systematic bias correction in source...

Systematic Bias Correction in

Source Localization

YIMING JI

CHANGBIN YU, Senior Member, IEEE

BRIAN D. O. ANDERSON, Life Fellow, IEEE

The Australian National University

A novel analytical approach is proposed to approximate and

correct the bias in localization problems in n-dimensional space

(n= 2 or 3) with N (N >= n) independently usable measurements

(such as distance, bearing, time difference of arrival (TDOA),

etc.). Here, N is often but not always the same as the number

of sensors. This new method mixes Taylor series and Jacobian

matrices to determine the bias and leads in the case when N =

n to an easily calculated analytical bias expression; however,

when N is greater than n, the nature of the calculation is more

complicated in that a further step is required. The proposed

novel method is generic, which means that it can be applied to

different types of measurements. To illustrate this approach we

analyze the proposed method in three situations. Monte Carlo

simulation results verify that, when the underlying geometry

is a good geometry (which allows the location of the target to

be obtained with acceptable mean square error (MSE)), the

proposed approach can correct the bias effectively in space of

dimension 2 or 3 with an arbitrary number of independent usable

measurements. In addition the proposed method is applicable

irrespective of the type of measurement (range, bearing, TDOA,

etc.).

Manuscript received August 16, 2010; revised April 17 and

September 21, 2011, and June 12, 2012; released for publication

November 14, 2012.

IEEE Log No. T-AES/49/3/944606.

Refereeing of this contribution was handled by C. Jauffret.

Y. Ji, C. Yu, and B. D. O. Anderson are supported by the Australian

Research Council under Grant DP-110100538. C. Yu is funded

through an ARC Queen Elizabeth II Fellowship and Overseas

Expert Program of Shandong Province. B. D. O. Anderson and Y. Ji

are also supported by National ICT Australia (NICTA). NICTA is

funded by the Australian Government as represented by the Dept.

of Broadband, Communications and the Digital Economy and the

Australian Research Council through the ICT Centre of Excellence

program. This material is based on research sponsored by the Air

Force Research Laboratory under Agreement FA2386-10-1-4102.

The U.S. Government is authorized to reproduce and distribute

reprints for Governmental purposes notwithstanding any copyright

notation thereon. The views and conclusions contained herein are

those of the authors and should not be interpreted as necessarily

representing the official policies or endorsements, either expressed

or implied, of the Air Force Research Laboratory or the U.S.

Government.

Authors’ address: The Australian National University, Building 115

(RSISE), Daley Road, Canberra, ACT, 0200, Australia, E-mail:

([email protected]).

0018-9251/13/$26.00 c° 2013 IEEE

I. INTRODUCTION

Localization–determining the geographical

position of a target using some form of measurements

related to its position is an old problem that comes in

many variations, and it has been widely investigated.

It continues to attract interest as the continuing

appearance of new ideas attests [1—6]. Application

areas include many in the military and environmental

spheres.

Understanding the limitations of localization

algorithms in recent years has come to be seen

as important in its own right. One of the primary

limitations arises from errors in the measurements,

which are virtually inevitable. The errors may

be caused by many different factors, including

particularly the accuracy of measurement equipment.

When errors exist in the measurements, almost all

current localization algorithms will be affected.

In other words the true position of targets cannot

normally be obtained with noisy measurements.

In order to improve the performance of existing

localization algorithms, many enhancement or

noise-mitigation techniques have been proposed. For

example, in [7], Bishop, et al. propose a new type

of algorithm to enhance the performance of target

position estimation. The novelty of the proposed

method is that it introduces a constraint on the passive

range-difference measurement errors to account

for the underlying geometry. Further, in [8], Cao,

et al. propose a novel method based on formulating

geometric relations among distances between nodes

as equality constraints by using the Cayley-Menger

determinant. These constraints can be further used

to formulate an optimization problem for estimation

of the measurement errors. The solution of the

optimization problem can be used to adjust noisy

distance measurements, which results in a more

precise estimation of the target position. Again, Liu,

et al. [9] focus on the error propagation problem that

can cause inaccurate localization estimation, especially

in large scale networks. In their paper they propose an

error-control mechanism based on characterization

of node uncertainties and discrimination between

neighboring nodes. The simulation results show that

the proposed mechanism can significantly reduce

the effect of error propagation, which enhances

localization accuracy and robustness.

Apart from the enhanced techniques mentioned

above, a further type of improvement technique that

aims at correcting the bias in localization algorithms

has attracted attention in recent years. Two types of

bias have, in fact, been investigated in localization

problems. The first one is measurement bias, which

means that bias (a systematic error in one direction)

exists in the measurement set. This is caused by

environmental conditions such as indoor/outdoors,

inaccurate calibration, registration error in measuring

1692 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 3 JULY 2013

equipment, or non-line-of-sight signal propagation,etc. In [10] Picard, et al. discuss several modelsfor handling bias in range measurements, presenta set of iterative algorithms that cope successfullywith the various bias models, and provide maximumlikelihood (ML) position estimates. Again, Lin,et al. [11] present a model of measurement biasin polar coordinates with an exact solution forthe measurement bias estimation also provided inthe paper. Moreover, in [12—15], the authors alsopresent different methods to reduce the errors in themeasurements in order to improve the localizationaccuracy. In our work we do not consider this typeof bias.The other type of bias is in estimation, which

is the subject of this paper. In such a case themeasurements themselves, though perturbed bynoise, are not biased. But the localization estimatesare obtained through nonlinear processing of themeasurements, and this gives rise to bias. Thisphenomenon is treated in a number of works.For example, Do³gancay, et al. [16] developa bias compensation algorithm to reduce theposition estimation bias based on a comprehensivebias analysis for a weighted least squares (LS)estimator using time difference of arrival (TDOA)measurements. The simulation examples illustrate thesignificant bias reduction of the proposed algorithm.Nevertheless, this bias compensation algorithm isnot generic: the method is only applicable to TDOAlocalization. Furthermore, it is restricted to LS andML localization algorithms. Furthermore, in [17],[18], the authors also deal with the bias in TDOAlocalization with ML or LS algorithms.In [19] an introduction to tensor algebra is given

with a few examples in estimation theory. One ofthe applications of tensor algebra addressed in thepaper treats the bias in nonlinear systems with anoisy observable. The method expands the nonlinearfunction that maps measurements to target positionsto second-order in the noise using a Taylor series.The expected value of the second order term isconsidered as the analytic expression of bias, andthe concepts are illustrated to obtain the bias in theCartesian coordinates of a target where noisy rangeand bearing measurements (from a single point) aregiven. However, the main focus of [19] is how to usetensor algebra rather than bias analysis. Therefore,there is no systematic analysis nor detailed simulationfor the bias problem.

