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Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2013 Article ID 646287 11 pageshttpdxdoiorg1011552013646287

Research ArticleTarget Tracking with NLOS Detection and Mitigation inWireless Sensor Networks

Xiaoping Wu Junguo Hu Yujia Jiang Enbin Liu and Guoying Wang

United Laboratory of Low Carbon and Internet of Things Technology Zhejiang Agriculture amp Forestry UniversityHangzhou 311300 China

Correspondence should be addressed to Junguo Hu hawkhjg163com

Received 5 July 2013 Revised 28 August 2013 Accepted 13 September 2013

Academic Editor Yuan He

Copyright copy 2013 Xiaoping Wu 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

RN and SSR approaches are designed to detect NLOS propagation paths in this paper When the NLOS propagation paths aredetected an estimation approach for the NLOS range errors is proposed by using residual-error decomposition The approachcan estimate the NLOS range errors quickly and effectively even if there are multiple NLOS propagation paths Combining themovement equation with observation position the position of mobile target node can be tracked precisely with Kalman filteralgorithmUsing the estimatedNLOS range errors we correct the localization result andmodify Kalman filter tomitigate theNLOSpropagationsThe simulations demonstrate the validity with RN and SSR detectionmethods and analyze the impacts ofNLOS rangeerrors and number of NLOS anchor nodes The estimated NLOS range errors are proved to be close to the true especially whenthe NLOS range errors are much bigger than LOS range errors The simulation results show that the position of target node can betracked precisely and effectively when the NLOS mitigation is used to track the target node position with modified Kalman filter

1 Introduction

Recent advances in wireless communications microelectromechanical systems electronics and distributed processingtechnology have enabled the deployment of large numbers ofcheap and smart sensor nodes to construct wireless sensornetworks (WSNs) Deployed in themonitoring region a largenumber of sensor nodes form a multihop ad hoc networksystem through wireless communication These networkedsensors are able to process sensed data locally and extractrelevant information to collaborate with other sensors on theapplication specific task and to provide the resultant infor-mation about the monitored events for a number of poten-tial applications ranging from battlefield monitoring andenvironmental surveillance to health care [1ndash3] To make thedata collected from sensor nodesmeaningful it often requiresrelated node positions Target localization and tracking ofmobile nodes are important research directions inWSNs [4ndash10]

It is often the casewith a general assumption that the posi-tions of nodes deployed in the monitoring region are known(called anchor nodes) so that it is possible to track the

positions of the mobile target with a sensor node [11] Totrack and locate the mobile target node some measurementapproaches are proposed such as time of arrival (TOA) [1213] angle of arrival (AOA) [14] time difference of arrival(TDOA) and received signal strength indication (RSSI) [15]Some hybrid approaches of TOA AOA TDOA and RSSIhave also been proposed for target localization and trackingThe measured results are often transmitted to base stationwhich computes the current position of mobile target nodebased on the received measurements and the history Thenthe base station reports the tracking results to each sensornode or target node

Since the rangemeasurements are usually prone to errorsthe localization result of target node will be far from the trueposition [16ndash18] Some studies are focused on line-of-sight(LOS) assumption so the target node can be located preciselywith the traditional localization algorithms [19] Howeversince the direct path between nodes can be blocked by build-ings and other obstacles the transmitted signal could onlyreach the receiver through reflected diffracted or scatteredpaths called nonline-of-sight (NLOS) propagation pathsMost of previous researches on NLOS propagation have

2 International Journal of Distributed Sensor Networks

focused on the NLOS identification and mitigation [20ndash22] In [23] a positioning algorithm in severe NLOS prop-agation path scenarios is proposed to enhance positionalaccuracy of network-based positioning systems when theposition receiver does not perform well due to the complexpropagation environment In the event that the statisticsof the NLOS errors and measurement noise are knownsuch as those based on field trials statistical processing cansignificantly reduce the NLOS effect [24] When a database isestablished in advance signature matching can be employedto greatly improve position accuracy in NLOS scenarios Byexerting constraints or introducing an NLOS error-relatedparameter into the cost function optimization algorithmscan be developed to mitigate the NLOS effect [25 26] Someof researches take advantage of the NLOS propagation pathsrather than canceling them [27]

Using its movement velocities the position of target nodecan be approximately estimated when the initial positionof target node is known However the position of iterativeestimation with movement equation is prone to error whenthe movement velocities include noises In WSNs the posi-tion of target node also can be located by range measurementbetween the target node and anchor nodes known positionsWhen the range measurements include noises the localiza-tion result also would be imprecise [28] Kalman filter (KF)to deal with the linear function and its nonlinear extensionextended Kalman filter (EKF) provide a feasible solution tomitigate the position error of the mobile target node andtherefore improving accuracy of mobile target tracking [29ndash31]

To track the position of mobile target node precisely wepropose to mitigate the NLOS propagations and make gooduse of the LOS range measurements by detecting and identi-fying the NLOS propagation paths In this paper we firstlyintroduce two approaches to detect the NLOS propagationpaths when the range measurements conform to Gaussiandistribution The first detection approach utilizes residual ofnode (RN) which conforms to zero-mean Gaussian distribu-tion when there are no NLOS propagation paths The secondapproach to detect NLOS propagation paths utilizes sum ofsquare residual (SSR) which conforms to chi-square distribu-tion when no NLOS propagation paths exist The anchornodes with NLOS propagation paths are called NLOS anchornodes and the anchor nodes with LOS propagation paths arealso called LOS anchor nodes To identify the NLOS anchornodes we propose a residual-error decompositionmethod toestimate the NLOS range errors The localization result canbe improved when estimated NLOS range errors are usedto amend the residuals Then the corrected position is con-sidered as observation and used to track the mobile targetposition preciselyThe contributions of our work are summa-rized as follows

(1) RN and SSR Approaches to Detect the NLOS Anchor NodesUsing the Jacobian matrix we derive the residuals comingfrom range errors between the target node and each anchornode Applying the analysis method of multiple factor staticswe conclude that RN conforms toGaussian distributionwhenthe range measurements are Gaussian distributed Then we

demonstrate that the SSR of all anchor nodes conforms tochi-square distribution when no NLOS anchor nodes existSo the probability of RN and SSR can be used to detect theNLOS propagation paths precisely

(2) An Estimation Method for NLOS Range Errors Is ProposedWhen Multiple NLOS Anchor Nodes Simultaneously Exist Byanalyzing the relationships between the errors and the resid-uals in the process of nonlinearminimization localization wepropose an estimation method for NLOS range errors Sincethe residuals are caused by the range errors the range errorsof NLOS anchor nodes can be estimated approximately byignoring the assigned residuals from LOS anchor nodes tothe NLOS anchor nodes Using the estimated range errorsof anchor nodes the NLOS anchor nodes can be identifiedcorrectly

(3) To Tracking A Mobile Target Node Precisely the ModifiedKF Is Put Forward by Applying the NLOS Mitigation MethodBy mitigating the NLOS propagation paths the localizationaccuracy can be improved so the covariance of target nodeposition would be reduced The covariance of localized posi-tion is derived from the range noises between the target nodeand each anchor node Considering the localized positionas observation the position of target node is tracked withmodified KF based on the movement equation

This paper presents an effective tracking method ofmobile target node with modified KF in NLOS environmentThe rest of this paper is structured as follows Section 2presents two detection approaches for the NLOS propagationpaths Section 3 describes the estimation method of NLOSrange errors Section 4 introduces the target tracking algo-rithm with modified KF in NLOS environment Section 5analyzes the simulation results The conclusion is presentedin Section 6

2 Detection of NLOS Propagation Path

For simplicity we only focus on the case of tracking a singletarget node in a two-dimensional field covered with multipleanchor nodes Since most localization systems of wirelesscommunications may suffer from the NLOS propagationpaths and dense multipath situation it is an important issueto obtain higher accuracy in determining range informationIn dealing with the NLOS propagation path effects the rangemeasurement 119889

119894119895between target node position 119894 and the posi-

tion of anchor node 119895 corresponding to the TOA measure-ment metrics can be modeled as

119889

119894119895= 119889

119894119895+ 119899

119894119895+ 119887

119894119895 (1)

where 119894 means the time instant of tracked mobile node and119894 = 1 119872 119895 = 1 119873 represents the fact that there are119873measurable anchor nodes corresponding to the target nodeAs mentioned earlier usually the measurement noise values119899

119894119895are modeled as zero-mean Gaussian random variables

with variance 1205752119894119895 119887119894119895caused by NLOS propagation path is

a positive random variable There will be no NLOS errorcomponent if the LOSpropagation path exists and 119887

119894119895= 0We

International Journal of Distributed Sensor Networks 3

define Δ119889119894119895= 119899

119894119895+119887

119894119895 then Δ119889

119894119895simN(119887

119894119895 120575

2

119894119895) HereN(119887

119894119895 120575

2

119894119895)

denotes the Gaussian distribution with mean 119887119894119895and variance

120575

2

119894119895 119889119894119895represents the true distance between the target node

position 119894 and fixed position of anchor node 119895 and can bewritten as

119889

119894119895=

radic

(119909

119894minus 119909

119895)

2

+ (119910

119894minus 119910

119895)

2

(2)

where z119894= (119909

119894 119910

119894) is the true position of mobile target node

at time instant 119905119894 whereas z

119895= (119909

119895 119910

119895) is the position of

anchor node 119895 If the positions of anchor nodes are assumedto be accurate the localization problem can be performedin different ways including the mixed norm [32] Here theoptimization-based nonlinear minimization is consideredThe cost function 119891(z

119894) called as sum of square residual at

time instant 119905119894is defined as

119891 (z119894) =

119873

sum

119895=1

120596

119894119895Δ119889

2

119894119895=

119873

sum

119895=1

120596

119894119895(

119889

119894119895minus 119889

119894119895)

2

(3)

The weights 120596119894119895are selected to emphasize the contribution of

smaller error terms among Δ119889119894119895 Note that in minimization

or optimization when the weights are properly selected withrespect to the quality of each measurement better estima-tion results will be expected When the optimal weights areemployed optimal estimation accuracy would be producedIf the range 119889

119894119895are supposed to be independent respectively

120596

119894119895can be chosen to be inversely proportional to the variances

of the distance measurement errors using the classical max-imum likelihood (ML) estimator when Δ119889

119894119895sim N(0 1205752

119894119895) In

the event of unknown or very similar statistics of 120596119894119895 equal

weights can be simply used In this paper we simplify theoptimization problem with equal weights and consider 120596

