (119883lowast) (10)

where 119883lowast is the local minimum point of 120590(119883) The elements

of the tunneling function 1205911(119883) and their purpose are given in

Table 1 The final tunneling function 1205911(119883) can be expressed

as follows

1205911 (

119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582 prod119903

119896=1(120596 + 119875

119896 (119883))1119903

prod119903

119896=1119875119896 (

119883)1119903

(11)

where 119875119896(119883) is

119875119896 (

119883) =1003817100381710038171003817119863119896(119883lowast) minus 119863119896 (

119883)1003817100381710038171003817

2

119908

= sum

119894lt119895

120596119894119895(119889119894119895(119883lowast) minus 119889119894119895 (

119883))

2

(12)

where 119903 represents the number of119883119896rsquos available

1205911(119883) mentioned above meets all conditions of the tun-

neling function The tunneling function is shown as formula(11) which can be treated as the ratio of two functions basedon119883 Assume that 119875(119883) gt 0 120591

1(119883) can be written as

1205911 (

119883) =

119872 (119883)

119875 (119883)

(13)

where119872(119883) and119873(119883) are respectively as follows

119872(119883) = 119873 (119883) (120596 + 119875 (119883)) (14)

119873(119883) =1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816

120582 (15)

1205911(119883) can get a certain pole through (14) and (15) Assume

that we can find 119884 which makes formula (16) right

1205911 (

119883) =

119872 (119883)

119875 (119883)

le

119872 (119884)

119875 (119884)

= 1205911 (

119884) (16)

Let us multiply both sides with 119875(119883) Then we can obtain

119872(119883) minus

119872(119884)

119875 (119884)

119875 (119883) le 0 (17)

or

119865 (119902119883) = 119872 (119883) minus 119902119875 (119883) le 0 (18)

We find 119883 that meets 119865(119902119883) lt 0 and then 1205911(119883) lt

1205911(119884) is right As a result the iterative optimization algorithm

will be used to optimize 119865(119902119883) Namely it minimizesoptimization 120591

1(119883)

We carry out iterative minimization operations onradic1205911(119883) since 120591

1(119883) have the same zero and stagnation point

Journal of Sensors 5Ite

ratio

ns

86

42

Pole width2

6

10

Pole strength

200

600

1000

(a)

Itera

tions

86

42

Pole width2

6

10

Pole strength

100

200

(b)

Figure 1 Strength parameters and width parameters configuration effect chart

and radic1205911(119883) is easier to build its optimization framework

For every optimization step and optimization function usingformulas (13) (16) (17) and (18) 119865(119902119883) can be got bystep-by-step operation According to formula (18) optimize119865(119902119883) or optimize

119872(119883) = radic119873 (119883)radic(

119903

prod

119896=1

(119875119896 (

119883) + 120596)1119903

)

minusradic(

119903

prod

119896=1

119875119896 (

119883)1119903

)

(19)

Specific optimization steps are as follows

Step 1 119873(119883) will be optimized by optimizing the root ofpositive function |120590(119883) minus 120590(119883

lowast)| Assume that there is 119883

lowast

which meets 120590(119883) gt 120590(119883lowast) namely

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816= 120590 (119883) minus 120590 (119883

lowast) (20)

We verify the update data for each iteration If the assumptionis wrong then there is 120590(119883) le 120590(119883

lowast) and the tunneling step

will be stopped Finally119883lowast can be found and optimization of

119873(119883) can be accomplished

Step 2 Optimize the output of 119875119896(119883) through 119889

119894119895(119883119896) to be

instead of 120575119894119895in formulas (4) (5) (6) and (7) and then realize

the optimization of119903

prod

119896=1

(119875119896 (

119883) + 120596)1119903

(21)

Step 3 Optimize the output of 119875119896(119883) and minusradic119875

119896(119883) to realize

the optimization of

minusradic(

119903

prod

119896=1

119875119896 (

119883)1119903

) (22)

The optimization step of 119875119896(119883) is the same as Step 2

According to Cauchy-Schwarzrsquos inequality the optimizationfor 119889119894119895(119883) and minus119889

119894119895(119883) is to realize the optimization for

minusradic119875119896(119883) The second optimization methods will be used

to optimize 119889119894119895(119883) [10] Then minus119889

119894119895(119883) will be optimized by

Holder inequality

119865(119902119883) will be optimized by the steps mentioned aboveand 119883 will be got which satisfies 119865(119902119883) lt 0 and makesformula (16) right Such cycle can get global minimum andrealize global optimization

33 Parameters of TunnelingMethod TheTunnelingMethodneeds a pole strength parameter and a pole width param-eter Figure 1 is the three-dimensional effect graph ofthe STRESS function under different strength parame-ters and width parameters The strength parameter 1120582 is115 12 13 110 whose number adds up to 18 Thewidth parameters 120596 are 05 1 2 3 4 5 6 7 8 whose numberadds up to 9 The number of iterations is 1000 We can drawa conclusion from Figure 1 that when strength parameterand width parameter are larger more iterations are neededto get the same result But when the strength parametersand width parameters are too small the result is not whatwe want Heiser carried out a lot of experiments and finallyconcluded that when the strength parameters are 13 andwidth parameter is 1198994 in Tunneling Method the result ofiteration is the best where 119899 is the number of iterationelements [8]

4 Simulation Results and Analysis

The TMDS 3D node localization algorithm simulation isoperated on Matlab 2009(a) (64 bits) to research its per-formance Analyze TMDS from three aspects the nodecommunication radius the number of nodes and the pro-portion of anchor node The simulation environment is

6 Journal of Sensors

0 2 4 6 8 10

0

5

10

0

2

4

6

8

10

y-axisx-axis

z-a

xis

Anchor node Unknown node

Figure 2 The distribution map in 3D for unknown node andanchor

described as follows the size of cube geometric model is10m times 10m times 10m and the nodes are placed in this model(the node communication radius is the same and all thenodes are stationary) The model is shown in Figure 2 Setthe simulation environments as follows when the iterationcalculation rule is the same node communication radius isalso the same Repeated optimization algorithm iterative timeis 60 The iterative computation error threshold is 00001Node localization error threshold 119890119903119903119900119903 119901 is 005mThe errorin distancemeasurement is taken as 15When the unknownnode location error meets 119890119903119903119900119903 le 119890119903119903119900119903 119901 the node will beregarded as the accurate localization of nodes

41 Node Communication Radius Influence on the Perfor-mance of the Algorithm Figure 3 shows the impact of nodecommunication radius on the performance of TMDS algo-rithmThe simulation environment is as follows the numberof unknown nodes 119873 is 50 which randomly distribute in a10m lowast 10m lowast 10m cube modelThe number of anchor nodes119872 is 8 which distribute in the 8 vertex positions of the cubemodel Change the node communication radius to analyze itsimpact on the performance of the algorithm

We can draw a conclusion form Figure 3(a) that theproportion of accurate positioning nodes will increase withthe increase of node communication radius When the nodecommunication radius increases to 8m the number ofaccurate positioning nodes reaches 50 And with the contin-uous increase of the node communication radius accuratepositioning of unknown nodes rate will be maintained at1 We can draw from Figure 3(b) that the average error ofaccurate positioning of node localization gradually decreaseswith the increase of node communication radius Whennode communication radius increases from 4m to 8m theaverage error decreases fast When node communicationradius changes from 8m to 12m the error is stable

When the communication radius is small network con-nectivity is low and the number of unknown nodes that cancommunicate with the node is reduced and a small node

communication radius (119877 = 8m) will get higher accuratepositioning node and low node localization error rate in theTMDS test The positioning result will be stable when theradius is increased

