Anchor Global Position Accuracy Enhancement Based on Data Fusion

1616 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 3, MARCH 2009

[8] Q. Liu, S. Zhou, and G. B. Giannakis, “Queuing with adaptive modulationand coding over wireless links: Cross-layer analysis and design,” IEEETrans. Wireless Commun., vol. 4, no. 3, pp. 1142–1153, May 2005.

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[11] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.:Cambridge Univ. Press, 2004.

Anchor Global Position Accuracy EnhancementBased on Data Fusion

Kegen Yu, Member, IEEE, and Y. Jay Guo, Senior Member, IEEE

Abstract—The location information of sensor nodes in wireless sensornetworks is crucial for various applications such as emergency rescue op-erations, environmental monitoring, home automation, and traffic control.In this paper, a new method to improve the anchor location accuracy inwireless sensor networks is proposed based on fusion of Global PositioningSystem (GPS) measurements and anchor-to-anchor parameter estimates.Novel algorithms are derived to increase the accuracy of anchor loca-tions using both anchor-to-anchor distance and angle-of-arrival (AOA)estimates for both line-of-sight (LOS) and non-LOS (NLOS) scenarios.When using anchor-to-anchor distance estimates in LOS conditions, anoptimization-based algorithm is developed. The Cramer–Rao lower bound(CRLB) is derived to benchmark positioning accuracy in 3-D environmentswith both GPS measurements and anchor-to-anchor parameter estimates,which has not been studied in the literature. Simulation results demon-strate that the proposed algorithms can substantially improve anchor posi-tion accuracy, and the performance of the proposed algorithms approachesthe CRLB.

Index Terms—Anchor-location accuracy improvement, anchor-to-anchor cooperation, Cramer–Rao lower bound (CRLB), Global Position-ing System (GPS) measurements, location error analysis.

I. INTRODUCTION

Localization in wireless sensor networks has drawn significantattention in the recent years. Various localization schemes and algo-rithms have been proposed. In many circumstances, such as in rescueoperations, asset and people tracking, and fleet management, a keyfunction of the sensor networks is to locate and track the target ofinterest. In environmental monitoring, the measured data may losetheir significance when no position information is provided.

The Global Positioning System (GPS) has widely been used fortracking and navigation in many applications and services [1]–[6].Some advanced GPS receivers such as the differential GPS (DGPS)can achieve accuracy ranging from a few meters down to a fewdecimeters [7], [8], whereas other low-cost commercial GPS receiversmay only achieve the accuracy that might be as poor as 20 m [9],

Manuscript received February 14, 2008; revised April 26, 2008 and June 9,2008; accepted June 30, 2008. First published July 16, 2008; current versionpublished March 17, 2009. This paper was presented in part at the IEEEInternational Conference on Communications (ICC), Beijing, China, May2008. The review of this paper was coordinated by Dr. T. Taniguchi.

The authors are with the Wireless Technologies Laboratory, CommonwealthScientific and Industrial Research Organisation (CSIRO) Information and Com-munications Technologies (ICT) Centre, Marsfield, N.S.W. 2122, Australia(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2008.928636

[10]. In addition, when a nonline-of-sight (NLOS) signal propagationexists between the satellite and the GPS receiver due to the adverseenvironments such as in urban canyons, under heavy foliage, orindoors, the GPS measurement accuracy may significantly degrade.

In wireless sensor networks, it is usually the case that only a smallpercentage of the nodes are equipped with GPS receivers, and theyare often called anchor nodes, whereas the other nodes are ordinarysensor nodes. The global positions of the ordinary sensor nodes maybe obtained by directly using anchor GPS measurements and a mul-tilateration method under the global coordinate system. Alternatively,ordinary node locations can first be estimated under a local coordinatesystem. Then, node global positions are produced by transformingthe local systems into a global system based on the anchor globallocations. The analysis of the impact of anchor location errors on thelocalization accuracy of ordinary nodes can be found in [11].