119894119895as

oneEquation (3) can be solved by Gauss-Newton method

Based on a linear approximation to the components of 119891(z119894)

(a linear model of 119891(z119894)) Gauss-Newton method may fail

when trapped in a local optimum Levenberg-Marquardt(L-M) method is recommended for the global optimum[33] Because L-M algorithm uses the approximate secondderivative information the convergence of L-M is muchfaster than the gradient descent method of Gauss-Newton Itis proved that L-Malgorithm can increase the speed of dozensor even hundreds of times of the original gradient descentmethod of Gauss-Newton A compact matrix form of rangeerrors can be written as follows

Δd119894= [Δ119889

1198941 Δ119889

1198942 Δ119889

119894119873]

119879

(4)

With the linearization of the system using Taylor seriesapproximation the optimization problem of (3) can be trans-form to

119891 (z119894) = (Δd

119894minus J119894Δz119894)

119879(Δd119894minus J119894Δz119894)

(5)

where Δz119894is the incremental matrix of the true z

119894 J119894is the

Jacobian matrix of d119894at the true position of target node

J119894=

[

[

[

[

120597119889

1198941

120597119909

119894

120597119889

1198942

120597119909

119894

sdot sdot sdot

120597119889

119894119873

120597119909

119894

120597119889

1198941

120597119910

119894

120597119889

1198942

120597119910

119894

sdot sdot sdot

120597119889

119894119873

120597119910

119894

]

]

]

]

119879

(6)

Using the principle of least square method

Δz119894= (J119879119894J119894)

minus1

J119879119894sdot Δd119894

(7)

r119894= Δd119894minus J119894Δz119894is called residual which can be rewritten as

r119894= [I minus J

119894(J119879119894J119894)

minus1

J119879119894]Δd119894= A119894Δd119894 (8)

where

A119894= I minus J

119894(J119879119894J119894)

minus1

J119879119894

(9)

Apparently A119894represents the distribution relationship

between range errors and residuals As observed from (9)the positions of target node and anchor nodes determine A

119894

which can be represented as

A119894=

[

[

[

[

[

119886

11119886

12sdot sdot sdot 119886

1119873

119886

21119886


2119873

sdot sdot sdot sdot sdot sdot

119886

1198731119886


119873119873

]

]

]

]

]

(10)

r119894can be rewritten as follows

r119894= [119903

1198941 119903

1198942 119903

119894119873]

119879

(11)

Here two approaches are introduced to detect the NLOSpropagation path residual of node (RN) and sum of squareresiduals (SSR)

21 Residual of Node (RN) Since Δ119889119894119895simN(119887

119894119895 120575

2

119894119895) the mean

120583

119894119895and variance ]2

119894119895of 119903119894119895can be written as

120583

119894119895=

119873

sum

119896=1

119886

119894119896119887

119896119895

]2119894119895=

119873

sum

119896=1

119886

2

119894119896120575

2

119896119895

(12)

The RN 119903

119894119895will conform to Gaussian distribution N(120583

119894119895 ]2119894119895)

When 119887119894119895is equal to zero a test result of RN probability den-

sity function (PDF) is plotted in Figure 1 which shows thatthe residual approximately conforms toGaussian distributionwith zero-mean Based on probability theory we obtain that

119875(

1003816

1003816

1003816

1003816

1003816

119903

119894119895minus 120583

119894119895

1003816

1003816

1003816

1003816

1003816

]119894119895

lt 120572) = 120573(13)


minus4 minus3 minus2 minus1 0 1 2 3 40

001

002

003

004

005

006

007

Residual

PDF(X) based on 50000 data samples 50 bins

PDF

(times100

)

Figure 1 A test of RN PDF

where 120573 is the probability of observing a measurement 119903119894119895

Typically when 120573 is equal to 999 120572 is approximately 33Then

119903

119894119895minus 120572]119894119895lt 120583

119894119895lt 119903

119894119895+ 120572]119894119895 (14)

If there are no NLOS range errors 119887119894119895must be equal to zero

and 120583119894119895= 0 According to (14) 119903

119894119895minus 120572]119894119895lt 120583

119894119895 So we can

conclude that if 119903119894119895minus 120572]119894119895gt 0 that is

119903

119894119895gt 120572]119894119895 (15)

120583

119894119895gt 0 and theremust be at least oneNLOSpropagation path

In most actual NLOS situations we can further assume that119887

119894119895≫ 120575

119894119895 As observed from (8) the residual 119903

119894119895is proportional

to the NLOS error 119887119894119895approximately so we obtain 119903

119894119895gt 120572]119894119895

when the NLOS propagation path existsIn the previous localization model since there are 119873

anchor nodes each range measurement is likely to be NLOSTo ensure the probability of observation 120573 can be ensuredwith

120573

119873= 120574 (16)

where 120574 is the probability of an observation when there areno NLOS propagation paths 120574 can be determined in prior120573 =

119873radic120574 Typically when 120574 = 995 and119873 = 5 120573 = 999

22 Sum of Square Residuals (SSR) The sum of square resid-uals 119891(z

119894) can be rewritten as

119891 (z119894) = r119879119894r119894=

119873

sum

119895=1

119903

2

119894119895 (17)

Here 119903119894119895simN(120583

119894119895 ]2119894119895) Assuming that there are no NLOS pro-

pagation paths and 120583119894119895= 0 119895 = 1 119873 the PDF of 1199032

119894119895can

be represented as

119875 (119911) =

1

radic2120587]119894119895

119911

minus(12)119890

minus(1199112]2119894119895)119911 ge 0

0 119911 lt 0

(18)

0 5 10 15 20 250

010203040506070809

1

Sum of square residual

CDF

(times100

)

N = 4

N = 6

N = 8

Figure 2 A test of SSR CDF

where 119911 = 119903

2

119894119895 So 119911]2

119894119895conforms to the distribution 1205942(1)

Here 1205942(1) denotes the chi-square distribution with freedomdegree one When there are no NLOS propagation paths 119903

119894119895

conforms to the Gaussian distribution N(0 ]2119894119895) and is inde-

pendent respectivelyWe relax the distribution of 119903119894119895

119895=12119873

to Gaussian distribution N(0 ]2max) where ]2max stands forthe maximum variance of all ]2

119894119895

119895=12119873 Then the PDF of

119891(z119894)]2max will conform to 1205942(119873) distribution with freedom

degree 119873 A test result of SSR cumulative distributionfunction (CDF) is plotted in Figure 2 which shows that moreanchor nodes will lead to the increasing of SSR Based onprobability theory we also have

119875 120594

2(119873) gt 120594

2

120579(119873) = 120579 (19)

Typically when 119873 = 5 and 120579 = 0995 119891(z119894) will be at least

1675 which is called SSR threshold denoted as 119878119897 So if

119891 (z119894) gt 119878

119897 (20)

there must be at least one NLOS propagation path

3 Identification of NLOS Anchor Nodes

The range measurements may be prone to potential NLOSerrors The NLOS range measurements result in a distortedposition whereas LOS measurements can reflect originallythe anticipation In the 2-dimensional plane node localiza-tion requires only three noncollinear anchor nodes In mostsituations the number of anchor nodes is more than threeand redundant Our idea is to identify the NLOS propagationpaths and make good use of LOS measurements When theother anchor nodes happened to beNLOS propagation pathsonly using the LOS anchor nodes can locate the target nodeprecise Our approach is to correct the localization result andmitigate NLOS with estimated NLOS range errors so the


mobile target node position can be tracked precisely withmodified KF

If multiple NLOS propagation paths exist it is necessaryto discern which anchor nodes are the NLOS ones In thissection we introduce a low rank residual-error decomposi-tion method to estimate the NLOS range errors Expanding(8) the residual between target node position 119894 and anchornode 119895 can be represented as

119903

119894119895=

119873

sum

119896=1

119886

119895119896sdot Δ119889

119894119896 (21)

where 119886119895119895gt 0

119895=1119873and Δ119889

119894119896represents the range error

between the target node position 119894 and anchor node 119896 Undermost NLOS conditions the NLOS range error Δ119889

119894119896≫ 0

If there is a NLOS propagation path between target nodeposition 119894 and anchor node 119896 the residual 119903

119894119895must be

overenlargedThe larger residuals of all anchor nodes are con-sidered as the more possible to be happening of NLOS prop-agation path To judge whether there are NLOS propagationpaths or not we firstly calculate the sum of square residuals(SSR) and compare with the threshold If the SSR is inthreshold there are no NLOS propagation paths Otherwisewe consider that theremust be at least oneNLOS propagationpath

In (8) A119894= I minus J

119894(J119879119894J119894)

minus1J119894 The localization method

represented by (3) locates the target node by L-M algorithmso A119894and r119894have been calculated out Since the matrix A

119894is

not full rank the range errorΔd119894cannot be directly calculated

out A119894in (8) reveals the relationship between residuals and

errors When the number of NLOS anchor nodes is less therange errors of NLOS anchor nodes are assigned to the LOSanchor nodes evenly so a larger residual in r

119894will tend to

be a larger error in Δd119894 The NLOS anchor nodes can be

distinguished from the LOS anchor nodes with the residualsResorting A

119894according to the residuals A

119894is decomposed

as

A119894= [

A11

A12

A21

A22

] (22)

where A11shows that the range errors of LOS anchor nodes

are assigned to the residuals of LOS anchor nodesA12shows

that the range errors of LOS anchor nodes are assigned to theresiduals of NLOS anchor nodes A

21shows that the range

errors of NLOS anchor nodes are assigned to the residuals ofLOS anchor nodes A

22shows that the range errors of NLOS

anchor nodes are assigned to the residuals of NLOS anchornodesWhen the range errors of LOS anchor nodes are muchless than those of NLOS anchor nodes the range errors ofNLOS anchor nodes can be roughly estimated by ignoring theimpact of LOS anchor nodes To identify the NLOS anchornodes correctly at least two anchor nodes are firstly chosento be LOS ones so the matrix A

11would be 2 times 2 one

The anchor nodes with less residuals are considered as morepossible to be LOS ones so the two anchor nodes with leastresiduals are chosen to be LOS ones Similarly the residual r

119894

and the estimated range error Δd119894can also be decomposed

as

r119894= [

r1

r2

]

Δd119894= [

Δd1

Δd2

]

(23)

Therefore the NLOS range errors can be approximatelyestimated with

Δd2= Aminus122sdot r2 (24)

