1
Accuracy of RSS-based Centroid Localization
Algorithms in Indoor Environment
Paolo Pivato, Graduate Student Member, IEEE, Luigi Palopoli, Member, IEEE,
and Dario Petri, Fellow, IEEE
Abstract—In this paper we analyse the accuracy of
Global Positioning System (GPS) solves many localiza-
indoor localization measurement based on a Wireless Sen-
tion problems outdoor, where the devices can receive
sor Network (WSN). The position estimation procedure is
the signals coming from the satellites, but the system
based on the Received Signal Strength (RSS) measurements
collected in a real indoor environment. Two different
is hardly usable indoor. Moreover, the wireless nodes
classes of low computational effort algorithms based on
present some advantages in terms of system miniatur-
the centroid concept are considered, namely the Weighted
ization, scalability, quick and easy network development,
Centroid Localization (WCL) method and the Relative
cost and reduced energy consumption.
Span Exponential Weighted Localization (REWL) method.
Most of the proposed localization solutions rely on the
In particular, different sources of measurement uncertainty
are analysed by means of theoretical simulations and
experimental results.
Index Terms—Wireless Sensor Networks, localization,
centroid algorithm, propagation model.
Received Signal Strength (RSS) measurements. In fact,
the RSS can be used to estimate the distance between
the unknown node (called target node) and a number of
reference nodes with known coordinates (called anchors
or beacons). The location of the target node is then
I. I NTRODUCTION
determined by multilateration [5].
In the last few years the use of Wireless Sensor
Unfortunately, some studies showed the large vari-
Networks (WSNs) has become commonplace in different
ability of the RSS, due to the degrading effects of
fields, ranging from environmental monitoring in harsh
reflections, shadowing and fading of the radio waves [6],
and hostile areas to precision agriculture, from security
[7], [8], [9]. As a result, localization methods using the
and surveillance to medicine and industry and – more
RSS are affected by large errors and lack of accuracy.
recently – home automation and assisted living [1], [2],
However, RSS-based techniques remain an appealing
[3], [4].
approach [10]. This is mainly due to the fact that RSS
A recent indoor application for Wireless Sensor Net-
measurements can be obtained with minimal effort and
works is the localization of moving targets. This ap-
do not require extra circuitry, with remarkable savings
plication is motivated primarily by the low cost of
in cost and energy consumption of the sensor node. In
this solution and the lack of effective positioning and
fact, most of the WSN transceiver chips have a built-in
tracking systems working inside buildings. Indeed, the
Received Signal Strength Indicator (RSSI), that provides
June 15, 2011
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2
RSS measurement without any extra cost.
In the literature there exist many works about RSSbased outdoor localization, most of which analyse the
allows us to provide an insightful interpretation of the
limitations of the approach, which can be useful for
future developments.
problem through simulations and experimental data [11],
The remainder of this paper, which extends what
[12]. Conversely, to the best of our knowledge, less at-
has been presented in [15], is organized as follows.
tention has been given to RSS-based indoor localization.
Section II contains the related work. Section III describes
The available results are obtained mainly by means of
the experimental framework, detailing the measurement
simulations. The models adopted in these simulations
scenario. The characterization of the indoor propagation
use either the same or different path-loss exponents for
channel is presented in Section IV, dealing with the
the each link but they usually do not account for the
adopted channel model and the related channel param-
different non-idealities of radio transmissions in indoor
eters estimation. In Section V, after a brief overview
environments [13]. In fact, there is a lack of experimental
on the centroid localization approach, we analyse the
data, which are necessary to adequately validate the
algorithms used in our experiments mainly on the base
proposed solutions [14].
of both meaningful simulation and experimental results.
The aim of our work is to investigate the accuracy
In Section VI we conclude the paper.
of RSS-based indoor localization. Thus, we propose a
deep analysis of the impact on the measurement accuracy
II. R ELATED W ORK
of different disturbing phenomena such as reflections,
As stated above, the vast majority of studies on RSS-
diffraction and scattering, and the influence of the error
based localization has been performed in outdoor envi-
introduced by low computational complexity localization
ronment. Conversely, the effects of spurious disturbances
algorithms recently proposed in the literature. Although
on the accuracy of RSS-based indoor localization have
the study is carried out for particular classes of algo-
received little attention in the literature. Moreover, a
rithms, we believe the proposed methodology is by a
small number of published papers is based on experi-
large extent generalizable.
mental results obtained from real indoor test beds.
At first, we consider the log-distance path loss model,
There exist several algorithms that can be used to
widely used for the analysis of indoor wireless channels,
determine the position of a target through RSS mea-
and characterize it with respect to a specific measure-
surements – some of them are geometric methods, like
ment context. In this case our goal is the identification
Lateration or Min-Max, whereas some others are based
of the channel parameters by applying linear regression
on statistical approaches, like Maximum Likelihood.
to a significant set of measurements.
