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Non-line-of-sight Node Localization Based on
Semi-Definite Programming in Wireless Sensor
Networks
Hongyang Chen, Gang Wang, Zizhuo Wang, H. C. So, and H. Vincent Poor, Fellow, IEEE
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Abstract
An unknown-position sensor can be localized if there are three or more anchors making time-ofarrival (TOA) measurements of a signal from it. However, the location errors can be very large due
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to the fact that some of the measurements are from non-line-of-sight (NLOS) paths. In this paper, a
semi-definite programming (SDP) based node localization algorithm in NLOS environments is proposed
for ultra-wideband (UWB) wireless sensor networks. The positions of sensors can be estimated using
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the distance estimates from location-aware anchors as well as other sensors. However, in the absence
of line-of-sight (LOS) paths, e.g., in indoor networks, the NLOS range estimates can be significantly
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biased. As a result, the NLOS error can remarkably decrease the location accuracy, and it is not easy
to accurately distinguish LOS from NLOS measurements. According to the information known about
the prior probabilities and distributions of the NLOS errors, three different cases are introduced and
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the related localization problems are addressed respectively. Simulation results demonstrate that this
algorithm achieves high location accuracy even for the case in which NLOS and LOS measurements are
not identifiable.
Index Terms
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IEEE Transactions on Wireless Communications
Wireless sensor networks, non-line-of-sight (NLOS), time-of-arrival (TOA), semi-definite programming (SDP).
This research was supported in part by the U. S. Office of Naval Research under Grant N00014-09-1-0342. H. Chen is
with the Institute of Industrial Science, the University of Tokyo (e-mail: hongyang@mcl.iis.u-tokyo.ac.jp). G. Wang is with the
ISN Lab, Xidian University, Xian, China. Z. Wang is with the Department of Management Science and Engineering, Stanford
University, Stanford, CA, USA (e-mail: zzwang@stanford.edu). H. C. So is with the Department of Electronic Engineering,
City University of Hong Kong, Kowloon, Hong Kong (e-mail: hcso@ee.cityu.edu.hk). H. V. Poor is with the Department of
Electrical Engineering, Princeton University, Princeton, NJ, USA (e-mail: poor@princeton.edu).
IEEE Transactions on Wireless Communications
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I. I NTRODUCTION
Development of localization algorithms for wireless sensor networks (WSNs) to find node positions
is an important research topic because position information is a major requirement in many WSN
applications. Examples include animal tracking, earthquake monitoring and location-aided routing.
Based on the type of information provided for localization, sensor protocols can be divided into two
categories: (i) range-based and (ii) range-free protocols [1]. Due to the coarse location accuracy of rangefree schemes [1], solutions based on range-based localization are often preferable. Range estimates from
anchors can be obtained using received signal strength (RSS), angle-of-arrival (AOA) or time-of-arrival
(TOA) observations of transmitted calibration signals [2]. Impulse-based ultra-wideband (UWB) is a
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promising technology that allows precise ranging to be embedded into data communication. It is robust
in dense multipath environments and it is able to provide accurate position estimation with low-datarate communication. In this paper, we focus on the investigation of range-based localization algorithms
for UWB WSNs. One of the main challenges for accurate node localization in range-based localization
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algorithms is non-line-of-sight (NLOS) propagation due to obstacles in the direct paths of beacon signals.
NLOS propagation results in unreliable localization and significantly decreases the location accuracy if its
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effects are not taken into account. This often occurs in an urban or indoor environment. Some localization
algorithms that cope with the existence of NLOS range measurement have been proposed in [3], [4] and
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[5], mostly for cellular networks. In those works, there are two approaches to deal with the localization
problem in the presence of NLOS propagation. The first approach identifies LOS and NLOS information
and discards the NLOS range information. The second approach uses all NLOS and LOS measurements
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and provides weighting or scaling to reduce the adverse impact of NLOS range errors on the accuracy
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of position estimates. In both methodologies, it is assumed that the NLOS range estimates have been
identified. In [6], Mazuelas et al. proposed the prior NLOS measurement correction (PNMC) method to
mitigate the effects of NLOS propagation in cellular networks. Furthermore, Bahillo et al. implemented
this interesting idea in a real indoor environment to alleviate the effect of severe NLOS propagation on
distance estimates [7].