42 Influence of Network Nodes on the Performance of TMDSFigure 4 shows that when the node communication radius119877 is 8 the network node influences the TMDS algorithmAccording to the analysis of 41 when the communicationradius is 8m it is easier to analyze other factors influence ofnodes on the performance of the algorithm So in this sectionthe simulation environment is set as follows The number ofanchor nodes 119872 is 8 which are respectively distributed onthe 8 vertex positions of the cubemodel Nodes have the sameradius (119877 = 8m) of communication Change the proportionof anchor nodes by changing the number of unknown nodesand then analyze the influence of network nodes on theperformance of the algorithm

We can draw a conclusion from the curve of Figure 4that with the number of nodes increasing from 38 to 78 thatis the numbers of unknown nodes increase from 30 to 70accurate positioning of unknownnodes rate has been 1Whenthe network nodes number is 83 that is the unknown nodesnumber is 75 accurate localization rate begins to declineThe results show that when TMDS algorithm works in anenvironment of few anchor nodes and many network nodesthe location accuracy of unknownnodes can bemaintained at1 So when the TMDS algorithmworks in environment of fewanchor nodes the unknown nodes can be located accuratelyand the result is significant

43 Influence of the Proportion of Anchor Nodes on thePerformance of TMDS Figure 5 shows the proportion ofanchor nodes influence on the performance of algorithmThe simulation environment is as follows the number ofunknown nodes 119873 is 100 which are randomly distributed inthe space model Eight anchor nodes are distributed in the 8vertex positions of the cubemodel and others are distributedrandomly in the space model All the nodes have the samecommunication radius (119877 = 8m) Change the number of theanchor nodes to analyze the proportion of anchor impact onthe performance of the algorithm

We can draw form Figure 5(a) that the number ofaccurate positioning nodes will increase with the increaseof the proportion of anchor nodes When the proportion ofanchor nodes is 01 the number of accurate positioning nodesreaches 100 And the proportion of anchor nodes changesfrom 015 to 05 the rate of accurate positioning of unknownnodeswill bemaintained at 100Namelywhen the proportionof anchor nodes is greater than or equal to 015 TMDSalgorithm can locate the unknown nodes accurately

We can draw from Figure 5(b) that the unknown nodelocalization error gradually decreases with the increase of theproportion of anchor nodes When the proportion of anchornodes changes from 01 to 016 the unknown nodes locationerror decreases fast When the proportion of anchor nodeschanges from 016 to 05 the average location error fluctuatesa little But the positioning error is within the threshold rangeThe result of accurate positioning of unknownnodes is stable

Journal of Sensors 7

3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

50

Node communication radius

Num

ber o

f acc

urat

e pos

ition

ing

node

s

(a)

3 4 5 6 7 8 9 10 11 1200

01

02

03

04

05

06

07

08

Node communication radius

Aver

age e

rror

(m)

(b)

Figure 3 Node communication radius influence on the performance of TMDS

35 40 45 50 55 60 65 70 75 80 8520

25

30

35

40

45

50

55

60

65

70

Num

ber o

f acc

urat

e pos

ition

ing

node

s

Numbers of network nodes(a)

35 40 45 50 55 60 65 70 75 80 85000

002

004

006

008

010

012

Aver

age e

rror

(m)

Numbers of network nodes(b)

Figure 4 Influence of network nodes on the performance of TMDS

44 The Comparison between MDS Algorithm and TMDSAlgorithm Tables 2 and 3 represent MDS algorithm andTMDS algorithm for 10 times the continuous operationunder the same simulation environment respectively Thecontent of the table includes the average positioning errorof unknown nodes number of accurate positioning nodesnode location accuracy rate and the program running timeresults data The same simulation environment settings areas follows the number of anchor nodes 119872 is 8 which arerespectively distributed on the 8 vertex positions on a cubemodel all nodes have the same radius of communication(119877 = 8m)

From the accuracy of positioning algorithm TMDSalgorithm can achieve 9 times accurate positioning of all the

unknownnode in 10 times continuous operation whose loca-tion accuracy rate was 1 only 1 time accurately positioningnode number is 49 and its accurate localization rate is 098We can draw from Tables 2 and 3 that TMDS algorithmcan basically realize accurate positioning of the node MDSalgorithm can achieve 4 times the accurate positioning of allthe unknown node in 10 times the continuous operation andhave 2 times the low positioning rate respectively 008 and01

From the running time of the algorithm TMDS andMDS can complete the algorithm in a relatively short periodof time Time of MDS algorithm that completes the basicalgorithm within 13 ssim14 s is relatively stable TMDS takesa long time because of iterative computation of repeated

8 Journal of Sensors

005 010 015 020 025 030 035 040 045 05040

50

60

70

80

90

100N

umbe

r of a

ccur

ate p

ositi

onin

g no

des

Proportion of anchor nodes

(a)

005 010 015 020 025 030 035 040 045 050000

001

002

003

004

005

006

007

008

Aver

age e

rror

(m)

Proportion of anchor nodes

(b)

Figure 5 Influence of the proportion of anchor nodes on the performance of TMDS

Table 2 MDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 02881 00238 00031 00144 00360 00274 00162 02670 00152 00226Number of accurate positioning nodes 4 47 50 50 33 45 50 5 50 48Node location accuracy rate 008 094 1 1 066 09 1 01 1 096Program running time (s) 0142 0140 0137 0136 0131 0132 0137 0133 0137 0138

Table 3 TMDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 00101 00147 00079 00104 00197 00065 00090 00064 00070 00054Number of accurate positioning nodes 50 50 50 50 49 50 50 50 50 50Node location accuracy rate 1 1 1 1 098 1 1 1 1 1Program running time (s) 15754 34418 17085 33918 36783 33358 36259 36219 36137 36363

optimization The selection of the first local minimum valuefor repeating optimization of the algorithm of iterativecalculation is random and the global optimal iterations timeis different The distribution of 50 unknown nodes in modelspace is random So the time to complete the positioning ofall nodes will be different To sum up the Tunneling Methodis applied to solve the problem of local minimum value ofrepeated optimization algorithm and achieves good results inthe positioning of WSN

5 Conclusion

This paper uses the Tunneling Method technology in theWSN 3D node localization and proposes a new 3D wirelesssensor network node positioningmethod based onTunnelingMethod algorithm Innovation points of the TMDS localiza-tion algorithm are as follows

(i) TMDS algorithm is a kind of WSN distributednodes localization algorithm based on anchor node

multidimensional scaling technique The locationalgorithm directly calculates the absolute location ofthe unknown nodes by Euclidean distance betweenanchor nodes andnodes without coordinate transfor-mation and network fusion operation So it avoids thecumulative error

(ii) The core of TMDS is a kind of repeated optimizationmethod based on the new anchor nodes The methodis deduced from STRESS algorithm using TunnelingMethod to solve the local minimum problem ofrepeated optimization method to improve the accu-racy of the optimization results

(iii) The TMDS algorithm is an improved algorithm basedon MDS which makes use of the difference betweenthe nodes The calculation results are stable and thestability of the location effect is also better

The tunneling function algorithm is a certain globaloptimization method It can search the global minimum

Journal of Sensors 9

point in the minimization problem search and avoid fallinginto local minimum to get a smaller minimum point Repeatthis process to obtain the global optimal solution It improvesthe convergence speed of the algorithm and enhances theability of the global optimization The algorithm provides anewmethod in 3DWSN node positioningThe experimentalresult shows that the application of the repeated optimiza-tion algorithm and global optimization theory in 3D nodelocalization algorithm in WSN improves the rate of accuratepositioning of unknown nodes reduces the unknown nodelocation error and ensures that the nodeswhich are randomlydistributed in a three-dimensional wireless sensor networkcan be positioned accurately and stably In addition there areseveral limitations in the proposed method The tunnelingfunction requires its target function to be continuouslydifferentiable and the search time and computation willincrease when WSNs have large scale and high node density

