This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings
Mapping Techniques for UWB Positioning
Eva Arias-de-Reyna
Umberto Mengali
Department of Signal Theory and Communications
University of Seville
Camino de los Descubrimientos s/n, 41092 Seville, Spain
Email: earias@us.es
Department of Information Engineering
University of Pisa
Via Caruso 16, 56122 Pisa, Italy
Email: umberto.mengali@iet.unipi.it
Abstract—This paper deals with a wireless indoor positioning
problem in which the location of a tag is estimated from range
measurements taken by fixed beacons. The measurements may
be affected by non-line-of-sight (NLOS) errors that must be
mitigated. We discuss a maximum likelihood (ML) positioning
technique that assumes a realistic model for the range errors and
a signature database providing information on the propagation
conditions at every hypothesized tag spot. The database can
be gathered from knowledge of the service area infrastructure
and through pre-measurements. It is given in the form of a
map indicating, at any node of a close-mesh grid, the nature
(LOS/NLOS) of the link between that node and each beacon.
The performance of the positioning algorithm is assessed by
simulation and is compared with other methods available in
literature. The results show that the proposed technique provides significant improvements and is robust against mismatches
between true and assumed values of the parameters in the
range error model. Comparisons with the Cramer-Rao Bound
are made.
I. I NTRODUCTION
In the last few years wireless indoor geolocation has become
an important technology for use in a variety of civil and military applications. In the commercial world, for example, there
is an increasing need for identifying the location of specific
items in warehouses and cargo ships, or for tracking people
with special needs in residential environments, hospitals and
nursing homes [1]–[3]. As recognized in the IEEE standard
802.15.4a, ultra-wideband (UWB) impulse radio can play a
significant role in this context as it allows centimeter accuracy
in ranging and low-power and low-cost communications.
In this paper we concentrate on indoor geolocation with
UWB techniques. In particular we consider a network of
beacons deployed inside a building and emitting UWB signals.
Their location is known while the unknown coordinates of a
tag must be determined from a set of time-of-arrival (TOA)
measurements. We envisage a scenario in which the tag can
move in a specific service area. For example we want to
estimate the position of a robot as is moves inside a laboratory.
For simplicity we concentrate on two-dimensional positioning.
As is known, three beacons would be sufficient to accurately
solve the problem if the links between tag and beacons were
This work has been supported by the Ministerio de Educación y Ciencia (Spanish Government) and the European Union (FEDER) under grant
TEC2006-13514-C02-02/TCM, by the Ministerio de Ciencia e Innovación
(Spanish Government) through the Project COMONSENS of the ConsoliderIngenio 2010 Program, and by the MINERVA Project 2C/039.
all in line-of-sight (LOS) [4]. Unfortunately such a condition is
seldom met in a typical indoor multipath environment where
the propagation is likely to be non-LOS (NLOS), meaning
that the direct path (DP) is totally obstructed by large objects
or attenuated to such a degree that its energy falls below the
threshold detection. When this happens a ray arriving from a
reflection can be erroneously declared as the DP, so giving
rise to a range measurement with a positive bias [2], [5].
The simple least squares (LS) location estimator suffers
severe degradations in the presence of NLOS propagation
[6]- [7] and several strategies have been proposed to protect
location estimates from NLOS errors. They take advantage of
prior information that may be available in various forms: (a)
knowledge on the range error statistics; (b) knowledge of the
service area layout; (c) location fingerprints of the service area
[1], [8]. Point (a) refers to the probability density functions
(PDFs) of the range errors at any given spot under LOS and
NLOS conditions and to the prior probabilities of LOS/NLOS
occurrence. Mathematical models for the range errors in a
typical UWB indoor environment are given in [9], [10], [11].
Point (b) refers to the geometry of the service area (perimeter,
inside walls, etc.) and to the presence of metallic objects in the
building (elevator shafts, chambers etc.) that can obstruct the
direct connection between tag and beacon. NLOS mitigation
strategies may be ranked according to the amount of prior
information they exploit. The more information they use, the
more efficient they are.
A simple NLOS mitigation scheme is proposed in [12]. No
information on the range error statistics is assumed. Only the
range measurements and the beacon locations are available.