RADAR [14] provided one of the first experimental
Besides the characterization of the adopted channel
works on indoor localization using IEEE 802.11 radios
propagation model, we carry on analysing the accuracy
for wireless LAN. It was based on a RSS mapping tech-
of the so called Weighted Centroid Localization (WCL)
nique, in which a set of RSS measurements with known
and Relative Span Exponential Weighted Localization
coordinates was collected a priori in the observation
(REWL) algorithms. The proposed metrological charac-
area. Then the location of a target node was computed
terization of the RSS-based indoor localization system
on-line by searching for the RSS values nearest to the
June 15, 2011
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3
0
current measurement in terms of some defined metrics.
Test points
Anchors
50
100
The average location error reported by RADAR was
150
The authors of [16] tested a RSS-based outdoor lo-
Distance [cm]
200
approximately 3 m.
1
4
7
10
2
5
8
11
3
6
9
12
250
300
350
400
calization methodology exploiting the Minimum Least
450
500
Squares algorithm, after modelling the propagation chan-
550
600
nel. The deployment of anchors had density of one node
0
50
100
150
200
250
300
350
400
Distance [cm]
450
500
550
600
650
700
over 25 m2 on an area of 500 m2 . The average distance
Fig. 1.
Location test points within the room.
error obtained was about 3 m.
In [17] an extensive indoor RSS measurement campaign was carried out in order to tune the parameters
equipped with the IEEE 802.15.4 compliant Chipcon
of the assumed channel model. Then, the collected RSS
CC2420 radio module. The antenna is a 2.4 GHz planar
data have been used off-line as inputs for two localization
inverted-F (PIFA) printed on the dielectric substrate of
methods, that is the Min-Max and the Bayesian filtering
the circuit board, feed by a coplanar waveguide (CPW).
algorithms. A distance error of the order of 5 m and of
It is located on the border of one of the short sides of
2 m was achieved for the two algorithms respectively.
the node. We used this type of platform because of its
In order to avoid the creation of RSS maps, complicate probability models or high computational effort
remarkable popularity in the academic community and
the wealth of software available.
algorithms, the authors of [18] proposed an approxi-
The experimental test-bed was a rectangular room of
mated indoor localization based on a weighted centroid
size 5.8 m × 4 m, furnished with a couch, a bookcase put
approach [19] combined with RSS measurements in
on the wall, a table and a settle. Therefore the considered
an IEEE 802.15.4 sensor network . The weights were
environment well emulated a real ambient living room.
defined as inversely proportional to the RSS values
The system infrastructure was composed of 1 mobile
measured between the target and each anchor node. The
node (the target), 1 base station node (the sink) and 12
solution was tested in a square room with side length
fixed nodes (the anchors). The mobile node was set on
of 3 m, using 4 anchor nodes displaced at the corners
a dielectric support 50 cm high and stood upright. The
and 1 target node located in 13 different positions. The
linoleum floor was divided into 68 test point located on
obtained relative localization error varied between 7.8%
a grid with a resolution of 50 cm. They are represented
and 26%. The weighted centroid approach is one of the
with circles in Fig. 1. The 12 fixed nodes were hanged
localization algorithm examined in the next sections.
on the ceiling, 260 cm high. They are represented with
squares in Fig. 1 and are labelled with a number. The
III. T HE MEASUREMENT CONTEXT
displacement of the fixed nodes formed a rectangular
The sensing platform used in our experimental set-
grid covering all the monitored environment. In the
up was a TelosB wireless sensor node produced by
test bed deployment phase we experimented different
Crossbow Inc. It is an open-source wireless node,
placements of the nodes on the roof and on the ceiling,
based on Texas Instruments MSP430 microprocessor and
in order to verify the existence of an optimal cover-
June 15, 2011
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4
age pattern assuring the communications among all the
± 1 dBm.
nodes. As a matter of fact, we verified that the radio link
was good in any configurations. We tested also several
relative antenna orientation between target and anchor
IV. C HANNEL CHARACTERIZATION
A. Indoor radio channel propagation model
pairs, without noticing remarkable differences in the RSS
values measured. Then we arranged the mobile node
and the fixed nodes so as their antennas were mutually
oriented one towards the others.
The system performs the RSS measurements from the
messages exchanged between the mobile node and each
anchor node. The low level communications between
the nodes are carried out by the services provided by
TinyOS. The mobile node sends a ping to the 12 anchor
nodes requesting a response. Then each anchor node
replies in turn with a message containing the node ID
and the transmitted power level. When the mobile node
receives a reply message, it measures the signal strength
through the built-in Received Signal Strength Indicator
(RSSI) and reads the other information contained in the
message. The process is repeated several times. All the
measured RSS data are sent to the base station, which
is connected to a laptop personal computer. Finally the
collected RSS values are processed and analyzed to
extract statistical information and to evaluate the result
of the localization algorithms described in Sec. V.
In order to minimise the exchange of messages and
the energy consumption, data collection and processing
should be performed on the sensor node. We made
a different choice because using a personal computer
allows for an exhaustive statistical analysis, which is the
main purpose of this paper, and an easier implementation
of different localization algorithms.