In the sensor network localization problem, the number of anchors is typically limited by practical
considerations. Therefore, it is a waste of resources to discard NLOS range measurements. To make
best use of all range measurements, we propose in this paper a computationally efficient semi-definite
programming (SDP) approach for this problem which effectively incorporates both LOS and NLOS range
information into the estimates.
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In the ideal scenario, by assuming that the NLOS error is exponentially distributed with known
parameters, the sum of the NLOS error and the measurement noise can be approximated as a Gaussian
distribution. Given this fact, the NLOS measurement model can be approximately expressed in a form
similar to the LOS measurement model. Combining the LOS and the approximate NLOS models, the
approximate maximum likelihood (AML) estimation problem can be naturally formulated. However, the
AML formulation is non-convex. We then relax this problem to an SDP and thus solve the relaxed
problem.
For the case in which only the probability of NLOS propagation and distribution parameters are
known, the AML formulation is quite complex. To facilitate SDP relaxation, we further approximate the
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estimation problem. After this step, the SDP relaxation technique can be easily applied to estimate the
node positions in this NLOS environment.
The last case we focus on is the problem of NLOS mitigation without the requirement of accurately
distinguishing between LOS and NLOS range estimates and without prior NLOS error information.
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Given a mixture of LOS and NLOS range measurements, our method is applicable in both cases without
discarding any range information.
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To our knowledge, this method is the first SDP based approach to reduce the impact of NLOS in
WSNs. The main advantages of this approach are given as follows.
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1) The statistics of the NLOS bias errors are not assumed to be known a priori for our method.
NLOS range estimates are not required to be readily distinguished from LOS range estimates
through channel identification. Thus, it makes use of all measurements.
2) No range information is discarded.
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3) SDP is efficiently applied to address the NLOS node localization problem which achieves excellent
localization accuracy.
In our proposed approach, we assume the following features of UWB TOA-based range estimation: the
range bias errors in NLOS conditions are always positive and significantly larger in magnitude than the
range-measurement noises. We will show that the node localization problem, given range information,
can be cast into a nonlinear programming formalism. We then use SDP relaxation techniques and rely
on both LOS and NLOS range estimates to estimate sensors’ positions in the NLOS environment.
The rest of the paper is organized as follows. Section II derives the SDP based localization algorithms
for NLOS environments. In Section III, the distributed algorithm implementation for our SDP approach
is introduced. In Section IV, simulation results are reported. Section V includes our conclusions.
IEEE Transactions on Wireless Communications
4
II. BACKGROUND
In this section, we introduce the technical preliminaries. The basic settings of this paper are as follows.
We assume an asynchronous UWB sensor network. Each node has the ability to achieve TOA estimation
based on UWB channel model. We use a complex baseband-equivalent channel model which is adopted by
the IEEE 802.15.4a working group. As described in [8], the modeling and characteristics of the employed
channel models are available for residential, office, outdoor, and industrial environments. There are n
unknown-position sensors in R2 and m anchor sensors whose positions are known a priori. We use
xi ∈ R2 , i = 1, 2, ..., n, to denote the unknown-position sensors and xj ∈ R2 , j = n + 1, n + 2, ..., n + m,
to denote the anchors.
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We use ri,j to denote the actual distance between the ith sensor and the j th sensor or anchor, i.e.,
ri,j = kxi − xj k ,
∀ i = 1, 2, ..., n, j = 1, 2, ..., m + n.