In Δd2 the anchor nodes with the larger estimated error

are considered as theNLOS ones and theNLOS anchor nodesare identified

Recalculating the Jacobian matrix with all anchor nodesin (7) and letting F

119894= (J119879119894J119894)

minus1J119879119894 the covariance of increment

Δz119894can be written as

Cov (Δz119894) = F119894Σ119894F119879119894 (25)

where Σ119894is the covariance of range error Δd

119894 If Δ119889

119894119895is

independent for 119895 = 1 2 119873 Σ119894can be represented as

Σ119894= diag 1205752

1198941 120575

2

1198942 120575

2

119894119873 (26)

4 Tracking with Modified KF

In target tracking applications the most popular methodsfor updating target node position incorporate variationsof Kalman filter estimator Kalman filter assumes that thedynamics of the target can be modeled and that noise affectsthe target dynamics and sensor measurements Since thelocalization of target node is an optimization problemof non-linear function the measurement conversion method is pro-posed to transform the nonlinear measurement model intolinear one and estimate the covariances of the convertedmea-surement noises before applying the standard Kalman filter

41 Target Motion Model A standard target moving in atwo-dimensional field for the mobile target node is usuallydescribed by its position and velocity in the119883-119884 plane

x119894= [119909 (119894) V

119909 (119894) 119910 (119894) V

119910 (119894)]

119879

(27)

where 119909(119894) and 119910(119894) are the position coordinates of the targetnode along 119883 and 119884 axes at time 119905

119894 respectively V

119909(119894) and

V119910(119894) are the velocities of the target node along 119883 and 119884 axes

at time 119905119894 respectively The following nearly constant velocity

model is adopted to represent the motion of the target node

x119894+1= Φ119894x119894+ Γ119894w119894 (28)

where Φ119894is called state transition matrix which can be

written as

Φ119894=

[

[

[

[

1 Δ119905

1198940 0

0 1 0 0

0 0 1 Δ119905

119894

0 0 0 1

]

]

]

]

(29)


and Γi is called as noise transition matrix which can bewritten as

Γ119894=

[

[

[

[

[

[

[

[

[

[

[

[

Δ119905

2

119894

2

0

Δ119905

1198940

0

Δ119905

2

119894

2

0 Δ119905

119894

]

]

]

]

]

]

]

]

]

]

]

]

(30)

In the previous equations Δ119905119894= 119905

119894+1minus 119905

119894is the sampling time

interval between 119905119894and 119905119894+1

w119894= [119908

119909119908

119910] is awhiteGaussiannoise sequencewith zeromean and covariancematrixQ

119908119908119909

and 119908119910represent the correspondence to noisy accelerations

along the 119883 and 119884 axes respectively If we assume that 119908119909is

uncorrelated with 119908119910Q119908can be given by

Q119908=

[

[

120575

2

1199081199090

0 120575

2

119908119910

]

]

(31)

where 1205752119908119909

and 1205752119908119910

are the variances of noisy acceleration119908

119909and 119908

119910 respectively It is noted that our moving model

of target node does not consider the case where the movingtarget node follows a given trajectory which happens whenthe target node travels on a given road segment But if suchtrajectory is available as in the case when a road map isavailable the system model for the moving target node canbe easily modified and our approach is still applicable

42 Modified Observation Model The localization result by(3) is considered as the observation Here the localizationresult is denoted as z

119894by (3) The position of target node can

be modified as

z119894= Hx119894+ u119894 (32)

whereH is called measurement matrix which can be writtenas

H = [

1 0 0 0

0 0 1 0

] (33)

u119894is called as measurement noise which is equal to Δz

119894

determined by localization algorithm and range errors Itremains to specify the statistics for noise u

119894before the local-

ization result z119894can act as observation and be used in Kalman

filtering The covariance matrix of u119894is denoted as R

119894 which

will be used to evaluate the observation quality in Kalmanfiltering Apparently the NLOS propagation paths wouldmake the observation z

119894greatly far from the true position In

order to track the target node position precisely R119894should be

increased when there are NLOS propagation paths Observedfrom (7) Δz

119894also conforms to Gaussian distribution since it

is linear with range error Δd119894 So we obtain that

R119894= Cov (Δz

119894) = F119894Σ119894F119879119894 (34)

which evaluates the observation quality

If there are no NLOS propagation paths z119894= z119894in

(32) The NLOS propagation paths aggravate the localizationresult so the observation of target node position would be farfrom the true position The estimated NLOS range errors in(24) can be used to correct the observation The range errorsof LOS anchor nodes are assumed as zero then

Δd = [0 0Δd2] (35)

So if there are NLOS propagation paths the observation z119894

will be modified as

z119894= z119894+ F119894Δd (36)

whereF119894Δd represents the incremental position errors caused

by NLOS range errors

43 Kalman Filtering The iterative operations of the Kalmanfilter can be summarized as follows

x119894+1|119894

= Φ119894x119894|119894

P119894+1|119894

= Φ119894P119894|119894Φ119879

119894+ Γ119894Q119908Γ119879

119894

K119894+1= P119894+1|119894

H119879[HP119894+1|119894

H119879 + R119894+1]

minus1

x119894+1|119894+1

= x119894+1|119894

+ K119894+1[z119894+1minusHx119894+1|119894

]

P119894+1|119894+1

= P119894+1|119894

minus K119894+1

HP119894+1|119894

(37)

The initial estimates are given as x0|0

= x0and P

0|0=

P0 which is defined as a large positive definite value in

prior Under the LOS case unbiased smoothing is used forestimating the true position of target node When the NLOSstatus is detected the uncertainty of target node positionobservation will be increased Our scheme of target trackingwith KF in NLOS environment can be illustrated in Figure 3and Algorithm 1

5 Simulation Results

To track the target node in NLOS environment we firstlyidentify the NLOS anchor nodes based on statics model andestimate the NLOS range errors with the method of residual-error decompositionWe derive the covariance of localizationresult coming from range noises when the range errorsconform to Gaussian distribution By correcting the obser-vations with the estimated NLOS range errors the positionsof target node would be tracked precisely Then the iterativeKF algorithm is applied to improve the accuracy of mobiletarget node position The simulations firstly demonstrate thetwo detection approaches for NLOS propagation paths

51 Detection of NLOS Propagation Paths Residual of node(RN) and sum of square residuals (SSR) are used to judgewhether there are NLOS propagation paths or not whenthe range errors conform to Gaussian distribution In (8)A119894represents the relationship between residual of nodes and

range errors When single anchor node NLOS propagation


InputΦ119894 state transition matrix Γ

119894 noise transition matrixQ

119908 variances matrix of noisy acceleration

H measurement matrix x0and P

0 initial estimates 1205752

119894119895 range measurement variance

Output x = (x1 x

119872) positions of target node

(1) locate the target node with all anchor nodes by (3)(2) RN or SSR to detect the NLOS propagation paths(3) while there are NLOS propagation paths do(4) estimate the NLOS range errors with (24)(5) correct the observation with (36)(6) end while(7) let R

119894= Cov (Δz

119894) = F119894sum

119894F119879119894

(8) target tracking with KF with (37)

Algorithm 1 Target tracking with modified KF in NLOS environment

Localizationby equation (3)

RN or SSR to detectNLOS propagation NLOS identification

Judgment NLOS

LOS

Estimate NLOSrange errors

Tracking withmodified KF

Figure 3 NLOS identification for target tracking with modified KF

path the residual of the NLOS anchor node will remarkablybe bigger than that of the other LOS anchor nodes With theincreasing of NLOS range error the residual of NLOS anchornode will be much bigger than that of the other LOS anchornodes The RN approach to detect the NLOS propagationpath will bemore effective with the increasing of single NLOSrange error

The position of target node is set at (50 50) and fiveanchor nodes are deployed in 100m times 100m region All ofrange errors conform to Gaussian distribution N(0 1) Inparticular one of anchor nodes includes NLOS range erroradded from 0m to 10mThe RN CDF of NLOS anchor nodeis plotted in Figure 4(a) When the NLOS range error is zero(there are no NLOS propagation paths) the residual of theanchor node is distributed evenly about 119884-axis If the NLOSrange error is increased to 5m about 7 residual are morethan 2m However when the NLOS range error is increasedto 10m the residual of the NLOS anchor node is at least 4m

If there is only one NLOS propagation path of all anchornodes the sum of square residuals (SSR) will be increasedmonotonously with the increasing of NLOS range error Theprinciple of SSR is same as that of RN since the singleNLOS anchor node dominates most residual in all anchornodes The SSR CDF of all anchor nodes is plotted in Figure4(b) When NLOS range error is zero (there are no NLOSpropagation paths) only very few SSR aremore than 10 If theNLOS range error of the NLOS anchor node is increased to 5m most SSR are enlarged and about 40 SSR are more than20However when theNLOS range error is increased to 10mthe least SSR is 20m and about 95 SSR are more than 40

Apparently the number of NLOS anchor nodes alsoaffects RN and SSR The simulations show that more NLOSanchor nodes cannot ensure the increasing of RN and SSRObserved from the matrix A

119894 the exact position distribution

of NLOS anchor nodes would make the residuals offset eachother When there are multiple NLOS propagation paths RNand SSR are possible to be reduced In the situations thedetection approaches of RN and SSR would be invalid Afeasible approach is to reselect the less anchor nodes againand compare with the previous RN and SSR when multipleanchor nodes are involved in NLOS propagation paths Ifthe multiple RN and SSR are in accord with each other noNLOS propagation paths can be concluded Apparently thereselecting and detection with different anchor nodes needplenty of computation costs

Another concerned problem is the successful detectionratio of NLOS propagation path We assume that the vari-ances of range error Δ119889

119894119895are all equal to 120575

2 for 119895 =

1 2 119873 The variance 1205752 of LOS range error determinesthe detection threshold Less 1205752 will ensure to detect NLOSpropagation paths successfullywhen keeping theNLOS rangeerror invariable Similarly five anchor nodes are placed on100m times 100m region and the target node is set at (50 50)Firstly the range errors of all anchor nodes conform to theGaussian distribution N(0 1205752) and one of anchor nodesincludes NLOS range error tuned from 1m to 10m We set120574 = 995 whichmeans 120573 = 999 and 120572 = 33when119873 = 5120579 is also set to 995 and the detection threshold 119878