In order to characterize the indoor propagation channel
we assume that the signal strength follows the logdistance shadowing path loss model proposed by Rappaport [20] . This propagation model is widely used in
indoor wireless link budget and is given by:
d
RSS(d) = RSS(d0 ) − 10η log
+w
d0
(1)
where d is the transmitter-receiver distance, d0 is a
reference distance, η is the path loss exponent – the rate
at which the signal decays – and w is a space-stationary
zero-mean Gaussian random variable with variance σw 2 .
RSS(d) is the received signal strength and RSS(d0 )
is the signal received at the reference distance (both in
dBm). An alternative formulation for equation (1) is:
d
+w
(2)
RSS(d) = Ptx + K − 10η log
d0
where Ptx is the transmitted power (in dBm) and K is
the attenuation factor at the reference distance d0 .
Let d denote the true distance between mobile and
anchor node, and dˆ denote the distance estimated by
inverting (2) and using the measured value for the
RSS(d). Under the assumptions made in (1), applying
the law of uncertainty propagation [21] to (2), we obtain
that the distance estimator is biased and its relative bias
is given by:
b[ dˆ]
σ2
' 0.03 w2
d
η
(3)
while the relative standard deviation results:
each of the 68 test point locations, resulting in a total
σ[ dˆ]
σw
' 0.23
d
η
amount of 24480 RSS values collected. The achieved
Both these formulas have been validated by simulations
measurement repeatability was quite high, always within
not reported here for the sake of conciseness.
The measuring process was repeated 30 times for
June 15, 2011
(4)
DRAFT
5
−60
In particular, from (3) and (4) we have:
−63
(a)
−66
(5)
Considering η ranging between 1 and 4, as commonly
occurs in practice, expression (5) returns values in
−69
−72
−75
RSS [dBm]
b[ dˆ]
σw
' 0.11
ˆ
η
σ[ d ]
(0.03 σw , 0.11 σw ). Thus the distance estimator bias
−78
−81
−84
−87
−90
could be significant. For instance, for η = 2.3 and
−93
−96
σw = 6.1 dB, as occurs in our experimental results, we
−99
have b[ dˆ]/σ[ dˆ] ' 30%.
−102
200
300
400
Log−distance [cm]
500
600
700
−60
Moreover, according to (4), the relative standard de-
−63
(b)
−66
viation of the estimated distance increases of about 5%
−69
to 20% for each dBm of RSS standard uncertainty. Thus
−72
the model (1) is very sensitive to RSS uncertainty. To
RSS [dBm]
−75
we can conclude that any distance estimator based on
−78
−81
−84
the best of our knowledge this interesting result has not
−87
been reported in the literature before.
−93
−90
−96
−99
B. Channel parameters estimation
−102
200
300
400
Log−distance [cm]
500
600
700
The data set of RSS measurements, collected as described in Sec. III, was used to estimate the channel
parameters K and η. The transmission power Ptx was set
Fig. 2.
RSS measurements and related channel models considering
(a) the 4 anchors in the corners of the room, and (b) all the 12 anchors
reported in Fig. 1.
to −25 dBm in all nodes. This value was chosen because
it was the minimum available power level ensuring a
complete coverage of the room, thus allowing a good
to 0.42. Moreover, the standard deviation of the RSS
balance between coverage and energy consumption. The
measurements was σw = 6.1 dB, leading to a relative
reference distance d0 , which is related to the antenna far
bias on the estimated distance of 17%, and a relative
field region, was set equal to 10 cm.
standard deviation of 60%, as provided by (3) and (4)
The channel parameters were firstly estimated by ap-
respectively.
plying the linear Least Squares Method (LSM) to all the
As a second step, we analyzed the path loss model
24480 RSS values collected, obtaining K = −17.2 dB
considering each anchor node individually, while the
and η = 2.3. The achieved result is shown in Fig. 2(b),
mobile node is still moved in each of the 68 test points.
where the dots represent RSS measurements and the
The channel parameters K and η were still estimated by
solid line refers to the theoretical path loss model de-
using the LSM for each data set of 2040 collected RSS
rived by the linear regression. In order to determine if
values.
the data well fit the derived parameters we computed
Then we considered the 4 anchors (# 1, 3, 10, 12)
also the regression coefficient ρ, which resulted equal
located in the corners of the room and the 6 anchors
June 15, 2011
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6
TABLE I
configuration given by the 4 anchors in the corners and
L OG N ORMAL C HANNEL PARAMETERS
the 2 anchors (# 5, 8) in the middle of the room. We
collected 8160 and 12240 RSS values respectively and,
Anchors #
as described before, we used these values to estimate the
η
K [dB]
σw [dB]
ρ
1 − 12
2.3
−17.2
6.1
0.42
1
3.1
−6.8
5.1
0.58
2
2.2
−19.0
4.4
0.48
3
3.3
−0.4
6.4
0.53
4
0.5
−46.2
5.9
0.15
channel parameters η and K through the LSM.
Table I lists the log-distance channel parameters estimated in each case, together with the error standard
deviation and the related regression coefficient. As we
5
3.6
0.2
6.1
0.44
can see, the channel model error standard deviation is
6
2.4
−17
6.8
0.35
nearly constant for all the considered sets of anchors,
7
1.1
−36.7
5.7
0.17
while the resulting path loss exponents is quite changing.
