Efficient Closed-form Algorithms for AOA Based Self-localization of Sensor Nodes Using Auxiliary Variables

1

Efficient Closed-form Algorithms for AOA

Based Self-localization of Sensor Nodes Using

Auxiliary VariablesHua-Jie Shao, Xiao-Ping Zhang, Senior Member, IEEE, and Zhi Wang

Abstract

Node self-localization is one of the most important research topics for wireless sensor networks (WSNs). There

are two main algorithms, the triangulation method and the maximum likelihood (ML) estimator, for angle of arrival

(AOA) based self-localization in the WSNs. However, the ML estimator requires a good initialization close to the true

location to avoid the divergence problem, while the triangulation method cannot obtain the closed-form solution with

high localization accuracy and low computational complexity. In this paper, we develop a set of efficient closed-form

AOA based self-localization algorithms derived from effective auxiliary variables based method. First, we formulate

the self-localization problem as a linear least squares problem using auxiliary variables. Based on its closed-form

solution, a new auxiliary variables based pseudo-linear estimator (AVPLE) is developed. By analyzing the estimation

error of the AVPLE algorithm, we present a bias compensated AVPLE (BCAVPLE) to reduce the estimation error.

Then we develop a novel BCAVPLE based weighted instrumental variable (BCAVPLE-WIV) estimator to achieve

asymptotically unbiased estimation of locations and orientations of unknown nodes based on the prior knowledge of

the AOA noise variance. In the case that the AOA noise variance is unknown, a new AVPLE based WIV (AVPLE-

WIV) estimator is developed to localize the unknown nodes. Also, we develop an autonomous coordinate rotation

(ACR) method to overcome the tangent instability of the proposed algorithms when the orientation of the unknown

node is near π/2. To analyze the theoretical performance, we derive the Cramer-Rao lower bound (CRLB) of the

maximum likelihood (ML) estimator. Extensive simulation results demonstrate that the new algorithms achieve much

higher localization accuracy than the triangulation method and also avoid local minima and divergence problems in

iterative ML estimators.

Index Terms

Angle of arrival, node self-localization, auxiliary variables, closed-form pseudo-linear estimator, bias compensated

auxiliary variable based pseudo-linear estimator, weighted instrumental variables, wireless sensor networks

H.-J. Shao is with the Department of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China. E-mail:

[email protected].

X.-P. Zhang is with the Department of Electrical & Computer Engineering, Ryerson University, 350 Victoria Street, Toronto, Ontario,

CANADA, M5B 2K3. E-mail: [email protected].

Z. Wang* is with the Department of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China. E-mail:

[email protected].

January 23, 2014 DRAFT

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I. INTRODUCTION

Location information of sensor nodes in wireless sensor networks (WSNs) is essential since it is the basis to

perform a number of tasks such as target localization and tracking [1]–[4], and animal behavior monitoring [5], etc.

In practice, random deployment of sensor nodes (e.g., dropped from aircraft) is commonly used since regions of

interest (ROI) usually reside in the harsh environment. As a result, the location information of sensor nodes is often

unknown. Obtaining accurate location information for sensor nodes becomes an important task. A possible method

is to equip each sensor node with the GPS receivers [6], but it is expensive especially for large-scale networks.

With the budget and energy constraints [7], [8], only limited number of sensor nodes can be equipped with GPS

receivers. These nodes are called beacons as their accurate locations are known by the GPS receivers. Other nodes

without GPS receivers are called unknown nodes, whose locations have to be estimated.

Many techniques have been proposed for sensor node localization, such as received signal strength indicator

(RSSI), time of arrival (TOA), angle of arrival (AOA), and time difference of arrival (TDOA) [6], [9]. In this paper,

we focus on the AOA technique [10], [11] for node self-localization in WSNs. One common method to obtain AOA

measurements is to use an antenna array on each sensor node [12]. Different from target localization, AOA based

node self-localization needs to estimate both the location and orientation of each unknown node. The orientation

in the paper refers to the reference direction against which the AOAs are measured.

A number of estimation algorithms have been proposed for AOA based self-localization method over the past

years, such as triangulation [10], [12], maximum likelihood (ML) estimator [13], [14], [15] and the semidefinite

programming (SDP) method [11]. The triangulation method uses two beacons and their AOAs to find the center

and radius of the circumscribed circle for the triplet of two beacons and one unknown node. Then the trilateration

approach is used to obtain the locations of nodes based on the known radiuses and centers of these circumscribed

circles. However, the localization accuracy of the triangulation method is low because of the error accumulation in

the process of obtaining the center and radius of the circumscribed circle. The ML estimator is an asymptotically

unbiased estimator for AOA based self-localization problem. However, it is a nonlinear least squares estimator

for Gaussian angle measurement noise. The ML method either requires a reasonable initialization close to the

true solution or may suffer from local minima and even divergence problems [16], [17]. Recently, Biswas et

al. [18], [11] proposed a SDP method to estimate the locations of nodes for large scale sensor networks. However,

the SDP relaxation method needs high connectivity among nodes and beacons so that the size of the triplets that

have mutual angle information is large enough for the set of equations to have a unique solution [11]. It has a

complicated structure and high computational complexity [6]. Also note that in the SDP for AOA based node

localization [11], the beacons and unknown nodes need to measure each other’s AOAs mutually, consuming much

energy of the nodes and beacons. Moreover, the SDP cannot estimate both the locations and orientations of unknown

nodes simultaneously. Its localization accuracy depends on the size of the triplets with relative angle information

and the locations of beacons. Some beacons are required to be placed on the perimeter of the WSNs area for the

SDP method to be effective.


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In [19], [20], closed-form solutions of the bearings only target tracking are presented. Those closed-form solutions

of bearing-only (i.e., AOA) target tracking only estimate the target location (x, y), and cannot be applied to estimate

both the locations and orientations of unknown nodes simultaneously.

In this paper, we develop an effective auxiliary variables based method to obtain the closed-form solution of

AOA based self-localization problem. It improves the localization accuracy compared with the triangulation method

and avoids the problems of iterative ML methods. First, we develop an auxiliary variables based pseudo-linear

estimator (AVPLE) algorithm, which employs auxiliary variables to formulate the self-localization problem as a

linear least squares problem and gives a closed-form solution. By analyzing the estimation error of the AVPLE

algorithm, we then present a novel bias compensated AVPLE (BCAVPLE) scheme to reduce the estimation error.

Furthermore, we develop a BCAVPLE based weighted instrumental variable (BCAVPLE-WIV) estimator to achieve

asymptotically unbiased estimation of locations and orientations of unknown nodes based on the prior knowledge of

the AOA noise variance. In the case that the AOA noise variance is unknown, an AVPLE based WIV (AVPLE-WIV)

estimator is developed to localize the unknown nodes. Also, we further investigate the impact of the orientation

on estimation performance of the new algorithms and present an autonomous coordinate rotation (ACR) method

to overcome the tangent instability of our proposed algorithms when the orientation of the unknown node is near

π/2. To analyze the theoretical performance, we derive the Cramer-Rao lower bound (CRLB) of the ML estimator.

Extensive simulation results demonstrate that the new algorithms achieve much higher localization accuracy than

the triangulation method and also avoid local minima and divergence problems in iterative ML estimators.

