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Abstract—
This paper presents an analysis of the localization
accuracy of indoor positioning systems using Cramer’s rule via IEEE
802.15.4 wireless sensor networks. The objective is to study the
impact of the methods used to convert the received signal strength
into the distance that is used to compute the object location in the
wireless indoor positioning system. Various methods were tested and
the localization accuracy was analyzed. The experimental results
show that the method based on the empirical data measured in the
non line-of-sight (NLOS) environment yield the highest localization
accuracy; with the minimum error distance less than 3 m.
Keywords—
Indoor positioning systems, Localization accuracy,
Wireless networks, Cramer’s rule
I. INTRODUCTION
CCURACY to define object’s location is the main concern
of the wireless indoor positioning systems (WIPS).
Although Global Positioning System (GPS) have been
deployed widely and could provide large service coverage, it
could not use in the indoor environment because there is no
line-of-sight path between the indoor receivers and the
satellites and the satellite signal cannot penetrate the building’s
walls [1]. Comparing with the outdoor environments, the
indoor structure is more complex due to varieties of obstacles
such as walls and furniture which cause the multipath effects
on the wireless signal. In addition, the human mobility and
interference from other wireless networks in the building could
impair the signal quality [2]. These indoor issues bring
challenges to the deployment of the WIPs.
Deploying positioning systems in the indoor environment,
one needs to install wireless networks that generate the
referencing signal used to define the object location. Several
wireless technologies have been adopted such as the IEEE
802.11 wireless LAN and the IEEE 802.15.4 wireless sensor
networks.
The effects of indoor environment on the performance of
WIPS have received little attention in the research study. Most
of the existing works in literature focused on developing
techniques to define the object positions. In [6], the authors
Manuscript received Oct 20, 2011. This work was supported in part by
Suranaree University of Technology, the Office of the Higher Education
Commission under NRU project of Thailand and the National Research
Council of Thailand (NRCT).
Kriangkrai Maneerat and Chutima Prommak*, are with the School of
Telecommunication Engineering, Suranaree University of Technology,
Nakhon Ratchasima, 30000 Thailand (phone: 66-44-224393; fax: 66-44-
224603; e-mail: cprommak@sut.ac.th). *corresponding author
proposed the indoor positioning systems that estimate the
position of the objects by analyzing the received signal
strengths in which the service area is divided into zones and
sub-zones. In [7], the authors proposed the algorithm based on
the decent gradient iteration to define the object position. In
[8], the RSS-based techniques were used to estimate the
distance between the object and the referencing nodes and the
triangulation techniques were used to define the object
position. In [9] and [10], the 3-D positioning systems were
simulated. To compute the object position, the Angle of
Arrival algorithms was used in [9] whereas the Time
Difference of Arrival algorithm (TDOA) was used in [10].
From the literature review, most works in literature focused
mainly on developing techniques to define the object positions
whereas he effects of indoor environment on the performance
of WIPS have received little attention. In order to improve the
localization accuracy of the WIPS, we need a suitable method,
which consider the effects of indoor environment, for
converting the received signal strength to the distance that is
used to compute the object location in WIPS. Therefore, in
this paper we present the study and comparison of using three
methods which are developed by three different ways,
including the method based on the empirical data measured in
the line-of-sight (LOS) environment, the method based on the
non line-of-sight (NLOS) empirical data and the method based
on the path-loss model using the empirical path-loss exponent.
Specifically, we apply these methods to the WIPS using
Cramer’s rule.
The rest of the paper is organized as followed. Section II
presents the Cramer’s rule approach. Experimental designs are
explained in Section III. Section IV shows the empirical data
measured in various environments and the analysis of the
location accuracy of WIPS. Finally, we conclude the paper in
section V.
II. CRAMER’S RULE APPROACH
Cramer’s rule has been widely applied in the localization
applications. It uses the principle concept of the linear
equation systems of which the number of equations is equal to
the number of variables and formats the linear equations
systems in the form of matrix. Then, the determinant is applied
and the variables’ values are derived [11]. Fig. 1 shows the
structure and components used by Cramer’s rule.