Gavish and Weiss [20] examine the performance

of two well-known bearing-only location algorithms,

viz., the ML and the Stansfield estimators. Analytical

expressions are derived for the covariance matrix

of the estimation error and the bias, which permit

performance comparison for any case of the

two algorithms. In order to obtain the analytical

expressions for bias, the first derivative of the ML

cost function is expanded by a Taylor series. Three

expansions of different orders are obtained separately.

The final expression for the bias involves the variance

of the measurement noise and various derivatives

of the cost function. However, the derivation of the

expression for the bias involves truncating three

different Taylor series expansions, which may lead

to imprecise results. Further, the process to obtain the

bias expression is not direct nor obvious.

In this paper a generic approach that is

independent of the type of measurements is presented

to correct the bias in two-dimensional (2D) and

three-dimensional (3D) localization algorithms with an

arbitrary number of independent usable measurements.

We first expand the localization mapping g (which

maps from the measurements to produce position

estimates) by a Taylor series to second order in the

measurement noise, and we consider the expected

values of the second-order term, expressible using

the derivatives of g, as bias. However, it is often veryhard to calculate the derivatives of g analytically.

In contrast the inverse mapping of g (call it f) thatmaps the target position to a (noiseless) set of

measurements can be obtained, together with its

derivatives, much more easily. Therefore, we introduce

the Jacobian matrix of f to compute the derivatives ofthe localization mapping g in terms of the derivatives

of f, which results in a simple calculation of bias.To demonstrate the performance of the proposed

bias-correction method, Monte Carlo simulations

have been carried out. Though the proposed method

is generic, in the simulation for ease of exposition, we

only apply the bias-correction method in localization

problems with distance-based, bearing-only, and

TDOA measurements.

Some preliminary results are proposed in

several conference papers. In [21] we restrict the

bias-correction method in to a 2D ambient space and

deal only with range measurements. In [22] we extend

our method to 3D space with an arbitrary number

of measurements. However, the simulation results

analysis in [22] is very limited. None of the previous

conference papers have investigated the influence

of different levels of noise. More importantly the

scan-based localization problem [23] with TDOA

measurements1 that is investigated in this paper has

not been studied in previous conference papers. In

this paper we generalize our method in both 2D and

3D space with an arbitrary number of measurements;

the types of measurements are not restricted to a

single type (and can include a mixture of range

measurements, bearing-only measurements, and

TDOA measurements). Finally, within the systematic

1Here, the TDOA measurements in the scan-based problem are

unlike conventional TDOA measurements that are caused by the

range difference to the emitter. In this case the time difference is

mainly caused by the mechanical rotation of a scanning emitter.

More details can be obtained in [23].

JI, ET AL.: SYSTEMATIC BIAS CORRECTION IN SOURCE LOCALIZATION 1693

presentation of simulation results, we study the effect

of changes of noise level and the existence of a

threshold for validity of the bias-correction method.

The rest of the paper is organized as follows.

In Section II high-level views of localization and

bias are summarized. We analyze the proposed

method in n-dimensional (n= 2 or 3) space with

N measurements in Section III. The results of

Monte Carlo simulation are provided in Section IV.

Section V summarizes the paper and comments on

future work.

II. PROBLEM STATEMENT

A. Notation

Some notational definitions are collected here for

convenience.

1) n denotes the number of dimensions of the

ambient space.

2) N denotes the number of independent usable

measurements that is often but not always the same as

the number of sensors.2

3) x= (x1,x2, : : : ,xn)T denotes the coordinate vector

of a target.3

4) £ = (μ1,μ2, : : : ,μN)T denotes a set of

measurements obtained from N sensors whose

location is known.

5) ±£ = (±μ1,±μ2, : : : ,±μN)T is the measurement

noise with entries generally assumed to be

independent Gaussian random variables with zero

mean and N £N covariance matrix S = ¾2 or

sometimes S = diag(¾2μ1 ,¾2μ2, : : : ,¾2μN ). The covariance

matrix will not be diagonal when the measurements

are TDOA [24].

6) f= (f1,f2, : : : ,fN)T is the mapping from the

target position to the measurements.

7) g= (g1,g2, : : : ,gn)T is the localization mapping

from the (noisy) measurements to the target position

(estimates).

Note that f is generally analytically available,while g may not be analytically available; quite ofteng is only defined implicitly, through the posing of aminimization problem.

B. High-Level View of Localization

Localization refers to the process of estimating the

location of a target using certain measurements. For

example, in wireless sensor networks, the position

2Normally, we assume each single sensor can provide one usable

measurement. For TDOA sensing in a scan-based localization

problem, N independent pieces of sensed data require N +1

physical sensors [23]. We continue to refer to this as an N sensor

situation; the sensors can be thought of in the abstract or as

pseudosensors. More details are shown in Section IV.3The units of positions and measurements will not influence the

performance of our method. Therefore, for ease of exposition, we

do not specify the units in this paper.

of an unlocalized node can often be estimated by

gathering the distance or bearing information from

neighboring nodes whose position has already been

estimated or is known a priori. In this subsection a

brief description of localization is presented. All the

analysis is done in n-dimensional (n= 2 or 3) space,

with N ¸ n usable measurements obtained from N

sensors whose locations are assumed known.

In the noiseless case the localization problem

can be formulated as follows. Suppose there is an

emitter or target whose coordinate vector is x=

(x1,x2, : : : ,xn)T. Further, a set of measurements £ =

(μ1,μ2, : : : ,μN)T can be obtained from N (generally

N ¸ n) sensors, where μj (j = 1,2, : : : ,N) denotesthe measurement obtained from sensor j. Here, the

measurements can be of any form, such as distance,

TDOA, or bearing, etc. Now, in the noiseless case, we

have

£ = f(x) (1)

where f= (f1,f2, : : : ,fN)T denotes the mapping

from the target position to the measurements. The

function f is assumed (as is reasonable) to be obtained

analytically according to the geometry of the target

and sensors.

However, in practice, measurement errors are

inevitable. Therefore, the mapping from the target

position to the measurements can be described by

a nonlinear equation as follows (where we use £ =

(μ1, μ2, : : : , μN) to denote the noisy measurements):

£ = f(x) + ±£ (2)

where ±£ = (±μ1,±μ2, : : : ,±μN)T has already been

defined in Subsection II-A.