When considering localization in wireless sensor networks, oneusually assumes that the anchor locations are error free. In reality,anchor locations are provided by GPS receivers in general. The GPSreceivers may not provide satisfactory location information due to costand complexity constraints applied on nodes. The research presentedin this paper was motivated by the following question: In the eventthat the accuracy of low-cost GPS receivers is not satisfactory, isit possible to obtain better location accuracy by making use of theanchor-to-anchor signal measurements? With the wideband and theultra wideband technologies, the accuracy of ranging can be submeters[12]–[15]. In addition, very accurate angle-of-arrival (AOA) estimatescan be obtained by using high-resolution AOA estimation approaches,including the MUSIC [16], [17] and ESPRIT [18], [19] algorithms.The essence of the work is to make use of the relatively more accurateanchor-to-anchor parameter estimates to improve the anchor locationaccuracy. Accordingly, the location accuracy of ordinary nodes can beincreased. To our knowledge, the issue of enhancing anchor locationaccuracy by fusion of both global and anchor-to-anchor measurementshas not been addressed in the literature. The main contributions of thispaper are summarized in the list that follows.

1) New methods to improve anchor global position accuracy byuse of anchor-to-anchor parameter estimates (either distance ordistance and AOA). To the best of our knowledge, this anchor-to-anchor, cooperation-based, accuracy-enhancement approachhas not been studied in the literature.

2) Novel accuracy improvement algorithms in the presence ofeither line-of-sight (LOS) or NLOS radio propagation betweenpairs of anchors. Specifically, with the anchor-to-anchor dis-tance and AOA estimates, a low-complexity but efficient al-gorithm is derived to enhance the anchor location accuracyin LOS conditions. In NLOS conditions, by introducing thescatterer location variables, we derive another algorithm toachieve better accuracy. In the case that only anchor-to-anchordistance estimates are available, a Lenvenberg–Marquardt (LM)optimization-based algorithm is developed.

3) Derivation of Cramer–Rao lower bounds (CRLBs). In the lit-erature, the study of CRLB in positioning is generally focusedon 2-D environments. To our knowledge, this is the first time toderive the CRLB in 3-D localization with both anchor-to-anchorparameter estimates and GPS coordinate measurements.

The remainder of this paper is organized as follows: Section II de-scribes the system and signal model. Section III derives the accuracy-enhancement algorithms in the presence of anchor-to-anchor distanceand AOA estimates. Section IV develops an iterative optimizationalgorithm based on anchor-to-anchor distance estimates. Sections Vderives the CRLBs when either both anchor-to-anchor distance and

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 3, MARCH 2009 1617

Fig. 1. Three-dimensional anchor positions and angles in LOS conditions.

Fig. 2. Three-dimensional anchor positions and angles in NLOS conditions.

AOA estimates or distance estimates are available. Section VI presentssome simulation results to show the effectiveness of the proposedalgorithms. Finally, Section VII concludes this paper.

II. SIGNAL MODEL

Consider a sensor network in which there are N anchors, eachof which is equipped with a GPS receiver. Each anchor is able tocommunicate with some other anchors within the radio range to obtaindistance estimates, for example, by measuring the round-trip time. Inaddition, each anchor is able to measure the AOA (both the azimuthand elevation angles) by using a directional antenna or an antennaarray. In this paper, we do not deal with the time and angle parameterestimation. Instead, we make use of the parameter estimates foranchor-location accuracy improvement. As shown in Fig. 1, anchorsi and j are within the radio range of each other in LOS propagationconditions, whereas in Fig. 2, the radio propagation is NLOS.

Let us define(xi, yi, zi) global position coordinates of anchor i;(x

(i,j)s , y

(i,j)s , z

(i,j)s ) position coordinates of the scatterer be-

tween anchors i and j under NLOSconditions;

di,j LOS distance between anchors i and j inFig. 1 and NLOS distance in Fig. 2;

φi,j azimuth angle of the signal transmitted fromanchor j and received at anchor i;

αi,j elevation angle of the signal transmittedfrom anchor j and received at anchor i.