The key idea is that, if the NLOS measurements are relatively
few, they can be identified and discarded (or downgraded)
and the tag can be accurately located. In this spirit the
measurements from all the beacons are taken in combinations
of three or more and, for each combination, an intermediate LS
tag location estimate is computed. Its residual over the range
measurements provides an indicator of the estimate reliability:
the smaller the residual, the more trustworthy the estimate. A
similar NLOS mitigation scheme (but easier to implement) is
discussed in [13].
In reference [14] the NLOS measurements are seen as
data affected by gross errors, inconsistent with the LOS error
distribution. As such, they are regarded as outliers. Several
robust algorithms exist that can cope with a certain percentage
978-1-4244-3435-0/09/$25.00 ©2009 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings
of outliers [15]. Among them the authors choose the least
median of squares (LMedS) estimator.
In [16] a wideband CDMA cellular system is envisioned
in which location measurements are based on time-differenceof-arrivals (TDOAs) and angle-of-arrivals (AOAs). While the
scenario exhibits significant differences in error statistics compared with an indoor UWB environment, the methods in [16]
are easily adapted to the problem investigated in the present
paper.
Mapping methods referred to as location fingerprinting offer
another way to counteract NLOS measurements. Instances of
these techniques are reported in [1], [17]. Assuming that the
service area is relatively small, the idea is to perform premeasurements at each possible tag position and to compile
a database of range signatures. Each signature consists of a
vector providing the measured distances from tag to beacons.
A nearest-neighbor algorithm determines the signature in the
database with the minimum Euclidean distance to the actual
range metrics.
Hereafter we discuss an alternative mapping technique in
which a signature represents the beacon states at any tag
location (not the beacon ranges to the tag). In particular, the
nth signature component is either zero or one, depending on
whether the link from tag to the nth beacon is LOS or NLOS.
The signatures can be extracted from: (a) the infrastructure
of the service area, taking into account size and position of
the major metallic objects [10]; (b) field measurements when
necessary. This substantial reduction in fingerprint information
is compensated by providing range error statistics. As opposed
to [1], [17] where no error statistics are needed, we assume
that the conditional range error PDFs are known from field
measurements [9]- [10] and we exploit them, along with the
signature database, to implement a maximum likelihood (ML)
position estimator.
We believe that the major advantage of storing LOS/NLOS
conditions rather than TOA-measured distances is in the
robustness of the database against perturbations due to mobile
objects in the environment. For example, suppose that the
DP from the nth beacon is blocked by an elevator shaft, so
creating a shadow zone where the tag is only reached through
NLOS propagation. Also, assume that some people are moving
around the elevator. As long as the tag lies in the shadow
zone, the nth component of a LOS/NLOS signature is unity,
independently of the presence of people. On the contrary,
in a signature containing TOA-measured distances, the nth
component may vary, depending on the positions of the people.
The main goals of the paper are as follows. First, we
compare the performance of the the ML positioning estimator
with that of other localization algorithms available in literature.
Second, we establish the standing of all these methods with
respect to the conventional (non-Bayesian) Cramer-Rao bound
(CRB). Third, we assess the robustness of the ML estimator
against mismatches between assumed and actual range error
statistics.
The rest of the paper proceeds as follows. In the next Section
we overview the range error model in [9] and we compute
the corresponding CRB in positioning estimation. Section
III describes the ML estimator and introduces a specific
service area that is used as a study case to assess estimation
performance. Section IV summarizes the positioning schemes
in [12], [14], [16] while Section V provides simulation results
and comparisons. Finally, Section VI offers some conclusions.
II. R ANGE ERROR MODEL , LIKELIHOOD FUNCTION AND
CRB
The following notations are adopted throughout the paper.
The coordinates of the tag are indicated as z = (x, y)T . There
are N beacons located at known positions z n = (xn , yn )T
(n = 1, 2, · · · , N ) and the nth beacon provides a range
measurement rn which is modelled as
rn = dn + n
where dn is the true distance from the tag, i.e.,
dn = (x − xn )2 + (y − yn )2
(1)
(2)
while n is the range error. Several statistical models have
been suggested for n . In this paper we concentrate on the
one reported in [9] but the discussion is easily extended to the
more complex model in [10].