When N = n one obtains a target position

estimate in effect by solving £ = f(x). However,

generally when N ¸ n+1, this equation will haveno solution in the noisy case. In order to obtain an

approximate position estimate, various methods have

been proposed, such as ML, LS, etc. [25, 26]. The

main idea of these approaches is similar: convert the

localization problem to an optimization problem as

follows.

x= argminxFcost-function(x,£) (3)

where x= (x1, x2, : : : , xn)T denotes the inaccurate target

position estimate. By solving the above equation,

which is often computationally difficult, we obtain

the estimated position.

C. High-Level View of Bias

Bias is a term in estimation theory and is defined

as the difference between the expected value of a

parameter estimate and the true value of the parameter

[27]. As mentioned in the Introduction, two kinds of

bias can arise in localization problems. The first one is

measurement bias, which is caused by environmental


conditions. In this paper we assume there is no bias

in the measurements. Our concern is with the second

type of bias, viz., estimation bias. In this subsection a

brief view of estimation bias is presented.

Assume that g= (g1,g2, : : : ,gn) denotes the

localization mapping from the measurements to the

target position estimates. In the noiseless case we have

x= g(£) (4)

where x and £ have the same meaning as above.

As mentioned in the last subsection, in practice,

noise will always exist in the measurements.

Therefore, in the noisy case, we have

x= g(£+ ±£) = g(£) (5)

where x, £, and ±£ have been defined above. In this

paper we assume that ±£ is a vector of independent

Gaussian random variables with zero mean and known

variance ¾2.4

In order to understand why this bias occurs,

contrast the above estimate of target position with one

obtained in a thought experiment where we repeat the

estimation process M times. For each measurement set

we would obtain an estimated position of the target,

which gives M target position estimates. Suppose that

we were to average these M target position estimates

to obtain a single position, which might then be

considered as the estimated position of target. We

would expect the ith entry of the estimate to go to

E[xi] = E[gi(£)]: (6)

Now, note that, if gi is nonlinear, we have

E[xi] = E[gi(£)]

6= gi(E[£])= gi(£)

= xi:

Therefore, bias appears in the one-shot estimation

process given by

Biasxi = E[xi]¡ xi, i= 1,2, : : : ,n: (7)

If computable the bias can be used to

systematically correct any single estimate from any

single measurement set. From the above analysis

we can see that once 1) the localization mapping is

nonlinear and 2) the measurements are noisy, bias

is to be expected. In practice these two factors are

mostly present. The desirability of bias correction

is the motivation, and the means to do so is the

contribution of this paper.

4In a scan-based localization problem, if TDOA measurements

arise, a variation on this assumption is necessary [24]. More details

are shown in Section IV.

III. A NOVEL GENERIC METHOD–TAYLOR-JACOBIAN METHOD

A novel generic bias-correction method is

proposed in this section. The analysis is done in

three situations: 1) N = n, 2) N = n+1, 3) N >

n+1. All the time we assume that there is only one

target of interest.5 In the first situation the number

of usable measurements is equal to the ambient

space dimension. The key step is to formulate the

bias in a simple way by combining a Taylor series

expansion and a Jacobian matrix, which is discussed

in Subsection III-A. Then, one more measurement

is introduced, which results in an overdetermined

equation set. When the measurements are noisy,

there will be no solution to the equation set, and

the Jacobian matrix idea cannot immediately be

used because no inverse mapping exists. In order to

resolve these problems, we introduce an extra variable

into the equation set. The details are described in

Subsection III-B. In Subsection III-C the situation

in which the number of usable measurements is at

least two greater than ambient space dimension is

described briefly. For ease of exposition in presenting

the method, we restrict attention to Cartesian position

coordinates. However, it is easy to extend the method

to other coordinate systems.

A. N = n Situation

In this situation the ambient space dimension

equals the number of obtained usable measurements.

At that time, in the noisy case, we can obtain the

position estimate x of the target by solving the

following equation.6

£ = f(x) (8)

where £ =£+ ±£ and x= x+ ±x. Here, f can be

easily written down analytically from the geometry.

For example, Fig. 1(a) depicts the situation in

2D space with two measurements. If the type of

measurement is distance (d1 and d2), f= (f1,f2)T has

the following form.

d1 = f1(x,y) =

q(x¡ x1)2 + (y¡ y1)2

d2 = f2(x,y) =

q(x¡ x2)2 + (y¡ y2)2:

5In a multiple-target situation, a preliminary data association or

de-interleaving step is required to associate signals with individual

targets. Following that, one can estimate the bias one by one for

each target by using the proposed method.6When the measurement is the range between target and sensor,

an ambiguity problem may be encountered: we may obtain two

estimated target positions. At that time we need to assume the

availability of further information, such as a priori area restriction

(e.g., the target is a ship, and one of the ambiguous positions is on

land), to resolve the ambiguity problem.


Fig. 1. Scenario with two measurements in 2D space.

(a) Range measurement and bearing-only measurement situations.

(b) TDOA measurement situation.

If we apply the bearing-only localization

algorithms, the analytical expression of f= (f1,f2)T

is as follows.

μ1 = f1(x,y) = ¼+actan

μx¡ x1y¡ y1

¶(mod2¼)

μ2 = f2(x,y) = actan

μx¡ x2y¡ y2

¶(mod2¼):

In the scan-based localization problem [23] by

using TDOA measurements, at least three physical

sensors are required to locate a single target in 2D

space. Though the number of sensors is greater

than the ambient space dimension, we still classify

TDOA situation of the three sensors as N = n

because, normally, we can obtain only two usable

measurements from the arrangement of Fig. 1(b) as

follows.

t2¡ t1 = f1(x,y)

= arccos

"(x¡ x1)2 + (y¡ y1)2 + (x¡ x2)2 + (y¡ y2)2¡ d2122p(x¡ x1)2 + (y¡ y1)2

p(x¡ x2)2 + (y¡ y2)2

#(mod2¼)=!

t3¡ t1 = f2(x,y)

= arccos

"(x¡ x1)2 + (y¡ y1)2 + (x¡ x3)2 + (y¡ y3)2¡ d2132p(x¡ x1)2 + (y¡ y1)2

p(x¡ x3)2 + (y¡ y3)2

#(mod2¼)=!

where ti denote the time when the mainlobe of the

radar crosses the sensor i. The distance between two

sensors are denoted by dij . ! is a known constant

scan-rate of the target.