In the presence of estimation error, the distance and angle estimates,respectively, are modeled as

di,j = di,j + ndi,j, 1 ≤ i; j ≤ N ; i �= j

φi,j =φi,j + nφi,j

αi,j =αi,j + nαi,j(1)

where ndi,jis the distance estimation error, nφi,j

is the azimuthangular estimation error, and nαi,j

is the elevation angular estimationerror. In LOS conditions, the distance and angles are described as

di,j =√

(xi − xj)2 + (yi − yj)2 + (zi − zj)2

φi,j =tan−1 yj − yi

xj − xi

αi,j =cos−1 zj − zi

di,j

. (2)

On the other hand, under NLOS conditions, the distance and anglesare defined as

di,j = d(i,j)s + d(j,i)

s

φi,j =tan−1 y(i,j)s − yi

x(i,j)s − xi

αi,j =cos−1 z(i,j)s − zi

d(i,j)s

d(i,j)s =

√(x

(i,j)s − xi

)2

+(y(i,j)s − yi

)2

+(z(i,j)s − zi

)2

d(j,i)s =

√(x

(i,j)s − xj

)2

+(y(i,j)s − yj

)2

+(z(i,j)s − zj

)2

(3)

where d(i,j)s is the LOS distance between anchor i and the scatterer that

is associated with anchors i and j. Please note the difference betweend(i,j)s and d

(j,i)s . The GPS coordinate estimates are modeled as

xi = xi + nxi, i = 1, 2, . . . , N

yi = yi + nyi

zi = zi + nzi(4)

where nxi, nyi

, and nziare the GPS coordinate estimation errors.

Note that the GPS errors come from various sources, including thesatellite geometry, satellite orbits, multipath effect, atmospheric ef-fects, and clock inaccuracies. Using the model in (4), the analysis cangreatly be simplified; however, this simplified model may overestimateor underestimate the GPS errors. Therefore, it would be interesting toconsider a more accurate model with multiple variables to describe theGPS errors in the future.

As shown in Fig. 3, the aim is to reestimate the anchor globalpositions by making use of anchor-to-anchor distance estimates or bothdistance and angle estimates in (1), and the GPS coordinate measure-ments in (4) as well. Once the refined anchor position estimates areobtained, they are employed to determine the global locations of theordinary sensor nodes. A number of node localization algorithms canbe found in [20].


Fig. 3. Block diagram of the anchor position accuracy improvement scheme.

III. ENHANCING ACCURACY USING DISTANCE

AND ANGLE ESTIMATES

A. Algorithm for LOS Scenarios

From (2) and Fig. 1, we can readily obtain the equivalent ob-servation equations related to the anchor-to-anchor distance andangle estimates as

xj − xi =(di,j − ndi,j) sin(αi,j − nαi,j

) cos(φi,j − nφi,j)

≈ di,j sin αi,j cos φi,j , i, j = 1, 2, . . . , N ; i �= j

yj − yi =(di,j − ndi,j) sin(αi,j − nαi,j

) sin(φi,j − nφi,j)

≈ di,j sin αi,j sin φi,j

zj − zi =(di,j − ndi,j) cos(αi,j − nαi,j

)

≈ di,j cos αi,j (5)

where the approximations result from dropping the estimation errors.Since (4) and (5) are linear in terms of the anchor coordinates, we canwrite them in a compact form as

Ap ≈ r (6)

where p is the position vector, which is defined as

p=[x1, x2, . . . , xN , y1, y2, . . . , yN , z1, z2, . . . , zN ]T ∈R3N×1 (7)

A ∈ RN(N−1)×3N is a constant matrix with elements equal to either0, 1, or −1, and r ∈ RN(N−1)×1 is a constant vector. To save space,the elements of A and r are not given.

Applying the weighted least-squares (WLS) estimator, we obtain therefined anchor location estimates as

ˆp = (AT WA)−1AT Wr (8)

where we use double hats to distinguish it from the origi-nal GPS coordinate measurements, and a weighting matrix W ∈RN(N−1)×N(N−1) is employed to emphasize the more reliable esti-mates [21]. For example, the received signal-to-noise ratio may be usedin choosing the weights. The matrix inverse is assumed to exist, andthis is usually true for an overdetermined system. It would be feasibleto run this algorithm at any anchor due to its low complexity, althoughan anchor with the most computational power might be chosen to runthe algorithm.