(1)
In practice, we get measurement information for a subset of pairs of nodes, which is denoted by
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E . We use Elos (and respectively Enlos ) to denote the set of index pairs such that the measurement
between the nodes is LOS (and respectively NLOS). Similarly, we use E1 to denote the set of index pairs
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such that the measurements come from unknown-position sensors and anchors, and E2 to denote the
set of index pairs such that the measurements come from unknown-position sensors [22]. By definition,
S
S
E = Elos Enlos = E1 E2 . Note that only the index pairs of the nodes that can communicate with each
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other are included in E . Notice that each of measurements could be either LOS or NLOS.
In this paper, we assume that the LOS range measurement is
di,j = ri,j + ni,j ,
(2)
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2 ) is the measurement error which follows a zero-mean Gaussian distribution with
where ni,j ∼ N (0, σi,j
standard deviation σi,j .
Similarly, the NLOS range measurement is assumed to be
Di,j = ri,j + ni,j + δi,j ,
(3)
where δi,j is the error of the NLOS measurement.
III. NLOS L OCALIZATION U SING SDP R ELAXATION
According to the information we have on NLOS error prior probabilities and distributions, three
different cases are introduced and the related problems are addressed respectively in the following
subsections.
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1) The ideal scenario in which we know which ranges are in NLOS conditions and the distribution
parameters for NLOS error.
2) The scenario with limited prior information in which we do not know which ranges are in NLOS
conditions, but we have a priori information on the NLOS probability and the distribution parameters at each anchor.
3) The worst case in which we do not know any a priori information on NLOS errors, but we know
the measurement noise power. This scenario is obviously the most interesting and practical case,
but also the most difficult one for which to address the effect of NLOS errors.
A. Known NLOS status and Distribution Parameters
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In this subsection, we consider the ideal scenario in which we know which ranges are in NLOS
conditions and the distribution parameters for the NLOS error. We regard the sum of the NLOS error
and the measurement noise as a new “measurement noise”. Given this fact, the NLOS measurement
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model can be approximately written as the form similar to the LOS measurement model. Unfortunately,
the distribution of the new “measurement noise” is generally difficult to obtain. However, the mean and
variance may be easily obtained. Based on this fact, we propose an AML formulation to estimate the
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sensor node positions. The non-convexity nature of the AML formulation makes the problem very difficult
to deal with. Similarly to an approach used in the literature [13], we relax the AML problem to an SDP,
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which can be solved very efficiently and can achieve reasonable estimation accuracy.
The NLOS measurements are given by
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Di,j = kxi − xj k + ni,j + δi,j , i < j, (i, j) ∈ Enlos .
(4)
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Without loss of generality, we here assume that the NLOS error, δi,j , is exponentially distributed with
mean λi,j and variance λ2i,j [23].
2 by assuming the independence
Let ci,j = ni,j + δi,j , which has mean λi,j and variance λ2i,j + σi,j
between ni,j and δi,j [18]. Then (4) can be written as
Di,j =kxi − xj k + ci,j = kxi − xj k + λi,j + vi,j ,
(5)
i < j, (i, j) ∈ Enlos ,
02 = λ2 + σ 2 is introduced. In this
where a new random variable vi,j with zero mean and variance σi,j
i,j
i,j
ideal scenario (known NLOS status), we can subtract the mean of the NLOS error from the measurements
such that (5) is equivalent to
0
Di,j = kxi − xj k + vi,j , i < j, (i, j) ∈ Enlos ,
(6)
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0
where Di,j = Di,j − λi,j .
On the other hand, the LOS measurements are given by
di,j = kxi − xj k + ni,j , i < j, (i, j) ∈ Elos .