Assume that the localization mapping g is well

defined for each point and that there are derivatives

of any order of g. Because N = n, g can be consideredas the inverse mapping of f. Thus

x= g(£): (9)

To determine the bias consider xi = gi(μ). Because

the localization mapping g is well defined, we canexpand the function gi by a Taylor series. Truncating

at second order:

xi+ ±xi = gi(μ1, μ2, : : : , μN)

= gi(μ1 + ±μ1,μ2 + ±μ2, : : : ,μN + ±μN)

¼ gi(μ1,μ2, : : : ,μN) +NXj=1

@gi@μj

±μj

+1

2!

NXj=1

NXl=1

±μj±μl@2gi@μj@μl

:

Because the measurement errors are independent

Gaussian distribution with zero mean and known

variance,7 the approximate bias expression

is immediate for range and bearing-only

measurements [28]:

E(±xi) =1

2!

NXj=1

¾2@2gi@μ2j

: (10)

When the scan-based localization problem with

TDOA measurements is considered, the above

equation (10) requires adjustment. In order see

why, note that noise in the time-of-arrival (TOA)

measurements can be modeled as follows (here we

take three physical sensors in 2D space for example):

ti = ti+ ±ti, i= 1,2,3 (11)

where the ±ti are normally assumed to be independent

and identically distributed (IID) Gaussian random

variables with zero mean and known variance ¾2.

However, in a TDOA measurements scenario, the

input measurements are t12 = t2¡ t1 and t23 = t3¡ t2.

7In practice the variance of measurement errors in the sensors

would have to be obtained from manufacturer and/or a priori

calibration.


Therefore, in practice with noisy measurements, we

can obtain

t12 = t2¡ t1 = t12 + ±t12 (12)

t23 = t3¡ t2 = t23 + ±t23 (13)

where ±t12 and ±t23 are no longer independent and

have covariance matrix § given by

§ = 2¾2·1 ¡0:5

¡0:5 1

¸(14)

where 2¾2 is the variance of an individual

range-difference measurement. Note that the mean of

±t12 and ±t23 still remain zero.

Now, the approximate bias expression for three

receivers is as follows (here, we take the bias of x for

example):

E(±x) =1

2!

·2¾2

@2g1@t212

¡ 2¾2 @2g1@t12@t23

+2¾2@2g1@t223

¸:

(15)

From the above analysis we see that, for the

scan-based localization problem, (10) should be

adjusted as follows.

E(±xi) =1

2!

24 NXj=1

2¾2@2gi@μ2j

¡NXl=1

NXk=1

2¾2@2gi@μl@μk

35 :(16)

For range-measurement localization it is not very

difficult to compute the derivatives of g. However,

when considering, e.g., a scenario in R3, obtaining the

analytical expression of g becomes very challenging.

In contrast f can be easily written down according

to the geometric relationship between the target and

sensors. Therefore, we consider how to use f and its

derivatives to calculate the derivatives of g, which

results in an easy calculation of the bias. Because

f and g are inverse mappings, the Jacobian identity

holds (with, recall, n=N):26666666664

@f1@x1

¢ ¢ ¢ @f1@xn

¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢@fN@x1

¢ ¢ ¢ @fN@xn

37777777775

26666666664

@g1@μ1

¢ ¢ ¢ @g1@μN

¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢@gn@μ1

¢ ¢ ¢ @gn@μN

37777777775= In:

(17)

By solving the equation set (17), we can obtain the

analytical expression for @gi=@μj (i= 1,2, : : : ,n; j =

1,2, : : : ,N = n) in terms of @fi=@xj (i= 1,2, : : : ,N = n;

j = 1,2, : : : ,n). For ease of exposition we use giμj to

denote the expressions of @gi=@μj as functions of

x1,x2, : : : ,xn. Here, we take @g1=@μ1 for example. We

can obtain the following equation.

@g1@μ1

= g1μ1 : (18)

Differentiating (18) in respect to x1 first, we can

obtain

@g1@μ21

@f1@x1

+ ¢ ¢ ¢+ @g1@μ1@μi

@fi@x1

+ ¢ ¢ ¢+ @g1@μ1@μN

@fN@x1

=@g1μ1@x1

:

If we further differentiate (18) in respect to x2, : : : ,xnrespectively, we can obtain an equation set as follows.26666666664

@f1@x1

¢ ¢ ¢ @fN@x1

¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢¢ ¢ ¢ ¢ ¢@f1@xn

¢ ¢ ¢ @fN@xn

37777777775

26666666664

@2g1@μ21¢¢¢

@2g1@μ1@μN

37777777775=

266666666664

@g1μ1@x1¢¢¢

@g1μ1@xn

377777777775: (19)

Note that the quantities on the right side of this

equation are all expressible analytically in terms

of derivatives of the fi, and so, as functions of

x1,x2, : : : ,xn. Hence, by solving the equation set

(19), we can obtain a formula for @2g1=@μ21 that

contains derivatives of only fi. The formulas for

@2gi=@μ2j for all i,j can be obtained in the same way.

Substituting the formulas into (10), we can finally

obtain the easily-calculated expressions for the bias.

The equation (16) for the TDOA situation can be

obtained analytically in the same way.

In practice we can obtain the inaccurate estimated

position of the target by using existing localization

algorithms. Then, we can input the inaccurate target

location into the obtained analytical expression of

bias. Finally, we can improve the accuracy of the

localization by subtracting the obtained bias, viz.,

xi¡ biasxi (i= 1,2, : : : ,n).

B. N = n+1 Situation

One more measurement is introduced in this

situation. In the noiseless case a single well-defined

position of the target can be obtained by solving

(1). However, in the noisy case generally, there will

be no solution for (8). Further, (8) will become

overdetermined, which means there are more scalar

measurements than there are unknowns. Because N 6=n we cannot obtain (17). In other words we cannot

straightforwardly express the bias using the derivatives

of f. At the same time, to calculate the localization,

mapping g directly becomes even harder. Therefore,

we adopt a method based on the LS approach to

introduce an extra variable into (8).

Consider N-dimensional space, with axes

corresponding to the N measurements. Assume that

a surface (shown in Fig. 2) consists of points which


Fig. 2. Introduce one extra variable ". (Here, N = 3 and n= 2).