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundations of China (Grant no 51305407)

References

[1] M Demirbas X Lu and P Singla ldquoAn in-network queryingframework for wireless sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 20 no 8 pp 1202ndash12152009

[2] V K Chaurasiya N Jain and G C Nandi ldquoA novel distanceestimation approach for 3D localization in wireless sensornetwork using multi dimensional scalingrdquo Information Fusionvol 15 no 1 pp 5ndash18 2014

[3] J-K Lee Y Kim J-H Lee and S-C Kim ldquoAn efficient three-dimensional localization scheme using trilateration in wirelesssensor networksrdquo IEEE Communications Letters vol 18 no 9pp 1591ndash1594 2014

[4] M Shon M Jo and H Choo ldquoAn interactive cluster-basedMDS localization scheme for multimedia information in wire-less sensor networksrdquo Computer Communications vol 35 no15 pp 1921ndash1929 2012

[5] H-B Chen D-Q Wang F Yuan and R Xu ldquoA MDS-basedlocalization algorithm for underwater wireless sensor networkrdquoinProceedings of theMTSIEEE SanDiegoConference AnOceanin Common (OCEANS rsquo13) San Diego Calif USA September2013

[6] 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

[7] E Kim S Lee C Kim and K Kim ldquoMobile beacon-based3D-localization with multidimensional scaling in large sensornetworksrdquo IEEE Communications Letters vol 14 no 7 pp 647ndash649 2010

[8] P J Groenen W J Heiser and J J Meulman ldquoGlobal opti-mization in least-squares multidimensional scaling by distancesmoothingrdquo Journal of Classification vol 16 no 2 pp 225ndash2541999

[9] P J F Groenen and W J Heiser ldquoThe tunneling method forglobal optimization in multidimensional scalingrdquo Psychome-trika vol 61 no 3 pp 529ndash550 1996

[10] J De Leeuw and W Heiser ldquoMultidimensional scaling withrestrictions on the configurationrdquo in Multivariate Analysis pp501ndash522 North-Holland Publishing 1980

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Control Scienceand Engineering

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Journal of Sensors 5Ite

ratio

ns

86

42

Pole width2

6

10

Pole strength

200

600

1000

(a)

Itera

tions

86

42

Pole width2

6

10

Pole strength

100

200

(b)

Figure 1 Strength parameters and width parameters configuration effect chart

and radic1205911(119883) is easier to build its optimization framework

For every optimization step and optimization function usingformulas (13) (16) (17) and (18) 119865(119902119883) can be got bystep-by-step operation According to formula (18) optimize119865(119902119883) or optimize

119872(119883) = radic119873 (119883)radic(

119903

prod

119896=1

(119875119896 (

119883) + 120596)1119903

)

minusradic(

119903

prod

119896=1

119875119896 (

119883)1119903

)

(19)

Specific optimization steps are as follows

Step 1 119873(119883) will be optimized by optimizing the root ofpositive function |120590(119883) minus 120590(119883

lowast)| Assume that there is 119883

lowast

which meets 120590(119883) gt 120590(119883lowast) namely

1003816100381610038161003816120590 (119883) minus 120590 (119883

lowast)1003816100381610038161003816= 120590 (119883) minus 120590 (119883

lowast) (20)

We verify the update data for each iteration If the assumptionis wrong then there is 120590(119883) le 120590(119883

lowast) and the tunneling step

will be stopped Finally119883lowast can be found and optimization of

119873(119883) can be accomplished

Step 2 Optimize the output of 119875119896(119883) through 119889

119894119895(119883119896) to be

instead of 120575119894119895in formulas (4) (5) (6) and (7) and then realize

the optimization of119903

prod

119896=1

(119875119896 (

119883) + 120596)1119903

(21)

Step 3 Optimize the output of 119875119896(119883) and minusradic119875

119896(119883) to realize

the optimization of

minusradic(

119903

prod

119896=1

119875119896 (

119883)1119903

) (22)

The optimization step of 119875119896(119883) is the same as Step 2

According to Cauchy-Schwarzrsquos inequality the optimizationfor 119889119894119895(119883) and minus119889

119894119895(119883) is to realize the optimization for

minusradic119875119896(119883) The second optimization methods will be used

to optimize 119889119894119895(119883) [10] Then minus119889

119894119895(119883) will be optimized by

Holder inequality

119865(119902119883) will be optimized by the steps mentioned aboveand 119883 will be got which satisfies 119865(119902119883) lt 0 and makesformula (16) right Such cycle can get global minimum andrealize global optimization

33 Parameters of TunnelingMethod TheTunnelingMethodneeds a pole strength parameter and a pole width param-eter Figure 1 is the three-dimensional effect graph ofthe STRESS function under different strength parame-ters and width parameters The strength parameter 1120582 is115 12 13 110 whose number adds up to 18 Thewidth parameters 120596 are 05 1 2 3 4 5 6 7 8 whose numberadds up to 9 The number of iterations is 1000 We can drawa conclusion from Figure 1 that when strength parameterand width parameter are larger more iterations are neededto get the same result But when the strength parametersand width parameters are too small the result is not whatwe want Heiser carried out a lot of experiments and finallyconcluded that when the strength parameters are 13 andwidth parameter is 1198994 in Tunneling Method the result ofiteration is the best where 119899 is the number of iterationelements [8]

4 Simulation Results and Analysis

The TMDS 3D node localization algorithm simulation isoperated on Matlab 2009(a) (64 bits) to research its per-formance Analyze TMDS from three aspects the nodecommunication radius the number of nodes and the pro-portion of anchor node The simulation environment is

6 Journal of Sensors

0 2 4 6 8 10

0

5

10

0

2

4

6

8

10

y-axisx-axis

z-a

xis

Anchor node Unknown node

Figure 2 The distribution map in 3D for unknown node andanchor

described as follows the size of cube geometric model is10m times 10m times 10m and the nodes are placed in this model(the node communication radius is the same and all thenodes are stationary) The model is shown in Figure 2 Setthe simulation environments as follows when the iterationcalculation rule is the same node communication radius isalso the same Repeated optimization algorithm iterative timeis 60 The iterative computation error threshold is 00001Node localization error threshold 119890119903119903119900119903 119901 is 005mThe errorin distancemeasurement is taken as 15When the unknownnode location error meets 119890119903119903119900119903 le 119890119903119903119900119903 119901 the node will beregarded as the accurate localization of nodes

41 Node Communication Radius Influence on the Perfor-mance of the Algorithm Figure 3 shows the impact of nodecommunication radius on the performance of TMDS algo-rithmThe simulation environment is as follows the numberof unknown nodes 119873 is 50 which randomly distribute in a10m lowast 10m lowast 10m cube modelThe number of anchor nodes119872 is 8 which distribute in the 8 vertex positions of the cubemodel Change the node communication radius to analyze itsimpact on the performance of the algorithm

We can draw a conclusion form Figure 3(a) that theproportion of accurate positioning nodes will increase withthe increase of node communication radius When the nodecommunication radius increases to 8m the number ofaccurate positioning nodes reaches 50 And with the contin-uous increase of the node communication radius accuratepositioning of unknown nodes rate will be maintained at1 We can draw from Figure 3(b) that the average error ofaccurate positioning of node localization gradually decreaseswith the increase of node communication radius Whennode communication radius increases from 4m to 8m theaverage error decreases fast When node communicationradius changes from 8m to 12m the error is stable