(7)
The joint likelihood function of Xs can be written as
Y
0
p(di,j , Di,j |Xs ) =
pni,j (di,j − kxi − xj k)
i<j;(i,j)∈Elos
Y
0
pvi,j (Di,j − kxi − xj k),
(8)
i<j;(i,j)∈Enlos
where pni,j and pvi,j denote the probability density functions (pdfs) of ni,j and vi,j , respectively. Evidently,
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vi,j (and also ci,j ) is not Gaussian distributed. To simplify the analysis, we use a Gaussian distribution
02 ) to approximate the distribution of v , where σ 02 is the variance of v . (Note that this
N (0, σi,j
i,j
i,j
i,j
approximation has the potential risk of the long tail problem when the error is large. But it makes sense
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in most cases. )
Using this approximation, the AML estimation of the sensor positions can be formulated as
X
min
Xs
i<j;(i,j)∈Elos
X
i<j;(i,j)∈Enlos
1
0
2
02 (Di,j − kxi − xj k) ,
σi,j
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+
1
2
2 (di,j − kxi − xj k)
σi,j
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(9)
where Xs = [x1 , . . . , xn ]. It is worth noting that (9) can also be regarded as the weighted least squares
2 for the LOS
(WLS) estimation of the sensor node positions, where the weights are chosen as 1/σi,j
The AML formulation (9) can be equivalently written as
min
Y,Xs ,g,h
X
i<j;(i,j)∈Elos
1 2
2 (di,j − 2hi,j di,j + gi,j ) +
σi,j
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02 for the NLOS measurements.
measurements and 1/σi,j
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X
i<j;(i,j)∈Enlos
1
0
0
2
(Di,j
− 2hi,j Di,j + gi,j )
02
σi,j
s.t. gi,j = Yi,i + ||xj ||2 − 2xTj xi , (i, j) ∈ E1 ,
gi,j = Yi,i + Yj,j − 2Yi,j , (i, j) ∈ E2 ,
gi,j = h2i,j , ∀(i, j) ∈ E,
Y = XTs Xs ,
where hi,j = kxi − xj k, gi,j = kxi − xj k2 , g = [gi,j ], h = [hi,j ], and Y = [Yi,j ].
(10)
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The last constraint in (10) can be equivalently written as Y º XTs Xs and rank(Y) = 2. By dropping
the rank-2 constraint, (10) can be relaxed to an SDP:
min
X
Y,Xs ,g,h
i<j;(i,j)∈Elos
1 2
2 (di,j − 2hi,j di,j + gi,j ) +
σi,j
X
i<j;(i,j)∈Enlos
1
0
0
2
(Di,j
− 2hi,j Di,j + gi,j )
02
σi,j
s.t. gi,j = Yi,i + ||xj ||2 − 2xTj xi , (i, j) ∈ E1 ,
gi,j = Yi,i + Yj,j − 2Yi,j , (i, j) ∈ E2 ,
gi,j ≥ h2i,j , ∀(i, j) ∈ E,
Y º XTs Xs .
(11)
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B. Known Probability of NLOS Propagation and Distribution Parameters
In this subsection, we consider the scenario in which we do not know which ranges are in NLOS
conditions, but we have a priori information on the NLOS probability of each edge and the corresponding
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distribution parameters. In this case, the approximate ML formulation is much more complex since it
contains logarithms. To facilitate SDP relaxation, we replace the algorithmic average by a geometric
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average in the maximum likelihood formula. After this step, the SDP relaxation technique can be easily
applied similarly to that in last subsection. In this case, the joint likelihood function can be written as
0
p(di,j , Di,j
|Xs ) =
Y
©
αi,j pni,j [(di,j − kxi − xj k)|LOS]
i<j;(i,j)∈E
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+ (1 − αi,j )pvi,j [(Di,j
− kxi − xj k)|N LOS]
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(12)
where αi,j is the probability that anchor j or sensor j has an LOS link with sensor i, and pni,j and pvi,j
have been defined earlier.
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Using a similar approximation to the distribution of vi,j , the AML estimation of the sensor positions
can be obtained based on (12):
0
X̂s = arg max p(di,j , Di,j
|Xs )
Xs
0
= arg max log p(di,j , Di,j
|Xs )
Xs
!#
("
Ã
X
(di,j − kxi − xj k)2
αi,j
≈ arg max
log
c1 exp −
2
Xs
2σi,j
i<j;(i,j)∈E
"
Ã
!#
)
0 − kx − x k)2
(Di,j
i
j
+ c2 exp −
(1 − αi,j ) .