Surface is set of points (μ1,μ2,μ3) = (f1(x,y),f2(x,y),f3(x,y))

obtained as x, y vary.

correspond to all sets of noiseless measurements

(μ1,μ2, : : : ,μN), i.e., μi = f(x,y) for i= 1,2, : : : ,N when

n= 2. According to the LS method,8 the cost function

in (3) has the following form:

Fcost-function(x,£) =

NXi=1

(fi¡ μi)2 =

NXi=1

±μ2i : (20)

In fact the LS method attempts to find a point

(μ1,μ2, : : : ,μN) (the white point in Fig. 2) on the

surface that corresponds to an obtained set of noisy

measurements (μ1, μ2, : : : , μN) (the black point in Fig. 2,

which is generically off the surface) to minimize

the distance between the two points. Hence, the

white point must be the orthogonal projection of the

black point onto the surface, or the black point must

be on the normal vector to a tangent plane of the

surface passing through the white point. Therefore, the

distance between the two points can be formulated.

Dmin =

vuut NXi=1

±μ2i = "kuk (21)

where u denotes the normal vector at the white point

and " is a coefficient to set the distance. We now

explain how the normal vector u can be calculated

as follows.

8The LS approach is equivalent to an ML approach when all noise

intensities are the same (and measurement noises are subject to

Gaussian distribution). Weighted LS can capture variations on this.

At the white point we can obtain n tangent vectors

as follows.

vi =

·@f1@xi,@f2@xi, : : : ,

@fN@xi

¸T, i= 1,2, : : : ,n: (22)

By cross multiplying the n tangent vectors, we can

obtain the normal vector u [29]:

u= [u1,u2, : : : ,uN]T = v1£ v2 ¢ ¢ ¢ £ vn: (23)

Note that, in the noiseless case, £ = f, where f canbe written down easily according to the geometry

of the sensors and target. Therefore, for the black

point, we can obtain a new analytical mapping F=(F1,F2, : : : ,FN)

T by moving from f along the normalvector for a distance "kuk. The new mapping F is nolonger overdetermined because an extra variable " has

been introduced into the mapping.

Now, we have a new mapping F : RN ! RN as

follows.£ = F(x,") = f(x) + "u: (24)

After introducing the extra variable ", now (24)

is no longer an overdetermined equation set; F is

invertible, and (24) is analogous to (8). Therefore, we

can consider the localization mapping (call it G) asthe inverse mapping of F. We can then proceed alongthe same lines as previously.

C. N > n+1 Situation

In this situation the number of usable

measurements is at least two greater than the ambient

spatial dimension. At that time, similar to the N =

n+1 situation, an overdetermined problem will arise.

However, in this situation, we need to introduce

more than one extra variable in order to solve the

overdetermined problem. Here, we take the situation

in 2D space with four sensors (n= 2, and N = 4) by

way of example to give a detailed description.

Consider a four-dimensional (4D) space, with

axes that correspond to the four measurements.

Assume that a (2D) surface consists of points which

correspond to all sets of noiseless measurements

£ = (μ1,μ2,μ3,μ4)T. Similarly, according to the LS

method, for each set of noisy measurements £ =

(μ1, μ2, μ3, μ4)T, we attempt to find a corresponding

point on the surface to minimize the distance between

the two points. Hence, the point on the surface

must be the orthogonal projection of the point off

the surface that corresponds to the set of noisy

measurements. Therefore, we still need to calculate

an associated normal vector u. Now, for each point

on the surface, we can obtain two tangent vectors as

follows.

v1 =

·@f1@x1

,@f2@x1

,@f3@x1

,@f4@x1

¸T(25)

v2 =

·@f1@x2

,@f2@x2

,@f3@x2

,@f4@x2

¸T: (26)


These two tangent vectors define a tangent plane

Ptangent.

According to the geometry we can obtain that:

uTv1 = 0, uTv2 = 0: (27)

In the N = n+1 situation, the point on the surface

corresponds to a set of points off the surface that

are all on a straight line. However, in the N > n+1

situation a point on the surface corresponds to a

set of points off the surface that together define a

plane; call it Pnormal. According to simple geometry

properties, the plane Pnormal is orthogonal to the

plane Ptangent. Assume that uT1 = [u11,u12,u13,u14] and

uT2 = [u21,u22,u23,u24] are two independent vectors,

i.e., not collinear, in the plane Pnormal. Now, the

normal vector u for each point in the plane Pnormalthat corresponds to the point on the surface can be

expressed as follows:

u= e1u1 + e2u2 (28)

where e1 and e2 are the two coefficients to vary the

u in order to correspond to the different points in

the plane Pnormal. Once the two vectors u1 and u2are fixed (analytically expressed), we can obtain a

new mapping F with four variables x, y, e1, and e2by moving the mapping f along the normal vector u

similar to the N = n+1 situation.

Here, we use ai to denote @fi=@x1 and bi to denote

@fi=@x2. Following is one possible set of u1 and u2:

u1 =

26666664

(b2a3¡b3a2)a1a1b3¡ a3b1

a1

(b1a2¡ a1b2)a1a1b3¡ a3b1

0

37777775

u2 =

26666664

(b4a3¡b3a4)b1a1b3¡ a3b1

0

(b1a4¡b4a1)b1a1b3¡ a3b1

b1

37777775 :(29)

When extra variables are introduced into the

equation set, the following processes are the same

as in the previous situation, and we do not show the

details here. When more sensors are used, a higher

degree of localization accuracy may be achieved. In

the simulation discussion below, we show that in 2D

space when four sensors are used, the accuracy of

localization is usually at a very high level. Therefore,

it may not be advantageous or even necessary to use

as many sensors as potentially available to locate the

target when the complexity of calculation of the target

position is taken into consideration.

IV. SIMULATION

In this section the results of Monte Carlo

simulation are presented. Simulation results are

provided in 2D space corresponding to the three

different types of situations in Section IV: 1) N = n

situation, 2) N = n+1 situation, and 3) N > n+1

situation. Three different types of measurements are

applied here: distance, bearing-only, and TDOA for

scan-based. The simulation results in the 3D case are

qualitatively similar to the results in 2D space in the

paper. Due to the space limitation, we do not present

the details in the 3D scenario here.

First, some assumptions are noted.

1) The units in the scan-based TDOA are seconds.

The scan-rate of the target in the scan-based TDOA

situation is set as 4¼=5 rad (144 deg) per second, with

the target scanning clockwise.