On the Analysis of Localization Accuracy of Wireless
Indoor Positioning Systems using Cramer’s Rule
Kriangkrai Maneerat and Chutima Prommak*
A
World Academy of Science, Engineering and Technology 60 2011
202
Fig. 1 Cramer’s rule
The structure of positioning system using Cramer’s rule
consists of three referencing nodes, namely Node 1, Node 2
and Node 3 in fig.1. The object location is the intersection of
the three circles centered at the referencing nodes. To derive
formulas for computing the coordinate of the object, we define
the following notations:
(xi, yi) = the coordinate of the referencing node i,
(xu, yu) = the coordinate of the object, and
Ri = the distance between the node i and the object.
We have the circle equation written
(
)
(
)
2
22
iuiui Ryyxx =−+− for i= 1,2,3 (1)
Then we have the relationship:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
2
3
2
1
2
3
2
1
2
3
2
1RRyyyyxxxx uuuu −=−−−+−−− (2)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
2
3
2
2
2
3
2
2
2
3
2
2RRyyyyxxxx uuuu −=−−−+−−− (3)
Arranging the equation (2) and (3), we obtain the following
(
)
(
)
(
)
(
)
(
)
2
3
2
1
2
3
2
1
2
3
2
11313 22 yyxxRRyyyxxx uu −−−−−=−−− (4)
(
)
(
)
(
)
(
)
(
)
2
3
2
2
2
3
2
2
2
3
2
22323 22 yyxxRRyyyxxx uu −−−−−=−−− (5)
Putting the equations (4) and (5) in the matrix form, we can
write determinant matrix (7), (8) and (9).
(
)
(
)
(
)
( ) ( ) ( )
−−−−−
−−−−−
=
−−
−−
2
3
2
2
2
3
2
2
2
3
2
2
2
3
2
1
2
3
2
1
2
3
2
1
2323
1313
2yyxxRR
yyxxRR
y
x
yyxx
yyxx
u
u (6)
( )
2*)(2*)(
2*)(2*)(
det
2323
1313
yyxx
yyxx
A−−
−−
= (7)
( )
(
)
(
)
(
)
(
)
( ) ( ) ( )( )
2*)(
2*)(
det
23
2
3
2
2
2
3
2
2
2
3
2
2
13
2
3
2
1
2
3
2
1
2
3
2
1
1yyyyxxRR
yyyyxxRR
A−−−−−−
−−−−−−
= (8)
( )
(
)
(
)
(
)
(
)
( ) ( ) ( )( )
2
3
2
2
2
3
2
2
2
3
2
223
2
3
2
1
2
3
2
1
2
3
2
113
22*)(
2*)(
det yyxxRRxx
yyxxRRxx
A−−−−−−
−−−−−−
= (9)
Finally, the coordinate of the object can be computed
from (10)
[
]
[ ]
A
A
xudet
det 1
= and
[
]
[ ]
A
A
yudet
det 2
= (10)
III. EXPERIMENTAL SETUP
This is paper aims to analyze of the localization accuracy of
indoor positioning systems using Cramer’s rule. The
experiments were performed using XBee Pro mudule based on
IEEE 8002.15.4 (ZigBee) standard. Structure of system consist
reference nodes and target node. The reference nodes are
configured as Coordinator and define on different place. The
target node are configured as End device and connect to laptop
can be mobile. The process of indoor positioning systems
starting from, the target node request received signal strengths
from each reference nodes. Next, we are convert the received
signal strength into the distance for compute coordinates of the
target node by Cramer’s rule, as show in Fig. 2. This paper, we
select the experimental area on the fourth floor of the C-
building at Suranaree University of Technology. The
dimension of floor is approximately 75 m x 75 m and we are
install 4 reference nodes which are A, B, C, and D. We define
40 sample points. As show in Fig. 3 In the experimental, we
propose 3 method of the convert received signal strength into
the distance, the correlation of Line of Sight method, the
correlation of Non Line of Sight method, and the correlation of
path loss model method.