B. Algorithm for Mitigating NLOS Impact

As shown in Fig. 2, the radio propagation between anchors i andj is NLOS, and the signal is reflected only once before arriving atthe receiver. φi,j is the azimuth angle of the signal transmitted fromanchor j, reflected by the scatterer, and received at anchor i; αi,j is theelevation angle of the signal transmitted from anchor j, reflected bythe scatterer, and received at anchor i. The purpose is to introduce thescatterer locations to reduce the NLOS effect. From (3) and Fig. 2, wecan readily obtain the equivalent observation equations related to theanchor-to-anchor distance and angle estimates as

x(i,j)s − xi ≈ d(i,j)

s sin αi,j cos φi,j

i = 1, . . . , N − 1; j = i + 1, . . . , N

y(i,j)s − yi ≈ d(i,j)

s sin αi,j sin φi,j

z(i,j)s − zi ≈ d(i,j)

s cos αi,j

x(i,j)s − xj ≈

(di,j − d(i,j)

s

)sin αj,i cos φj,i

y(i,j)s − yj ≈

(di,j − d(i,j)

s

)sin αj,i sin φj,i

z(i,j)s − zj ≈

(di,j − d(i,j)

s

)cos αj,i. (9)

Let N1 = N(N − 1)/2, and define the parameter vector as

θ =[xT ,yT , zT ,xT

s ,yTs , zT

s ,ds

]T ∈ RN(2N+1)×1 (10)

where

x = [x1, x2, . . . , xN ]T

y = [y1, y2, . . . , yN ]T

z = [z1, z2, . . . , zN ]T

xs =[x(1,2)

s , . . . , x(1,N)s , x(2,3)

s , . . . , x(2,N)s

. . . , x(N−1,N)s

]T ∈ RN1×1

ys =[y(1,2)

s , . . . , y(1,N)s , y(2,3)

s , . . . , y(2,N)s

. . . , y(N−1,N)s

]T ∈ RN1×1

zs =[z(1,2)

s , . . . , z(1,N)s , z(2,3)

s , . . . , z(2,N)s

. . . , z(N−1,N)s

]T ∈ RN1×1

ds =[d(1,2)

s , . . . , d(1,N)s , d(2,3)

s , . . . , d(2,N)s

. . . , d(N−1,N)s

]T ∈ RN1×1. (11)


Then, we can write (4) and (9) in a compact form as

Bθ ≈ g (12)

where B ∈ RN(N−1)×N(2N+1) is a constant matrix, and g ∈RN(N−1)×1 is a constant vector. In addition, to save space, the detailsof the elements of B and g are not provided. Similarly to (8), the WLSsolution for (12) is given by

θ = (BT QB)−1BT Qg (13)

where Q ∈ RN(N−1)×N(N−1) is a weighting matrix similar to W.Due to the fact that the first 3N1 components of g are zero,1 the refinedanchor position estimates can be computed according to

ˆp =[ˆx

T, ˆy

T, ˆz

T]T

= Vg (14)

where V ∈ R3N×3N1 is the top-right subblock matrix of(BT QB)−1BT Q, and g ∈ R3N1×1 is the column vector thatis a subset of vector g starting from the (3N1 + 1)th component to thelast component. Other NLOS identification and mitigation algorithmscan be found in [22] and [23].

IV. ENHANCING ACCURACY USING ANCHOR-TO-ANCHOR

DISTANCE ESTIMATES

In some situations, the anchors may not have an antenna array ora directional antenna; thus, the angle information is not available.In this case, the least-squares algorithms described in the previoussection are no longer applicable. In this section, we employ an iterativeoptimization method to improve anchor location accuracy based onanchor-to-anchor distance measurements.

Making use of all the anchor-to-anchor distance measurements, thecost function is defined as

ε(p) =

N∑i=1

N∑j=1j �=i

wi,j(di,j − di,j)2 (15)

where wi,j are the weights. The anchor global positions are refinedby minimizing the aforementioned cost function with respect to theanchor coordinates, i.e.,

ˆp = arg minp

ε(p) (16)

where the GPS measurements are used as the initial position coordi-nate estimates.