8
1.9
−25.1
7.5
0.24
In particular, considering the channels related to each
9
2.9
−8.4
7.5
0.42
10
2.8
−9.3
5.9
0.50
11
2.0
−22.7
5.3
0.45
single anchor, it ranges from a minimum of 0.5 for the
anchor #4 to a maximum of 4.1 for the anchor #12. The
12
4.1
11.7
6.1
0.63
1, 3, 10, 12
3.4
0
6.0
0.56
1, 3, 5, 8, 10, 12
2.7
−11.3
6.4
0.48
corresponding regression coefficients behave in a similar
way, ranging from 0.15 to 0.63 respectively. From a
distance estimation point of view, these anchors represent
the worst and the best case respectively. Indeed a higher
value of the regression coefficient means that the data
received by the related anchor carry more information
in Fig. 3 for the case of 4 and 12 anchors respectively.
about the unknown distance. It is worth noticing that the
Both distributions show a behaviour far from Gaussian,
4 anchors located in the corners of the room provided
conversely to the assumption commonly made in the
the highest value of the regression coefficient, thus they
literature [20].
can be considered as the more informative ones. To the
Moreover, we considered the RSS error histograms
best of our knowledge, no previous work has reported
obtained for different distance intervals of equal ampli-
remarkable differences of the channel parameters when
tude (i.e. 50 cm and 100 cm). The obtained histograms
considering each anchor node singularly. However, given
noticeably differ each other, suggesting a non-stationary
the different locations of the anchors, the received signal
behaviour of the RSS error with respect to the distance,
is expected to be not affected by the same reflections,
unlike we would expect from the model suggested
fading and multi-path interference, thus leading to a
in [20].
significant difference in the channel models. It is worth
noticing that similar observations on the irregularity
V. L OCALIZATION ALGORITHMS
of the wireless communication channel were presented
The localization problem can shortly be for-
in [22], in which an extension to the isotropic radio
malised as follows. Consider a set of nodes N =
model for outdoor environment was proposed.
{A1 , A2 , . . . , An }, each one with a fixed and known
The histograms hw of the RSS error w are depicted
June 15, 2011
position (hence the name anchors). In this paper we are
DRAFT
7
30
anchor distance estimations obtained through the RSS
measurements, and range-free, which determine the po-
(a)
25
sition of the target node without performing distance
20
estimation [5], [23].
hw 15
In the following subsections we analyse two different
methods, respectively the Weighted Centroid Localiza-
10
tion (WCL) algorithm and the Relative Span Exponential
5
Weighted Localization (REWL) algorithm. The former
0
−20
−15
−10
−5
0
5
10
15
belongs to the class of range-based solutions, while the
w
latter is a range-free approach. Both algorithms are char-
120
100
acterized by a low computational effort. This, combined
(b)
with low transmission power, allows to significantly limit
80
hw
the node energy consumption.
60
A. Weighted Centroid Localization
40
The WCL algorithm is based on the so called Cen20
troid Localization (CL) proposed in [24]. This solution
0
−30
−25
−20
−15
−10
−5
0
5
10
15
w
approximates the location p of the target node by calculating the centroid of the coordinates ai = (xi , yi ) of
RSS error histograms obtained considering (a) the 4 anchors
the so called visible anchor nodes, that is the nodes for
in the corners of the room, and (b) all the 12 anchors reported in Fig. 1.
which a communication has been established during the
Fig. 3.
measurement. More specifically, the estimated position
of the target node is given by:
tion, since the third dimension usually is not of primary
m
1 X
·
ai
p̂ =
m i=1
interest in indoor environment. Thus the position of an
where m is the cardinality of the subset N of visible
anchor is a 2-tuple ai = (xi , yi ), where xi and yi are
anchors. It is worth noticing that when the target node
evaluated with respect to an origin O. Let p denote the
communicates with all the anchors, that is all anchors
position of a target moving node of unknown coordinates
are visible, the centroid results the centre of the anchors
(x, y), and RSSi denote the measured intensity of the
coordinates.
working with the common assumption of 2-D localiza-
(6)
signal strength from anchor ai experienced by the target.
Notice that the CL approach assumes all the visible
The goal of a RSS-based localization algorithm is to
anchors equally near the target node. Since this as-
provide an estimate p̂ = (x̂, ŷ) of the position p given
sumption is most likely not satisfied in practice, in [19]
the vector [RSS1 , RSS2 , . . . , RSSn ].
the introduction of a function which assigns a greater
We can recognize two class of RSS-based localization
weight to the anchors closest to the target was proposed.
algorithms: range-based, which use several target-to-
The result is the WCL algorithm, which estimates the
June 15, 2011
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8
position of the target node as:
(a)
n
X
(dˆi−g · ai )
i=1
n
X
(7)
(dˆi−g )
140
80
120
70
100
60
ealg [cm]
p̂ =
90
i=1
80
60
50
40
40
20
30
where dˆi is the distance between the target and the
0
536
490
20
445
anchor ai , estimated through the RSSi of the visible
399
353
307
262
anchors. The exponent g > 0 determines the weight
216
[cm]
170
217
264
311
358
405
451
498
545
592
639
10
[cm]
of the contribution of each anchor. If g = 0 then p̂ is
(b)
120
simply the sample mean of the ai and the WCL reduces
140
100
120
to the CL approach. Increasing the value of g causes the
ealg [cm]
anchors to reduce the range of their “attraction field”
100
80
80
60
60
40
with respect to the mobile node, thus increasing the
20
40
0
536
490
relative weight of the nearest anchors.