A. Range error model
The authors in [9] express n as the sum of two terms1
n = n,LOS + ξn n,N LOS
(3)
where n,LOS is caused by multipath propagation and is
always present in an indoor environment. In case of LOS
propagation it is the only source of error considered. Instead
n,N LOS corresponds to a situation in which the direct path
is undetected (NLOS). Finally ξn is a binary variable taking
value 0 with LOS and 1 with NLOS propagation.
The term n,LOS is modelled as n,LOS = γ log(1 + dn ),
where γ is a Gaussian random variable with mean mLOS and
standard deviation σLOS . It follows that n,LOS is Gaussian
with mean
(4)
mn,LOS = mLOS log(1 + dn )
and variance
2
2
= σLOS
[log(1 + dn )]2
σn,LOS
(5)
Similarly, n,N LOS is modelled as a Gaussian RV (independent of n,LOS ) but, as opposed to n,LOS , its mean and
variance do not depend on dn
mn,N LOS = mN LOS
2
σn,N
LOS
=
2
σN
LOS
(6)
(7)
Putting these facts together and bearing in mind (3), we
conclude that n is Gaussian with mean
mn = mLOS log(1 + dn ) + ξn mN LOS
(8)
1 For convenience we have renamed the original parameters in [9] and have
suppressed a subscript W indicating dependence on the signal bandwidth.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings
and variance
2
2
σn2 = σLOS
[log(1 + dn )]2 + ξn σN
LOS
(9)
Empirical values of mLOS , mN LOS , σLOS and σN LOS are
given in Table I of [9] as a function of the signal bandwidth.
From the above it follows that the PDF of the range
measurement rn can be written as
1
(rn − dn − mn )2
p(rn ; z) = √
(10)
exp −
2σn2
2πσn
Also, as the range measurements are all independent and
Gaussian distributed, the PDF of r = (r1 , r2 , · · · , rN )T is
given by
N
(rn − dn − mn )2
1
√
(11)
exp −
p(r; z) =
2σn2
2πσn
n=1
from which the log-likelihood function Λ(r; z) ln p(r; z)
is computed as
Λ(r; z) = −N ln
√
2π −
N
ln σn
n=1
−
N
(rn − dn − mn )2
2σn2
n=1
(12)
B. Cramer-Rao Bound
The CRB for the estimate ẑ = (x̂, ŷ)T of the tag location
is computed from the Fisher information matrix
∂Λ ∂Λ
2
E{( ∂Λ
∂x ) } E{ ∂x ∂y }
(13)
J
∂Λ
∂Λ 2
E{ ∂Λ
∂x ∂y } E{( ∂y ) }
where the expectations are taken with respect to p(r; z) and
the z value corresponds to the true tag location. The CRB is
expressed in terms of the diagonal elements of J−1 as
CRB(z) [J−1 ]11 + [J−1 ]22
(14)
The explicit expression of CRB(z) is easily derived from (14)
but is not given here for space limitations.
Two remarks are in order. The first is that (14) gives a
bound to the variance of any unbiased estimator when the
tag occupies the specific location z. In practice, however,
we are interested in an average bound that accounts for all
the possible tag locations. In the simulations shown later this
average is computed as the arithmetic mean of CRB(z) over
the random tag locations involved in the simulation.
The second remark stems from the observation that the
computation of CRB(z) ignores the plan of the service area,
i.e., its perimeter and possible internal obstacles. For example,
in the simulations we assume a rectangular service area limited
by external walls and containing two obstacles that the tag
cannot go through. Now, suppose the tag is close to an
obstacle. While this fact is irrelevant in the calculation of
CRB(z), it can be taken into account by a location estimator.
In particular, some of the algorithms considered later involve
the minimization of a cost function. As the minimization is
performed over the useful service area, it follows that the
locations inside the obstacles or outside the service area’s
perimeter are never tested nor selected as final estimates. In
conclusion, the CRB must be viewed as a performance bound
for estimators that ignore restrictions on z due to the building
layout. It need not be a bound for estimators that take such
restrictions into account.