When the communication radius is small network con-nectivity is low and the number of unknown nodes that cancommunicate with the node is reduced and a small node

communication radius (119877 = 8m) will get higher accuratepositioning node and low node localization error rate in theTMDS test The positioning result will be stable when theradius is increased

42 Influence of Network Nodes on the Performance of TMDSFigure 4 shows that when the node communication radius119877 is 8 the network node influences the TMDS algorithmAccording to the analysis of 41 when the communicationradius is 8m it is easier to analyze other factors influence ofnodes on the performance of the algorithm So in this sectionthe simulation environment is set as follows The number ofanchor nodes 119872 is 8 which are respectively distributed onthe 8 vertex positions of the cubemodel Nodes have the sameradius (119877 = 8m) of communication Change the proportionof anchor nodes by changing the number of unknown nodesand then analyze the influence of network nodes on theperformance of the algorithm

We can draw a conclusion from the curve of Figure 4that with the number of nodes increasing from 38 to 78 thatis the numbers of unknown nodes increase from 30 to 70accurate positioning of unknownnodes rate has been 1Whenthe network nodes number is 83 that is the unknown nodesnumber is 75 accurate localization rate begins to declineThe results show that when TMDS algorithm works in anenvironment of few anchor nodes and many network nodesthe location accuracy of unknownnodes can bemaintained at1 So when the TMDS algorithmworks in environment of fewanchor nodes the unknown nodes can be located accuratelyand the result is significant

43 Influence of the Proportion of Anchor Nodes on thePerformance of TMDS Figure 5 shows the proportion ofanchor nodes influence on the performance of algorithmThe simulation environment is as follows the number ofunknown nodes 119873 is 100 which are randomly distributed inthe space model Eight anchor nodes are distributed in the 8vertex positions of the cubemodel and others are distributedrandomly in the space model All the nodes have the samecommunication radius (119877 = 8m) Change the number of theanchor nodes to analyze the proportion of anchor impact onthe performance of the algorithm

We can draw form Figure 5(a) that the number ofaccurate positioning nodes will increase with the increaseof the proportion of anchor nodes When the proportion ofanchor nodes is 01 the number of accurate positioning nodesreaches 100 And the proportion of anchor nodes changesfrom 015 to 05 the rate of accurate positioning of unknownnodeswill bemaintained at 100Namelywhen the proportionof anchor nodes is greater than or equal to 015 TMDSalgorithm can locate the unknown nodes accurately

We can draw from Figure 5(b) that the unknown nodelocalization error gradually decreases with the increase of theproportion of anchor nodes When the proportion of anchornodes changes from 01 to 016 the unknown nodes locationerror decreases fast When the proportion of anchor nodeschanges from 016 to 05 the average location error fluctuatesa little But the positioning error is within the threshold rangeThe result of accurate positioning of unknownnodes is stable

Journal of Sensors 7

3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

50

Node communication radius

Num

ber o

f acc

urat

e pos

ition

ing

node

s

(a)

3 4 5 6 7 8 9 10 11 1200

01

02

03

04

05

06

07

08

Node communication radius

Aver

age e

rror

(m)

(b)

Figure 3 Node communication radius influence on the performance of TMDS

35 40 45 50 55 60 65 70 75 80 8520

25

30

35

40

45

50

55

60

65

70

Num

ber o

f acc

urat

e pos

ition

ing

node

s

Numbers of network nodes(a)

35 40 45 50 55 60 65 70 75 80 85000

002

004

006

008

010

012

Aver

age e

rror

(m)

Numbers of network nodes(b)

Figure 4 Influence of network nodes on the performance of TMDS

44 The Comparison between MDS Algorithm and TMDSAlgorithm Tables 2 and 3 represent MDS algorithm andTMDS algorithm for 10 times the continuous operationunder the same simulation environment respectively Thecontent of the table includes the average positioning errorof unknown nodes number of accurate positioning nodesnode location accuracy rate and the program running timeresults data The same simulation environment settings areas follows the number of anchor nodes 119872 is 8 which arerespectively distributed on the 8 vertex positions on a cubemodel all nodes have the same radius of communication(119877 = 8m)

From the accuracy of positioning algorithm TMDSalgorithm can achieve 9 times accurate positioning of all the

unknownnode in 10 times continuous operation whose loca-tion accuracy rate was 1 only 1 time accurately positioningnode number is 49 and its accurate localization rate is 098We can draw from Tables 2 and 3 that TMDS algorithmcan basically realize accurate positioning of the node MDSalgorithm can achieve 4 times the accurate positioning of allthe unknown node in 10 times the continuous operation andhave 2 times the low positioning rate respectively 008 and01

From the running time of the algorithm TMDS andMDS can complete the algorithm in a relatively short periodof time Time of MDS algorithm that completes the basicalgorithm within 13 ssim14 s is relatively stable TMDS takesa long time because of iterative computation of repeated

8 Journal of Sensors

005 010 015 020 025 030 035 040 045 05040

50

60

70

80

90

100N

umbe

r of a

ccur

ate p

ositi

onin

g no

des

Proportion of anchor nodes

(a)

005 010 015 020 025 030 035 040 045 050000

001

002

003

004

005

006

007

008

Aver

age e

rror

(m)

Proportion of anchor nodes

(b)

Figure 5 Influence of the proportion of anchor nodes on the performance of TMDS

Table 2 MDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 02881 00238 00031 00144 00360 00274 00162 02670 00152 00226Number of accurate positioning nodes 4 47 50 50 33 45 50 5 50 48Node location accuracy rate 008 094 1 1 066 09 1 01 1 096Program running time (s) 0142 0140 0137 0136 0131 0132 0137 0133 0137 0138

Table 3 TMDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 00101 00147 00079 00104 00197 00065 00090 00064 00070 00054Number of accurate positioning nodes 50 50 50 50 49 50 50 50 50 50Node location accuracy rate 1 1 1 1 098 1 1 1 1 1Program running time (s) 15754 34418 17085 33918 36783 33358 36259 36219 36137 36363

optimization The selection of the first local minimum valuefor repeating optimization of the algorithm of iterativecalculation is random and the global optimal iterations timeis different The distribution of 50 unknown nodes in modelspace is random So the time to complete the positioning ofall nodes will be different To sum up the Tunneling Methodis applied to solve the problem of local minimum value ofrepeated optimization algorithm and achieves good results inthe positioning of WSN

5 Conclusion

This paper uses the Tunneling Method technology in theWSN 3D node localization and proposes a new 3D wirelesssensor network node positioningmethod based onTunnelingMethod algorithm Innovation points of the TMDS localiza-tion algorithm are as follows

(i) TMDS algorithm is a kind of WSN distributednodes localization algorithm based on anchor node

multidimensional scaling technique The locationalgorithm directly calculates the absolute location ofthe unknown nodes by Euclidean distance betweenanchor nodes andnodes without coordinate transfor-mation and network fusion operation So it avoids thecumulative error

(ii) The core of TMDS is a kind of repeated optimizationmethod based on the new anchor nodes The methodis deduced from STRESS algorithm using TunnelingMethod to solve the local minimum problem ofrepeated optimization method to improve the accu-racy of the optimization results

(iii) The TMDS algorithm is an improved algorithm basedon MDS which makes use of the difference betweenthe nodes The calculation results are stable and thestability of the location effect is also better

The tunneling function algorithm is a certain globaloptimization method It can search the global minimum