445
399
The plots in Fig. 4 show the results of simulations
353
307
262
running the WCL algorithm with 4, 6, and 12 anchors
216
[cm]
170
217
264
311
358
405
451
498
545
592
20
[cm]
(c)
positioned as described in Section III and IV and choos-
70
ing g = 1.8. Each surface represents the algorithm error
140
ealg in terms of the distance between the true position
100
60
120
ealg [cm]
and the position estimated using the WCL algorithm with
639
50
80
40
60
40
a grid resolution of 5 cm. The algorithm inputs were
the true distances between the target and the anchors,
30
20
0
536
490
20
445
399
calculated from their known coordinates. As we can see,
353
307
262
[cm]
passing from 4 to 12 anchors the error globally decreases
216
170
217
264
311
358
405
451
498
545
592
639
10
[cm]
while it is drastically reduced in the proximity of anchor
Fig. 4. WCL distance error for g = 1.8 considering (a) the 4 anchors
locations. Furthermore, we can observe that using 6
anchors the error tends to increase with respect to the 4
in the corners of the room, (b) 6 anchors, and (c) all the 12 anchors
reported in Fig.1.
anchors configuration. This is likely due to the presence
of the 2 additional anchors placed in the centre of the
room, which increases the centre clustering behaviour
Otherwise, the noise affects significantly areas with small
featured by algorithms based on the CL approach.
error values, usually located near the centre of the room,
Similar error surfaces were obtained assuming that an
Additive White Gaussian Noise (AWGN) with different
so increasing the centre clustering behaviour of the
algorithm.
values of standard deviation affects the RSS measure-
Table II and III respectively show the mean and the
ments. We noticed that, on average, the maximum values
root mean square (RMS) values of the total distance
of the algorithm error are little sensitive to the noise.
error in the estimated position etot , achieved by running
June 15, 2011
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9
the WCL algorithm on the experimental data, with 4,
TABLE II
WCL A LGORITHM M EAN D ISTANCE E RROR
6 and 12 anchors, and considering different values of
the exponent g within the range (1.0, 1.8). In particular,
the algorithm inputs were the distances between the
target and each anchor, estimated by inverting (2) and
4 anchors
using the RSS values measured in each of the 68 test
points, as described in Section III and IV. The same
6 anchors
Tables also summarize the mean and the RMS of the
algorithm distance error ealg and the experimental noise
distance error ew determined for the same sets of anchors
and values of the exponent g. The experimental noise
12 anchors
g
1.0
1.2
1.4
1.6
1.8
µtot [cm]
113
113
116
120
124
µalg [cm]
67
52
42
38
40
µw [cm]
78
89
99
107
114
µtot [cm]
116
112
110
109
110
µalg [cm]
94
83
75
69
64
µw [cm]
55
63
71
78
84
µtot [cm]
123
117
114
111
109
µalg [cm]
87
73
60
50
41
µw [cm]
50
59
68
76
83
distance error was obtained as the difference between
the position estimates determined by running the WCL
algorithm using both the true and the estimated distances.
Notice that this latter error is due to the noise component
w in (2). In particular, Table II and III show that, on
first approximation, the mean µalg and the RM Salg of
the algorithm error depend little on the anchor number
and decrease with a raising exponent g. This behaviour
is partially compensated by the noise error, whose mean
µw and RM Sw values increase with an increasing value
of g and decrease with a raising number of anchors. As a
result, the effect of the algorithm error on the total error
in negligible for the 4 anchors, while it counts for 6 and
decreasing of the regression coefficient, using more
anchors reduces the effect of noise by a factor which is
smaller than the square root of the number of anchors.
In any case, expression (8) can provide some useful
hints on the expected effect of noise in different system
configurations with changing number of anchors.
Fig. 5 depicts the cumulative histograms H of the
WCL estimation errors considering the 4 anchors in
the corners of the room and all the 12 anchors. As
expected the median total estimation error is about 1 m
using both 4 or 12 anchors. Indeed the 4 anchors in the
12 anchors.