119897= 1675

120575

2 is set to 025 1 and 4 respectivelyThe curves in Figure 4(c) compare the successful detec-

tion ratio of NLOS propagation path with RN and SSR


NLOS range error = 0 mNLOS range error = 5 mNLOS range error = 10 m

minus6 minus4 minus2 0 2 4 6 8 100

01

02

03

04

05

06

07

08

09

1

Residual of NLOS anchor node

CDF

(times100

)

(a) RN CDF of NLOS anchor node


0 20 40 60 80 100 1200

02

04

06

08

1


CDF

(times100

)

(b) SSR CDF of all anchor nodes

Succ

essfu

l det

ectio

n ra

tio (times

100

)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

NLOS range error (m)

SSR and 1205752 = 1SSR and 1205752 = 4SSR and 1205752 =025

RN and 1205752 = 1RN and 1205752 = 4RN and 1205752 = 025

(c) Comparison of NLOS detection ratio

Figure 4 NLOS propagation detection with SSR and RN approaches

approaches It can be seen that the successful detection ratioof NLOS propagation path increases with larger NLOS rangeerror and smaller variance of LOS range error When theNLOS range error is equal to 4mand 1205752 = 025 the successfuldetection ratio ofNLOSpropagation path is almost 100withthe approach of RN and SSRHowever when theNLOS rangeerror is equal to 4m and 1205752 = 1 the successful detection ratioof NLOS propagation path is decreased to 599 with SSRapproach or 456 with RN approach However when theNLOS range error is equal to 4m and 1205752 = 4 the successfuldetection ratio ofNLOSpropagation path is decreased to 77with SSR approach or 38 with RN approach Comparingthe successful detection ratio with two different approachesthe performance of SSR approach is slightly better than thatof RN approach

52 Estimation of NLOS Range Errors The RN and SSNapproaches can judge whether there are NLOS propagationpaths When the NLOS propagation paths are identified theresidual-error decomposition method is used to estimate theNLOS range errors which correct the observation Equation(24) illustrates the estimated NLOS range errors with themethod of low rank residual-error decomposition Withthe estimated NLOS range errors the NLOS anchor nodescan be identified The estimation method of residual-errordecomposition can estimate multiple NLOS range errorssimultaneously The simulations test the performance of ourNLOS range errors estimation method

Let the geographical region bemarked by a 100m times 100mregion There are 10 anchor nodes placed randomly in theregion and a target node is placed randomly in the area Each


1 2 3 4 5 6 7 8 9 10minus4

minus2

0

2

4

6

8

10

12

Range error of three NLOS anchor nodes (m)

Estim

ated

rang

e err

or (m

)

First NLOS anchor nodeSecond NLOS anchor nodeThird NLOS anchor node

One of LOS anchor nodesTrue NLOS range error

Figure 5 Estimated range errors and true NLOS range errors

distance between the target node and each anchor node canbe measured We let the range errors on all links conformto a Gaussian distribution N(0 1) To simulate the NLOSrange measurements on the links we assume that the NLOSrange errors of three NLOS links are added from 1m to 10msimultaneously

The curves in Figure 5 plot the relationships betweenestimated NLOS range errors and true NLOS range errorsWhen the NLOS range errors of the three NLOS anchornodes are small the residuals caused by LOS anchors takemost parts in the total residuals and the estimated NLOSrange errors of three NLOS anchor nodes are impreciseenough due to the Gaussian errors of LOS anchor nodesWith the increasing of non-Gaussian NLOS range errorsof three NLOS anchor nodes the residuals caused by non-Gaussian NLOS range errors of three NLOS anchor nodesdominate in the total residuals Ignoring the impact of LOSanchor nodes the NLOS range errors of NLOS anchor nodescan be estimated approximately with (24) When the NLOSrange errors of three NLOS anchor nodes are set to 10m theestimatedNLOS range errors ofNLOS anchor nodes are closeto the true However the estimated LOS range error of LOSanchor node is still slightly fluctuated around zero when theNLOS range errors of three NLOS anchor nodes vary from1m to 10m

In order to evaluate the accuracy of estimated NLOSrange errors to a mobile target node the NLOS range errorsare estimated along the tracking path In the simulation sixanchor nodes are randomly deployed in a 200m times 200mregion Range errors of five LOS anchor nodes conform toGaussian distribution N(0 1) but one of anchor nodes hasNLOS propagation path The target node walks forward atthe velocity of 1ms 1ms in the direction of axis 119883 and 119884respectively from the origin That is to say x

0= [0 1 0 1]

The NLOS range error varies from 5m to 15m along thesimulated trajectory We keep the sample interval Δ119905 = 1 and

0 50 100 150 200minus5

0

5

10

15

20

25

Times (s)

Rang

e err

or (m

)

True NLOS range errorEstimated NLOS range error

Estimated LOS range errorTrue LOS range error

Figure 6 Estimated NLOS and LOS range error with time instant

sample 200 seconds The true NLOS range error Δ119889 of theNLOS anchor node varies as follows

Δ119889 =

5 +

15119905

200

times rand (1 1) 119905 le 70 or 119905 ge 130

5 +

15119905

200

70 lt 119905 lt 130

(38)

The simulation results are plotted in Figure 6 Since theresidual of LOS anchor node is also affected by the NLOSrange error the estimated LOS error is close to zero Theestimated NLOS range error is far from the estimated LOSrange error and fluctuated with the true slightly

53 Tracking with Modified KF If the movement equationand observation can be represented with the linear functionsthe position of mobile target node can be tracked more pre-cisely with KFThe movement equation is simulated as linearone affected by a white Gaussian noise w

119894 Correcting the

observationwith the estimatedNLOS range errors theNLOSpropagations will be mitigated The simulations also demon-strate the performance of target tracking with modified KF

There are six anchor nodes placed in 200m times 200mregion on which a target node is moving at the velocities of1ms 1ms in the direction of axis 119883 and 119884 The velocitiesare affected by the noise acceleration with the covariance ofQ119908= [004 0 0 004] The sampling time is Δ119905

119894= 1 s All

of range errors between the target node and each anchornodes conform to Gaussian distribution N(0 1) but one ofanchor nodes includes NLOS range error of 10m Let x

0=

[0 1 0 1] and 1198750= [004 0 0 004]

Assuming that the true location of target node is (119909119894119904 119910

119894119904)

the root mean square error (RMSE) at the time instant 119894 isdefined as

RMSE119894=radic

1

119879

119879

sum

119896=1

[(119909

119894119896minus 119909

119894119904)

2+ (119910

119894119896minus 119910

119894119904)

2]

(39)


10 20 30 40 50 60 70 80 90 10005

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)

0 NLOS and no corrected1 NLOS and corrected1 NLOS and no corrected

(a) RMSE comparison of three different conditions

0 20 40 60 80 1000

05

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)

Our modified KFKFCRLB

(b) Comparison of RMSE in NLOS conditions

Figure 7 Tracking with modified KF in NLOS environments

In order to evaluate the precision obtained by the correctedlocalization result we have computed the RMSE of three dif-ferent conditionsThenumber119879 ofMonteCarlo (MC) testingis set to 200 Figure 7(a) plots the RMSE of three differentconditions If the localization result is not corrected withthe estimated NLOS range error the RMSE of target nodeposition is fluctuated around 33m Due to the NLOS propa-gation path the localized position of target node is far awayfrom the true If there are no NLOS propagation paths theRMSE of target node position is about 07m By correctingthe localization result with the estimated NLOS range errorthe RMSE of target node position is fluctuated around 12m

The localization result is considered as the observationSince the modified KF algorithm utilizes the corrected targetnode position its position estimation error is much smallerthan that of the original KF method Computer simulationshave been conducted to evaluate the tracking performanceof the proposed methods by comparing with Cramer-Raolower bound (CRLB) when the range errors are Gaussiandistributed The curves in Figure 7(b) compare the RMSE ofKF algorithm modified KF algorithm and CRLB of targetnode position Due to the imprecise observation the RMSEof KF algorithm is much larger than that of modified KFalgorithm The RMSE of modified KF algorithm is almostclose to that of CRLB

6 Conclusion

Wehave studied themobile target tracking for wireless sensornetworks in NLOS environment and proposed a novel NLOSidentification and mitigation method which are applied totrack the mobile target node Firstly we provide RN and SSRdetection approaches for NLOS propagation path when therange errors conform to Gaussian distribution The RN andSSR approach are effective to detect the NLOS propagation

path when there is only one NLOS anchor node Morethan one NLOS propagation paths would make the residualsoffset and cannot ensure to detect the NLOS propagationpaths successfully When there are multiple propagationpaths simultaneously selecting the anchor nodes over againand rejudging with RN and SSR can identify the NLOSpropagation paths effectively Apparently the reselectionwould improve the performance of the NLOS propagationdetection but it adds the computation costs

If there aremultipleNLOS propagation paths we proposean estimation method for NLOS range errors with the lowrank residual-error decomposition The method of residual-error decomposition can estimate NLOS range errors quicklyeven if there are multiple NLOS propagation paths Since theNLOS range errors are much larger than LOS range errorsthe NLOS anchor nodes can be identified with the estimatedrange errors Using the estimated NLOS range errors wecorrect the localization result and improve the observationConsidering the corrected result as the observation theposition of mobile target node can be tracked precisely Ourapproaches to detect and identify the NLOS propagationpaths provide a novel idea for tracking themobile target nodefor wireless sensor networks

Acknowledgments

This study is supported by the NSF China Major Pro-gram 61190114 and NSF China Program 61174023 EducationDepartment of Zhejiang province Scientific Research ProjectY201328700 ZAFU Advanced Research Foundation Project2010FK045 NSYF China Programs 31300539 and 61303236Zhejiang provincial Natural Science Foundation LY12F02016and Zhejiang Province Key Science and Technology Innova-tion Team 2012R10023-02


References

[1] M Li and Y Liu ldquoUnderground structure monitoring withwireless sensor networksrdquo in Proceedings of the 6th InternationalSymposium on Information Processing in Sensor Networks (IPSNrsquo07) pp 69ndash78 April 2007

[2] L Mo Y He Y Liu et al ldquoCanopy closure estimates withGreenOrbs sustainable sensing in the forestrdquo in Proceedings ofthe 7th ACM Conference on Embedded Networked Sensor Sys-tems (SenSys rsquo09) pp 99ndash112 November 2009

[3] X Wu S Tan T Chen X Yi and D Dai ldquoDistributeddynamic navigation for sensor networksrdquo Tsinghua Science andTechnology vol 16 no 6 pp 648ndash656 2011