There are a number of optimization algorithms that can be used tosolve the minimization problem in (16) [24], [25]. In the simulation,we will apply the LM method [26], [27]; thus, it is worth giving a briefdescription of the algorithm as follows. Define

f(p) = [f1,2, f2,1, . . . , f1,N , fN,1, f2,3, f3,2, . . . , f2,N , fN,2

. . . , fN−1,N , fN,N−1]T ∈ R2N1×1 (17)

where

fi,j =√

wi,j(di,j − di,j), i, j = 1, 2, . . . , N ; i �= j. (18)

Let p be the vector of the current position estimates and p + s be thevector of the next (new) position estimates. Clearly, the main issueis about how to determine the position increment vector s, which is

1This can be observed from the first three equations in (9), in which theconstant terms are zeros.

also called the direction vector. In the Gauss–Newton method, theincrement vector is determined according to

s = −(JT (p)J(p)

)−1JT (p)f(p) (19)

where J(p) is the Jacobian matrix of f(p) at p. The key point of theLM method is introducing a damping factor into (19), such that s iscomputed based on

s = −(JT (p)J(p) + λI

)−1JT (p)f(p) (20)

where the (nonnegative) damping factor λ is adjusted at each iteration.If the reduction of ε(p) is rapid, a smaller value of λ can be used toincrease the magnitude of the increment vector so that the search canbe sped up. On the other hand, if an iteration gives insufficient reduc-tion in the cost function, λ should be increased so that the gradientdescent direction can be approached. As a result, the algorithm runsat a speed between that of the Gauss–Newton method and that of thesteepest descent method.

Specifically, the implementation of the LM algorithm can be de-scribed as follows. Given a position estimate at time k − 1 as pk−1

and a positive damping factor as λk−1, one uses (20) to calculate theincrement vector sk−1. Then, based on the given position estimate andthe increment vector, the predicted cost function becomes

εe(pk) = ‖J(pk−1)sk−1 + f(pk−1)‖2 . (21)

In addition, the predicted position is given by

pk = pk−1 + sk−1. (22)

Then, using (15), one obtains the cost function as ε(pk−1) andε(pk) at pk−1 and pk, respectively. By cubically interpolating thetwo cost function points and using gradient and function evaluation,the minimum cost function is produced as εk(p∗). In addition, acorresponding step-length parameter is obtained as α∗. Then, theinterpolated minimum cost function is compared with the predictedcost function as follows. If εe(pk) > εk(p∗), the damping factor isreduced by

λk =λk−1

1 + α∗. (23)

Meanwhile, if εe(pk) < εk(p∗), the damping factor is increased by

λk = λk−1 +εk(p∗) − εe(pk)

α∗. (24)

The updated damping factor is used to update the increment vectorby using (20). Accordingly, the position estimate is updated accord-ing to (22). The procedure continues until the position increment issufficiently small.

V. ANCHOR POSITION ACCURACY ANALYSIS

The CRLB sets a lower bound on the variance of any unbiasedestimator and has widely been used as a performance measure inparameter estimation [21]. The vector parameter CRLB places a boundon the variance of each element, and the CRLB for the kth parameterof the position estimation vector ˆp is defined by

CRLB([ˆp]k)

=[F−1(ˆp)

]k,k

(25)


where F(ˆp) is the Fisher information matrix (FIM) whose elementsare defined by

[F(ˆp)

]k,�

= −E

[∂2 ln p(r|p)

∂pk∂p�

], k, � = 1, 2, . . . , N (26)

where the log-likelihood function ln p(r|p) under LOS conditions canbe derived as

ln p(r|p) = − 1

2

N∑i=1

N∑j=1j �=i

[1

σ2di,j

(di,j − di,j)2 +

1

σ2φi,j

×(

φi,j − tan−1 yi,j

xi,j

)2

+1

σ2αi,j

(αi,j − cos−1 zi,j

di,j

)2]

− 1

2

N∑k=1

[1

σ2xk

(xk − xk)2 +1

σ2yk

(yk − yk)2

+1

σ2zk

(zk − zk)2]

(27)

where

xi,j = xi − xj

yi,j = yi − yj

zi,j = zi − zj . (28)

To work out the CRLB, we first need to determine the FIM, which canbe derived as

F(ˆp) =

[Fxx Fxy Fxz

Fyx Fyy Fyz

Fzx Fzy Fzz

]∈ R3N×3N (29)

where the subblocks of the matrix are derived as follows. Define

ui =1

2

N∑j=1j �=i

[1

σ2di,j

(di,j − di,j)2 +

1

σ2dj,i

(dj,i − di,j)2

+1

σ2φi,j

(φi,j − tan−1 yj,i

xj,i

)2

+1

2σ2φj,i

(φj,i − tan−1 yi,j

xi,j

)2

+1

σ2αi,j

(αi,j − cos−1 zj,i

di,j

)2

+1

2σ2αj,i

(αj,i − cos−1 zi,j

di,j

)2]