Considering any two different anchor configurations
(e.g. 4 and 12 anchors) it is interesting to note that the
ratio of the correspondent mean µw and root mean square
RSSw values of the noise error reported in Table II
and III tend to be inversely proportional to the square
corners carry most of the information about the unknown
distance, as shown in Section IV. Notice also that the use
of 12 anchors, although does not produce a significant
reduction of the average estimation error, has a beneficial
effect on the maximum error.
root of the product between the number of anchors n
and the regression coefficient ρ given in Table I. That
B. Relative Span Exponential Weighted Localization
The REWL is a RSS-based range-free localization
is:
r
µw ∝
1
n·ρ
r
RM Sw ∝
1
n·ρ
(8)
Since a growing number of anchors results in a
June 15, 2011
algorithm recently proposed in the literature [25]. This
algorithm is inspired by the the WCL method. The
weights are obtained by the relative placement of the
DRAFT
10
TABLE III
1
WCL A LGORITHM RMS D ISTANCE E RROR
0.9
g
1.0
1.2
1.4
1.6
1.8
RM Stot [cm]
129
130
134
139
144
0.8
0.7
4 anchors
6 anchors
12 anchors
Halg
Htot
Hnoise
0.6
RM Salg [cm]
73
60
51
47
46
RM Sw [cm]
92
105
117
128
137
RM Stot [cm]
129
126
125
125
126
0.3
RM Salg [cm]
101
90
82
75
71
0.2
RM Sw [cm]
64
74
83
90
97
0.1
RM Stot [cm]
134
130
126
124
122
RM Salg [cm]
95
80
67
55
46
RM Sw [cm]
57
67
77
85
92
H 0.5
0.4
(a)
0
0
50
100
150
200
250
300
350
Error [cm]
1
0.9
0.8
0.7
Halg Hnoise
Htot
0.6
anchor RSS value within the span of all the RSS values
H 0.5
measured by the target node. In the estimation of the
0.4
target position, the REWL algorithm favours the anchors
0.3
which exhibits higher RSS values and therefore are likely
0.2
(b)
0.1
to be closer to the target node. This is obtained using a
0
0
weighting factor λ according the exponentially moving
50
100
150
200
250
300
350
Error [cm]
average concept [25]. The estimated target node position
Fig. 5.
is given by [25]:
n
X
p̂ =
Cumulative histograms for the WCL localization algorithm
errors considering (a) the 4 anchors in the corners of the room, and
(b) all the 12 anchors reported in Fig. 1.
[(1 − λ)RSSmax −RSSi × ai ]
i=1
n
X
(9)
RSSmax −RSSi
(1 − λ)
i=1
the REWL algorithm for λ = 0.15, with 4, 6, and 12
where RSSmax is the maximum value in the span of
anchors, with a grid resolution of 5 cm. The inputs of
the RSS values measured by the target node. Suggested
the algorithm were the theoretical RSS values that might
values for λ, experimentally determined, range from 0.10
be measured by the target in each point of the grid
to 0.20 [25].
in the absence of noise. These values were evaluated
In fact, assuming the path loss model (1), it can be
using the path loss model (2), considering for each set of
shown that in case of no noise the REWL algorithm
anchors the related channel parameters η and K reported
reduces to expression (7), where g ranges between 0.5
in Table I, and the true distances d between the target and
and 3.9 when η and λ assume values in (1, 4) and
the anchors, calculated from their known coordinates.
(0.1, 0.2) intervals respectively (see Appendix).
The transmission power was assumed to be Ptx =
Fig. 6 shows the surfaces representing the algorithm
−25 dB and the reference distance was d0 = 10 cm.
distance error ealg , obtained by running simulations of
Clearly, the error tends to decrease with the increasing
June 15, 2011
DRAFT
11
(a)
90
As previously highlighted for the WCL, the 6 anchors
80
configuration exhibits higher algorithm error values with
ealg [cm]
140
120
70
100
60
Moreover, we analysed also the error surfaces obtained
80
50
60
40
20
0
536
490
445
399
353
307
262
216
[cm]
170
217
264
311
358
405
451
498
545
592
40
by assuming the RSS values affected by AWGN. Assess-
30
ment on these are similar to the ones drawn for the WCL
20
algorithm.
639
10
120
estimated position etot , achieved by running the REWL
algorithm with 4, 6, and 12 anchors, and considering
140
100
120
λ = 0.10, λ = 0.15, and λ = 0.20. The algorithm inputs
100
ealg [cm]
Tables IV and V list the mean and the root mean
square (RMS) values of the total distance error in the
[cm]
(b)
80
80
were the RSS values measured by the target in each of
60
40
60
the 68 test point, as described in Section III and IV. The
40
same Tables report also the mean and the RMS values
20
of the algorithm distance error ealg and the experimental
20
0
536
490
445
399
353
307
262
216
[cm]
170
217
264
311
358
405
451
498
545
592
639
noise distance error ew . This latter error is determined
[cm]
as the difference between the position estimates achieved
(c)
140
80
by running the REWL algorithm on both the theoretical
70
and the measured RSS values. As we can see, the mean
60
µalg and RM Salg of the algorithm error depend on the
120
100
ealg [cm]
respect to the 4 anchors configuration.