[4] Z Yang and Y Liu ldquoUnderstanding node localizability of wire-less AdHoc and sensor networksrdquo IEEETransactions on Paralleland Distributed Systems vol 11 no 8 pp 1249ndash1260 2012

[5] X Wu S Tan and Y He ldquoEffective error control of iterativelocalization for wireless sensor networksrdquo International Journalof Electronics and Communications vol 67 no 5 pp 397ndash4052013

[6] N Patwari J N Ash and S Kyperountas ldquoCooperative local-ization in wireless sensor networksrdquo IEEE Signal ProcessingMagazine vol 22 no 4 pp 54ndash68 2005

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

[8] Y He Y Liu X Shen L Mo and G Dai ldquoNoninteractive loca-lization of wireless camera sensors with mobile beaconrdquo IEEETransactions on Mobile Computing vol 12 no 2 pp 333ndash3452013

[9] S Rallapalli L Qiu Y Zhang and Y-C Chen ldquoExploitingtemporal stability and low-rank structure for localization inmobile networksrdquo in Proceedings of the 16th Annual Conferenceon Mobile Computing and Networking (MobiCom rsquo10) pp 161ndash172 September 2010

[10] Y Liu Z Yang X Wang and L Jian ldquoLocation localizationand localizabilityrdquo Journal of Computer Science and Technologyvol 25 no 2 pp 274ndash297 2010

[11] S Capkun M Hamdi and J-P Hubaux ldquoGPS-free positioningin mobile ad hoc networksrdquo Cluster Computing vol 5 no 2 pp157ndash167 2001

[12] Y-T Chan W-Y Tsui H-C So and P-C Ching ldquoTime-of-arrival based localization under NLOS conditionsrdquo IEEETransactions on Vehicular Technology vol 55 no 1 pp 12ndash242006

[13] I Guvenc and C-C Chong ldquoA survey on TOA based wirelesslocalization andNLOSmitigation techniquesrdquo IEEE Communi-cations Surveys and Tutorials vol 11 no 3 pp 107ndash124 2009

[14] L Cong and W Zhuang ldquoHybrid TDOAAOA mobile userlocation for wideband CDMA cellular systemsrdquo IEEE Trans-actions on Wireless Communications vol 1 no 3 pp 439ndash4472002

[15] Z Zhong and T He ldquoAchieving range-free localization beyondconnectivityrdquo in Proceedings of the 7th ACM Conference onEmbedded Networked Sensor Systems (SenSys rsquo09) pp 281ndash294November 2009

[16] H T Kung C-K Lin T-H Lin and D Vlah ldquoLocalizationwith snap-inducing shaped residuals (SISR) coping with errorsin measurementrdquo in Proceedings of the 15th Annual ACMInternational Conference on Mobile Computing and Networking(MobiCom rsquo08) pp 333ndash344 September 2009

[17] L Jian Z Yang and Y Liu ldquoBeyond triangle inequality siftingnoisy and outlier distance measurements for localizationrdquo inProceedings of IEEE INFOCOM March 2009

[18] D Moore J Leonard D Rus and S Teller ldquoRobust dis-tributed network localization with noisy range measurementsrdquoin Proceedings of the 2nd International Conference on EmbeddedNetworked Sensor Systems (SenSys rsquo04) pp 50ndash61 November2004

[19] J Lee K Cho S Lee T Kwon and Y Choi ldquoDistributed andenergy-efficient target localization and tracking in wireless sen-sor networksrdquoComputer Communications vol 29 no 13-14 pp2494ndash2505 2006

[20] C K Seow and S Y Tan ldquoNon-Line-of-Sight localization inmultipath environmentsrdquo IEEE Transactions on Mobile Com-puting vol 7 no 5 pp 647ndash660 2008

[21] J-F Liao and B-S Chen ldquoRobust mobile location estimatorwith NLOS mitigation using interacting multiple model algo-rithmrdquo IEEE Transactions on Wireless Communications vol 5no 11 pp 3002ndash3006 2006

[22] X Wang M Fu and H Zhang ldquoTarget tracking in wirelesssensor networks based on the combination of KF and MLEusing distance measurementsrdquo IEEE Transactions on MobileComputing vol 11 no 4 pp 567ndash576 2012

[23] C Ma R Klukas and G Lachapelle ldquoA nonline-of-sight error-mitigation method for TOAmeasurementsrdquo IEEE Transactionson Vehicular Technology vol 56 no 2 pp 641ndash651 2007

[24] M Nezafat M Kaveh H Tsuji and T Fukagawa ldquoStatisticalperformance of subspace matching mobile localization usingexperimental datardquo in Proceedings of the IEEE 6th Workshopon Signal Processing Advances in Wireless Communications(SPAWC rsquo05) pp 645ndash649 June 2005

[25] K Yu and Y J Guo ldquoImproved positioning algorithms fornonline-of-sight environmentsrdquo IEEE Transactions on Vehicu-lar Technology vol 57 no 4 pp 2342ndash2353 2008

[26] K Yu and E Dutkiewicz ldquoGeometry and motion-based posi-tioning algorithms formobile tracking inNLOS environmentsrdquoIEEE Transactions on Mobile Computing vol 11 no 2 pp 254ndash263 2012

[27] H Miao K Yu and M J Juntti ldquoPositioning for NLOS propa-gation algorithm derivations and Cramer-Rao boundsrdquo IEEETransactions on Vehicular Technology vol 56 no 5 pp 2568ndash2580 2007

[28] AN Campos E L Souza F GNakamura E F Nakamura andJ J P C Rodrigues ldquoOn the impact of localization and densitycontrol algorithms in target tracking applications for wirelesssensor networksrdquo Sensors vol 12 pp 6930ndash6952 2012

[29] B L Le K Ahmed and H Tsuji ldquoMobile location estimatorwith NLOSmitigation using kalman filteringrdquo in Proceedings ofthe Wireless Communications and Networking 2003

[30] W Ke and L Wu ldquoMobile location with NLOS identificationand mitigation based on modified Kalman filteringrdquo Sensorsvol 11 no 2 pp 1641ndash1656 2011

[31] B-S Chen C-Y Yang F-K Liao and J-F Liao ldquoMobile loca-tion estimator in a rough wireless environment using extendedKalman-based IMM and data fusionrdquo IEEE Transactions onVehicular Technology vol 58 no 3 pp 1157ndash1169 2009

[32] W-Y Chiu and B-S Chen ldquoMobile location estimation inurban areas using mixed ManhattanEuclidean norm and con-vex optimizationrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 1 pp 414ndash423 2009

[33] K Madsen H Nielsen and O Tingleff ldquoOptimization withconstraintsrdquo Tech Rep IMM DTU 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



focused on the NLOS identification and mitigation [20ndash22] In [23] a positioning algorithm in severe NLOS prop-agation path scenarios is proposed to enhance positionalaccuracy of network-based positioning systems when theposition receiver does not perform well due to the complexpropagation environment In the event that the statisticsof the NLOS errors and measurement noise are knownsuch as those based on field trials statistical processing cansignificantly reduce the NLOS effect [24] When a database isestablished in advance signature matching can be employedto greatly improve position accuracy in NLOS scenarios Byexerting constraints or introducing an NLOS error-relatedparameter into the cost function optimization algorithmscan be developed to mitigate the NLOS effect [25 26] Someof researches take advantage of the NLOS propagation pathsrather than canceling them [27]

Using its movement velocities the position of target nodecan be approximately estimated when the initial positionof target node is known However the position of iterativeestimation with movement equation is prone to error whenthe movement velocities include noises In WSNs the posi-tion of target node also can be located by range measurementbetween the target node and anchor nodes known positionsWhen the range measurements include noises the localiza-tion result also would be imprecise [28] Kalman filter (KF)to deal with the linear function and its nonlinear extensionextended Kalman filter (EKF) provide a feasible solution tomitigate the position error of the mobile target node andtherefore improving accuracy of mobile target tracking [29ndash31]

To track the position of mobile target node precisely wepropose to mitigate the NLOS propagations and make gooduse of the LOS range measurements by detecting and identi-fying the NLOS propagation paths In this paper we firstlyintroduce two approaches to detect the NLOS propagationpaths when the range measurements conform to Gaussiandistribution The first detection approach utilizes residual ofnode (RN) which conforms to zero-mean Gaussian distribu-tion when there are no NLOS propagation paths The secondapproach to detect NLOS propagation paths utilizes sum ofsquare residual (SSR) which conforms to chi-square distribu-tion when no NLOS propagation paths exist The anchornodes with NLOS propagation paths are called NLOS anchornodes and the anchor nodes with LOS propagation paths arealso called LOS anchor nodes To identify the NLOS anchornodes we propose a residual-error decompositionmethod toestimate the NLOS range errors The localization result canbe improved when estimated NLOS range errors are usedto amend the residuals Then the corrected position is con-sidered as observation and used to track the mobile targetposition preciselyThe contributions of our work are summa-rized as follows

(1) RN and SSR Approaches to Detect the NLOS Anchor NodesUsing the Jacobian matrix we derive the residuals comingfrom range errors between the target node and each anchornode Applying the analysis method of multiple factor staticswe conclude that RN conforms toGaussian distributionwhenthe range measurements are Gaussian distributed Then we

demonstrate that the SSR of all anchor nodes conforms tochi-square distribution when no NLOS anchor nodes existSo the probability of RN and SSR can be used to detect theNLOS propagation paths precisely

(2) An Estimation Method for NLOS Range Errors Is ProposedWhen Multiple NLOS Anchor Nodes Simultaneously Exist Byanalyzing the relationships between the errors and the resid-uals in the process of nonlinearminimization localization wepropose an estimation method for NLOS range errors Sincethe residuals are caused by the range errors the range errorsof NLOS anchor nodes can be estimated approximately byignoring the assigned residuals from LOS anchor nodes tothe NLOS anchor nodes Using the estimated range errorsof anchor nodes the NLOS anchor nodes can be identifiedcorrectly

(3) To Tracking A Mobile Target Node Precisely the ModifiedKF Is Put Forward by Applying the NLOS Mitigation MethodBy mitigating the NLOS propagation paths the localizationaccuracy can be improved so the covariance of target nodeposition would be reduced The covariance of localized posi-tion is derived from the range noises between the target nodeand each anchor node Considering the localized positionas observation the position of target node is tracked withmodified KF based on the movement equation