+1

2σ2xi

(xi − xi)2 +

1

2σ2yi

(yi − yi)2 +

1

2σ2zi

(zi − zi)2

(30)

and let

d2ij =√

(xj,i)2 + (yj,i)2, j �= i

ρ(vi,j) =σ−2vi,j

+ σ−2vj,i

v = d, φ, α. (31)

Then, we obtain the components of the FIM in (29) as

[Fxx]i,i =E

[∂2ui

∂x2i

]

=

N∑j=1j �=i

(x2

i,jρ(di,j)

d2i,j

+y2

i,jρ(φi,j)

d42ij

+x2

i,jz2i,jρ(αi,j)

d4i,jd

22ij

)+

1

σ2xi

[Fxx]i,j =E

[∂2ui

∂xi∂xj

]

= −(

x2i,jρ(di,j)

d2i,j

+y2

i,jρ(φi,j)

d42ij

+x2

i,jz2i,jρ(αi,j)

d4i,jd

22ij

)

[Fxy]i,i =E

[∂2ui

∂xi∂yi

]

=

N∑j=1j �=i

xi,jyi,j

(ρ(di,j)

d2i,j

− ρ(φi,j)

d42ij

+z2

i,jρ(αi,j)

d4i,jd

22ij

)

[Fxy]i,j =E

[∂2ui

∂xi∂yj

]

= − xi,jyi,j

(ρ(di,j)

d2i,j

− ρ(φi,j)

d42ij

+z2

i,jρ(αi,j)

d4i,jd

22ij

)

[Fxz]i,i =E

[∂2ui

∂xi∂zi

]=

N∑j=1j �=i

xi,jzi,j

d2i,j

(ρ(di,j) − ρ(αi,j)

d2i,j

)

[Fxz]i,j =E

[∂2ui

∂xi∂zj

]= −xi,jzi,j

d2i,j


d2i,j

)

[Fyy]i,i =E

[∂2ui

∂y2i

]=

N∑j=1j �=i

(y2

i,jρ(di,j)

d2i,j

+x2

i,jρ(φi,j)

d42ij

+(yi,jzi,j)

2ρ(αi,j)

d4i,jd

22ij

)+

1

σ2yi

[Fyy]i,j =E

[∂2ui

∂yi∂yj

]= −

(y2

i,jρ(di,j)

d2i,j

+x2

i,jρ(φi,j)

d42ij

+(yi,jzi,j)

2ρ(αi,j)

d4i,jd

22ij

)

[Fyz]i,i =E

[∂2ui

∂yi∂zi

]=

N∑j=1j �=i

yi,jzi,j

d2i,j


d2i,j

)

[Fyz]i,j =E

[∂2ui

∂yi∂zj

]= −yi,jzi,j

d2i,j


d2i,j

)

[Fzz]i,i =E

[∂2ui

∂z2i

]=

N∑j=1j �=i

(z2

i,jρ(di,j)

d2i,j

+d22ijρ(αi,j)

d4i,j

)+

1

σ2zi

[Fzz]i,j =E

[∂2ui

∂zi∂zj

]=

(z2

i,jρ(di,j)

d2i,j

+d22ijρ(αi,j)

d4i,j

).

Therefore, the CRLBs for the coordinate estimates are

CRLB(ˆxi) =[F−1(ˆp)

]i,i

, i = 1, 2, . . . , N

CRLB(ˆyi) =[F−1(ˆp)

]N+i,N+i

CRLB(ˆzi) =[F−1(ˆp)

]2N+i,2N+i

. (32)


Fig. 4. Impact of distance accuracy on the LS algorithms when σα = σφ =

2.1◦, the STD of the GPS coordinate estimates is equal to either 10 or 18 m,and there are six anchors.

When dropping the angle-related terms in the aforementioned CRLB,we obtain the CRLB in the presence of anchor-to-anchor distanceestimates and GPS coordinate measurements. It is worth noting thatin the absence of GPS measurements, the derived FIM in (29) isassociated with the localization in anchor-free sensor networks. In thiscase, the FIM is singular so that the standard CRLB does not exist.