The paper is organized as follows. Section II states the problem of node self-localization. Section III presents

an effective auxiliary variables based method to estimate the unknown node with closed-form solutions. Section IV

analyzes the estimation error of node self-localization using the AVPLE algorithm. Section V provides a BCAVPLE

estimator to reduce the estimation error and further presents a BCAVPLE-WIV estimator and an AVPLE-WIV

estimator to estimate the unknown node. Section VI presents the ML estimator to localize the unknown node for

AOA based self-localization and derives the CRLB of it. Section VII evaluates the performances of our proposed

algorithms based on extensive simulations. Finally, Section VIII concludes the paper.

The main notation in this paper is listed as follows:

N number of beacons

p = [x, y, θ]T position and orientation vector of the unknown node

qj = [aj , bj ]T position vector of jth beacon

βj measured AOA with noise for the jth beacon

βj true AOA for the jth beacon

∆βj AOA measurement noise for the jth beacon

σ2j AOA noise variance

β = [β1, . . . , βN ]T vector of AOAs with noise

β = [β1(p), . . . , βN (p)]T vector of AOAs as a function of position vector of the unknown node


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U = [u1, u2, u3]T accurate auxiliary variables related to position vector of the unknown node

Uavple = [u1, u2, u3]T estimated auxiliary variables for the AVPLE

∆U = [∆u1,∆u2,∆u3]T estimation bias of auxiliary variables for the AVPLE

f(β|β) bearing-only likelihood function

J ML cost function

Qj = [aj + u3bj , bj − u3aj ]T assumed observer location so as to analyze the estimation bias

U12 = [u1, u2]T assumed true target location

U12 = [u1, u2]T estimated target location for U12 using AVPLE

sj noisy AOA vector for the observer Qj

rj = [sin βj ,− cos βj ]T orthogonal unit vector for sj

dj the range from observer Qj to target U12

η2j = E{sin2 ∆βj} Approximate AOA noise variance when the AOA noise is small

ξ = [ξx, ξy, ξθ]T estimation errors of position vector p

F Fisher information matrix for the ML estimator

drj Euclidean distance from the unknown node and jth beacon

Cavple error covariance of the AVPLE

∆U approximate estimation bias of ∆U for large N

∆Û approximate estimation bias of ∆U

I identity matrix

U bcavple = [ub1, ub2, ub3]T estimated auxiliary variables for BCAVPLE

L instrumental variable (IV) matrix

βj approximated AOA obtained from 6 (U12 − Qj)

νj = [sin βj ,− cos βj ]T approximated orthogonal unit vector for sj

U bcavple−wiv estimated auxiliary variables for the BCAVPLE-WIV

Uavple−wiv estimated auxiliary variables for the AVPLE-WIV

II. PROBLEM STATEMENT OF NODE SELF-LOCALIZATION

As shown in Fig. 1, assume there are N beacons equipped with GPS receivers and an unknown node without

GPS receivers in a region. Each beacon knows its accurate location, denoted by qj = [aj , bj ]T , j = 1, 2, . . . , N ,

but does not know its orientation. That is to say, we do not use the orientation information of the beacon in the

estimation. The nodes in black color are beacons. The unknown node, in yellow color, does not know its location

and orientation information. The location and orientation of the unknown node, denoted by p = [x, y, θ]T , are to

be estimated.

The beacons are signal emitters and the unknown node is a signal receiver. Hence, the AOAs of the beacons can

be measured by the neighboring unknown node with respect to its own orientation. Let βj denote the measured

AOA by the unknown node with respect to its own orientation θ from the jth beacon. The AOA measurements are


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Beacon

Unknown Node

1

Orientation

( , )j jj a bq

)( , ,x yp

1 11( , )a bq

j

Fig. 1: Illustration of the AOA based node self-localization in WSNs.

given as follows [21]:

βj = βj + ∆βj , ∆βj ∼ N (0, σ2j ), (1)

where βj is the true AOA measurement from the jth beacon, ∆βj ∼ N (0, σ2j ) is the measurement noise following

a Gaussian distribution with mean zero and variance of σ2j . Note that in [21], it has been shown that the errors of

AOA estimation by the MUSIC algorithm are asymptotically Gaussian distributed with zero means.

In this paper, the objective is to estimate the location and orientation of the unknown node based on the known

locations of beacons and their AOAs measured by the unknown node.

III. AUXILIARY VARIABLE BASED NODE SELF-LOCALIZATION IN CLOSED-FORM

As shown in Fig. 1, the relationship between the unknown node and its neighbor beacons is:

tan(θ + βj) =bj − yaj − x

, j = 1, 2, . . . , N, (2)

where (aj , bj) is the location of the jth beacon, (x, y) and θ denote the location and orientation of the unknown

node, respectively.

After mathematical derivation, we get

sinβj(x+ y tan θ)− cosβj(y − x tan θ)

− (aj cosβj + bj sinβj) tan θ = aj sinβj − bj cosβj .(3)

See Appendix A for the detailed mathematical derivation from (2) to (3).

To solve the unknown node location estimation problem, we introduce the following auxiliary variables:

u1 = x+ y tan θ,

u2 = y − x tan θ,

u3 = tan θ.

(4)


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Then (3) can be written as

sinβju1 − cosβju2 − (aj cosβj + bj sinβj)u3 = aj sinβj − bj cosβj . (5)

As the true AOA, βj , cannot be obtained in the outdoor environment, we replace it with the measured AOA with

noise, βj , to formulate the linear least squares problem. Thus, (5) could be approximately expressed as

AU ≈ b, (6)

where

A =

sin β1 − cos β1 −a1 cos β1 − b1 sin β1

sin β2 − cos β2 −a2 cos β2 − b2 sin β2...

......

sin βN − cos βN −aN cos βN − bN sin βN

, (7a)

b =

a1 sin β1 − b1 cos β1

a2 sin β2 − b2 cos β2...

aN sin βN − bN cos βN

, (7b)

U = [u1, u2, u3]T. (7c)

A least squares criteria can be used to solve (6). We can adopt the PLE to get a closed-form solution. The

estimated value of the auxiliary variables U , denoted by Uavple, is

Uavple = argminU||AU − b||22. (8)

Based on (8) and (4), the location and orientation of the unknown node is given byx

y

θ

=

u1−u3u2

1+u23

u2+u1u3

1+u23

arctan u3

. (9)

Remark 1 Note that the tangent becomes unstable when the orientation of the unknown node is near π/2 (or

−π/2). Thus, it may result in high estimation error of the auxiliary variables based PLE (AVPLE) method when

the orientation is near ±π/2.

To solve the tangent instability problem, we develop an autonomous coordinate rotation (ACR) algorithm to

autonomously rotate the coordinate system and therefore avoid the tangent instability as follows.

Based on the estimated location and orientation of the unknown node, p1 = [x1, y1, θ1]T , the locations of

beacons, qj = [aj , bj ]T , and the measured AOAs, βj , we will test whether to choose the original coordinate system

or the alternative rotated (by π/2) coordinate system. Specifically, in the original coordinate system, we compute

an error αj between the estimated AOAs based on the j-th beacon and estimated unknown node, denoted by

arctan(y1−bjx1−aj

)− θ1, and the measured AOAs βj , i.e., αj = arctan

(y1−bjx1−aj

)− θ1− βj . Similarly, we can compute


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this error in the rotated coordinate system, denoted by α′j . Now for N beacons, we compare the total square error,∑Nj=1 α

2j and

∑Nj=1 α

′2j , and then we adopt the coordinate system with lower total square error.