It is worth mentioning that knowledge on z could be
incorporated by modelling z as a random variable with a PDF
p(z) that accounts for the constraints imposed by the service
area layout. This would pave the way to a number of Bayesian
bounds, including the Bayesian CRB [18]. Unfortunately the
computation of such bounds involves restrictive conditions on
p(r;z) that are not satisfied when z is constrained within finite
regions.
III. ML LOCATION ESTIMATOR
We propose an ML location estimator that exploits knowledge of the propagation conditions (LOS/NLOS) for every
beacon at every possible tag position. The specific setting
taken as a study case is a rectangular area of size 20 m × 10
m, with two internal square metallic objects of side 4 m. The
obstacles completely block the UWB signals, thus generating
shadow zones where NLOS conditions take place.
We assume that such conditions only occur when the
direct path is obstructed by the metallic obstacles2 . Thus,
since geometry and location of the obstacles are known, for
each beacon it can be established whether the propagation
is LOS or NLOS at any possible tag spot. Collecting these
data together generates a signature database for use in the
estimation process. Figure 1 shows the service area layout,
with red squares representing the metallic obstacles and the
dark blue area indicating the shadow zone for a beacon lying
in the origin of the coordinates.
The ML estimator maximizes the log-likelihood function
Λ(r; z) in (12) over all the possible tag locations z. Actually
the maximization is carried out over the nodes of a grid with
a mesh size of 10 cm. For each hypothesized tag location the
state (LOS/NLOS) of the beacons is provided by the signature
database. Maximization is performed over the useful service
area. Sub-areas the tag cannot reach are excluded. In particular,
the maximum of Λ(r; z) is looked for within the perimeter of
the service area and outside the metallic obstacles.
IV. OTHER POSITIONING SCHEMES
In this section we overview other localization methods with
the aim of comparing them with the ML algorithm.
A. Method in [12]
All the possible combinations of three or more range meaN
surements are considered. Their number is K = i=3 Ni .
Each combination is represented by the set of indices Sk
2 The scenario is readily extended to the case in which the NLOS conditions
are caused by a large distance from tag to beacon, such that the DP is
attenuated below the detection threshold.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings
C. Method in [16]
One of the methods in [16] assumes a PDF as in (10)
but does not exploit any precise knowledge of the state
(LOS/NLOS) of the tag-to-beacon links. The only informa(N LOS)
(z) the
tion available is of statistical nature. Call Pn
probability that the signal from the nth beacon undergoes
NLOS propagation at the tag location z. Obviously, the LOS
(LOS)
(N LOS)
(z) = 1 − Pn
(z). Then, the
probability is Pn
likelihood function of z is
10
shadow zone
y (m)
8
6
4
metallic
obstacle
metallic
obstacle
2
p(r; z) =
0
0
Fig. 1.
5
10
x (m)
15
20
+p(rn ; z)|ξn =1 Pn(N LOS) (z) (19)
(k = 1, 2, · · · , K) of the corresponding beacons. For each
set an intermediate LS estimate of z is computed as
z
(15)
where Res(z; Sk ) is the sum of the squared differences
between the range measurements and the distances of the
beacons to the hypothesized position z
2
[rn − z − z n ]
(16)
Res(z; Sk ) =
n∈Sk
Res(z; Sk ) is calculated at z = ẑ k and is normalized to the
number of elements in Sk
(ẑ k ; Sk )
Res(ẑ k ; Sk )
Size of Sk
(17)
The inverse of (ẑ k ; Sk ) is viewed as a measure of the
reliability of the estimate ẑ k and the final estimate of z is
taken as a weighted combination of the type
ẑ =
K
−1
k=1 ẑ k ((ẑ k ; Sk ))
K
−1
k=1 ((ẑ k ; Sk ))
p(rn ; z)|ξn =0 Pn(LOS) (z)
n=1
Service area and shadow zone for a beacon lying at (0,0).
ẑ k = arg min{Res(z; Sk )}
N
(18)
B. Method in [14]
The method in [14] is similar to that in [12] except that
only combinations of three beacons at a time are considered
and, instead of minimizing the sum of the squared residues,
the median of the residues is minimized.