Fig. 2 Structure of the indoor positioning system
Fig. 3 Experimental setup: the indoor environment, locations of four
referencing nodes and 40 test points
World Academy of Science, Engineering and Technology 60 2011
203
A. Correlation of Line of Sight method
The correlation of Line of Sight method there is
characteristics of transmission is Line of Sight. We collecting
RSS value form each test points, 10 times to find the average
RSS value of the 62 test points. Next, we are using the average
RSS value to build the relationship with the Matlab simulation
by using the cftool command. We will have correlation
equation between the received signal strength and the distance
for the LOS case as equation (12) and the Fit Curve in Matlab
is plotted in Fig. 5
Correlation equation of LOS
f(x) = p1*x4 + p2*x3 + p3*x2 + p4*x + p5 (11)
then
f(x) =(-1.123x10-4*x4) + (-0.0239*x3) + (-1.833*x2) + (-61.58*x) + (- 770.9)
(12)
When p1 - p5 refer the coefficient of linear model poly4, the
distance function f(x) is a distance between the reference node
and target node (m) and the variable x represent the RSS value
(dBm).
Fig. 4 Equipment used in the experiment: the referencing nodes and
the object node
-75 -70 -65 -60 -55 -50 -45 -40
0
10
20
30
40
50
60
RSS (dB)
Distance (m)
RSS Sample; LOS
Poly4 of RSS Sample
Fig. 5 LOS correlation
B. Correlation of Non Line of Sight method
The correlation of Non Line of Sight method there is no
LOS path between transmitter and receiver. Because, there are
effects of the indoor environment such as wall, door. We
collecting RSS value form each test points, 10 times to find the
average RSS value of the 57 test points. Then, we perform to
build the relationship between the average RSS value and the
distance for the NLOS case as follows (14)
Correlation equation of NLOS
f(x) = p1*x4 + p2*x3 + p3*x2 + p4*x + p5 (13)
then
f(x) =(1.255x10-6*x4) + (4.863x10-4*x3) + (0.0515*x2) + (1.133*x) + (-0.922)
(14)
When p1 - p5 represent the coefficient of linear model poly4,
the distance function f(x) is a distance between the reference
node and target node (m) and the variable x is the RSS value
(dBm).
C. Correlation of path loss model method
The correlation of path loss model method utilizes the
property of signal radio propagation within a building [12].
The equation of the relationship for the path loss model case
can be represented by equation (15)
Pr = Pt + K + Gt + Gr - 10αlog10(d/d0) (15)
K= 20log10 (λ/4π d0) 2 (16)
The variable Pr refers to the received signal strength value
(dBm), the transmit power Pt assume is 18 dBm. K that
depends on the average channel attenuation can be compute by
equation (16). The λ is signal wavelength (m), d0 is typically
assumed to be 1 m. The variable α can be obtained to
approximate either an analytical or empirical model, our
experiment using path loss exponent is 3.45 [12]. Gt is the
transmit antenna gain and Gr is the receive antenna gain, we
determine these to parameters equal 1.5 dBi. The d refers to
distance between the reference node and the target node (m).
Note that the using the 3 relationship above, we can be
converting received signal strength value into the distance. By
the condition, if the RSS value not in the range -80 dBm to -40
dbm was assume to be equal the boundary. For instance,
measuring the received signal strength value is -95 dBm. We
define a new received signal strength value are -80 dBm.
IV. EXPERIMENTAL RESULTS AND ANALYSIS
In our experiments, we performed measurement of the 40
sample points. Each test points measuring 10 times in order to
find the average RSS value of the 40 sample points. Use the 3
approach for convert the received signal strength into the
distance as mentioned in Section III. Next, we are using the
distance value to location estimate of the target node by
World Academy of Science, Engineering and Technology 60 2011
204
Cramer’s rule. Examples of the position systems show as
follows. We assume the method of the convert the received
signal strength into the distance are LOS case. At sample point
S1, starting that the target node measured RSS value form the
4 reference nodes are RA = -97.9 dBm, RB = -52.75 dBm, RC =
-65.9 dBm, and RD = -89.65 dBm. Select the RSS value to the
best RSS value of the 3 in 4 reference node (RB=-52.75 dBm,
RC = -65.9 dBm, and RD o ver bound = -80 dBm). Next, converts
the RSS value into the distance by the correlation equation of
LOS method as equation (11). We obtained the distances are
f(RB) = 12.62 m, f(RC) = 43.12 m, and f(RD) = 51.05 m. Then,
using the distances value to compute the coordinate of the
target node (X, Y) by the Cramer’s rule. Final, we will be the
position of the target node are (15.47, 28.30).Table I shows
the error distance between the actual target location and the
experiment target location when used the 3 method of the
correlation equation. We can see that the correlation of path
loss model method be provided the least accuracy positioning.