2) The measurement error for each sensor is

produced by an independent Gaussian distribution

with zero mean and known variance ¾2. The variance

is adjusted in the simulation set as 1 or 0.5 (the

corresponding standard deviation is 1 or 0.7071)

for distance measurements. For bearing-only

measurements the variance is set as 0.1 and 0.05

(the corresponding standard deviation is 0.3162

or 0.2236 rad (18.1169 or 12.8113 deg). In the

scan-based TDOA situation, we choose two values

of variance, viz., 0.1 and 0.05 (the corresponding

standard deviation is 0.3162 or 0.2236). More details

on the effect of the level of measurement noise on

localization and bias can be obtained in [30].

3) All the simulation results are obtained from

5000 Monte Carlo experiments.

4) In the simulation the bias is considered as

the absolute distance (average of 5000 experimental

results) between the true target position and the

estimated target position. In the simulation figure it

is designated as average absolute distance error.9

5) Analytical bias denotes the bias value computed

by using the analytical expression derived from the

proposed bias-correction method.

6) Experimental bias denotes the bias value

without using any bias-correction method.

7) “Without bias-correction method” denotes the

bias arising in localization without using any bias

correction. In fact this is the same as the experimental

bias.

8) After bias correction denotes the bias value that

is equal to the experimental bias minus the analytical

bias computed by the proposed bias-correction

method.

9In practice the bias is a vector whose entries can be negative or

positive. Here, we only focus on how large the bias is. Therefore,

the absolute distance between the estimated target position and the

true position is used to evaluate the bias.


Fig. 3. In 2D space with two anchors based on distance-based

localization algorithm (¾2 = 1=¾ = 1).

TABLE I

The Ratio of the Bias and the RMSE of x Component Expressed

as a Percentage (¾2 = 1=¾ = 1)

Value of x 6 8 10 12

Bias(x)

RMSE(x)(%) 16.26 9.53 6.63 6.16

Value of x 14 16 18 20

Bias(x)

RMSE(x)(%) 6.51 7.85 8.48 9.67

A. N = n Situation

In this situation the ambient space dimension

will be equal to the number of usable measurements.

We present the simulation results in 2D space with

two usable measurements. Though the proposed

method is generic, for ease of exposition, three types

of measurements are applied here, distance-based,

bearing-only, and TDOA.

First in 2D space we fix the two sensors at (0,8)

and (0,¡8). Further, we fix the value of y of the targetat zero while changing the x value from 6 to 20 in

steps of 2. We assume that a priori knowledge allows

elimination of the binary ambiguity.

Figure 3 illustrates the comparison of the

bias in two different situations based on distance

measurements: without any bias-correction method

(the dashed curve with circles) and after applying the

proposed bias-correction method (the solid line curve).

Here, we set the variance of measurement errors as 1

(¾2 = 1). Evidently, the proposed method can reduce

the localization bias for values of the x-coordinate of

the target ranging from 6 to 20, and we can notice

that the bias can reduce the bias up to 78%. Figure 4

Fig. 4. Comparison between experimental bias and analytical bias

computed by proposed method in 2D space with two anchors

based on distance-based localization algorithm (¾2 = 1=¾ = 1).

TABLE II

The Ratio of the Bias and the RMSE of x Component Expressed

as a Percentage (¾2 = 2=¾ = 1:4142)

Value of x 6 8 10 12

Bias(x)

RMSE(x)(%) 20.48 18.02 14.01 12.4

Value of x 14 16 18 20

Bias(x)

RMSE(x)(%) 13.62 15.06 16.99 18.58

depicts a comparison of the experimental bias and

the analytical bias. From the figure we can see that

the analytical bias computed by the proposed method

(the solid line curve) is almost always consistent with

the experimental bias (the dashed curve with circles).

This also verifies, from another standpoint, that the

proposed method is effective.

Moreover, Table I illustrates the bias of the x

component compared with the RMSE (root mean

square error) in estimating x.10 From the table we

can conclude that the bias is always greater than 6%

(may be 16%) of RMSE. Further, the percentage

will significantly increase (up to 20%) when the

noise level is increased (as illustrated in Table II

where the level of measurement errors is increased to

¾2 = 2=¾ = 1:4142). Therefore, it may be important

to consider the bias correction for improving the

accuracy of localization. Further, from Fig. 3 and

the two tables, we can conclude that the mean square

error (MSE) of x will also be reduced by using the

10Here, the bias and RMSE denote the experimental value as

determined by an ML estimator before bias correction.


proposed bias-correction method. However, when

a bad geometric relationship between the sensors

and the target is encountered, such as collinearity of

sensors and target, the proposed method will become

less effective or even noneffective. When the target

is very close to or far away from the line joining

the two sensors, we can consider that a collinearity

problem occurs.11 At that time the proposed method

becomes ineffective, which means the bias cannot be

reduced. Of course, in this situation, localization with

or without bias correction is known to be difficult.

In this paper we do not focus on the collinearity

problem. In other words we assume that an estimated

target position can always be obtained. More details

on the collinearity issue in localization algorithms

and bias-correction methods can be obtained in [21],

[30], [31].

In addition we adjust the level of noise over a

large range via changing the standard deviation ¾ of

measurement errors in order to see the influence made

by different levels of noise. Here, we set the target

at (14,0) and adjust the standard deviation ¾ from

0.5 to 4 in steps of 0.5. In Fig. 5(a) the simulation

results are divided into two parts: ¾ · 3 and ¾ > 3.From Fig. 5(a) we can see that the proposed method

can work very well when the standard deviation

is adjusted from 0.5 to 3, which demonstrates that

the proposed bias-correction method has a good

robustness ability to the level of noise. When the

standard deviation ¾ becomes greater than 3 (such

as ¾ = 3:5 and 4), it is possible for the localization

algorithm itself to fail. The measured distances (d1and d2) may fail to satisfy the triangle inequality

(as shown in Fig. 5(b)) because of the high level of

noise. In the simulation, when the standard deviation

¾ = 3:5 (or ¾ = 4), there is around 10 out of 1000

(or 20 out of 1000) sets of measurements which

cannot produce an estimated target position because

the triangle inequality cannot be satisfied. Therefore

the proposed bias-correction method, without doubt,

will become noneffective because no estimated target

position can be obtained. Now, if we reject the sets

of measurements which cannot satisfy the triangle

inequality, the proposed bias-correction method

still can work very well (see the dotted lines in the

Fig. 5(a)). Roughly speaking the threshold noise level

below which the proposed bias-correction method

is effective is much the same as the level below

which the localization algorithm itself is effective

(the triangle inequality can be satisfied). Because

localization algorithms have been investigated for a

long time but apparently without a simple analytical

expression or lower bound for this critical threshold

11Given collinearity and noisy measurements, it can easily be

the case that there is no intersection between the two circles that

correspond to the noisy measurements from the two sensors, or

such an intersection can be at a great distance from the true target.