02
2σi,j
(13)
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However, this problem is difficult to solve due to its non-convexity nature and the existence of logarithms.
To facilitate SDP relaxation, we replace the arithmetic average in (13) by the geometric average, i.e., we
replace
log
("
Ã
(di,j − kxi − xj k)2
c1 exp −
2
2σi,j
!#
"
Ã
αi,j + c2 exp −
0 − kx − x k)2
(Di,j
i
j
02
2σi,j
!#
)
(1 − αi,j )
(14)
by
Ã
(di,j − kxi − xj k)2
log c1 −
2
2σi,j
!
Ã
αi,j +
log c2 −
0 − kx − x k)2
(Di,j
i
j
02
2σi,j
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!
(1 − αi,j ).
(15)
Then the AML problem becomes
!
"Ã
X
(di,j − kxi − xj k)2
X̂s ≈ arg max
log c1 −
αi,j
2
Xs
2σi,j
i<j;(i,j)∈E
!
#
Ã
0 − kx − x k)2
(Di,j
i
j
(1 − αi,j )
+ log c2 −
02
2σi,j
"
#
0 − kx − x k)2
X
(Di,j
(di,j − kxi − xj k)2
i
j
= arg min
αi,j +
(1 − αi,j ) .
2
02
Xs
σi,j
σi,j
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i<j;(i,j)∈E
(16)
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Now we can apply SDP relaxation to (16) in a similar manner to (11) as follows:
#
"
X
(1 − αi,j ) 0 2
αi,j 2
0
(Di,j − 2hi,j Di,j + gi,j )
min
2 (di,j − 2hi,j di,j + gi,j ) +
02
Y,Xs ,g,h
σi,j
σi,j
i<j;(i,j)∈E
s.t. gi,j = Yi,i + ||xj ||2 − 2xTj xi , (i, j) ∈ E1 ,
gi,j = Yi,i + Yj,j − 2Yi,j , (i, j) ∈ E2 ,
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gi,j ≥ h2i,j , ∀(i, j) ∈ E,
Y º XTs Xs .
(17)
A close inspection of (16) reveals that it has the same form with the multidimensional scaling (MDS)
2 and (1 − α )/σ 02 can be seen as the weights [17]. Hence, it can give a
formulation, where αi,j /σi,j
i,j
i,j
reasonably good estimate.
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C. Known Measurement Noise Power Only
In this subsection, we consider the worst case that we only know the measurement noise power. The
idea of our approach is to get an upper bound as well as a lower bound on the true distance of each pair
of nodes (which could be either an anchor and an unknown-position sensor or a pair of unknown-position
sensors) based on the measurement noise power, without distinguishing whether it comes from an LOS
or NLOS measurement. These bounds will form a feasible region for possible positions of each sensor
and we then choose one “center” point from this region as our estimate.
First we show how we obtain an upper bound on the distance of a certain pair of nodes. Since in the
NLOS case, the measured distance is larger than the actual distance, the measurement itself is an upper
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bound. For the LOS case, we have the upper bound
ri,j ≤ di,j + 2σi,j
where 2σi,j is obtained from the upper bound on the measurement error, which can be chosen based on the
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tail probability of the measurement error (e.g., several standard deviations for the Gaussian distribution).
Therefore, we have a uniform upper bound on the distance between each pair of nodes as follows:
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ri,j = kxi − xj k ≤ di,j + 2σi,j ,
Next, the lower bound is given as follows:
∀(i, j) ∈ E.
(18)
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.
ri,j ≥ li,j = di,j − 4σi,j
vi
for any pair of nodes in range. The motivation for us to use this lower bound is as follows: Since in the
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LOS case, the measured distance can be larger or smaller than the actual distance, we include a minus
variable, −4σi,j , to confirm the measured distance is larger than the lower bound. For the NLOS case,
the NLOS error can be also decreased with using this minus variable.