This paper presents an effective tracking method ofmobile target node with modified KF in NLOS environmentThe rest of this paper is structured as follows Section 2presents two detection approaches for the NLOS propagationpaths Section 3 describes the estimation method of NLOSrange errors Section 4 introduces the target tracking algo-rithm with modified KF in NLOS environment Section 5analyzes the simulation results The conclusion is presentedin Section 6

2 Detection of NLOS Propagation Path

For simplicity we only focus on the case of tracking a singletarget node in a two-dimensional field covered with multipleanchor nodes Since most localization systems of wirelesscommunications may suffer from the NLOS propagationpaths and dense multipath situation it is an important issueto obtain higher accuracy in determining range informationIn dealing with the NLOS propagation path effects the rangemeasurement 119889

119894119895between target node position 119894 and the posi-

tion of anchor node 119895 corresponding to the TOA measure-ment metrics can be modeled as

119889

119894119895= 119889

119894119895+ 119899

119894119895+ 119887

119894119895 (1)

where 119894 means the time instant of tracked mobile node and119894 = 1 119872 119895 = 1 119873 represents the fact that there are119873measurable anchor nodes corresponding to the target nodeAs mentioned earlier usually the measurement noise values119899

119894119895are modeled as zero-mean Gaussian random variables

with variance 1205752119894119895 119887119894119895caused by NLOS propagation path is

a positive random variable There will be no NLOS errorcomponent if the LOSpropagation path exists and 119887

119894119895= 0We


define Δ119889119894119895= 119899

119894119895+119887

119894119895 then Δ119889

119894119895simN(119887

119894119895 120575

2

119894119895) HereN(119887

119894119895 120575

2

119894119895)


120575

2



119889

119894119895=

radic

(119909

119894minus 119909

119895)

2

+ (119910

119894minus 119910

119895)

2

(2)

where z119894= (119909

119894 119910



119895= (119909

119895 119910





119891 (z119894) =

119873

sum

119895=1

120596

119894119895Δ119889

2

119894119895=

119873

sum

119895=1

120596

119894119895(

119889

119894119895minus 119889

119894119895)

2

(3)





120596



119894119895sim N(0 1205752

119894119895) In



119894119895as





Δd119894= [Δ119889

1198941 Δ119889

1198942 Δ119889

119894119873]

119879

(4)


119891 (z119894) = (Δd

119894minus J119894Δz119894)

119879(Δd119894minus J119894Δz119894)

(5)


119894 J119894is the


J119894=

[

[

[

[

120597119889

1198941

120597119909

119894

120597119889

1198942

120597119909

119894

sdot sdot sdot

120597119889

119894119873

120597119909

119894

120597119889

1198941

120597119910

119894

120597119889

1198942

120597119910

119894

sdot sdot sdot

120597119889

119894119873

120597119910

119894

]

]

]

]

119879

(6)


Δz119894= (J119879119894J119894)

minus1

J119879119894sdot Δd119894

(7)


r119894= [I minus J

119894(J119879119894J119894)

minus1

J119879119894]Δd119894= A119894Δd119894 (8)

where

A119894= I minus J

119894(J119879119894J119894)

minus1

J119879119894

(9)



119894


A119894=

[

[

[

[

[

119886

11119886


1119873

119886

21119886


2119873


119886

1198731119886


119873119873

]

]

]

]

]

(10)


r119894= [119903

1198941 119903

1198942 119903

119894119873]

119879

(11)



119894119895 120575

2

119894119895) the mean

120583



120583

119894119895=

119873

sum

119896=1

119886

119894119896119887

119896119895

]2119894119895=

119873

sum

119896=1

119886

2

119894119896120575

2

119896119895

(12)

The RN 119903


119894119895 ]2119894119895)



119875(

1003816

1003816

1003816

1003816

1003816

119903

119894119895minus 120583

119894119895

1003816

1003816

1003816

1003816

1003816

]119894119895

lt 120572) = 120573(13)



001

002

003

004

005

006

007

Residual


PDF

(times100

)




119903

119894119895minus 120572]119894119895lt 120583

119894119895lt 119903

119894119895+ 120572]119894119895 (14)


and 120583119894119895= 0 According to (14) 119903

119894119895minus 120572]119894119895lt 120583

119894119895 So we can


119903

119894119895gt 120572]119894119895 (15)

120583



119894119895≫ 120575




119894119895gt 120572]119894119895



120573

119873= 120574 (16)





119891 (z119894) = r119879119894r119894=

119873

sum

119895=1

119903

2

119894119895 (17)

Here 119903119894119895simN(120583



119894119895can

be represented as

119875 (119911) =

1

radic2120587]119894119895

119911

minus(12)119890

minus(1199112]2119894119895)119911 ge 0

0 119911 lt 0

(18)

0 5 10 15 20 250

010203040506070809

1


CDF

(times100

)

N = 4

N = 6

N = 8


where 119911 = 119903

2

119894119895 So 119911]2



119894119895



119895=12119873


119894119895

119895=12119873 Then the PDF of



119875 120594

2(119873) gt 120594

2

120579(119873) = 120579 (19)



119891 (z119894) gt 119878

119897 (20)







119903

119894119895=

119873

sum

119896=1

119886

119895119896sdot Δ119889

119894119896 (21)

where 119886119895119895gt 0

119895=1119873and Δ119889



119894119896≫ 0


119894119895must be



119894(J119879119894J119894)



119894is




119894will tend to




119894is decomposed

as

A119894= [

A11

A12

A21

A22

] (22)










119894


as

r119894= [

r1

r2

]

Δd119894= [

Δd1

Δd2

]

(23)






119894= (J119879119894J119894)



Cov (Δz119894) = F119894Σ119894F119879119894 (25)


119894 If Δ119889

119894119895is


Σ119894= diag 1205752

1198941 120575

2

1198942 120575

2

119894119873 (26)




x119894= [119909 (119894) V

119909 (119894) 119910 (119894) V

119910 (119894)]

119879

(27)



119909(119894) and




x119894+1= Φ119894x119894+ Γ119894w119894 (28)


written as

Φ119894=

[

[

[

[

1 Δ119905

1198940 0

0 1 0 0

0 0 1 Δ119905

119894

0 0 0 1

]

]

]

]

(29)



Γ119894=

[

[

[

[

[

[

[

[

[

[

[

[

Δ119905

2

119894

2

0

Δ119905

1198940

0

Δ119905

2

119894

2

0 Δ119905

119894

]

]

]

]

]

]

]

]

]

]

]

]

(30)


119894+1minus 119905



w119894= [119908

119909119908


119908119908119909




Q119908=

[

[

120575

2

1199081199090

0 120575

2

119908119910

]

]

(31)

where 1205752119908119909

and 1205752119908119910


119909and 119908





be modified as

z119894= Hx119894+ u119894 (32)


H = [

1 0 0 0

0 0 1 0

] (33)


119894





119894 which







R119894= Cov (Δz

119894) = F119894Σ119894F119879119894 (34)




Δd = [0 0Δd2] (35)


will be modified as

z119894= z119894+ F119894Δd (36)




x119894+1|119894

= Φ119894x119894|119894

P119894+1|119894

= Φ119894P119894|119894Φ119879

119894+ Γ119894Q119908Γ119879

119894

K119894+1= P119894+1|119894

H119879[HP119894+1|119894

H119879 + R119894+1]

minus1

x119894+1|119894+1

= x119894+1|119894

+ K119894+1[z119894+1minusHx119894+1|119894

]

P119894+1|119894+1

= P119894+1|119894

minus K119894+1

HP119894+1|119894

(37)


= x0and P

0|0=














Output x = (x1 x



119894= Cov (Δz

119894) = F119894sum

119894F119879119894





Judgment NLOS

LOS











119894119895are all equal to 120575

2 for 119895 =


119897= 1675

120575






01

02

03

04

05

06

07

08

09

1


CDF

(times100

)



0 20 40 60 80 100 1200

02

04

06

08

1


CDF

(times100

)


Succ

essfu

l det

ectio

n ra

tio (times

100

)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1










1 2 3 4 5 6 7 8 9 10minus4

minus2

0

2

4

6

8

10

12


Estim

ated

rang

e err

or (m

)







0= [0 1 0 1]


0 50 100 150 200minus5

0

5

10

15

20

25

Times (s)

Rang

e err

or (m

)





Δ119889 =

5 +

15119905

200


5 +

15119905

200

70 lt 119905 lt 130

(38)






119894= 1 s All


0=

[0 1 0 1] and 1198750= [004 0 0 004]


119894119904)


RMSE119894=radic

1

119879

119879

sum

119896=1

[(119909

119894119896minus 119909

119894119904)

2+ (119910

119894119896minus 119910

119894119904)

2]

(39)


10 20 30 40 50 60 70 80 90 10005

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)



0 20 40 60 80 1000

05

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)






6 Conclusion




Acknowledgments



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










define Δ119889119894119895= 119899

119894119895+119887

119894119895 then Δ119889

119894119895simN(119887

119894119895 120575

2

119894119895) HereN(119887

119894119895 120575

2

119894119895)


120575

2



119889

119894119895=

radic

(119909

119894minus 119909

119895)

2

+ (119910

119894minus 119910

119895)

2

(2)

where z119894= (119909

119894 119910



119895= (119909

119895 119910





119891 (z119894) =

119873

sum

119895=1

120596

119894119895Δ119889

2

119894119895=

119873

sum

119895=1

120596

119894119895(

119889

119894119895minus 119889

119894119895)

2

(3)





120596



119894119895sim N(0 1205752

119894119895) In



119894119895as





Δd119894= [Δ119889

1198941 Δ119889

1198942 Δ119889

119894119873]

119879

(4)


119891 (z119894) = (Δd

119894minus J119894Δz119894)

119879(Δd119894minus J119894Δz119894)

(5)


119894 J119894is the


J119894=

[

[

[

[

120597119889

1198941

120597119909

119894

120597119889

1198942

120597119909

119894

sdot sdot sdot

120597119889

119894119873

120597119909

119894

120597119889

1198941

120597119910

119894

120597119889

1198942

120597119910

119894

sdot sdot sdot

120597119889

119894119873

120597119910

119894

]

]

]

]

119879

(6)


Δz119894= (J119879119894J119894)

minus1

J119879119894sdot Δd119894

(7)


r119894= [I minus J

119894(J119879119894J119894)

minus1

J119879119894]Δd119894= A119894Δd119894 (8)

where

A119894= I minus J

119894(J119879119894J119894)

minus1

J119879119894

(9)