Journal of Sensors 9

point in the minimization problem search and avoid fallinginto local minimum to get a smaller minimum point Repeatthis process to obtain the global optimal solution It improvesthe convergence speed of the algorithm and enhances theability of the global optimization The algorithm provides anewmethod in 3DWSN node positioningThe experimentalresult shows that the application of the repeated optimiza-tion algorithm and global optimization theory in 3D nodelocalization algorithm in WSN improves the rate of accuratepositioning of unknown nodes reduces the unknown nodelocation error and ensures that the nodeswhich are randomlydistributed in a three-dimensional wireless sensor networkcan be positioned accurately and stably In addition there areseveral limitations in the proposed method The tunnelingfunction requires its target function to be continuouslydifferentiable and the search time and computation willincrease when WSNs have large scale and high node density

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundations of China (Grant no 51305407)

References

[1] M Demirbas X Lu and P Singla ldquoAn in-network queryingframework for wireless sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 20 no 8 pp 1202ndash12152009

[2] V K Chaurasiya N Jain and G C Nandi ldquoA novel distanceestimation approach for 3D localization in wireless sensornetwork using multi dimensional scalingrdquo Information Fusionvol 15 no 1 pp 5ndash18 2014

[3] J-K Lee Y Kim J-H Lee and S-C Kim ldquoAn efficient three-dimensional localization scheme using trilateration in wirelesssensor networksrdquo IEEE Communications Letters vol 18 no 9pp 1591ndash1594 2014

[4] M Shon M Jo and H Choo ldquoAn interactive cluster-basedMDS localization scheme for multimedia information in wire-less sensor networksrdquo Computer Communications vol 35 no15 pp 1921ndash1929 2012

[5] H-B Chen D-Q Wang F Yuan and R Xu ldquoA MDS-basedlocalization algorithm for underwater wireless sensor networkrdquoinProceedings of theMTSIEEE SanDiegoConference AnOceanin Common (OCEANS rsquo13) San Diego Calif USA September2013

[6] 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

[7] E Kim S Lee C Kim and K Kim ldquoMobile beacon-based3D-localization with multidimensional scaling in large sensornetworksrdquo IEEE Communications Letters vol 14 no 7 pp 647ndash649 2010

[8] P J Groenen W J Heiser and J J Meulman ldquoGlobal opti-mization in least-squares multidimensional scaling by distancesmoothingrdquo Journal of Classification vol 16 no 2 pp 225ndash2541999

[9] P J F Groenen and W J Heiser ldquoThe tunneling method forglobal optimization in multidimensional scalingrdquo Psychome-trika vol 61 no 3 pp 529ndash550 1996

[10] J De Leeuw and W Heiser ldquoMultidimensional scaling withrestrictions on the configurationrdquo in Multivariate Analysis pp501ndash522 North-Holland Publishing 1980

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Sensors

0 2 4 6 8 10

0

5

10

0

2

4

6

8

10

y-axisx-axis

z-a

xis

Anchor node Unknown node

Figure 2 The distribution map in 3D for unknown node andanchor

described as follows the size of cube geometric model is10m times 10m times 10m and the nodes are placed in this model(the node communication radius is the same and all thenodes are stationary) The model is shown in Figure 2 Setthe simulation environments as follows when the iterationcalculation rule is the same node communication radius isalso the same Repeated optimization algorithm iterative timeis 60 The iterative computation error threshold is 00001Node localization error threshold 119890119903119903119900119903 119901 is 005mThe errorin distancemeasurement is taken as 15When the unknownnode location error meets 119890119903119903119900119903 le 119890119903119903119900119903 119901 the node will beregarded as the accurate localization of nodes

41 Node Communication Radius Influence on the Perfor-mance of the Algorithm Figure 3 shows the impact of nodecommunication radius on the performance of TMDS algo-rithmThe simulation environment is as follows the numberof unknown nodes 119873 is 50 which randomly distribute in a10m lowast 10m lowast 10m cube modelThe number of anchor nodes119872 is 8 which distribute in the 8 vertex positions of the cubemodel Change the node communication radius to analyze itsimpact on the performance of the algorithm

We can draw a conclusion form Figure 3(a) that theproportion of accurate positioning nodes will increase withthe increase of node communication radius When the nodecommunication radius increases to 8m the number ofaccurate positioning nodes reaches 50 And with the contin-uous increase of the node communication radius accuratepositioning of unknown nodes rate will be maintained at1 We can draw from Figure 3(b) that the average error ofaccurate positioning of node localization gradually decreaseswith the increase of node communication radius Whennode communication radius increases from 4m to 8m theaverage error decreases fast When node communicationradius changes from 8m to 12m the error is stable

When the communication radius is small network con-nectivity is low and the number of unknown nodes that cancommunicate with the node is reduced and a small node

communication radius (119877 = 8m) will get higher accuratepositioning node and low node localization error rate in theTMDS test The positioning result will be stable when theradius is increased

42 Influence of Network Nodes on the Performance of TMDSFigure 4 shows that when the node communication radius119877 is 8 the network node influences the TMDS algorithmAccording to the analysis of 41 when the communicationradius is 8m it is easier to analyze other factors influence ofnodes on the performance of the algorithm So in this sectionthe simulation environment is set as follows The number ofanchor nodes 119872 is 8 which are respectively distributed onthe 8 vertex positions of the cubemodel Nodes have the sameradius (119877 = 8m) of communication Change the proportionof anchor nodes by changing the number of unknown nodesand then analyze the influence of network nodes on theperformance of the algorithm

We can draw a conclusion from the curve of Figure 4that with the number of nodes increasing from 38 to 78 thatis the numbers of unknown nodes increase from 30 to 70accurate positioning of unknownnodes rate has been 1Whenthe network nodes number is 83 that is the unknown nodesnumber is 75 accurate localization rate begins to declineThe results show that when TMDS algorithm works in anenvironment of few anchor nodes and many network nodesthe location accuracy of unknownnodes can bemaintained at1 So when the TMDS algorithmworks in environment of fewanchor nodes the unknown nodes can be located accuratelyand the result is significant

43 Influence of the Proportion of Anchor Nodes on thePerformance of TMDS Figure 5 shows the proportion ofanchor nodes influence on the performance of algorithmThe simulation environment is as follows the number ofunknown nodes 119873 is 100 which are randomly distributed inthe space model Eight anchor nodes are distributed in the 8vertex positions of the cubemodel and others are distributedrandomly in the space model All the nodes have the samecommunication radius (119877 = 8m) Change the number of theanchor nodes to analyze the proportion of anchor impact onthe performance of the algorithm

We can draw form Figure 5(a) that the number ofaccurate positioning nodes will increase with the increaseof the proportion of anchor nodes When the proportion ofanchor nodes is 01 the number of accurate positioning nodesreaches 100 And the proportion of anchor nodes changesfrom 015 to 05 the rate of accurate positioning of unknownnodeswill bemaintained at 100Namelywhen the proportionof anchor nodes is greater than or equal to 015 TMDSalgorithm can locate the unknown nodes accurately

We can draw from Figure 5(b) that the unknown nodelocalization error gradually decreases with the increase of theproportion of anchor nodes When the proportion of anchornodes changes from 01 to 016 the unknown nodes locationerror decreases fast When the proportion of anchor nodeschanges from 016 to 05 the average location error fluctuatesa little But the positioning error is within the threshold rangeThe result of accurate positioning of unknownnodes is stable

Journal of Sensors 7

3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

50

Node communication radius

Num

ber o

f acc

urat

e pos

ition

ing

node

s

(a)

3 4 5 6 7 8 9 10 11 1200

01

02

03

04

05

06

07

08

Node communication radius

Aver

age e

rror

(m)