For later convenience, we uniformly write the upper and lower bound for the distance between nodes
i and j as ui,j and li,j , respectively.
Remark 1: In some circumstances, we do not have communication between unknown-position sensors.
In that case, we simply remove the constraint between them, keeping only those between unknownposition sensors and anchors.
Remark 2: This approach can also be applied to the cases when we have prior information on which
measurements are from LOS paths and which are from NLOS paths. If we know that a certain measurement is from an NLOS path, then we can compute the upper bound by using the measurement, or if
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we know the error follows a certain distribution, then we can again adjust the upper and lower bounds
accordingly. The same thing applies when we know a certain measurement is from an LOS path.
Furthermore, we present a convex optimization algorithm for node localization based on the bounds
we have obtained in the previous subsection.
As shown in Fig. 1, for the single constraint case (s < kx − ak 6 R), it is easy to see that one
heuristic position estimate lies on the circle with center a and radius
R+s
2 .
(The square in Fig. 1 indicates
the possible node position.)
This can be determined by minimizing the following expression:
µ
¶
s+R 2
kx − ak −
.
2
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(19)
Expanding (19) yields
µ
2
kx − ak − (s + R)kx − ak +
where ((s + R)/2)2 is a constant.
s+R
2
¶2
(20)
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Therefore, the optimization problem for locating the sensors can be formulated as
x
h
i
kxi − xj k2 − (li,j + ui,j ) kxi − xj k .
X
min
er
i<j:(i,j)∈E
(21)
Obviously, (21) is nonconvex, and so cannot be solved easily. However, we can relax the problem to
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two convex optimization problems by using the SDP relaxation techniques as proposed in [10] and [11],
which are referred to as FullSDP and ESDP, respectively.
We can rewrite (21) as
X
Y,Xs ,g,h
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[gi,j − (li,j + ui,j )hi,j ]
i<j;(i,j)∈E
s.t. gi,j = Yi,i + ||xj ||2 − 2xTj xi , (i, j) ∈ E1 ,
gi,j = Yi,i + Yj,j − 2Yi,j , (i, j) ∈ E2 ,
gi,j = h2i,j , ∀(i, j) ∈ E,
Y = XTs Xs .
(22)
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11
By performing SDP relaxation, we relax (22) to an SDP
min
X
Y,Xs ,g,h
[gi,j − (li,j + ui,j )hi,j ]
i<j;(i,j)∈E
s.t. gi,j = Yi,i + ||xj ||2 − 2xTj xi , (i, j) ∈ E1 ,
gi,j = Yi,i + Yj,j − 2Yi,j , (i, j) ∈ E2 ,
gi,j ≥ h2i,j , ∀(i, j) ∈ E,
Y º XTs Xs .
(23)
When the network is large, solving the SDP problem might be slow [12]. Based on the work of [11],
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we further relax (23) into an ESDP formulation:
min
Z,g,h
X
[gi,j − (li,j + ui,j )hi,j ]
i<j;(i,j)∈E
s.t. gi,j = Yi,i + ||xj ||2 − 2xTj xi , (i, j) ∈ E1 ,
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gi,j = Yi,i + Yj,j − 2Yi,j , (i, j) ∈ E2 ,
er
gi,j ≥ h2i,j , ∀(i, j) ∈ E,


I2 Xs
,
Z=
T
Xs Y
Z(1,2,i,j) º 0,
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∀(i, j) ∈ E,
(24)
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where Z(1,2,i,j) denotes the principal submatrix of Z consisted of rows and columns 1, 2, i, j .
Remark 3: In practice, we might want to add different weights to different terms in the objective
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function according to the confidence of each measurement. For instance, we can give a lower weight to
the NLOS part if we have prior statistics on NLOS measurements.