The algorithm proceeds as follows. The number of beacon
combinations is now K = N3 . For a given combination Sk ,
an intermediate estimate of z is obtained as in (15). Next the
2
set of squared residues [rn − z − z n ] (n = 1, 2, · · · , N )
is computed and its median Mk is calculated, so that at the
end of the process we have a set of K intermediate estimates
ẑ k and the corresponding medians Mk . The final estimate is
taken as the intermediate estimate with the minimum median.
The authors in [16] look for the maximum of a modified
(N LOS)
(LOS)
(z) and Pn
(z) are
version of (19) in which Pn
(N LOS)
(LOS)
and Pn
independent of
replaced by quantities Pn
z. One difficulty in going further is to assign sensible values
(N LOS)
(LOS)
and Pn
. We explored two routes. In one
to Pn
(LOS)
case we put Pn
equal to the fraction of the service area
in LOS connection with the nth beacon. In the other we set
(LOS)
(N LOS)
= Pn
= 1/2. As experimental evidence shows
Pn
minimal difference in the results, in the following we only
report on the second case.
V. S IMULATION RESULTS
Simulations have been run with the specific service area
introduced in Section III. Here, 2000 tag positions are randomly chosen and, for each position, 10 different realizations
of the range measurements are generated. The ML estimator
(henceforth referred to as ML with signature database, shortly
MLSD) is compared with the algorithms in [12], [14] and
[16]. The parameters of the range error model are taken from
Table I of [9] for a signal bandwidth of 500 MHz and have
the following values: mLOS = 0.21 m, mN LOS = 1.62 m,
σLOS = 26.9 cm and σN LOS = 80.9 cm.
The performance of the algorithms is expressed as the root
mean square error (RMSE) in the position estimates. The
number of beacons N varies from 3 to 8. The first four are
placed at the corners of the service area’s perimeter. The 5th
and 6th are at the mid points of the longer walls. The 7th
and 8th are at the mid points of the shorter walls. The CRB
is computed as the square root of the arithmetic average of
CRB(z) over the set of random tag locations tested in the
simulation. All the algorithms exploit knowledge of the service
area plan in the way mentioned in Section II-B.
Figure 2 illustrates performance comparisons. As expected,
MLSD and the algorithm in [16] have superior performance
as they exploit range error statistics. On the other hand MLSD
outperforms the algorithm in [16] because it takes advantage
of the signature database and, in consequence, has precise
information on the propagation conditions. Finally, MLSD
is occasionally better than the CRB, in particular when the
number of beacons is small. This confirms that CRB is a bound
only for estimators with no knowledge of the service area.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings
3.5
Ref. [12]
3
Ref. [14]
Ref. [16]
RMSE (m)
2.5
MLSD
CRB
2
1.5
1
0.5
0
3
4
5
6
7
8
N
Fig. 2. RMSE of the positioning error for different location algorithms as a
function of the number of beacons.
2.5
Ref. [16] no mismatch
Ref. [16] with mismatch
2
MLSD no mismatch
RMSE (m)
MLSD with mismatch
CRB
1.5
1
0.5
0
3
4
5
6
7
8
N
Fig. 3. RMSE of the positioning error in the presence of a mismatch between
true and assumed parameters.
As perfect knowledge of the range error statistics is unrealistic, the MLSD performance has been checked in the
presence of a mismatch between true and assumed values of
the parameters in the range error model. Figure 3 compares
MLSD with the method in [16] with and without mismatch.
The mismatch consists of taking mN LOS and σN LOS 30%
greater than their true values. As is seen, the impact of the
mismatch is marginal.
VI. C ONCLUSIONS
We have investigated a TOA-based geolocation algorithm
that exploits information on the range error statistics and the
layout of the service area. In particular it makes use of a
signature database where the the propagation conditions at
every possible tag spot are recorded. The database is in the
spirit of the standard location fingerprinting approach but is
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simpler to implement and store. Each signature provides in fact
just the states (LOS/NLOS) of the beacons, not the detailed
range measurements.
The performance of the proposed algorithm has been compared with that of other methods available in literature. As expected, a tradeoff exists between performance and complexity.
The simplest algorithms have knowledge neither of the range
error statistics nor of the environment layout and have the
worst performance. The MLSD is the most complex and, as
expected, has the best performance. With a sufficient number
of beacons its performance achieves the CRB. Occasionally
it can be better. An explanation of this apparent paradox has
been given.
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