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:
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
1694 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 3 JULY 2013
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.
JI, ET AL.: SYSTEMATIC BIAS CORRECTION IN SOURCE LOCALIZATION 1695
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.
1696 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 3 JULY 2013
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
JI, ET AL.: SYSTEMATIC BIAS CORRECTION IN SOURCE LOCALIZATION 1697
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)
1698 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 3 JULY 2013
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.
JI, ET AL.: SYSTEMATIC BIAS CORRECTION IN SOURCE LOCALIZATION 1699
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.
1700 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 3 JULY 2013
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,
JI, ET AL.: SYSTEMATIC BIAS CORRECTION IN SOURCE LOCALIZATION 1701
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).
Fig. 7. Comparison between experimental bias and analytical bias
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).
Fig. 9. Comparison between experimental bias and analytical 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).
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.
1702 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 3 JULY 2013
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
In this situation the number of usable
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
localization algorithm (¾2 = 0:05=¾ = 0:2236 rad or 12.8113 deg).
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
JI, ET AL.: SYSTEMATIC BIAS CORRECTION IN SOURCE LOCALIZATION 1703
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).
Fig. 16. Comparison between experimental bias and analytical
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
1704 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 3 JULY 2013
Fig. 17. In 2D space with two measurements based on
scan-based localization algorithm (TDOA measurements and
¾2 = 0:1=¾ = 0:3162).
Fig. 18. Comparison between experimental bias and analytical
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
In this situation the number of usable
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.
JI, ET AL.: SYSTEMATIC BIAS CORRECTION IN SOURCE LOCALIZATION 1705
Fig. 20. In 2D space with three anchors based on bearing-only
localization algorithm (¾2 = 0:05=¾ = 0:2236 rad or 12.8113 deg).
Fig. 21. In 2D space with two measurements based on
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
localization algorithm (¾2 = 1=¾ = 1).
Fig. 23. In 2D space with four sensors based on distance-based
localization algorithm (¾2 = 1=¾ = 1).
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
1706 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 3 JULY 2013
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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1708 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 3 JULY 2013
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.
JI, ET AL.: SYSTEMATIC BIAS CORRECTION IN SOURCE LOCALIZATION 1709