Remark 4: The heuristic of trying to put the sensor into the middle of the upper and lower bound is
of course an approximation. There is certainly some bias associated with this approach. However, given
upper and lower bounds which form a feasible region of the sensor position, we can obtain a feasible
solution from this region. Numerical results shown later show that this approach can give accurate sensor
position estimates.
In the next section, we discuss one extension of above model, i.e., the situation in which there are
uncertainties in the anchor positions. We show that a similar SDP model can be formulated to solve this
problem [13].
IEEE Transactions on Wireless Communications
12
D. Complexity Analysis
The SDPs in Sections III-A-III-C (except the ESDP (24) in Section III-C) have a similar complexity.
Since the SDPs have a similar form to the SDP in [19], we can borrow the result therein. According
√
to [19], the worst-case complexity of solving the SDPs is O( n + k(n3 + n2 k + k 3 ) log (1/²)), where
k = 3+|E| is the number of equality constraints (|E| represents the cardinality of E ), and ² is the solution
precision. Typically, k = O(n2 ), which implies that the worst-case complexity is of the order O(n6 ).
However, it is shown in [19] that the running time typically increases linearly as the number of sensor
nodes n increases, and hence the actual complexity is O(n3 ).
IV. N UMERICAL R ESULTS
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In this section, simulation results are presented and analyzed. In the simulations, all the SDPs above
are solved by the standard SDP solver SeDuMi [14], and we choose CVX [15] as the programming
interface. Since our algorithm is the first one to address the NLOS node localization in WSNs, we
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present simulation results only for our algorithms.
We consider a 2-dimensional square region of size 40 m by 40 m, where we randomly deploy
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40 sensors. There are a total of 18 anchors located in the region. Eight of them are located at the
boundary (20, 20) m, (-20,20) m, (20,-20) m, (-20,-20) m, (0,20) m, (20,0) m, (0,-20) m, and (-20,0)
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m, while the remaining ten anchors are randomly deployed in this region. In the simulations, they
are located at (4.3416,-19.3696) m, (-19.3458,-12.3970) m, (3.4767,-17.6967) m, (-5.2972,5.2580) m,
(8.7053,7.7067) m, (-16.6368,-1.8257) m, (-2.3268,-5.8699) m, (-13.8557,7.0257) m, (7.9685,9.1003) m
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and (-0.8646,2.1936) m. The configuration of this network is shown in Fig. 2. We here assume that the
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sensors and anchors are partially connected. The communication ranges of all sensors and anchors are
assumed to be identical and equal to 25 m. The measurements are generated according to either (4) or (7),
where δi,j is exponentially distributed with mean λi,j . For the noise level σi,j and the NLOS error level
λi,j , we assume that they are identical for all measurements, i.e., σi,j = σ and λi,j = λ. The performance
is evaluated using the root mean square error (RMSE), which is computed using 100 Monte Carlo runs.
A. Known NLOS status and Distribution Parameters
In this subsection, we provide simulation results when the NLOS status and distribution parameters
are perfectly known. The SDP (11) is used to obtain the estimate of the sensor node positions. We will
consider the following 3 scenarios to evaluate the performance of the SDP relaxation method.
Scenario 1:
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In this scenario, we fix the number of the anchors and the level of NLOS errors to show the performance
of the SDP relaxation method when the measurement noise level is varying. The first 8 anchors listed
above are used (i.e., the anchors at the boundary) and the mean of the NLOS error δi,j is λ = 4 m in
(4). The probabilities that the measurements are LOS and NLOS are set as 0.7 and 0.3, respectively,
i.e., αi,j = 0.7. In the following, we perform simulations using different values of σ . In our simulations,
the proposed SDP formulation (11) is applied to estimate the sensor positions. Fig. 3 plots the RMSE
versus the noise standard deviation σ . From the figure we see that the proposed method can provide
good estimation by mitigating the effect of NLOS measurements. Moreover, we have mentioned that
the SDP solution could also act as an initial guess for other numerical search methods to obtain better
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estimates. For the purpose of comparison, the RMSEs of the AML solutions that respectively use the SDP
solution and the true sensor positions as the initial guesses (labeled as “AML-SDP” and “AML-True”,
respectively) are also plotted in the figure. We see that “AML-SDP” and “AML-True” have very similar
RMSEs, indicating that SDP can provide a good estimate.