VI. SIMULATION RESULTS

In this section, we evaluate the performance of the proposed accu-racy improvement algorithms that are described in Sections III and IV.In addition, we examine the analytical results presented in Section V.Furthermore, we evaluate the impact of the anchor location errors onthe location accuracy of ordinary sensor nodes.

A. Accuracy Improvement of Anchor Locations

A cubic region of dimensions 500 m × 500 m × 500 m is consid-ered,2 where the anchors are randomly deployed. For each simulationrun, 1000 anchor position configurations are examined, and the perfor-mance is then averaged. The performance measure is the root-mean-square error (RMSE) of the anchor location estimation.

The same anchor position configurations are used to compute theaverage CRLB, which is defined as√√√√ 1

3NL

L∑�=1

3N∑k=1

[F−1(ˆp)

](�)k,k

(33)

where � indexes the anchor configurations, and L is set at 1000.Fig. 4 shows the accuracy of the linear least-squares (LS) algo-

rithms derived in Section III with respect to the distance estimationerrors. “LS-nlos” and “LS-los” denote results of the LS algorithmsunder NLOS and LOS conditions, respectively. Two different standarddeviations (STDs) of the original GPS measurement errors (10 and18 m) are examined, the STD of the angular estimation errors is setat 2.1◦, and there are six anchors. Although the accuracy degradationis considerable under the NLOS condition, the accuracy improvementis still substantial. In the event that there are both NLOS and LOSdistance and angular estimates, the accuracy improvement of the LS

2This dimensional size is only used to show the efficiency of the proposedaccuracy-enhancement algorithms. In practice, the vertical dimensional size ofthe location area may be much smaller than the horizontal dimensional size.

Fig. 5. Impact of AOA accuracy on the LS algorithms when σdi= 6 m, the

STD of the GPS coordinate estimates is equal to either 10 or 18 m, and thereare six anchors.

Fig. 6. Impact of distance estimation accuracy and number of anchors on theCRLB when the STD of the GPS coordinate estimates is equal to 10 m.

algorithm will be between the best (all LOS conditions) and the worst(all NLOS conditions) results, which are shown in the figure.

Fig. 5 shows the accuracy of the linear LS algorithms versus the ac-curacy of the angle (both azimuth and elevation) estimation. The STDof the distance estimation errors is set at 6 m; in addition, two differentSTDs of the GPS measurements are examined. Comparatively, theaccuracy is more sensitive to the angular estimation errors, particularlyin the NLOS conditions. Probably, the reason is that when the distanceis large, a small angular error would produce a larger distance error.In both Figs. 4 and 5, we can observe that the accuracy of the LSalgorithm approaches the corresponding CRLBs.

Fig. 6 illustrates the impact of the distance estimation error andthe number of anchors on the CRLB when the STD of the GPSmeasurement error is set at 10 m in the absence of AOA estimates.It is shown that as the number of anchors increases, the accuracyimproves. Perhaps, this is due to the fact that the addition of oneanchor introduces three more unknown position parameters; however,6(N − 1) more anchor-to-anchor measurements and three more GPSmeasurements are produced. Generally speaking, more independentmeasurements could result in a redundancy/diversity gain.

Fig. 7 shows the performance of the LM-based iterative optimiza-tion algorithm described in Section IV. The simulation setup is thesame as that for Fig. 6. Basically, there is a good match between Figs. 6and 7. Clearly, the distance-based algorithm can also provide a goodaccuracy gain.


Fig. 7. Impact of distance estimation accuracy and number of anchors on theaccuracy of the LS algorithm when the STD of the GPS coordinate estimates isequal to 10 m.

Note that it would be interesting to perform a performance compar-ison between the proposed method and the existing methods such asa low-cost DGPS-based solution in more realistic scenarios; however,this is beyond the scope of this paper.