80
50
60
number of anchors but they feature a different behaviour
40
40
as the weighting factor λ changes. In fact, with 4 anchors
20
30
they increase passing from λ = 0.10 to λ = 0.20,
0
536
490
20
445
399
353
307
262
[cm]
216
170
217
264
311
358
405
451
498
545
592
639
10
whereas with 6 and 12 anchors they decrease significantly when λ increases. The mean µw and RM Sw of
[cm]
the noise error decrease with an increasing number of
REWL distance error for λ = 0.15 considering (a) the 4
anchors, while they decrease with a raising value of λ.
anchors in the corners of the room, (b) 6 anchors, and (c) all the 12
As regard the mean µtot and RM Stot of the total error,
Fig. 6.
anchors reported in Fig.1
they depend little on both the anchors number and the
weighting factor λ.
Fig. 7 shows the cumulative histograms of the local-
number of anchors. The shape of the error surfaces is
ization errors resulting when running the REWL algo-
substantially similar to the one of the WCL algorithm,
rithm with λ = 0.15 and for the 4 anchors on the corners
with the exception of the 4 anchors configuration which
of the room and for all 12 anchor nodes respectively.
features an higher error around the centre of the grid.
Considerations similar to those expressed for the WCL
June 15, 2011
DRAFT
12
TABLE IV
1
REWL A LGORITHM M EAN D ISTANCE E RROR
0.9
λ
0.10
0.15
0.20
µtot [cm]
111
114
125
0.8
0.7
4 anchors
6 anchors
12 anchors
Halg
Hnoise
Htot
0.6
µalg [cm]
38
58
77
µw [cm]
95
117
130
µtot [cm]
116
106
105
µalg [cm]
82
62
59
0.2
µw [cm]
56
76
91
0.1
µtot [cm]
127
113
105
µalg [cm]
83
49
30
µw [cm]
51
75
92
H
0.5
0.4
0.3
(a)
0
0
50
100
150
200
250
300
350
Error [cm]
1
0.9
0.8
0.7
Halg
TABLE V
Hnoise
Htot
0.6
REWL A LGORITHM RMS D ISTANCE E RROR
H
0.5
0.4
λ
0.10
0.15
0.20
RM Stot [cm]
127
131
144
RM Salg [cm]
47
59
82
RM Sw [cm]
111
138
157
RM Stot [cm]
130
121
120
RM Salg [cm]
89
69
65
RM Sw [cm]
69
91
107
RM Stot [cm]
139
126
120
RM Salg [cm]
91
54
33
RM Sw [cm]
61
87
104
0.3
4 anchors
6 anchors
12 anchors
0.2
(b)
0.1
0
0
50
100
150
200
250
300
350
Error [cm]
Fig. 7.
Cumulative histograms for the REWL localization algorithm
errors for λ = 0.15 considering (a) the 4 anchors in the corners of the
room, and (b) all the 12 anchors reported in Fig. 1.
effort algorithms based on the centroid concept, called
WCL and REWL, was analysed. The measurement system was deployed in a real indoor environment and, by
algorithm can be done. In particular, for the 12 anchors
on-line running of the adopted algorithms, it provided
configuration the maximum total distance error etot is
the estimated position of a target node in different test
limited, while the algorithm distance error ealg is not
points inside the observation field.
negligible.
At first we characterized the wireless propagation
channel using a model largely adopted in the literature.
VI. C ONCLUSION
We showed that this model is affected by a quite high
In this paper we analysed the accuracy of indoor
relative bias and standard uncertainty. Thus, we might
localization based on RSS measurements collected by a
expect that the error increases as the distance from the
WSN. The accuracy of two classes of low computational
anchors increases. This is most likely due to the severe
June 15, 2011
DRAFT
13
propagation conditions of the indoor radio channel, that
A PPENDIX
is affected by reflections of the signals against the walls,
E QUIVALENCE BETWEEN REWL AND WCL
the floor and the ceiling.
Moreover, we noticed that the information carried by
each anchor strongly depends on the anchor position.
Thus, for all the considered localization algorithms, a
growth of the anchors number does not necessarily
improve the measurement accuracy, conversely to what
we would expect. Even tough the error introduced by the
analysed algorithms is not negligible, the measurement
uncertainty is mainly due to the noise associated to the
wireless propagation channel model. In any case, the
ALGORITHMS
Considering the path loss model given by (1), and
assuming the noise component w negligible, the received
signal strength experienced by the target at a distance di
from the anchor ai can be expressed as
di
RSSi = RSS0 − 10η log
d0
(A.1)
According to (A.1), the maximum RSSmax of the
received signal strength corresponds to the minimum
distance dmin between the target and the anchor node,
that is
measurement uncertainty is as high as few tens of percent
RSSmax = RSS0 − 10η log
of the size of the observed indoor environment.
dmin
d0
(A.2)
Subtracting (A.1) from (A.2) and replacing in (1) we
Although we cannot claim that this work provides the
obtain
ultimate answer to all the questions related to the RSS-
n 10η log
X
(1 − λ)
based localization in indoor environment, in our opinion
p̂ =
it offers some interesting insights on the topic. To the
(1 − λ)
as a relative distance uncertainty. Also, starting from the
experimental data, we suggest that the noise component
of the error is inversely proportional to the square root
of the product of the anchors number and the regression
coefficient of the channel propagation model.