The key of the algorithm is to decide when to use the rotated (by π/2) coordinate system. The algorithmic

procedures of autonomous coordinate rotation (ACR) algorithm are summarized in Algorithm 1.

Algorithm 1 ACR algorithm

1: Initialization: βj , qj = [aj , bj ]T , j = 1, . . . , N ;

2: Use AVPLE algorithm to estimate the unknown node based on Eqs. (2)-(9), p1 = [x1, y1, θ1]T ;

3: Adopt the estimated p1 and the locations of beacons qj = [aj , bj ]T to calculate the estimated AOAs, denoted

by arctany1−bjx1−aj − θ1, j = 1, 2, . . . , N ;

4: Calculate an error between the estimated AOAs and the measured AOAs βj , denoted by αj = arctan(y1−bjx1−aj

)−

θ1 − βj ;

5: Rotate coordinate system, counter-clockwise 90 degrees. Then, Eq. (2) becomes tan(θ + βj) =x−ajbj−y ;

6: Estimate the unknown node in Step 5 using AVPLE, p2 = [x2, y2, θ2]T ;

7: Similar to Step 3 and 4, calculate the error between the estimated AOAs and the measured AOAs based on p2,

denoted by α′j = arctan(y2−bjx2−aj

)− θ2 − βj ;

8: If∑Nj=1 α

2j >

∑Nj=1 α

′2j

Adopt the rotated coordinate system, return;

else

Adopt original coordinate system, return;

End If

IV. ERROR ANALYSIS FOR UNKNOWN NODE LOCALIZATION

Note that in (6), the noise contaminated AOA measurements are used instead of the true AOA in (5). It leads to

the estimation error for the unknown node localization. In this section, we analyze the theoretical estimation error

of node self-localization using the AVPLE algorithm. First we provide the bias analysis of the auxiliary variables

using the AVPLE, and then derive the location error of the unknown node based on the estimation error of the

auxiliary variables.

Different from target localization, the error analysis for AOA based node self-localization involves three variables

in the two-dimension plane. This analysis has not been done before. We convert the formulation of node self-

localization using auxiliary variables in (5) to a similar target localization geometrical relationship in [20] as

depicted in Fig. 2. We assume that Qj = [aj +u3bj , bj −u3aj ]T represents the observer location, U12 = [u1, u2]T

represents the estimated target location, U12 = [u1, u2]T denotes the true target location and εj represents the


8

jQ

12U 12U

j

j

1 2( , )u u

3 3( , )j j j ja u b b u a

jε

jr

js jsj

U

jζ

Target

Observer

jδ

Fig. 2: Geometrical relationship of node self-localization using auxiliary variables for equation (5).

orthogonal error vector. In addition, we define the orthogonal unit vector for noisy AOA vector sj as follows:

rj =

sin βj

− cos βj

. (10)

According to the target localization geometry described in Fig. 2, we have

U12 = U12 + ζj , (11)

where U12 denotes the estimated target location and ζj is the error vector.

Note that

U12 = Qj + sj , ζj = δj + εj . (12)

Substituting (12) into (11) yields

U12 = Qj + sj + δj + εj , (13)

where the orthogonal vector εj is defined by

εj = rjdj sin ∆βj , (14)

and dj = ‖sj‖ is the distance from Qj to U12.

By pre-multiplying both sides of (13) with rTj , we can get

rTj U12 = rTj Qj + rTj εj = rTj Qj + dj sin ∆βj . (15)

After expansion,

sin βju1 − cos βju2 − (aj cos βj + bj sin βj)u3

= aj sin βj − bj cos βj + dj sin ∆βj .(16)


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Concatenating (16) for j = 1, 2, . . . , N yields

AU = b+ φ, (17)

where

φ =

d1 sin ∆β1

d2 sin ∆β2...

dN sin ∆βN

, (18)

matrices A, U and b are the same as those in (7). Based on (17), the accurate solution of AVPLE is

U = (ATA)−1AT (b+ φ). (19)

Hence, the estimation bias of the auxiliary variables for AVPLE in (8) can be given by

∆U = E{Uavple −U

}= E

{(ATA)−1AT b− (ATA)−1AT (b+ φ)

}= −E

{(ATA)−1ATφ

}.

(20)

Then, based on Slutsky’s theorem [22], [2], for large N the AVPLE bias mentioned above can be approximated by

∆U ≈ −E

{ATA

N

}−1E

{ATφ

N

}. (21)

As N →∞, (21) becomes an equality. From (21), it can be seen that the AVPLE bias is determined by ATA as

well as ATφ.

In the following, we further analyze the relationship between ∆U with ATφ and ATA, and obtain a more

accurate theoretical description of the AVPLE bias. Note that

ATφ =

N∑j=1

djsin∆βj

sin βj

− cos βj

−aj cos βj − bj sin βj

=

N∑j=1

dj

12 sinβj sin 2∆βj

− 12 cosβj sin 2∆βj

− 12 (aj cosβj + bj sinβj) sin 2∆βj

+

N∑j=1

dj

cosβj sin2 ∆βj

sinβj sin2 ∆βj

(aj sinβj − bj cosβj) sin2 ∆βj

.

(22)


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ATA =

N∑j=1

sin2 βj − 1

2 sin 2βj − 12aj sin 2βj − bj sin2 βj

− 12 sin 2βj cos2 βj aj cos2 βj + 1

2bj sin 2βj

− 12aj sin 2βj − bj sin2 βj aj cos2 βj + 1

2bj sin 2βj (aj cos βj + bj sin βj)2

. (25)

Thus, the expectation of ATφ/N can be expressed by

E

{ATφ

N

}=

1

N

N∑j=1

η2j

dj cosβj

dj sinβj

dj(aj sinβj − bj cosβj)

=1

N

N∑j=1

η2j

u1 − bju3 − aju2 − bj + aju3

aju2 − bju1 + cju3

=

1

N

N∑j=1

η2j (GjU −Dj)

(23)

where

Gj =

1 0 −bj0 1 aj

−bj aj cj

,Dj =

aj

bj

0

, (24a)

η2j = E{sin2 ∆βj}, cj = a2j + b2j . (24b)

If the AOA measurement noise is sufficiently small, η2j ≈ E{∆β2j } = σ2

j , same as the variance of AOA noise

in (1).

Now we examine the autocorrelation matrix E{ATA/N}. Based on (7a), we have (25), please see it on the top

of next page. The elements in (25) are

sin2 βj = sin2 βj cos2 ∆βj + cos2 βj sin2 ∆βj

+1

2sin 2βj sin 2∆βj ,

(26a)

cos2 βj = cos2 βj cos2 ∆βj + sin2 βj sin2 ∆βj

− 1

2sin 2βj sin 2∆βj ,

(26b)

sin 2βj = sin 2βj cos 2∆βj + cos 2βj sin 2∆βj . (26c)

Thus for small AOA measurement noise we can get the expectation (27), please see it on the top of next page.