Scenario 2:
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In this scenario, we fix the number of the anchors and measurement noise level and vary the levels
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of NLOS errors. The measurement noise level is fixed as σ = 4 m, and other settings are the same
with those in Scenario 1. Fig. 4 plots the RMSE versus the mean of NLOS λ, from which we see that
“AML-SDP” and “AML-True” still have very similar RMSEs. Furthermore, we see that increasing NLOS
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errors does not result in dramatic increase in RMSE, indicating that the proposed method can effectively
mitigate the NLOS errors.
Scenario 3:
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In this scenario, we evaluate the performance of the SDP relaxation method when the number of
anchors is varying. We fix the measurement level and the NLOS error level as σ = 4 m and λ = 4 m,
respectively. The number of anchors N is varying from 8 to 18, and the first N anchors listed above are
used. The corresponding simulation results are shown in Fig. 5, from which we see that increasing the
number of anchors does not result in obvious performance improvement.
B. Known Probability of NLOS Propagation and Distribution Parameters
In this subsection, we provide simulation results for the case in which only the probability of the NLOS
status and distribution parameters are known. We also consider 3 scenarios similar to those in Section
IV-A. The SDP (16) is used to obtain the estimate of the sensor node positions. The corresponding
simulation results are shown in Figs. 6, 7, and 8, respectively. The three figures demonstrate the efficacy
IEEE Transactions on Wireless Communications
14
of our method which can be used to efficiently decrease the impact of NLOS errors.
C. Known Measurement Noise Power Only
In this subsection, we consider the localization case that only the measurement noise power σ 2 is
known. This parameter is used to determine the lower and upper bounds described in Section III-C.
We here consider the 3 scenarios similar to those in Section IV-A again. The SDP (23) is used to find
the estimated sensor positions and the simulation results are shown in Figs. 9, 10, and 11. From the
figures we see that the proposed method can provide good estimates by mitigating the effects of NLOS
measurements. Since only the noise power is known and the factors that affect the performance are
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complicated in this worst case, the localization accuracy is not linearly changed with the change of λ in
Fig. 10.
V. C ONCLUSIONS AND F UTURE W ORK
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In this paper, we have considered node location using UWB technology in wireless sensor networks.
Because of its power efficiency, fine delay resolution, and robust operation in harsh environments, UWB
transmission is promising as a means to address NLOS localization problem. A semi-definite programming
er
based node localization algorithm has been proposed for this purpose. In particular, the problem of node
localization in the NLOS environment has been approximated by a convex optimization problem using
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the SDP relaxation technique. Given a mixture of LOS and NLOS range measurements, our method is
applicable in both cases without discarding any range information. Simulation results demonstrate the
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effectiveness of our method.
In the future, we plan to implement our localization scheme in a testbed and verify its performance
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with a physical UWB sensor network. Moreover, the implementation of our method in a distributed way,
the NLOS location case in the presence of anchor position errors, and the derivation of performance
bound are some interesting research topics.
ACKNOWLEDGEMENT
We would like to thank Dr. W.-K. Ma from the Chinese University of Hong Kong, Dr. Kenneth W. K.
Lui and the anonymous reviewers for their valuable suggestions concerning this work.
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The configuration of the network. Triangle: anchor position; circle: true sensor position.
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2.3
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Fig. 4. The RMSE versus the mean of NLOS error λ when the NLOS status and distribution parameters are perfectly known.
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Fig. 10.
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The RMSE versus the mean of NLOS error λ when only distribution parameters is known.
IEEE Transactions on Wireless Communications
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Page 22 of 30
The RMSE versus the number of anchors when only distribution parameters is known.
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