Fig. 5. Influence of different levels of noise, showing collapse as

noise levels increase.

being found, the problem of finding such a threshold

below which bias correction will be effective is

evidently very challenging. Nevertheless, we aim to do

further research on this threshold in our future work.

Further, we now apply the proposed method to the

bearing-only localization algorithms, with the same

geometry as before. Figure 6 shows the simulation

results by using bearing-only measurements with

the measurement error variance ¾2 equal to 0.1 (¾ =

0:3162 rad or 18.1169 deg). From the figure we can

see that, from (6,0) to (20,0), the bias can be reduced

by using the proposed bias-correction method.

Figure 7 illustrates a comparison of the experimental

bias and the analytical bias. Again, the analytical

bias obtained from the proposed bias-correction

method (the solid line curve) is always close to the

experimental bias (the dashed curve with circles).

This also demonstrates, from another point of view,


Fig. 6. In 2D space with two anchors based on bearing-only

localization algorithm (¾2 = 0:1=¾ = 0:3162 rad or 18.1169 deg).


computed by proposed method in 2D space with two anchors

based on bearing-only localization algorithm

(¾2 = 0:1=¾ = 0:3162 rad or 18.1169 deg).

that the proposed method performs well in reducing

bias.

To demonstrate that the proposed method is

generic, it is now implemented in a scan-based

localization problem using TDOA measurements.

Different from the previous range and bearing-only

measurements situations, now we need three physical

sensors to obtain two usable measurements. Therefore,

we set an extra sensor at (26,0), thus three sensors

are at (0,8), (0,¡8), and (26,0). Figure 8 depicts thesimulation results from using TDOA measurements

Fig. 8. In 2D space with two measurements based on scan-based

localization algorithm (TDOA measurements and

¾2 = 0:1=¾ = 0:3162).


computed by proposed method in 2D space with two

measurements based on scan-based localization algorithm (TDOA

measurements and ¾2 = 0:1=¾ = 0:3162).

in a scan-based localization problem with the

measurement error variance ¾2 equal to 0.1 (¾ =

0:3162). From the figure, again, we can obtain that

the proposed method can correct the bias very well.

Figure 9 shows a comparison of the experimental bias

and the analytical bias. The analytical bias computed

by the proposed method (the solid line curve) is

always close to the experimental bias (the dashed

curve with circles), which verifies again that the

proposed method is effective in eliminating bias.


Fig. 10. In 2D space with two anchors based on distance-based

localization algorithm (¾2 = 0:05=¾ = 0:2236).

Next, we consider the simulation results with

different noise levels. Here, we set the variance of

the errors in measurement ¾2 = 0:05 (¾ = 0:2236).

Figure 10 illustrates the simulation results with

distance measurements. From the figure we can see

that, by using the proposed method, the bias or more

strictly, the residual average absolute distance error,

(the solid line curve) becomes very small compared

with the experimental bias (the dashed curve with

circles). The simulation results show the same

phenomenon in the situations with bearing-only and

TDOA measurements (shown in Fig. 11 and Fig. 12).

Compared with the ¾2 = 0:1=¾ = 0:3162 situation, we

can see that, when the variance of measurement errors

is reduced, the bias will become smaller. Further,

the simulation results also verify that the proposed

method can be effective with different noise levels.

From the above simulation results, we can observe

that, in most situations (except with collinearity

in 2D space or a coplanar situation in 3D space)

and for the noise levels postulated, the proposed

method can reduce the bias with different types of

measurements (distance, bearing-only, and TDOA).

Using more terms before truncation may lead to

improved precision. More analysis will be done in the

future.

B. N = n+1 Situation


measurements is one more than the dimension of the

ambient space. We present simulation results for 2D

space with three measurements. Again, three different

types of measurements are applied in this situation

with two different levels of noise: the distance, the

bearing-only, and the TDOA measurements.

Fig. 11. In 2D space with two anchors based on bearing-only


Fig. 12. In 2D space with two measurements based on

scan-based localization algorithm (TDOA measurements and

¾2 = 0:05=¾ = 0:2236).

First, in 2D space, we fix the three sensors at

(0,8), (0,¡8), and (8,0). Further, we fix the value ofy of the target at zero while changing the x value from

6 to 20 except 8 (because one sensor is at (8,0)) in

steps of 2. At the beginning we apply the high level of

noise (¾2 = 1=¾ = 1 for distance-based measurements

and ¾2 = 0:1=¾ = 0:3162 for bearing-only and TDOA

measurements).

Figure 13 shows the simulation results in 2D space

with three distance measurements. Again, from the

figure, we can see the proposed method (the solid line


Fig. 13. In 2D space with three measurements based on

distance-based localization algorithm (¾2 = 1=¾ = 1).

Fig. 14. Comparison between experimental bias and analytical

bias computed by proposed method in 2D space with three

measurements based on distance-based localization algorithm

(¾2 = 1=¾ = 1).

curve) can reduce the bias, which is always below

the dashed curve with squares corresponding to no

correction. From Fig. 14 we can conclude that the

analytical bias derived from the proposed method

(the solid line curve) is almost always consistent with

the experimental bias (the dashed curve with circles),

which from another standpoint, demonstrates the good

performance of the proposed method. (As before

absolute values are depicted.) Further, comparing with

the simulation results in the N = n situation (Fig. 3

Fig. 15. In 2D space with three measurements based on

bearing-only localization algorithm (¾2 = 0:1=¾ = 0:3162 rad or

18.1169 deg).


bias computed by proposed method in 2D space with three

measurements based on bearing-only localization algorithm

(¾2 = 0:1=¾ = 0:3162 rad or 18.1169 deg).

and Fig. 4), we can see that introducing one extra

measurement (assuming that no bad geometry such

as collinearity occurs) can improve the accuracy of the

localization.

Figures 15 and 16 illustrate the simulation results

of the proposed method on bearing-only localization

methods. Similar to the N = n situation, the proposed

method can reduce the bias in localization. In

Fig. 16 the analytical bias calculated by the proposed



scan-based localization algorithm (TDOA measurements and

¾2 = 0:1=¾ = 0:3162).


bias computed by proposed method in 2D space with two

measurements based on scan-based localization algorithm (TDOA

measurements and ¾2 = 0:1=¾ = 0:3162).

method (the solid line curve) is consistent with the

experimental bias (the dashed curve with circles).