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E

{ATA

N

}=

1

N

N∑j=1

sin2 βj − 12 sin 2βj − 1

2aj sin 2βj − bj sin2 βj

− 12 sin 2βj cos2 βj aj cos2 βj + 1

2bj sin 2βj

− 12aj sin 2βj − bj sin2 βj aj cos2 βj + 1

2bj sin 2βj (aj cosβj + bj sinβj)2

+η2j

cos 2βj sin 2βj aj sin 2βj − bj cos 2βj

sin 2βj − cos 2βj −aj cos 2βj − bj sin 2βj

aj sin 2βj − bj cos 2βj −aj cos 2βj − bj sin 2βj cos 2βj(b2j − a2j )− 2ajbj sin 2βj

.

(27)

Based on (23) and (27), the AVPLE bias in (21) can be rewritten as

∆U ≈ −E{ATA

N}−1

1

N

N∑j=1

η2j (GjU −Dj)

≈ −E{ATA}−1

N∑j=1

η2j (GjU −Dj),

(28)

where E{ATAN }−1 is the inverse of formulation (27). In the limit when N →∞, (28) would become an equality.

It can be seen from (28) that for large N the estimation bias is determined by E{ATA/N} as well as matrix

Gj and Dj , which are related to the locations of beacons. Moreover, (23) and (27) show that the estimation error

is proportional to η2j , which approximates the AOA noise variance when the AOA noise is small. Note that the

estimation bias ∆U does not vanish with the increase of N even if when N →∞, implying that the AVPLE is a

biased estimator.

Remark 2 We employ the chi-squared test to check the error distributions of U = [u1, u2, u3]T , and find that

their error distributions are not Gaussian. It is because the cosine and sine functions in (6) and (7) are nonlinear

functions that apply to the measured DOA with Gaussian noise βj . Thus, the algorithm to estimate the auxiliary

variables for node self-localization in our paper is a pseudo-linear estimator (PLE) similar to those defined in

[19], [23]–[25], and it is a biased estimator.

After obtaining the estimation bias of the auxiliary variables for the AVPLE, denoted by [∆u1,∆u2,∆u3]T =

∆U , we can get the estimation bias ξ of the unknown node as follows

ξx =u1 − u2u3

1 + u23− u1 − u2u3

1 + u23

={u1 + ∆u1 − (u2 + ∆u2)(u3 + ∆u3)}

1 + (u3 + ∆u3)2− u1 − u2u3

1 + u23,

(29a)

ξy =u2 + u1u3

1 + u23− u2 + u1u3

1 + u23

={u2 + ∆u2 + (u1 + ∆u1)(u3 + ∆u3)}

1 + (u3 + ∆u3)2− u2 + u1u3

1 + u23,

(29b)


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ξθ = arctan u3 − arctanu3

= arctan∆u3

1 + tan(u3 + ∆u3) tanu3,

(29c)

where ξ = [ξx, ξy , ξθ]T denotes the estimation error of the location and orientation for the unknown node,

[u1, u2, u3]T are the accurate values of the auxiliary variables for node self-localization.

V. BIAS COMPENSATION FOR AVPLE AND WEIGHTED INSTRUMENTAL VARIABLE ESTIMATOR

In this section, we provide a bias compensation scheme for the AVPLE algorithm based on the prior knowledge

of the AOA noise variance. To further reduce the estimation bias and approximate the CRLB accuracy, a new

BCAVPLE based WIV (BCAVPLE-WIV) estimator is proposed. In addition, when the AOA noise variance is

unknown, a new AVPLE based WIV (AVPLE-WIV) estimator is developed to localize the unknown node.

A. Bias Compensation for the AVPLE (BCAVPLE)

The position estimation bias of the unknown node is determined by the estimation bias of auxiliary variables as

shown in (29). Therefore, it is essential to estimate the auxiliary variables as accurately as possible. We show that

the AVPLE is a biased estimator, motivating us to develop a method to compensate the bias.

Note that in AVPLE estimation bias (20), the true ranges and AOA noises in φ are unavailable to calculate the

bias ∆U . For sufficiently large N , using the similar methodology as in [19], we can approximate the estimation

bias ∆U by

∆U = −(ATA)−1E{ATφ

}= −(ATA)−1

N∑j=1

η2j (GjU −Dj) .(30)

For large N , the expectation of the approximate estimation bias in (30) is

E{∆U} ≈ −E{ATA}−1N∑j=1

η2j (GjU −Dj) ≈ ∆U . (31)

Since the true value of U is unavailable, the approximate estimation bias ∆U cannot be obtained directly. To

calculate the approximate estimation bias ∆U of the auxiliary variables, the matrix U in (30) is replaced by the

estimation result of AVPLE, Uavple. Thus the estimation bias can be approximated as

∆Û = −(ATA)−1

N∑j=1

η2j (GjUavple −Dj) (32)

Note that Uavple = U + ∆U . The expectation of the approximate estimation bias in (32) can be expressed by

E{∆ Û} ≈ −E{ATA}−1

N∑j=1

η2j (Gj(U + ∆U)−Dj)

≈ ∆U − E{ATA}−1N∑j=1

η2jGj∆U

≈(I − E{ATA}−1H

)∆U ,

(33)


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where I is the identity matrix, and

H =

N∑j=1

η2jGj . (34)

Note that the approximate estimation bias in (31) is unbiased, while the estimation error in (33) is asymptotically

biased. The estimation bias in (33) could have a small bias only if meets one of the following conditions:

• E{ATA}−1H ≈ µI,∀ 0 ≤ µ� 1;

• η2j ≈ 0.

As we obtain the approximated estimation bias ∆Û , the BCAVPLE estimator can be written as

U bcavple = Uavple −∆Û

= Uavple + (ATA)−1N∑j=1

η2j (GjUavple −Dj)(35)

It can be seen in (33) that the approximated estimation bias is biased, thus the BCAVPLE is also biased. However,

we can see from (35) that the BCAVPLE has a reduced estimation bias compared with the AVPLE.

Based on (35), the estimate bias of BCAVPLE is given by

∆U bcavple = E{U bcavple} −U ≈ ∆U − E{∆ Û}

≈ E{ATA}−1H∆U .(36)

Remark 3 The BCAVPLE algorithm is effective based on the accurate prior knowledge of the transformed AOA

noise variance η2j = E{sin2 ∆βj}, which approximates the AOA noise variance σ2j when it is small. Without this

prior knowledge of the AOA measurement noise, the BCAVPLE cannot reduce the estimation bias effectively as

shown below.

Assuming that the inaccurate AOA noise variance parameter is η2j , the error between the BCAVPLE without

accurate AOA noise variance parameter and the BCAVPLE with accurate prior knowledge of AOA noise variance

is

εbcavple = (ATA)−1N∑j=1

(η2j − η2j )(GjUavple −Dj), (37)

and the error between the AVPLE and the BCAVPLE with accurate prior knowledge of AOA noise variance is

εavple,bcavple = −(ATA)−1N∑j=1

η2j (GjUavple −Dj). (38)

Comparing (37) and (38), we can see that BCAVPLE may have larger estimation bias for the node self-localization

compared with the AVPLE when the error between the accurate η2j and inaccurate η2j is large.

B. Asymptotically Unbiased Weighted Instrumental Variable Estimator

In general, the BCAVPLE is not an unbiased estimator though its estimation bias could be reduced compared

with the AVPLE. To construct an asymptotically unbiased estimator, based on the methodology in [19], [17], a new

BCAVPLE-WIV estimator is proposed to localize the unknown node based on the prior knowledge of the AOA


14

noise variance. In addition, in the case that the AOA noise variance is unknown, we present a new AVPLE-WIV

estimator to estimate the unknown node. We will elaborate these two new methods respectively below in this section.