The margin error of maximum error is 39.6306 m, average
error is 22.418428 m, minimum error is 5.1764 m, and the
standard deviation is approximately 8 m. And the method of
the correlation equation give the most accuracy positioning is
the correlation of NLOS method. A maximum error is 36.582
m, average error is 17.59365 m, minimum error is 2.7716 m,
and the standard deviation is about 8.5 m.The plot in Figure 6,
7, and 8 shows the actual target location and the experiment
target location all 40 sample points of the correlation method
of the LOS, NLOS, path loss model, respectively. In Fig. 8 the
correlation of path loss model method, it provides localization
accuracy less than when comparing the localization accuracy
of the correlation of LOS and NLOS method
ostensibly.Histogram analysis for comparison the distance
error of the 3 different methods as show in Figure 9. Notices,
that the correlation of NLOS method there error distance
distribution by spreading in the range 0 to 25 m that more than
the correlation of LOS and path loss model method. Moreover,
the Fig. 10 the cumulative distribution function (CDF) of the
error distance of the 3 different methods. Shows that the
correlation of NLOS method has a location precision of 80%
within 25 m (the CDF of the error distance of 25 m is 0.8).
Different from the correlation of LOS and path loss model
method that a location precision of 80% within 30 m.
Therefore, the estimation of the object within a building by
Cramer’s rule, using the correlation of NLOS method provided
localization accuracy of the positioning more than the
correlation of LOS and path loss model method.
TABLE I
ERROR DISTANCE OBTAINED BY DIFFERENT METHODS
Method
Error distance (m)
Min Mean Max SD
LOS 3.26 20.46 36.99 9.48
NLOS 2.77 17.59 36.58 8.53
Path Loss model 5.17 22.41 39.63 8.21
Actual location Estimated location
Fig. 6 the actual and estimated locations of 40 test points
using LOS method
Actual location
Estimated location
Fig. 7 the actual and estimated locations of 40 test points
using NLOS method
Actual location Estimated location
Fig. 8 the actual and estimated locations of 40 test points
using path-loss model
World Academy of Science, Engineering and Technology 60 2011
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Fig. 9 Histogram of the error distances
Fig. 10 Cumulative distribution function of the error distances
V. CONCLUSION
In this paper, we present the analysis of the localization
accuracy of the wireless indoor positioning systems using
Cramer’s rule. We analyze the use of different methods to
convert the received signal strength into the distance used in
the Cramer’s rule to compute the object location. Specifically,
we compare three methods, including the method based on the
empirical data measured in the line-of-sight (LOS)
environment, the method based on the non line-of-sight
(NLOS) empirical data and the method based on the path-loss
model using the empirical path-loss exponent. The
experimental study shows that the NLOS based method results
in the highest localization accuracy, with the minimum error
distance less than 3 m. Our ongoing work is to develop an
efficient indoor positioning framework that can apply to
various service environments including the single floor and
multiple floor area.
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Chutima Prommak received her Ph.D. and M.S. degree in
Telecommunications from the University of Pittsburgh in 2005 and from the
University of Colorado at Boulder in 1998, respectively and received her B.S.
from the University of Khon Kaen in 1992. She is now an assistant professor
at the School of Telecommunication Engineering, Suranaree University of
Technology, Thailand. Her research interests include wireless network design,
optical network design, network optimization and heuristic approaches for
network design.
Kriangkrai Maneerat received him B.S. degree in Telecommunication
Engineering from Suranaree University of Technology, Thailand in 2009.
Currently he is pursuing him M.S. degree in the school of Telecommunication
Engineering, Suranaree University of Technology. He is a member at the
Wireless Communication Research Lab at Suranaree University of
Technology.
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