(b)

Figure 3 Node communication radius influence on the performance of TMDS

35 40 45 50 55 60 65 70 75 80 8520

25

30

35

40

45

50

55

60

65

70

Num

ber o

f acc

urat

e pos

ition

ing

node

s

Numbers of network nodes(a)

35 40 45 50 55 60 65 70 75 80 85000

002

004

006

008

010

012

Aver

age e

rror

(m)

Numbers of network nodes(b)

Figure 4 Influence of network nodes on the performance of TMDS

44 The Comparison between MDS Algorithm and TMDSAlgorithm Tables 2 and 3 represent MDS algorithm andTMDS algorithm for 10 times the continuous operationunder the same simulation environment respectively Thecontent of the table includes the average positioning errorof unknown nodes number of accurate positioning nodesnode location accuracy rate and the program running timeresults data The same simulation environment settings areas follows the number of anchor nodes 119872 is 8 which arerespectively distributed on the 8 vertex positions on a cubemodel all nodes have the same radius of communication(119877 = 8m)

From the accuracy of positioning algorithm TMDSalgorithm can achieve 9 times accurate positioning of all the

unknownnode in 10 times continuous operation whose loca-tion accuracy rate was 1 only 1 time accurately positioningnode number is 49 and its accurate localization rate is 098We can draw from Tables 2 and 3 that TMDS algorithmcan basically realize accurate positioning of the node MDSalgorithm can achieve 4 times the accurate positioning of allthe unknown node in 10 times the continuous operation andhave 2 times the low positioning rate respectively 008 and01

From the running time of the algorithm TMDS andMDS can complete the algorithm in a relatively short periodof time Time of MDS algorithm that completes the basicalgorithm within 13 ssim14 s is relatively stable TMDS takesa long time because of iterative computation of repeated

8 Journal of Sensors

005 010 015 020 025 030 035 040 045 05040

50

60

70

80

90

100N

umbe

r of a

ccur

ate p

ositi

onin

g no

des

Proportion of anchor nodes

(a)

005 010 015 020 025 030 035 040 045 050000

001

002

003

004

005

006

007

008

Aver

age e

rror

(m)

Proportion of anchor nodes

(b)

Figure 5 Influence of the proportion of anchor nodes on the performance of TMDS

Table 2 MDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 02881 00238 00031 00144 00360 00274 00162 02670 00152 00226Number of accurate positioning nodes 4 47 50 50 33 45 50 5 50 48Node location accuracy rate 008 094 1 1 066 09 1 01 1 096Program running time (s) 0142 0140 0137 0136 0131 0132 0137 0133 0137 0138

Table 3 TMDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 00101 00147 00079 00104 00197 00065 00090 00064 00070 00054Number of accurate positioning nodes 50 50 50 50 49 50 50 50 50 50Node location accuracy rate 1 1 1 1 098 1 1 1 1 1Program running time (s) 15754 34418 17085 33918 36783 33358 36259 36219 36137 36363

optimization The selection of the first local minimum valuefor repeating optimization of the algorithm of iterativecalculation is random and the global optimal iterations timeis different The distribution of 50 unknown nodes in modelspace is random So the time to complete the positioning ofall nodes will be different To sum up the Tunneling Methodis applied to solve the problem of local minimum value ofrepeated optimization algorithm and achieves good results inthe positioning of WSN

5 Conclusion

This paper uses the Tunneling Method technology in theWSN 3D node localization and proposes a new 3D wirelesssensor network node positioningmethod based onTunnelingMethod algorithm Innovation points of the TMDS localiza-tion algorithm are as follows

(i) TMDS algorithm is a kind of WSN distributednodes localization algorithm based on anchor node

multidimensional scaling technique The locationalgorithm directly calculates the absolute location ofthe unknown nodes by Euclidean distance betweenanchor nodes andnodes without coordinate transfor-mation and network fusion operation So it avoids thecumulative error

(ii) The core of TMDS is a kind of repeated optimizationmethod based on the new anchor nodes The methodis deduced from STRESS algorithm using TunnelingMethod to solve the local minimum problem ofrepeated optimization method to improve the accu-racy of the optimization results

(iii) The TMDS algorithm is an improved algorithm basedon MDS which makes use of the difference betweenthe nodes The calculation results are stable and thestability of the location effect is also better

The tunneling function algorithm is a certain globaloptimization method It can search the global minimum

Journal of Sensors 9

point in the minimization problem search and avoid fallinginto local minimum to get a smaller minimum point Repeatthis process to obtain the global optimal solution It improvesthe convergence speed of the algorithm and enhances theability of the global optimization The algorithm provides anewmethod in 3DWSN node positioningThe experimentalresult shows that the application of the repeated optimiza-tion algorithm and global optimization theory in 3D nodelocalization algorithm in WSN improves the rate of accuratepositioning of unknown nodes reduces the unknown nodelocation error and ensures that the nodeswhich are randomlydistributed in a three-dimensional wireless sensor networkcan be positioned accurately and stably In addition there areseveral limitations in the proposed method The tunnelingfunction requires its target function to be continuouslydifferentiable and the search time and computation willincrease when WSNs have large scale and high node density

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundations of China (Grant no 51305407)

References

[1] M Demirbas X Lu and P Singla ldquoAn in-network queryingframework for wireless sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 20 no 8 pp 1202ndash12152009

[2] V K Chaurasiya N Jain and G C Nandi ldquoA novel distanceestimation approach for 3D localization in wireless sensornetwork using multi dimensional scalingrdquo Information Fusionvol 15 no 1 pp 5ndash18 2014

[3] J-K Lee Y Kim J-H Lee and S-C Kim ldquoAn efficient three-dimensional localization scheme using trilateration in wirelesssensor networksrdquo IEEE Communications Letters vol 18 no 9pp 1591ndash1594 2014

[4] M Shon M Jo and H Choo ldquoAn interactive cluster-basedMDS localization scheme for multimedia information in wire-less sensor networksrdquo Computer Communications vol 35 no15 pp 1921ndash1929 2012

[5] H-B Chen D-Q Wang F Yuan and R Xu ldquoA MDS-basedlocalization algorithm for underwater wireless sensor networkrdquoinProceedings of theMTSIEEE SanDiegoConference AnOceanin Common (OCEANS rsquo13) San Diego Calif USA September2013

[6] 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

[7] E Kim S Lee C Kim and K Kim ldquoMobile beacon-based3D-localization with multidimensional scaling in large sensornetworksrdquo IEEE Communications Letters vol 14 no 7 pp 647ndash649 2010

[8] P J Groenen W J Heiser and J J Meulman ldquoGlobal opti-mization in least-squares multidimensional scaling by distancesmoothingrdquo Journal of Classification vol 16 no 2 pp 225ndash2541999

[9] P J F Groenen and W J Heiser ldquoThe tunneling method forglobal optimization in multidimensional scalingrdquo Psychome-trika vol 61 no 3 pp 529ndash550 1996

[10] J De Leeuw and W Heiser ldquoMultidimensional scaling withrestrictions on the configurationrdquo in Multivariate Analysis pp501ndash522 North-Holland Publishing 1980

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Sensors 7

3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

50

Node communication radius

Num

ber o

f acc

urat

e pos

ition

ing

node

s

(a)

3 4 5 6 7 8 9 10 11 1200

01

02

03

04

05

06

07

08

Node communication radius

Aver

age e

rror

(m)

(b)

Figure 3 Node communication radius influence on the performance of TMDS

35 40 45 50 55 60 65 70 75 80 8520

25

30

35

40

45

50

55

60

65

70

Num

ber o

f acc

urat

e pos

ition

ing

node

s

Numbers of network nodes(a)