The key of the BCAVPLE-WIV is to design and construct the instrumental variable (IV) matrix L. Our new

method is described as follows.

Referring to (6) and (8), the solution of the AVPLE is obtained from the following normal equation

ATAUavple = AT b. (39)

To reduce the bias caused by the correlation between AT and φ when ATA is nonsingular, the IV matrix L is

introduced to replace the matrix A in (39), yielding

LTAU iv = LT b, (40)

where U iv is the IV estimator

U iv = (LTA)−1LT b. (41)

The most challenging task for the IV estimator is to choose a reasonable L such that E{LTA/N} is nonsingular

and E{LTφ/N} = 0, which can make the IV estimator an asymptotically unbiased estimator, i.e., E{U iv} = U iv

as N →∞. Usually, the noise-free A can be chosen as the optimal IV matrix [19], [22], i.e.,

A0 =

νT1 Φ1

...

νTNΦN

, (42)

where

νj =

sinβj

− cosβj

, j = 1, 2, . . . , N, (43a)

Φj =

1 0 −bj0 1 aj

, j = 1, 2, . . . , N. (43b)

However, the optimal IV matrix is not available since the true AOAs cannot be obtained in the open air environment

with noise. A simple method for constructing the IV matrix L is to replace the true AOA measurement βj in A0

with the approximated AOA obtained from the BCAVPLE. Here the approximated AOA is given by

βj = 6 (U b − Qj), j = 1, 2, . . . , N, (44)

where

U b = [ub1, ub2]T , j = 1, 2, . . . , N, (45a)

Qj = [aj + bj ub3, bj − aj ub3]T , j = 1, 2, . . . , N, (45b)

and

[ub1, ub2, ub3]T = U bcavple, (45c)

which is the estimation result of the BCAVPLE.


15

Based on the βj estimated in (44), the IV matrix can be constructed as

L =

νT1 Φ1

...

νTNΦN

, νj =

sin βj

− cos βj

. (46)

To reduce the estimation variance, a WIV estimator is proposed to estimate the unknown node based on the IV

estimator. Under the assumption that the AOA noise variance σ2j is known, the weighted matrix can be given by

W = diag(d21σ21 , d

22σ

22 , · · · , d2Nσ2

N ), (47)

where dj is the estimated range from the observer Qj to estimated target location U12 based on BCAVPLE. Using

the similar methodology in [26], [19], the closed-form solution of the WIV estimator, i.e., the BCAVPLE-WIV

estimator, is

U bcavple−wiv = (LTW−1A)−1LTW−1b. (48)

If the AOA noise is independent identically distributed (i.i.d.) with the variance σ2j = σ2, j = 1, 2, . . . , N , the

weighted matrix in (47) can be rewritten as

W = diag(d21, d22, . . . , d

2N ). (49)

However, sometimes the AOA noise variance σ2j is hard to estimate due to the complicated outdoor environments.

We present an AVPLE-WIV estimator, which does not need the prior knowledge of AOA noise variance, to localize

the unknown node. Similar to the BCAVPLE-WIV estimator, the AVPLE-WIV estimator is given by

Uavple−wiv = (L1TW1

−1A)−1L1TW1

−1b, (50)

where W1 is the same as W in (49), and

L1 =

ν′T

1 Φ1

...

ν′TNΦN

, ν′j =

sin β′j

− cos β′j

, (51)

where

β′j = 6 (U ′ − Q′j), j = 1, 2, . . . , N, (52a)

U ′ = [u′1, u′2]T , Q′j = [aj + bj u′3, bj − aj u′3]T , (52b)

and

[u′1, u′2, u′3]T = Uavple, (52c)

which is the estimation result of the AVPLE.

Remark 4 The BCAVPLE-WIV estimator is effective and feasible when the prior knowledge of AOA noise variance

σ2j is available, while the AVPLE-WIV does not need the prior knowledge of AOA noise variance. For the BCAVPLE-

WIV estimator, the approximated AOA βj used in IV matrix L is derived from the BCAVPLE, while for the AVPLE-

WIV estimator, the approximated AOA β′j used in IV matrix L1 is derived from the AVPLE. When the AOA noise


16

is i.i.d., we can use the AVPLE-WIV estimator to estimate the unknown node. However, while the AOA noise is not

i.i.d., its performance may degrade and we should use BCAVPLE-WIV if possible since it does not assume i.i.d.

The following Remark 5 gives the property of the BCAVPLE-WIV estimator and the AVPLE-WIV estimator:

Remark 5 For sufficiently large N , both the AVPLE-WIV estimator and the BCAVPLE-WIV estimator are asymp-

totically unbiased estimators. Moreover, assume that AOA noise variance is i.i.d. with variance σ2j = σ2 and the

prior knowledge of σ2 is known, for finite value of N , the BCAVPLE-WIV estimator has lower estimation error

than the AVPLE-WIV estimator.

Proof: See Appendix B.

The algorithmic procedures of both the BCAVPLE-WIV estimator and the AVPLE-WIV estimator are summarized

in Algorithm 2.

VI. MAXIMUM LIKELIHOOD ESTIMATOR AND CRAMER-RAO LOWER BOUND

We briefly introduce the ML estimator for AOA based node self-localization, and then derive its Cramer-Rao

Lower Bound (CRLB).

A. Maximum Likelihood Estimator

The ML is an asymptotically unbiased estimator. To compare the localization error with our new methods, the

ML estimator is also used as a benchmark. Under the assumption that the AOA measurements uncertainty is a

Gaussian distribution with zero mean and known covariance Σ, the likelihood function [27] is

f(β|β(p)) =1

(2π)N/2|Σ|1/2

× exp

{−1

2(β − β(p))TΣ−1(β − β(p))

},

(53)

where β = [β1, . . . , βN ]T is the N × 1 vector of AOA measurements, β(p) = [β1(p), . . . , βN (p)]T is the

N × 1 vector of AOAs as a function of the location and orientation with respect to the unknown nodes, Σ =

diag(σ21 , σ

22 , . . . , σ

2N ) is the N ×N diagonal covariance matrix of the AOA measurement errors, and |Σ| denotes

the determinant of Σ. Using the log-likelihood function to (53), we can obtain the ML cost function for estimating

the unknown node as

J =1

2(β − β(p))TΣ−1(β − β(p)) =

N∑j=1

1

2σ2j

(βj − βj(p))2

=

N∑j=1

1

2σ2j

(βj + θ − arctan

y − bjx− aj

)2

,

(54)

where N is the number of beacons within the sensing range of the unknown node, (aj , bj) is the location of the

jth beacon, βj is the measured AOA for the jth beacon, (x, y) and θ denote the location and orientation of the

unknown node, respectively.


17

Algorithm 2 WIV estimator (BCAVPLE-WIV and AVPLE-WIV)

1: Input: βj , aj , bj , and σ2j , j = 1, 2, . . . , N

2: If σ2j is known

• Compute U bcavple (Equation (35))

• U = U bcavple

• λ2j = σ2j , j = 1, 2, . . . , N

else

• Compute Uavple (Equation (8))

• U = Uavple

• λ2j = 1, j = 1, 2, . . . , N

End

3: Compute

• U b = [ub1, ub2]T , where [ub1, ub2, ub3]T = U

• Qj = [aj + bj ub3, bj − aj ub3]T , j = 1, 2, . . . , N

4: Compute

βj = 6 (U b − Qj), d2j =‖ Qj − U b ‖22, j = 1, 2, . . . , N

5: Compute

• νj =

sin βj

− cos βj

, L =

νT1 Φ1

...