Again, comparing with the N = n situation (Fig. 6 and

Fig. 7), the accuracy of the localization is enhanced by

introducing one more measurement.

Figures 17 and 18 show the simulation results

in a scan-based localization problem. Here, in order

to obtain three usable measurements, we need four

physical sensors. The four sensors are set at (0,8),

Fig. 19. In 2D space with three anchors based on distance-based

localization algorithm (¾2 = 0:5=¾ = 0:7071).

(0,¡8), (26,¡6), and (26,6). Again, from both

figures, we can conclude that the proposed method

performs well. Further, comparing with the N = n

situation (Figs. 8 and 9), the localization accuracy is

improved by using one more measurement.

Next, we consider the situation with a low level

of noise, where ¾2 = 0:5=¾ = 0:7071 (distance-based)

and ¾2 = 0:05=¾ = 0:2236 (bearing-only and TDOA).

Figure 19 shows the simulation results by using

distance measurements, while Fig. 20 and Fig. 21

illustrate the situation with bearing-only and TDOA

measurements respectively. From the two figures we

can see that our method still can correct the bias well

with a different noise level. Similarly, comparing with

Fig. 10, Fig. 11, and Fig. 12, we can conclude that,

when one extra sensor is introduced, the accuracy of

the localization is greatly improved.

From the above simulation results, we can see that,

when one extra sensor is introduced (and with the

assumed noise level,) the proposed method still can

correct the bias in localization almost perfectly with

distance measurements, bearing-only measurements,

or TDOA measurements. Further, one extra sensor

enhances the accuracy of localization.

C. N > n+1 Situation


measurements is at least two greater than the ambient

space dimension. For ease of exposition we only

present simulation results in 2D space with four

sensors. We fix the four sensors at (0,8), (0,¡8),(8,8), and (8,¡8). Further, we fix the value of y ofthe target at zero while changing the x value from 6

to 20 in steps of 2. Here, we set the level of noise in

measurement as ¾2 = 1=¾ = 1.


Fig. 20. In 2D space with three anchors based on bearing-only



scan-based localization algorithm with TDOA measurements

(¾2 = 0:05=¾ = 0:2236).

Figure 22 illustrates the simulation results in 2D

space with four sensors. From the figure we can see

that the analytical bias computed by the proposed

method (the dashed curve with squares) is always

consistent with the experimental bias (the dashed

curve with circles). Further, by using the proposed

method, the bias is reduced to a very low level (the

solid line curve), which demonstrates the performance

of the proposed method. Figure 23 shows the same

simulation results with the same scale as Fig. 3 and

Fig. 13. Compared with Fig. 3 and Fig. 13, the figures

Fig. 22. In 2D space with four anchors based on distance-based


Fig. 23. In 2D space with four sensors based on distance-based


illustrate the unsurprising conclusion that normally

having more measurements improves accuracy.

From the simulation results shown in three

different situations, we can conclude that the proposed

method with the assumed noise level can correct the

bias almost perfectly, expect for adverse geometries

(collinearity or coplanarity problem) in arbitrary

dimension space with arbitrary sensor count by

using different types of measurements (distance,

bearing-only, and TDOA measurements).

V. CONCLUSIONS

A novel generic method to reduce the bias in

localization algorithms is proposed in this paper. The

proposed method formulates the bias analytically in


an easy way by mixing Taylor series and Jacobian

matrices. We analyze the proposed method in three

different situations. In the first situation the ambient

space dimension is equal to the number of usable

measurements. In the second situation one more

measurement is introduced, which leads to an

overdetermined problem. To solve the overdetermined

problem, we introduce an extra variable into the

equation set that corresponds to adopting an LS

method. The number of usable measurements is at

least two greater than the ambient space dimension

in the third situation; similarly, we still use the LS

method to introduce extra variables into the equation

set in order to solve the overdetermined problem.

Monte Carlo experiments illustrate that the proposed

method can correct the bias very well for the noise

levels postulated with different types of measurements,

except for adverse geometries such as collinear

(coplanar situations in 3D space). Our future work

includes seeking to improve the performance of the

proposed method by using higher order terms of the

Taylor series; this may be important in high noise. A

further topic we plan to examine, identified earlier in

the paper, is the threshold problem for localization

algorithms.

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Yiming Ji is a NICTA endorsed Ph.D. candidate at the Australian NationalUniversity, Canberra, Australia, under the supervision of Professor Brian D. O.

Anderson. He received his B.S. degree in computer science and engineering

from Northwestern Polytechnical University, China in 2008. His current research

interests include consensus, localization, and security problems in wireless sensor

networks.

Changbin Yu (S’05–M’08–SM’11) received his B.Eng. degree with first classhonors from Nanyang Technological University, Singapore in 2004 and his Ph.D.

degree from the Australian National University, Canberra, Australia in 2008.

He has been a faculty member at the Australian National University and is

currently adjunct at NICTA, Australia and Shandong Computer Science Center,

China. He was a recipient of an ARC Australian Postdoctoral Fellowship in 2008,

a Queen Elizabeth II Fellowship in 2011, the Chinese Government Award for

Outstanding Chinese Students Abroad in 2006, the Australian Government’s

Endeavour Asia Award in 2005, and the Best Paper Award of the Asian Journal of

Control. His current research interests include control of autonomous formations,

multi-agent systems, sensor networks, and graph theory. He is a Member of IFAC

Technical Committee on Networked Systems.

Brian D. O. Anderson (M’66–SM’74–F’75–LF’07) was born in Sydney,Australia and educated at Sydney University in mathematics and electrical

engineering, with a Ph.D. in electrical engineering from Stanford University in

1966.

He is a distinguished professor at the Australian National University and

distinguished researcher in National ICT Australia. His awards include the

IEEE Control Systems Award of 1997, the 2001 IEEE James H. Mulligan, Jr.

Education Medal, and the Bode Prize of the IEEE Control System Society in

1992, as well as several IEEE and other best paper prizes. He is a Fellow of

the Australian Academy of Science, the Australian Academy of Technological

Sciences and Engineering, the Royal Society, and a foreign associate of the

U.S. National Academy of Engineering. He holds honorary doctorates from a

number of universities, including Universite Catholique de Louvain, Belgium and

ETH, Zurich. He is a Past President of the International Federation of Automatic

Control and the Australian Academy of Science. His current research interests are

in distributed control, sensor networks, and econometric modelling.


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