35 40 45 50 55 60 65 70 75 80 85000

002

004

006

008

010

012

Aver

age e

rror

(m)

Numbers of network nodes(b)

Figure 4 Influence of network nodes on the performance of TMDS

44 The Comparison between MDS Algorithm and TMDSAlgorithm Tables 2 and 3 represent MDS algorithm andTMDS algorithm for 10 times the continuous operationunder the same simulation environment respectively Thecontent of the table includes the average positioning errorof unknown nodes number of accurate positioning nodesnode location accuracy rate and the program running timeresults data The same simulation environment settings areas follows the number of anchor nodes 119872 is 8 which arerespectively distributed on the 8 vertex positions on a cubemodel all nodes have the same radius of communication(119877 = 8m)

From the accuracy of positioning algorithm TMDSalgorithm can achieve 9 times accurate positioning of all the

unknownnode in 10 times continuous operation whose loca-tion accuracy rate was 1 only 1 time accurately positioningnode number is 49 and its accurate localization rate is 098We can draw from Tables 2 and 3 that TMDS algorithmcan basically realize accurate positioning of the node MDSalgorithm can achieve 4 times the accurate positioning of allthe unknown node in 10 times the continuous operation andhave 2 times the low positioning rate respectively 008 and01

From the running time of the algorithm TMDS andMDS can complete the algorithm in a relatively short periodof time Time of MDS algorithm that completes the basicalgorithm within 13 ssim14 s is relatively stable TMDS takesa long time because of iterative computation of repeated

8 Journal of Sensors

005 010 015 020 025 030 035 040 045 05040

50

60

70

80

90

100N

umbe

r of a

ccur

ate p

ositi

onin

g no

des

Proportion of anchor nodes

(a)

005 010 015 020 025 030 035 040 045 050000

001

002

003

004

005

006

007

008

Aver

age e

rror

(m)

Proportion of anchor nodes

(b)

Figure 5 Influence of the proportion of anchor nodes on the performance of TMDS

Table 2 MDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 02881 00238 00031 00144 00360 00274 00162 02670 00152 00226Number of accurate positioning nodes 4 47 50 50 33 45 50 5 50 48Node location accuracy rate 008 094 1 1 066 09 1 01 1 096Program running time (s) 0142 0140 0137 0136 0131 0132 0137 0133 0137 0138

Table 3 TMDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 00101 00147 00079 00104 00197 00065 00090 00064 00070 00054Number of accurate positioning nodes 50 50 50 50 49 50 50 50 50 50Node location accuracy rate 1 1 1 1 098 1 1 1 1 1Program running time (s) 15754 34418 17085 33918 36783 33358 36259 36219 36137 36363

optimization The selection of the first local minimum valuefor repeating optimization of the algorithm of iterativecalculation is random and the global optimal iterations timeis different The distribution of 50 unknown nodes in modelspace is random So the time to complete the positioning ofall nodes will be different To sum up the Tunneling Methodis applied to solve the problem of local minimum value ofrepeated optimization algorithm and achieves good results inthe positioning of WSN

5 Conclusion

This paper uses the Tunneling Method technology in theWSN 3D node localization and proposes a new 3D wirelesssensor network node positioningmethod based onTunnelingMethod algorithm Innovation points of the TMDS localiza-tion algorithm are as follows

(i) TMDS algorithm is a kind of WSN distributednodes localization algorithm based on anchor node

multidimensional scaling technique The locationalgorithm directly calculates the absolute location ofthe unknown nodes by Euclidean distance betweenanchor nodes andnodes without coordinate transfor-mation and network fusion operation So it avoids thecumulative error

(ii) The core of TMDS is a kind of repeated optimizationmethod based on the new anchor nodes The methodis deduced from STRESS algorithm using TunnelingMethod to solve the local minimum problem ofrepeated optimization method to improve the accu-racy of the optimization results

(iii) The TMDS algorithm is an improved algorithm basedon MDS which makes use of the difference betweenthe nodes The calculation results are stable and thestability of the location effect is also better

The tunneling function algorithm is a certain globaloptimization method It can search the global minimum

Journal of Sensors 9

point in the minimization problem search and avoid fallinginto local minimum to get a smaller minimum point Repeatthis process to obtain the global optimal solution It improvesthe convergence speed of the algorithm and enhances theability of the global optimization The algorithm provides anewmethod in 3DWSN node positioningThe experimentalresult shows that the application of the repeated optimiza-tion algorithm and global optimization theory in 3D nodelocalization algorithm in WSN improves the rate of accuratepositioning of unknown nodes reduces the unknown nodelocation error and ensures that the nodeswhich are randomlydistributed in a three-dimensional wireless sensor networkcan be positioned accurately and stably In addition there areseveral limitations in the proposed method The tunnelingfunction requires its target function to be continuouslydifferentiable and the search time and computation willincrease when WSNs have large scale and high node density

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundations of China (Grant no 51305407)

References

[1] M Demirbas X Lu and P Singla ldquoAn in-network queryingframework for wireless sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 20 no 8 pp 1202ndash12152009

[2] V K Chaurasiya N Jain and G C Nandi ldquoA novel distanceestimation approach for 3D localization in wireless sensornetwork using multi dimensional scalingrdquo Information Fusionvol 15 no 1 pp 5ndash18 2014

[3] J-K Lee Y Kim J-H Lee and S-C Kim ldquoAn efficient three-dimensional localization scheme using trilateration in wirelesssensor networksrdquo IEEE Communications Letters vol 18 no 9pp 1591ndash1594 2014

[4] M Shon M Jo and H Choo ldquoAn interactive cluster-basedMDS localization scheme for multimedia information in wire-less sensor networksrdquo Computer Communications vol 35 no15 pp 1921ndash1929 2012

[5] H-B Chen D-Q Wang F Yuan and R Xu ldquoA MDS-basedlocalization algorithm for underwater wireless sensor networkrdquoinProceedings of theMTSIEEE SanDiegoConference AnOceanin Common (OCEANS rsquo13) San Diego Calif USA September2013

[6] 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

[7] E Kim S Lee C Kim and K Kim ldquoMobile beacon-based3D-localization with multidimensional scaling in large sensornetworksrdquo IEEE Communications Letters vol 14 no 7 pp 647ndash649 2010

[8] P J Groenen W J Heiser and J J Meulman ldquoGlobal opti-mization in least-squares multidimensional scaling by distancesmoothingrdquo Journal of Classification vol 16 no 2 pp 225ndash2541999

[9] P J F Groenen and W J Heiser ldquoThe tunneling method forglobal optimization in multidimensional scalingrdquo Psychome-trika vol 61 no 3 pp 529ndash550 1996

[10] J De Leeuw and W Heiser ldquoMultidimensional scaling withrestrictions on the configurationrdquo in Multivariate Analysis pp501ndash522 North-Holland Publishing 1980

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Sensors

005 010 015 020 025 030 035 040 045 05040

50

60

70

80

90

100N

umbe

r of a

ccur

ate p

ositi

onin

g no

des

Proportion of anchor nodes

(a)

005 010 015 020 025 030 035 040 045 050000

001

002

003

004

005

006

007

008

Aver

age e

rror

(m)

Proportion of anchor nodes

(b)

Figure 5 Influence of the proportion of anchor nodes on the performance of TMDS

Table 2 MDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 02881 00238 00031 00144 00360 00274 00162 02670 00152 00226Number of accurate positioning nodes 4 47 50 50 33 45 50 5 50 48Node location accuracy rate 008 094 1 1 066 09 1 01 1 096Program running time (s) 0142 0140 0137 0136 0131 0132 0137 0133 0137 0138