νTNΦN

6: Compute

• W = diag(d21λ21, d

22λ

22, . . . , d

2Nλ

2N )

• Uwiv = (LTW−1A)−1LTW−1b

The location of the unknown node can be obtained by minimizing J . The minimization of cost function J

involves a LS nonlinear problem, which does not have a closed-form solution. One can adopt search algorithms

such as Gauss-Newton algorithm [28], Nelder-Mead simplex algorithm [29] or Newton-Raphson [30] to seek the

optimal solutions by setting a reasonable initialization.

B. Cramer-Rao Lower Bound (CRLB) of ML Estimator

The CRLB is the theoretical lower bound of variance of an unbiased parameter estimate [4], [31], [32]. For the

AOA based self-localization, the lower bound can be defined as the inverse of the following Fisher information


18

matrix,

F = E

[∂ ln f(β|p)

∂p

]T [∂ ln f(β|p)

∂p

]= E

{[∂J

∂x

∂J

∂y

∂J

∂θ

]T [∂J

∂x

∂J

∂y

∂J

∂θ

]},

(55)

where J is the cost function of likelihood estimation in (54).

Provided that AOA measurements are subjected to the Gaussian white noise with variance σ2j . Let ∆xj = x−aj ,

∆yj = y − bj and drj represents the Euclidean distance from the unknown node to the jth beacon, denoted by

drj =√

∆x2j + ∆y2j . Then the CRLB on the variance of the ML estimator is given by

Fc =

N∑j=1

1

σ2jdr

4j

(∆yj)

2 −∆xj∆yj dr2j∆yj

−∆xj∆yj (∆xj)2 −dr2j∆xj

dr2j∆yj −dr2j∆xj dr4j

−1

. (56)

VII. PERFORMANCE EVALUATION

Matlab simulations are conducted to evaluate the performance of our new algorithms. First, without loss of

generality, the estimation errors of new algorithms, the triangulation method and the ML estimator are compared

in the original coordinate system. Next the impact of the prior knowledge of the AOA noise variance on the

BCAVPLE algorithm and the BCAVPLE-WIV estimator are evaluated. Then the performance of the new AVPLE-

WIV estimator is illustrated by comparing the estimation errors with other proposed new algorithms. Finally, we

illustrate the effectiveness of the ACR algorithm to overcome the instability problem (tangent instability) of our

new estimation algorithms when the orientation of the unknown node is near π/2.

Assume that there are no systematic biases for locations of the beacons. Also assume that all the AOA measure-

ments are subject to i.i.d. additive Gaussian white noise with the noise variance σ2. Root mean square error (RMSE)

and estimation bias are used to evaluate the estimation accuracy, denoted as RMSE =√∑M

m=1 ‖p− p‖22/M and

‖E{p}−p‖2, respectively, where p is the estimated location, p is the true location of the unknown node and M is

the number of the simulation runs. Simulation results are obtained with M = 2, 000 Monte Carlo simulation runs.

A. Algorithm Comparisons based on Accurate Prior Knowledge of AOA Noise Variance

According to Remark 3 and Remark 4, the BCAVPLE and the BCAVPLE-WIV estimator are feasible and effective

with the accurate prior knowledge of the AOA noise variance σ2j . Thus, to compare our proposed algorithms with

the triangulation method and the ML estimator, the AOA noise variance σ2j is supposed to be known in advance.

We first consider a scenario with ten beacons and one unknown node randomly deployed in the 100m × 100m

region. Each trial is a new random realization of the placements of the 10 beacons and one unknown node based

on a Monte Carlo method.

The ML estimator is implemented by the Nelder-Mead simplex algorithm as in [29]. The iterative ML algorithm

requires a good initialization. The estimation result of the BCAVPLE is selected as the initialization for the ML


19

1 2 3 4 5 6 7 8 9 100

10

20

30

40

AOA Noise Standard Deviation (degrees)

Loca

tion

Err

or (

m)

Triangulation AVPLE BCAVPLE ML BCAVPLE−WIV

1 2 3 4 5 6 7 8 9 100

10

20

30


Orie

natio

n E

rror

(de

gree

s)

Fig. 3: Estimation errors of the proposed new algorithms including AVPLE, BCAVPLE and BCAVPLE-WIV, in

comparison with the triangulation and the ML estimator (N = 10).

estimator to have a fair comparison between the BCAVPLE-WIV estimator and the ML estimator. Note that

BCAVPLE-WIV does not need initialization, but it uses the estimation result of the BCAVPLE to obtain the

approximated AOA βj used in IV matrix L for the BCAVPLE-WIV estimator.

Fig. 3 illustrates the comparisons of estimation error (i.e, RMSE) for the triangulation method, the AVPLE, the

BCAVPLE, the BCAVPLE-WIV and the ML estimator initialized to BCAVPLE. It can be seen that the AVPLE

estimator has much lower estimation error than the triangulation method, which verifies the effectiveness and

feasibility of our new auxiliary variables based algorithm. Moreover, the BCAVPLE has higher localization accuracy

than the AVPLE as the AOA noise variance is known. In addition, the BCAVPLE-WIV estimator has lower estimation

error than the BCAVPLE and the AVPLE, but slightly higher than the ML estimator. However, the ML estimator

is sensitive to initializations and may suffer from local convergence problem, which will be discussed in detail in

the following subsection VII-D.

To compare the actual estimation bias [19] of different algorithms, we conduct simulations with one unknown

node, p = [80, 90, π/4]T , and 10 beacons randomly deployed in the 100m× 100m region, as shown in Fig. 4. We

can see that the proposed new algorithms have much lower estimation bias than the triangulation method, while

the BCAVPLE-WIV and the ML estimator have nearly the same estimation bias.

To show the effectiveness of the presented new algorithms and demonstrate that the BCAVPLE-WIV estimator

is an asymptotically unbiased estimator for large N , we randomly place 100 beacons and one unknown node in the

100m×100m region. For each Monte Carlo trial, the locations of 100 beacons and the unknown node are randomly

deployed in the area. We can see from Fig. 5 that the BCAVPLE has lower estimation error than the AVPLE. In

addition, the BCAVPLE-WIV estimator has much lower estimation error than the AVPLE and the BCAVPLE, which


20

1 2 3 4 5 6 7 8 9 100

10

20

30

40


Loca

tion

Bia

s (m

)

Triangulation AVPLE BCAVPLE ML BCAVPLE−WIV

1 2 3 4 5 6 7 8 9 100

5

10

15

20


Orie

natio

n B

ias

(deg

rees

)

Fig. 4: Estimation bias of the proposed new algorithms including AVPLE, BCAVPLE and BCAVPLE-WIV, in

comparison with the triangulation and the ML(N = 10).