119894


A119894=

[

[

[

[

[

119886

11119886


1119873

119886

21119886


2119873


119886

1198731119886


119873119873

]

]

]

]

]

(10)


r119894= [119903

1198941 119903

1198942 119903

119894119873]

119879

(11)



119894119895 120575

2

119894119895) the mean

120583



120583

119894119895=

119873

sum

119896=1

119886

119894119896119887

119896119895

]2119894119895=

119873

sum

119896=1

119886

2

119894119896120575

2

119896119895

(12)

The RN 119903


119894119895 ]2119894119895)



119875(

1003816

1003816

1003816

1003816

1003816

119903

119894119895minus 120583

119894119895

1003816

1003816

1003816

1003816

1003816

]119894119895

lt 120572) = 120573(13)



001

002

003

004

005

006

007

Residual


PDF

(times100

)




119903

119894119895minus 120572]119894119895lt 120583

119894119895lt 119903

119894119895+ 120572]119894119895 (14)


and 120583119894119895= 0 According to (14) 119903

119894119895minus 120572]119894119895lt 120583

119894119895 So we can


119903

119894119895gt 120572]119894119895 (15)

120583



119894119895≫ 120575




119894119895gt 120572]119894119895



120573

119873= 120574 (16)





119891 (z119894) = r119879119894r119894=

119873

sum

119895=1

119903

2

119894119895 (17)

Here 119903119894119895simN(120583



119894119895can

be represented as

119875 (119911) =

1

radic2120587]119894119895

119911

minus(12)119890

minus(1199112]2119894119895)119911 ge 0

0 119911 lt 0

(18)

0 5 10 15 20 250

010203040506070809

1


CDF

(times100

)

N = 4

N = 6

N = 8


where 119911 = 119903

2

119894119895 So 119911]2



119894119895



119895=12119873


119894119895

119895=12119873 Then the PDF of



119875 120594

2(119873) gt 120594

2

120579(119873) = 120579 (19)



119891 (z119894) gt 119878

119897 (20)







119903

119894119895=

119873

sum

119896=1

119886

119895119896sdot Δ119889

119894119896 (21)

where 119886119895119895gt 0

119895=1119873and Δ119889



119894119896≫ 0


119894119895must be



119894(J119879119894J119894)



119894is




119894will tend to




119894is decomposed

as

A119894= [

A11

A12

A21

A22

] (22)










119894


as

r119894= [

r1

r2

]

Δd119894= [

Δd1

Δd2

]

(23)






119894= (J119879119894J119894)



Cov (Δz119894) = F119894Σ119894F119879119894 (25)


119894 If Δ119889

119894119895is


Σ119894= diag 1205752

1198941 120575

2

1198942 120575

2

119894119873 (26)




x119894= [119909 (119894) V

119909 (119894) 119910 (119894) V

119910 (119894)]

119879

(27)



119909(119894) and




x119894+1= Φ119894x119894+ Γ119894w119894 (28)


written as

Φ119894=

[

[

[

[

1 Δ119905

1198940 0

0 1 0 0

0 0 1 Δ119905

119894

0 0 0 1

]

]

]

]

(29)



Γ119894=

[

[

[

[

[

[

[

[

[

[

[

[

Δ119905

2

119894

2

0

Δ119905

1198940

0

Δ119905

2

119894

2

0 Δ119905

119894

]

]

]

]

]

]

]

]

]

]

]

]

(30)


119894+1minus 119905



w119894= [119908

119909119908


119908119908119909




Q119908=

[

[

120575

2

1199081199090

0 120575

2

119908119910

]

]

(31)

where 1205752119908119909

and 1205752119908119910


119909and 119908





be modified as

z119894= Hx119894+ u119894 (32)


H = [

1 0 0 0

0 0 1 0

] (33)


119894





119894 which







R119894= Cov (Δz

119894) = F119894Σ119894F119879119894 (34)




Δd = [0 0Δd2] (35)


will be modified as

z119894= z119894+ F119894Δd (36)




x119894+1|119894

= Φ119894x119894|119894

P119894+1|119894

= Φ119894P119894|119894Φ119879

119894+ Γ119894Q119908Γ119879

119894

K119894+1= P119894+1|119894

H119879[HP119894+1|119894

H119879 + R119894+1]

minus1

x119894+1|119894+1

= x119894+1|119894

+ K119894+1[z119894+1minusHx119894+1|119894

]

P119894+1|119894+1

= P119894+1|119894

minus K119894+1

HP119894+1|119894

(37)


= x0and P

0|0=














Output x = (x1 x



119894= Cov (Δz

119894) = F119894sum

119894F119879119894





Judgment NLOS

LOS











119894119895are all equal to 120575

2 for 119895 =


119897= 1675

120575






01

02

03

04

05

06

07

08

09

1


CDF

(times100

)



0 20 40 60 80 100 1200

02

04

06

08

1


CDF

(times100

)


Succ

essfu

l det

ectio

n ra

tio (times

100

)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1










1 2 3 4 5 6 7 8 9 10minus4

minus2

0

2

4

6

8

10

12


Estim

ated

rang

e err

or (m

)







0= [0 1 0 1]


0 50 100 150 200minus5

0

5

10

15

20

25

Times (s)

Rang

e err

or (m

)





Δ119889 =

5 +

15119905

200


5 +

15119905

200

70 lt 119905 lt 130

(38)






119894= 1 s All


0=

[0 1 0 1] and 1198750= [004 0 0 004]


119894119904)


RMSE119894=radic

1

119879

119879

sum

119896=1

[(119909

119894119896minus 119909

119894119904)

2+ (119910

119894119896minus 119910

119894119904)

2]

(39)


10 20 30 40 50 60 70 80 90 10005

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)



0 20 40 60 80 1000

05

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)






6 Conclusion




Acknowledgments



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











001

002

003

004

005

006

007

Residual


PDF

(times100

)




119903

119894119895minus 120572]119894119895lt 120583

119894119895lt 119903

119894119895+ 120572]119894119895 (14)


and 120583119894119895= 0 According to (14) 119903

119894119895minus 120572]119894119895lt 120583

119894119895 So we can


119903

119894119895gt 120572]119894119895 (15)

120583



119894119895≫ 120575




119894119895gt 120572]119894119895



120573

119873= 120574 (16)





119891 (z119894) = r119879119894r119894=

119873

sum

119895=1

119903

2

119894119895 (17)

Here 119903119894119895simN(120583



119894119895can

be represented as

119875 (119911) =

1

radic2120587]119894119895

119911

minus(12)119890

minus(1199112]2119894119895)119911 ge 0

0 119911 lt 0

(18)

0 5 10 15 20 250

010203040506070809

1


CDF

(times100

)

N = 4

N = 6

N = 8


where 119911 = 119903

2

119894119895 So 119911]2



119894119895



119895=12119873


119894119895

119895=12119873 Then the PDF of



119875 120594

2(119873) gt 120594

2

120579(119873) = 120579 (19)



119891 (z119894) gt 119878

119897 (20)







119903

119894119895=

119873

sum

119896=1

119886

119895119896sdot Δ119889

119894119896 (21)

where 119886119895119895gt 0

119895=1119873and Δ119889



119894119896≫ 0


119894119895must be



119894(J119879119894J119894)



119894is




119894will tend to




119894is decomposed

as

A119894= [

A11

A12

A21

A22

] (22)










119894


as

r119894= [

r1

r2

]

Δd119894= [

Δd1

Δd2

]

(23)






119894= (J119879119894J119894)



Cov (Δz119894) = F119894Σ119894F119879119894 (25)


119894 If Δ119889

119894119895is


Σ119894= diag 1205752

1198941 120575

2

1198942 120575

2

119894119873 (26)




x119894= [119909 (119894) V

119909 (119894) 119910 (119894) V

119910 (119894)]

119879

(27)



119909(119894) and




x119894+1= Φ119894x119894+ Γ119894w119894 (28)


written as

Φ119894=

[

[

[

[

1 Δ119905

1198940 0

0 1 0 0

0 0 1 Δ119905

119894

0 0 0 1

]

]

]

]

(29)



Γ119894=

[

[

[

[

[

[

[

[

[

[

[

[

Δ119905

2

119894

2

0

Δ119905

1198940

0

Δ119905

2

119894

2

0 Δ119905

119894

]

]

]

]

]

]

]

]

]

]

]

]

(30)


119894+1minus 119905



w119894= [119908

119909119908


119908119908119909




Q119908=

[

[

120575

2

1199081199090

0 120575

2

119908119910

]

]

(31)

where 1205752119908119909

and 1205752119908119910


119909and 119908





be modified as

z119894= Hx119894+ u119894 (32)


H = [

1 0 0 0

0 0 1 0

] (33)


119894





119894 which







R119894= Cov (Δz

119894) = F119894Σ119894F119879119894 (34)




Δd = [0 0Δd2] (35)


will be modified as

z119894= z119894+ F119894Δd (36)




x119894+1|119894

= Φ119894x119894|119894

P119894+1|119894

= Φ119894P119894|119894Φ119879

119894+ Γ119894Q119908Γ119879

119894

K119894+1= P119894+1|119894

H119879[HP119894+1|119894

H119879 + R119894+1]

minus1

x119894+1|119894+1

= x119894+1|119894

+ K119894+1[z119894+1minusHx119894+1|119894

]

P119894+1|119894+1

= P119894+1|119894

minus K119894+1

HP119894+1|119894

(37)


= x0and P

0|0=














Output x = (x1 x



119894= Cov (Δz

119894) = F119894sum

119894F119879119894





Judgment NLOS

LOS











119894119895are all equal to 120575

2 for 119895 =


119897= 1675

120575






01

02

03

04

05

06

07

08

09

1


CDF

(times100

)



0 20 40 60 80 100 1200

02

04

06

08

1


CDF

(times100

)


Succ

essfu

l det

ectio

n ra

tio (times

100

)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1










1 2 3 4 5 6 7 8 9 10minus4

minus2

0

2

4

6

8

10

12


Estim

ated

rang

e err

or (m

)







0= [0 1 0 1]


0 50 100 150 200minus5

0

5

10

15

20

25

Times (s)

Rang

e err

or (m

)





Δ119889 =

5 +

15119905

200


5 +

15119905

200

70 lt 119905 lt 130

(38)






119894= 1 s All


0=

[0 1 0 1] and 1198750= [004 0 0 004]