Table 3 TMDS algorithm result

Operation times 1 2 3 4 5 6 7 8 9 10Average error (m) 00101 00147 00079 00104 00197 00065 00090 00064 00070 00054Number of accurate positioning nodes 50 50 50 50 49 50 50 50 50 50Node location accuracy rate 1 1 1 1 098 1 1 1 1 1Program running time (s) 15754 34418 17085 33918 36783 33358 36259 36219 36137 36363

optimization The selection of the first local minimum valuefor repeating optimization of the algorithm of iterativecalculation is random and the global optimal iterations timeis different The distribution of 50 unknown nodes in modelspace is random So the time to complete the positioning ofall nodes will be different To sum up the Tunneling Methodis applied to solve the problem of local minimum value ofrepeated optimization algorithm and achieves good results inthe positioning of WSN

5 Conclusion

This paper uses the Tunneling Method technology in theWSN 3D node localization and proposes a new 3D wirelesssensor network node positioningmethod based onTunnelingMethod algorithm Innovation points of the TMDS localiza-tion algorithm are as follows

(i) TMDS algorithm is a kind of WSN distributednodes localization algorithm based on anchor node

multidimensional scaling technique The locationalgorithm directly calculates the absolute location ofthe unknown nodes by Euclidean distance betweenanchor nodes andnodes without coordinate transfor-mation and network fusion operation So it avoids thecumulative error

(ii) The core of TMDS is a kind of repeated optimizationmethod based on the new anchor nodes The methodis deduced from STRESS algorithm using TunnelingMethod to solve the local minimum problem ofrepeated optimization method to improve the accu-racy of the optimization results

(iii) The TMDS algorithm is an improved algorithm basedon MDS which makes use of the difference betweenthe nodes The calculation results are stable and thestability of the location effect is also better

The tunneling function algorithm is a certain globaloptimization method It can search the global minimum

Journal of Sensors 9

point in the minimization problem search and avoid fallinginto local minimum to get a smaller minimum point Repeatthis process to obtain the global optimal solution It improvesthe convergence speed of the algorithm and enhances theability of the global optimization The algorithm provides anewmethod in 3DWSN node positioningThe experimentalresult shows that the application of the repeated optimiza-tion algorithm and global optimization theory in 3D nodelocalization algorithm in WSN improves the rate of accuratepositioning of unknown nodes reduces the unknown nodelocation error and ensures that the nodeswhich are randomlydistributed in a three-dimensional wireless sensor networkcan be positioned accurately and stably In addition there areseveral limitations in the proposed method The tunnelingfunction requires its target function to be continuouslydifferentiable and the search time and computation willincrease when WSNs have large scale and high node density

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundations of China (Grant no 51305407)

References

[1] M Demirbas X Lu and P Singla ldquoAn in-network queryingframework for wireless sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 20 no 8 pp 1202ndash12152009

[2] V K Chaurasiya N Jain and G C Nandi ldquoA novel distanceestimation approach for 3D localization in wireless sensornetwork using multi dimensional scalingrdquo Information Fusionvol 15 no 1 pp 5ndash18 2014

[3] J-K Lee Y Kim J-H Lee and S-C Kim ldquoAn efficient three-dimensional localization scheme using trilateration in wirelesssensor networksrdquo IEEE Communications Letters vol 18 no 9pp 1591ndash1594 2014

[4] M Shon M Jo and H Choo ldquoAn interactive cluster-basedMDS localization scheme for multimedia information in wire-less sensor networksrdquo Computer Communications vol 35 no15 pp 1921ndash1929 2012

[5] H-B Chen D-Q Wang F Yuan and R Xu ldquoA MDS-basedlocalization algorithm for underwater wireless sensor networkrdquoinProceedings of theMTSIEEE SanDiegoConference AnOceanin Common (OCEANS rsquo13) San Diego Calif USA September2013

[6] 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

[7] E Kim S Lee C Kim and K Kim ldquoMobile beacon-based3D-localization with multidimensional scaling in large sensornetworksrdquo IEEE Communications Letters vol 14 no 7 pp 647ndash649 2010

[8] P J Groenen W J Heiser and J J Meulman ldquoGlobal opti-mization in least-squares multidimensional scaling by distancesmoothingrdquo Journal of Classification vol 16 no 2 pp 225ndash2541999

[9] P J F Groenen and W J Heiser ldquoThe tunneling method forglobal optimization in multidimensional scalingrdquo Psychome-trika vol 61 no 3 pp 529ndash550 1996

[10] J De Leeuw and W Heiser ldquoMultidimensional scaling withrestrictions on the configurationrdquo in Multivariate Analysis pp501ndash522 North-Holland Publishing 1980

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Sensors 9

point in the minimization problem search and avoid fallinginto local minimum to get a smaller minimum point Repeatthis process to obtain the global optimal solution It improvesthe convergence speed of the algorithm and enhances theability of the global optimization The algorithm provides anewmethod in 3DWSN node positioningThe experimentalresult shows that the application of the repeated optimiza-tion algorithm and global optimization theory in 3D nodelocalization algorithm in WSN improves the rate of accuratepositioning of unknown nodes reduces the unknown nodelocation error and ensures that the nodeswhich are randomlydistributed in a three-dimensional wireless sensor networkcan be positioned accurately and stably In addition there areseveral limitations in the proposed method The tunnelingfunction requires its target function to be continuouslydifferentiable and the search time and computation willincrease when WSNs have large scale and high node density

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is partially supported by the National NaturalScience Foundations of China (Grant no 51305407)

References

[1] M Demirbas X Lu and P Singla ldquoAn in-network queryingframework for wireless sensor networksrdquo IEEE Transactions onParallel and Distributed Systems vol 20 no 8 pp 1202ndash12152009

[2] V K Chaurasiya N Jain and G C Nandi ldquoA novel distanceestimation approach for 3D localization in wireless sensornetwork using multi dimensional scalingrdquo Information Fusionvol 15 no 1 pp 5ndash18 2014

[3] J-K Lee Y Kim J-H Lee and S-C Kim ldquoAn efficient three-dimensional localization scheme using trilateration in wirelesssensor networksrdquo IEEE Communications Letters vol 18 no 9pp 1591ndash1594 2014

[4] M Shon M Jo and H Choo ldquoAn interactive cluster-basedMDS localization scheme for multimedia information in wire-less sensor networksrdquo Computer Communications vol 35 no15 pp 1921ndash1929 2012

[5] H-B Chen D-Q Wang F Yuan and R Xu ldquoA MDS-basedlocalization algorithm for underwater wireless sensor networkrdquoinProceedings of theMTSIEEE SanDiegoConference AnOceanin Common (OCEANS rsquo13) San Diego Calif USA September2013

[6] 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

[7] E Kim S Lee C Kim and K Kim ldquoMobile beacon-based3D-localization with multidimensional scaling in large sensornetworksrdquo IEEE Communications Letters vol 14 no 7 pp 647ndash649 2010

[8] P J Groenen W J Heiser and J J Meulman ldquoGlobal opti-mization in least-squares multidimensional scaling by distancesmoothingrdquo Journal of Classification vol 16 no 2 pp 225ndash2541999

[9] P J F Groenen and W J Heiser ldquoThe tunneling method forglobal optimization in multidimensional scalingrdquo Psychome-trika vol 61 no 3 pp 529ndash550 1996

[10] J De Leeuw and W Heiser ldquoMultidimensional scaling withrestrictions on the configurationrdquo in Multivariate Analysis pp501ndash522 North-Holland Publishing 1980

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of