1 2 3 4 5 6 7 8 9 100

5

10

15

20


Loca

tion

Err

or (

m)

AVPLEBCAVPLEMLAVBCPLE−WIVCRLB

1 2 3 4 5 6 7 8 9 100

5

10

15

20


Orie

natio

n E

rror

(de

gree

s)

AVPLEBCAVPLEMLAVBCPLE−WIVCRLB

Fig. 5: Comparisons of estimation errors for the AVPLE, the BCAVPLE, the BCAVPLE-WIV, the ML and the

CRLB (N = 100).

approaches the CRLB for large N . Further, we illustrate the impact of the number of beacons on the estimation

errors of our new algorithms and the ML estimator. Fig. 6 shows that the estimation errors of all algorithms gradually

decrease with the increase of N . Note that the estimation errors of the AVPLE and the BCAVPLE decrease to a

constant larger than the CRLB, while the estimation errors of the BCAVPLE-WIV estimator and the ML estimator

decrease to approximate the CRLB as N increases.


21

10 60 110 160 210 260 310 360 4100

1

2

3

4

Number of Beacons (N)

Loca

tion

Err

or (

m)

AVPLE BCAVPLE ML BCAVPLE−WIV CRLB

10 60 110 160 210 260 310 360 4100

1

2

3

4


Orie

natio

n E

rror

(de

gree

s)

Fig. 6: Comparisons of estimation errors for our proposed algorithms and the ML as N increases (σ = 2).

B. Impact of the Prior Knowledge of the AOA Noise Variance on the BCAVPLE and the BCAVPLE-WIV

The effectiveness of the BCAVPLE and the BCAVPLE-WIV estimator are related to the prior knowledge of the

AOA noise variance σ2j . Assume that the predicted AOA noise standard deviation for the BCAVPLE, denoted by

φj , is four degrees worse than the actual σj , i.e., φj = σj + 4. In this simulation, with the same self-localization

scene as depicted in subsection VII-A, ten beacons and one unknown node are randomly deployed in the network

area. Fig. 7 shows that when the noise standard deviation φj is used by the BCAVPLE to compensate the estimate

bias of the AVPLE, the BCAVPLE has higher estimation error than the AVPLE for node self-localization when the

AOA noise standard deviation is less than 5 degrees. When the AOA noise standard deviation is beyond 4 degrees,

the estimation error of the BCAVPLE is lower than the AVPLE, but still higher than that when the prior knowledge

of σ2j is accurate as in Fig. 3. Thus, it can be seen that the BCAVPLE is sensitive to the accuracy of the prior

knowledge of the AOA noise variance σ2j . Note that the BCAVPLE-WIV estimator and AVPLE-WIV estimator

have nearly the same estimation errors that are much lower than the AVPLE. The BCAVPLE-WIV estimator is

more robust than the BCAVPLE.

C. Performance Evaluation for the AVPLE-WIV Estimator

In practice, the prior knowledge of the AOA noise variance σ2j is difficult to obtain owing to the complicated

outdoor environment for node self-localization. Therefore, the AVPLE-WIV estimator, which does not require prior

knowledge of σ2j , can be adopted to localize the unknown node. For the simulations in this subsection, we randomly

place N beacons and one unknown node in the 100m × 100m area. Fig. 8 shows the comparisons of estimation

errors for the AVPLE-WIV estimator, the AVPLE, the BCAVPLE and the BCAVPLE-WIV estimator. It can be seen

that the AVPLE-WIV estimator has lower estimation error than the AVPLE and the BCAVPLE, and it has almost

the same estimation error as the BCAVPLE-WIV estimator. However, for a large and finite N , the AVPLE-WIV


22

1 2 3 4 5 6 7 8 9 100

5

10

15

20

AOA Noise Standard Deviation (degree)

Loca

tion

Err

or (

m)

1 2 3 4 5 6 7 8 9 100

10

20

30

AOA Noise Standard Deviation (degree)

Orie

natio

n E

rror

(de

gree

s)

AVPLEBCAVPLEBCAVPLE−WIVAVPLE−WIV

AVPLEBCAVPLEBCAVPLE−WIVAVPLE−WIV

Fig. 7: The impact of AOA noise variance on the BCAVPLE and the BCAVPLE-WIV estimator (N = 10).

estimator has slightly higher estimation error than the BCAVPLE-WIV estimator as illustrated in Fig. 9. In addition,

we can see from Fig. 8 that the estimation errors of the AVPLE-WIV and the BCAVPLE-WIV gradually decrease

and approach the CRLB as N increases, verifying Remark 5.

10 60 110 160 210 260 310 360 4100

1

2

3

4


Loca

tion

Err

or (

m)

AVPLE BCAVPLE BCAVPLE−WIV AVPLE−WIV CRLB

10 60 110 160 210 260 310 360 4100

1

2

3

4


Orie

natio

n E

rror

(de

gree

s)

Fig. 8: Comparisons of estimation errors for the AVPLE-WIV and other algorithms as the number of beacons N

increases (σ =2).

D. Algorithm Comparisons: AVPLE-WIV and BCAVPLE-WIV VS. ML Estimator

The ML estimator is sensitive to initializations and may suffer from local convergence problem. To illustrate

the impact of different initializations on the localization accuracy for the ML estimator, we conduct the simulation


23

60 110 160 210 260 310 360 4100

0.2

0.4

The Number of Beacons

Loca

tion

Err

or (

m)

BCAVPLE−WIVAVPLE−WIVCRLB

60 110 160 210 260 310 360 4100

0.2

0.4

The Number of BeaconsOrie

natio

n E

rror

(de

gree

)

BCAVPLE−WIVAVPLE−WIVCRLB

Fig. 9: Comparisons of estimation errors for the BCLPLE-WIV and the AVPLE-WIV estimator as the number of

beacons N increases (σ =2).

with ten beacons randomly placed in the 100m × 100m area, and the location information of unknown node

is p = [100, 100, π/3]T . In Fig. 10, ML1 and ML2 represent the ML estimator initialized to the BCAVPLE

and other values, respectively. We can see that the ML estimator has different estimation biases under different

initializations, while the BCAVPLE-WIV and AVPLE-WIV estimator is not sensitive to initialization. Sometimes,

the ML estimator tends to diverge as the AOA noise increases as shown ML2 in Fig. 10, while the BCAVPLE-WIV

and the AVPLE-WIV remain stable. In addition, it should be pointed out that the BCAVPLE-WIV algorithm can

obtain the closed-form solutions that have much lower computational complexity than the multiple iterative ML

estimator.

E. Impact of the Orientation on Estimation Performance

For our auxiliary variables based closed-form algorithms, when the orientation θ of the unknown node is near

π/2, the tangent in (5) becomes unstable. In this paper, we further investigate the impact of the orientation on

estimation performance of the proposed algorithms. We randomly place 10 beacons and one unknown node in

the 100m× 100m region based on Monte Carlo method. Fig. 11 shows the localization accuracy of the proposed

algorithms without coordinate rotation as the orientation θ changes from 0 to 180 degrees. When the orientation

is between 80 degrees and 100 degrees, the estimation error of AVPLE and BCAVPLE is pretty large, especially

when θ is near 90 degrees, while the estimation error of the AVPLE-WIV and the BCAVPLE-WIV approximately

increases to 5 meters and 4 degrees when θ is near 90 degrees (i.e., π/2).