119894119904)


RMSE119894=radic

1

119879

119879

sum

119896=1

[(119909

119894119896minus 119909

119894119904)

2+ (119910

119894119896minus 119910

119894119904)

2]

(39)


10 20 30 40 50 60 70 80 90 10005

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)



0 20 40 60 80 1000

05

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)






6 Conclusion




Acknowledgments



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119903

119894119895=

119873

sum

119896=1

119886

119895119896sdot Δ119889

119894119896 (21)

where 119886119895119895gt 0

119895=1119873and Δ119889



119894119896≫ 0


119894119895must be



119894(J119879119894J119894)



119894is




119894will tend to




119894is decomposed

as

A119894= [

A11

A12

A21

A22

] (22)










119894


as

r119894= [

r1

r2

]

Δd119894= [

Δd1

Δd2

]

(23)






119894= (J119879119894J119894)



Cov (Δz119894) = F119894Σ119894F119879119894 (25)


119894 If Δ119889

119894119895is


Σ119894= diag 1205752

1198941 120575

2

1198942 120575

2

119894119873 (26)




x119894= [119909 (119894) V

119909 (119894) 119910 (119894) V

119910 (119894)]

119879

(27)



119909(119894) and




x119894+1= Φ119894x119894+ Γ119894w119894 (28)


written as

Φ119894=

[

[

[

[

1 Δ119905

1198940 0

0 1 0 0

0 0 1 Δ119905

119894

0 0 0 1

]

]

]

]

(29)



Γ119894=

[

[

[

[

[

[

[

[

[

[

[

[

Δ119905

2

119894

2

0

Δ119905

1198940

0

Δ119905

2

119894

2

0 Δ119905

119894

]

]

]

]

]

]

]

]

]

]

]

]

(30)


119894+1minus 119905



w119894= [119908

119909119908


119908119908119909




Q119908=

[

[

120575

2

1199081199090

0 120575

2

119908119910

]

]

(31)

where 1205752119908119909

and 1205752119908119910


119909and 119908





be modified as

z119894= Hx119894+ u119894 (32)


H = [

1 0 0 0

0 0 1 0

] (33)


119894





119894 which







R119894= Cov (Δz

119894) = F119894Σ119894F119879119894 (34)




Δd = [0 0Δd2] (35)


will be modified as

z119894= z119894+ F119894Δd (36)




x119894+1|119894

= Φ119894x119894|119894

P119894+1|119894

= Φ119894P119894|119894Φ119879

119894+ Γ119894Q119908Γ119879

119894

K119894+1= P119894+1|119894

H119879[HP119894+1|119894

H119879 + R119894+1]

minus1

x119894+1|119894+1

= x119894+1|119894

+ K119894+1[z119894+1minusHx119894+1|119894

]

P119894+1|119894+1

= P119894+1|119894

minus K119894+1

HP119894+1|119894

(37)


= x0and P

0|0=














Output x = (x1 x



119894= Cov (Δz

119894) = F119894sum

119894F119879119894





Judgment NLOS

LOS











119894119895are all equal to 120575

2 for 119895 =


119897= 1675

120575






01

02

03

04

05

06

07

08

09

1


CDF

(times100

)



0 20 40 60 80 100 1200

02

04

06

08

1


CDF

(times100

)


Succ

essfu

l det

ectio

n ra

tio (times

100

)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1










1 2 3 4 5 6 7 8 9 10minus4

minus2

0

2

4

6

8

10

12


Estim

ated

rang

e err

or (m

)







0= [0 1 0 1]


0 50 100 150 200minus5

0

5

10

15

20

25

Times (s)

Rang

e err

or (m

)





Δ119889 =

5 +

15119905

200


5 +

15119905

200

70 lt 119905 lt 130

(38)






119894= 1 s All


0=

[0 1 0 1] and 1198750= [004 0 0 004]


119894119904)


RMSE119894=radic

1

119879

119879

sum

119896=1

[(119909

119894119896minus 119909

119894119904)

2+ (119910

119894119896minus 119910

119894119904)

2]

(39)


10 20 30 40 50 60 70 80 90 10005

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)



0 20 40 60 80 1000

05

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)






6 Conclusion




Acknowledgments



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Γ119894=

[

[

[

[

[

[

[

[

[

[

[

[

Δ119905

2

119894

2

0

Δ119905

1198940

0

Δ119905

2

119894

2

0 Δ119905

119894

]

]

]

]

]

]

]

]

]

]

]

]

(30)


119894+1minus 119905



w119894= [119908

119909119908


119908119908119909




Q119908=

[

[

120575

2

1199081199090

0 120575

2

119908119910

]

]

(31)

where 1205752119908119909

and 1205752119908119910


119909and 119908





be modified as

z119894= Hx119894+ u119894 (32)


H = [

1 0 0 0

0 0 1 0

] (33)


119894





119894 which







R119894= Cov (Δz

119894) = F119894Σ119894F119879119894 (34)




Δd = [0 0Δd2] (35)


will be modified as

z119894= z119894+ F119894Δd (36)




x119894+1|119894

= Φ119894x119894|119894

P119894+1|119894

= Φ119894P119894|119894Φ119879

119894+ Γ119894Q119908Γ119879

119894

K119894+1= P119894+1|119894

H119879[HP119894+1|119894

H119879 + R119894+1]

minus1

x119894+1|119894+1

= x119894+1|119894

+ K119894+1[z119894+1minusHx119894+1|119894

]

P119894+1|119894+1

= P119894+1|119894

minus K119894+1

HP119894+1|119894

(37)


= x0and P

0|0=














Output x = (x1 x



119894= Cov (Δz

119894) = F119894sum

119894F119879119894





Judgment NLOS

LOS











119894119895are all equal to 120575

2 for 119895 =


119897= 1675

120575






01

02

03

04

05

06

07

08

09

1


CDF

(times100

)



0 20 40 60 80 100 1200

02

04

06

08

1


CDF

(times100

)


Succ

essfu

l det

ectio

n ra

tio (times

100

)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1










1 2 3 4 5 6 7 8 9 10minus4

minus2

0

2

4

6

8

10

12


Estim

ated

rang

e err

or (m

)







0= [0 1 0 1]


0 50 100 150 200minus5

0

5

10

15

20

25

Times (s)

Rang

e err

or (m

)





Δ119889 =

5 +

15119905

200


5 +

15119905

200

70 lt 119905 lt 130

(38)






119894= 1 s All


0=

[0 1 0 1] and 1198750= [004 0 0 004]


119894119904)


RMSE119894=radic

1

119879

119879

sum

119896=1

[(119909

119894119896minus 119909

119894119904)

2+ (119910

119894119896minus 119910

119894119904)

2]

(39)


10 20 30 40 50 60 70 80 90 10005

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)



0 20 40 60 80 1000

05

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)






6 Conclusion




Acknowledgments



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















Output x = (x1 x



119894= Cov (Δz

119894) = F119894sum

119894F119879119894





Judgment NLOS

LOS











119894119895are all equal to 120575

2 for 119895 =


119897= 1675

120575






01

02

03

04

05

06

07

08

09

1


CDF

(times100

)



0 20 40 60 80 100 1200

02

04

06

08

1


CDF

(times100

)


Succ

essfu

l det

ectio

n ra

tio (times

100

)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1










1 2 3 4 5 6 7 8 9 10minus4

minus2

0

2

4

6

8

10

12


Estim

ated

rang

e err

or (m

)







0= [0 1 0 1]


0 50 100 150 200minus5

0

5

10

15

20

25

Times (s)

Rang

e err

or (m

)





Δ119889 =

5 +

15119905

200


5 +

15119905

200

70 lt 119905 lt 130

(38)






119894= 1 s All


0=

[0 1 0 1] and 1198750= [004 0 0 004]


119894119904)


RMSE119894=radic

1

119879

119879

sum

119896=1

[(119909

119894119896minus 119909

119894119904)

2+ (119910

119894119896minus 119910

119894119904)

2]

(39)


10 20 30 40 50 60 70 80 90 10005

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)



0 20 40 60 80 1000

05

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)






6 Conclusion




Acknowledgments



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












01

02

03

04

05

06

07

08

09

1


CDF

(times100

)



0 20 40 60 80 100 1200

02

04

06

08

1


CDF

(times100

)


Succ

essfu

l det

ectio

n ra

tio (times

100

)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1










1 2 3 4 5 6 7 8 9 10minus4

minus2

0

2

4

6

8

10

12


Estim

ated

rang

e err

or (m

)







0= [0 1 0 1]


0 50 100 150 200minus5

0

5

10

15

20

25

Times (s)

Rang

e err

or (m

)





Δ119889 =

5 +

15119905

200


5 +

15119905

200

70 lt 119905 lt 130

(38)






119894= 1 s All


0=

[0 1 0 1] and 1198750= [004 0 0 004]


119894119904)


RMSE119894=radic

1

119879

119879

sum

119896=1

[(119909

119894119896minus 119909

119894119904)

2+ (119910

119894119896minus 119910

119894119904)

2]

(39)


10 20 30 40 50 60 70 80 90 10005

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)



0 20 40 60 80 1000

05

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)






6 Conclusion




Acknowledgments



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 4 5 6 7 8 9 10minus4

minus2

0

2

4

6

8

10

12


Estim

ated

rang

e err

or (m

)







0= [0 1 0 1]


0 50 100 150 200minus5

0

5

10

15

20

25

Times (s)

Rang

e err

or (m

)





Δ119889 =

5 +

15119905

200


5 +

15119905

200

70 lt 119905 lt 130

(38)






119894= 1 s All


0=

[0 1 0 1] and 1198750= [004 0 0 004]


119894119904)


RMSE119894=radic

1

119879

119879

sum

119896=1

[(119909

119894119896minus 119909

119894119904)

2+ (119910

119894119896minus 119910

119894119904)

2]

(39)


10 20 30 40 50 60 70 80 90 10005

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)



0 20 40 60 80 1000

05

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)






6 Conclusion




Acknowledgments



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10 20 30 40 50 60 70 80 90 10005

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)



0 20 40 60 80 1000

05

1

15

2

25

3

35

4

45

5

Time (s)

RMSE

(m)






6 Conclusion




Acknowledgments



References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Target Tracking with NLOS Detection …downloads.hindawi.com/journals/ijdsn/2013/646287.pdf · Target Tracking with NLOS Detection and Mitigation in ... of the NLOS