× ai
10η log
di
(A.3)
dmin
i=1
relationship between absolute and relative distance error,
highlighting how the RSS uncertainty tends to propagate
dmin
i=1
n
X
best of our knowledge, our study first points out the
di
from which, using the properties of logarithms, we have
10η log(1−λ)
n X
di
× ai
dmin
p̂ = i=1 n (A.4)
10η log(1−λ)
X
di
dmin
i=1
Expression (A.4) reduces to (7) defining g = 10η log(1−
λ).
We also analysed the uncertainty introduced by the
use of approximated localization algorithms, such as
WCL and REWL. In particular, we found that the
ACKNOWLEDGMENT
error introduced by the algorithm is usually negligible
The research activities described in this paper are co-
compared to the uncertainty due to propagation channel
funded by the Autonomous Province of Trento, Italy,
model noise, nevertheless it can become significant when
within the project titled “ACube - Ambient Aware As-
the number of anchors increases.
sistance” and by the Italian Ministry of University and
June 15, 2011
DRAFT
14
Research within the PRIN 2008 project titled “Method-
Ad Hoc Communications and Networks, 2009. SECON ’09. 6th
ologies and measurement techniques for spatio-temporal
Annual IEEE Communications Society Conference on, June 2009,
pp. 1 –9.
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Outdoor Wireless Sensor Networks,” SIGMOBILE Mob. Comput.
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[22] T. He, C. Huang, B. M. Blum, J. A. Stankovic, and T. Abdelzaher,
Dario Petri received the M.Sc. degree
“Range-Free Localization Schemes for Large Scale Sensor Net-
(summa cum laude) and the Ph.D. degree
works,” in Proceedings of the 9th annual international conference
in Electronics Engineering from the Uni-
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respectively. At present he is the head of
[23] H. Chen, Q. Shi, R. Tan, H. Poor, and K. Sezaki, “Mobile
the Department of Information Engineering
Element Assisted Cooperative Localization for Wireless Sen-
and Computer Science at the University of
sor Networks with Obstacles,” Wireless Communications, IEEE
Trento. He has chaired the Italy Chapter of the IEEE Instrumentation
Transactions on, vol. 9, no. 3, pp. 956 –963, 2010.
and Measurement Society from 2006 to 2010. Currently he is the Vice
[24] N. Bulusu, J. Heidemann, and D. Estrin, “GPS-less Low-Cost
Chair of the IEEE Italy Section, which gather almost 5000 researchers
Outdoor Localization for Very Small Devices,” Personal Com-
from Italian Universities and Companies in the area of Electrical and
munications, IEEE, vol. 7, no. 5, pp. 28–34, Oct 2000.
Information Engineering. Also, he has been a Co-founder and General
[25] C. Laurendeau and M. Barbeau, “Relative Span Weighted Local-
Chair of the Ph.D Schol International Measurement University (IMU)
ization of Uncooperative Nodes in Wireless Networks,” in Pro-
of the IEEE Instrumentation & Measurement Society and General
ceedings of the International Conference on Wireless Algorithms,
Chair of various international Conferences and Workshops. He has
Systems and Applications. WASA 2009, Aug. 2009.
chaired the Italian research line on Measurement for the Information
Society of the Italian Society of Electrical and Electronic Measurement (GMEE), from 2002 to 2008 and the Italian Ph.D. School on
Measurement and Information Society of the same Society from 2002
to 2005. He is currently the Vice-chair of the GMEE, which gather
researchers from Universities, Metrological Institutes and Companies
Paolo Pivato received the M.Sc. degree
in the field of measurement and instrumentation. Dario Petri is an
in Telecommunication Engineering from the
Associate Editor of the IEEE Transactions on Instrumentation and
University of Trento, Italy, where he is cur-
Measurement, a member of the AdCom of the IEEE Instrumentation
rently working toward the Ph.D. degree in
and Measurement Society and a Fellow member of the IEEE. Dr. Petri
Embedded Electronics and Computing Sys-
has authored or co-authored over two hundred papers published in
tems at the Department of Information Engi-
international journals or in proceedings of peer reviewed international
neering and Computer Science. His principal
conferences. Research activities of Dario Petri are in the area of
research interests focus on localization in Wireless Sensor Networks
measurement science and technology, and they are focused on data
and embedded systems design and development.
acquisition systems design and testing, embedded systems design
and characterization, fundamentals of measurement theory, uncertainty
evaluation methods, statistical inference methods, application of digital
signal processing to measurement problems, measurement for quality
management systems.
Luigi Palopoli graduated with a degree in
computer engineering from the University of
Pisa, Pisa, Italy, in 1992, and received the
Ph.D. degree in computer engineering from
Scuola Superiore SantAnna, Pisa, in 2002.
He is an Assistant Professor of Computer
Engineering at the University of Trento. His
main research activities are in embedded system design with a particular focus on resource-aware control design and adaptive mechanisms
for QoS manage- ment. He has served in the program committee of
different conferences in the area of real-time and control systems.
June 15, 2011
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