Based on the ACR algorithm, we evaluate the performance of our proposed algorithm again as shown in the

following Fig. 12. Fig. 12 shows that after coordinate rotation, the location error and the orientation error of AVPLE

have decreased from about 23 meters (in Fig. 11) to 2.2 meters and from 30 degrees to 2.3 degrees, respectively,


24

1 2 3 4 5 6 7 8 9 100

50

100

150


Loca

tion

Err

or (

m)

1 2 3 4 5 6 7 8 9 100

50

100

150

200


Orie

natio

n E

rror

(de

gree

s)

ML1ML2AVPLE−WIVBCAVPLE−WIV

ML1ML2AVPLE−WIVBCAVPLE−WIV

Fig. 10: Comparisons of estimation errors of the AVPLE-WIV, the BCAVPLE-WIV and the ML with different

initializations (N = 10).

0 20 40 60 80 100 120 140 160 1800

10

20

30

Orientation of Unknown Node (degrees)

Loca

tion

Err

or (

m)

0 20 40 60 80 100 120 140 160 1800

10

20

30

40


Orie

natio

n E

rror

(de

gree

s)

AVPLEBCAVPLEMLBCAVPLE−WIVAVPLE−WIV


Fig. 11: Impact of the orientation angle on estimation performance of our proposed algorithms in the original

coordinate system (N = 10, σ = 2).

when the orientation is near 90 degrees, and those of BCAVPLE have decreased from 13 meters to 2 meters

and from 16.2 degrees to 2 degrees, while the estimation errors of the AVPLE-WIV and the BCAVPLE-WIV are

approximately the same as that of ML estimator. Therefore, our proposed ACR algorithm is effective to solve the

tangent instability problem.


25

0 20 40 60 80 100 120 140 160 1800

5

10


Loca

tion

Err

or (

m)

0 20 40 60 80 100 120 140 160 1800

5

10


Orie

natio

n E

rror

(de

gree

s)



Fig. 12: Impact of the orientation angle on estimation performance of our proposed algorithms with autonomous

coordinate rotation (ACR) (N = 10, σ = 2).

VIII. CONCLUSION

In this paper, we present a set of novel auxiliary variables based close-form algorithms for the efficient AOA

based self-localization. First we obtain the closed-form AVPLE solution for AOA based self-localization problem

based on auxiliary variables. Based on the analysis of the theoretical estimation error of auxiliary variables for

the AVPLE, we further develop a bias compensation scheme for AVPLE to reduce the estimation error of node

self-localization, namely BCAVPLE method. When the prior knowledge of the AOA noise variance is known,

we present a BCAVPLE-WIV estimator to reach the approximate CRLB accuracy for node self-localization. In

the case that the AOA noise variance is unknown, a AVPLE-WIV estimator is presented to localize the unknown

node. Compared with the triangulation method, our proposed auxiliary variables based algorithms can improve the

localization accuracy significantly with lower complexity. The simulation results verify that the BCAVPLE reduces

the estimation error compared with the AVPLE, and for large number of beacons, N , the BCAVPLE-WIV estimator

can approximately reach the CRLB. We also show that when the AOA noise variance is unknown, the AVPLE-WIV

estimator has lower estimation error than the AVPLE and the BCAVPLE, which approaches the CRLB for the large

N . However, compared with the BCAVPLE-WIV estimator, the AVPLE-WIV estimator has higher estimation error

for a finite value of N .

In summary, the new AOA based self-localization algorithms, AVPLE, BCAVPLE, AVPLE-WIV and BCAVPLE-

WIV, have higher localization accuracy than the triangulation and can avoid the local minima and divergence

problems in iterative ML estimators. The AVPLE has lower localization accuracy than AVPLE-WIV, BCAVPLE,

and BCAVPLE-WIV when the prior knowledge of the AOA noise variance is known. Note that the effectiveness

of BCAVPLE and BCAVPLE-WIV estimator depends on the accurate prior knowledge of the AOA noise variance.


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The BCAVPLE-WIV estimator and the AVPLE-WIV estimator are both unbiased estimators for large N , but

BCAVPLE-WIV estimator has lower estimation error than AVPLE-WIV estimator for finite value of N . However,

the AVPLE-WIV estimator is more robust when we have no prior knowledge of the AOA noise variance and the

AOA noise is i.i.d. In addition, when the orientation is near 90 degrees, the proposed autonomous coordinate rotation

(ACR) method could solve the tangent instability problem of the presented new algorithms effectively.

APPENDIX A

MATHEMATICAL DERIVATION OF EQUATION (3)

Referring to (2), we have

(aj − x) tan(θ + βj) = bj − y, j = 1, 2, . . . , N. (57)

According to a trigonometric identity, (57)becomes

tan θ + tanβj1− tan θ tanβj

(aj − x) = bj − y, j = 1, 2, . . . , N. (58)

Rewrite (58) as

(tan θ + tanβj)(aj − x) = (bj − y)(1− tan θ tanβj). (59)

Rearrange the aforementioned formulation (59),

tanβj(x+ y tan θ)− (y − x tan θ)

− (aj + bj tanβj) tan θ = aj tanβj − bj .(60)

Pre-multiply both sides of (60) with cos βj , we obtain

sinβj(x+ y tan θ)− cosβj(y − x tan θ)

− (aj cosβj + bj sinβj) tan θ = aj sinβj − bj cosβj .(61)

APPENDIX B

PROOF OF REMARK 5

Similar to the prior work in [33], the correlation between the Uavple and the noise vector φ vanishes as the number

of beacons N → ∞. Thus the approximated AOA obtained from the AVPLE, denoted by β′j , is asymptotically

uncorrelated with φ. As a result, the elements of the IV matrix L1 for the AVPLE-WIV estimator in (51) are

independent of the elements of the matrix φ for sufficient large N . Moreover, the elements of L1 are uniformly

bounded. Based on the Theorem 2 in [22], we have P limN→∞1NL

T1 φ = 0, where “P lim” denotes the probability

limit [22].

In addition, if the unknown node is observable, P limN→∞ATA/N is always nonsingular, thus P limN→∞L

T1A/N

is nonsingular as well. Therefore, according to the two sufficient and necessary conditions in [22], we can conclude

that the AVPLE-WIV estimator is asymptotically unbiased. Using the same methodology, we can also justify that

the BCAVPLE-WIV estimator is asymptotically unbiased.


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Next, we will verify that the BCAVPLE-WIV estimator has lower estimation error than the AVPLE-WIV estimator

for finite value of N when the AOA noise variance σ2j is known. Firstly, since the BCAVPLE estimator has higher

localization accuracy than the AVPLE when the prior knowledge of σ2j is known, the approximated AOA used in

IV matrix L obtained from the BCAVPLE, denoted by βj , is more accurate than β′j obtained from the AVPLE,

i.e.,

|βj − βj | < |β′j − βj |, (62)

where βj is the true AOA measurement.

According to (62) and (46), for finite value of N , we can see that the IV matrix for BCAVPLE-WIV estimator,

denoted by L, is closer to the optimal IV matrix A0 than that of the AVPLE-WIV. When the AOA noise subjects

to i.i.d., the BCAVPLE-WIV and the AVPLE-WIV have the same formulation as shown in (48) and (50), except

for the different IV matrix. Generally, the more closely the estimated IV matrix approaches the optimal IV matrix

A0, the higher is the localization accuracy of the estimator. Therefore, we can conclude that the BCAVPLE-WIV

estimator has lower estimation error than the AVPLE-WIV for finite value of N .

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Efficient Closed-form Algorithms for AOA Based Self-localization of Sensor Nodes Using Auxiliary Variables

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