B. Accuracy Improvement of Sensor Node Locations

As demonstrated through simulation in Section VI-A, the proposedmethods can substantially enhance the anchor location accuracy. Itwould be interesting to further evaluate how the anchor location accu-racy affects the performance of the ordinary sensor node localization.For this purpose, we conduct simulations under the following setup.The monitored area has a cubic shape with the edge length equal to200 m. There are six anchors, whose locations, denoted by (xi, yi, zi),are set at (200, 0, 0), (200, 200, 0), (0, 200, 0), (200, 0, 200), (0, 0, 200),and (0, 200, 200) (the unit is in meters). The location of the sensornode, which is denoted by (x, y, z), is randomly generated in themonitored area, and 5000 different sensor node locations are examinedfor each simulation run. The propagation between the sensor node andeach anchor node is assumed LOS. Let di be the measurement of thedistance between the sensor node and the ith anchor, and let φi andαi be the azimuth and elevation angle measurements made at the ithanchor, respectively. Then, the measurement equation is given by

Mp ≈ f (34)

where

M =

[eN 0 00 eN 00 0 eN

]p =

[xyz

]

f =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x1 + d1 sin α1 cos φ1

...xN + dN sin αN cos φN

y1 + d1 sin α1 sin φ1

...yN + dN sin αN sin φN

z1 + d1 cos α1

...xN + dN cos αN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(35)

where eN is a column vector of N ones. The LS solution for (34) isgiven by

p = (MT ΦM)−1MT Φf (36)

Fig. 8. Impact of anchor location accuracy on sensor node localization. TheSTD of the AOA errors is set at 2◦ and 5◦, and the STD of the distance errorsis set at 2 and 8 m.

where Φ is a weighting matrix, which may be determined based on theaccuracy of the distance and angle measurements at each anchor. Here,it is simply set at an identity matrix.

Fig. 8 shows the accuracy (RMSE) of the sensor node locationestimation. The STD of the AOA (both the azimuth and elevationangle) estimation errors is set at two different values, i.e., 2◦ and 5◦.Meanwhile, the STD of the distance measurement errors is set at 2 and8 m. The STD of the anchor coordinate errors ranges from 0 to 18 m.From the figure, it can be observed that when the anchor coordinateerror goes from 10 m down to 5 m, the sensor node location errorsare reduced from about 1.9 to 0.9 m under σa = 2◦ and σd = 2 m.That is, the sensor location error is decreased by about 50%. Whenσa = 5◦ and σd = 8 m, the accuracy improvement of the sensorlocation estimation is also considerable if the anchor location errorsis reduced by about 50%. For instance, the sensor location error goesfrom 2.85 m down to 2.45 m when the anchor location error goes from10 m down to 5 m, or the sensor location error goes from 3.78 m downto 2.73 m when the anchor location error goes from 18 m down to 9 m.It can also be observed that when the anchor location error is large, thesensor location error is comparatively much smaller. The main reasonmay be due to the fact that at each anchor, the sensor location estimateis the sum of two terms, one of which is the anchor coordinate estimate,and the other is related to the distance and angle measurements. Afterapplying the LS estimation, the effect of the anchor location errorswould be largely averaged out.

VII. CONCLUSION

In this paper, we have made use of anchor-to-anchor cooperationto improve the anchor-location accuracy in 3-D environments. Itwas considered for scenarios where the measurements of basic GPSreceivers are not satisfactory, and more accurate anchor-to-anchor pa-rameter estimates can be obtained. Novel low-complexity algorithmswere derived when GPS coordinate measurements and both anchor-to-anchor distance and angle estimates were provided. In the casethat GPS measurements and anchor-to-anchor distance estimates areavailable, an optimization-based algorithm was developed to improveanchor location accuracy under LOS conditions. Furthermore, wederived the CRLBs to benchmark the algorithms. Simulation resultsdemonstrated that the proposed algorithms can substantially improvethe anchor location accuracy. In addition, the accuracy of the devel-oped algorithms approaches the CRLBs. It was also demonstratedthat the accuracy of ordinary sensor node location estimation can


considerably be improved when the location accuracy of anchor nodesis enhanced by the proposed algorithms.

In the proposed method, both the distance and AOA informationmust be obtained in the anchors. This requirement may not be desirablefor some applications. Therefore, additional research is needed to solvethe restrictions and find alternative techniques to improve the locationaccuracy of the anchors. In addition, the GPS-related delay issues arenot considered in the proposed method. To make the proposed methodsuited for practical applications, the delay issues must be considered,particularly when the anchors are moving. It would be interesting toaddress the problem in future work.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers for theirvaluable comments and suggestions.

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Anchor Global Position Accuracy Enhancement Based on Data Fusion