Cubature Kalman Filter-based Performance Enhancement of
Wireless Indoor Localization using Ultra-wideband
Seong Yun Cho
School of Robotics Engineering, Kyungil University, 50 Gamasilgil, Hayangup, Kyungsan, Republic of Korea
Keywords: Indoor Wireless Localization, Ultra-wideband, Cubature Kalman Filter.
Abstract: Ultra-wideband has been widely used for accurate wireless indoor localization systems due to accurate
ranging measurement capability. There are several methods for ranging-based localization systems: iterative
methods, linear closed-form solutions, model-based filters, etc. These methods have their advantages and
disadvantages. In this paper, the characteristics of these methods are analysed and a cubature Kalman filter-
based localization method is presented to improve the localization performance in various indoor
environments.
1 INTRODUCTION
In the indoor space, location information is used as
key information for robot control and distribution
control as well as various location-based services.
The location estimation technique can be divided
into a sensor based and a communication based, and
the communication based localization is performed
using distance data, angle data, and signal strength
data obtained through communication. In this paper,
distance measurement based localization techniques
are discussed (Kolodziej and Hjelm, 2006; Banani et
al., 2013; Silva and Hancke, 2016).
In order to measure the distance based on
communication, time-of-arrival (ToA) technique is
used when the nodes are synchronized with each
other, and two-way-ranging (TWR) technique is
used when the time synchronization is not achieved.
Recently, accurate localization systems have been
developed by using Ultra-Wideband (UWB) which
can measure accurate distance easily by TWR
technique. The UWB can acquire distance
measurements with a resolution of 30cm or less by
using microwave having a bandwidth of 500Mhz or
more (Oh et al., 2009; Cho, 2014). In addition, the
UWB signal has a higher obstacle transparency,
which is higher than other signals in the indoor
space. However, additional distance measurement
errors due to multipath signals can not be avoided in
indoor space (Lee and Scholtz, 2002; Lee et al.,
2013; Yan et al., 2013; Cho, 2014; Silva and Hancke,
2016). In this environment, various localization
algorithms show different performance.
In this paper, various localization methods are
summarized; iterative least squares (ILS) method,
direct solution (DS) method, and difference of
squared ranging measurements (DSRM) method.
The advantages and disadvantages of each method
are analyzed and a cubature Kalman filter (CKF)-
based localization filter is designed to avoid the
disadvantages. The CKF-based localization filter
enables state variable expansion to estiamte the
channel-specific error and is left as a future study.
After analyzig the properties of the methods in
equation expansion, it is shown that the performance
of the CKF-based localization method is superior to
other methods in indoor environment through
simulation results.
2 WIRELESS LOCALIZATION
METHODS
For wireless localization, the following ranging
measurement equation is most basic (Cho et al.,
2017).
22
()()
iiM iMii
rxx yy bw

 

(1)
where
i
r

is the ranging measurement between an
anchor node (AN) i and a mobile node (MN),
[]
T
ii
x
y
and
[]
T
MM
xy
are the locations of the AN
Cho, S.
Cubature Kalman Filter-based Performance Enhancement of Wireless Indoor Localization using Ultra-wideband.
DOI: 10.5220/0006839704050410
In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, pages 405-410
ISBN: 978-989-758-321-6
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
405

i and MN, respectively, b
i
is the non-Gaussian error,
and w
i
is the white Gaussian noise.
Various methods for estimating the location of
MN based on this equation have been studied
(Mendel, 1995; Biton et al., 1998; Arasaratnam and
Haykin, 2009; Cho and Kim, 2013). In this section,
the advantages and disadvantages of these methods
are analysed and finally a location estimation filter
based on the CKF is designed.
2.1 Iterative Least Squares
To linearize the nonlinear equation (1), in the ILS
method, the first order Taylor series expansion of
equation (1) is performed using the nominal point
that is initially set. The linearized equation can be
yield as following matrix form.
RHXW
(2)
where
[]
T
MM
Xxy  
(3a)
11
[]
T
nn
Rrr rr 


(3b)
11 11
()/()/
()/()/
MM
M
nn M nn
x
xr y yr
H
x
xr y yr









(3c)
11
nn
bw
W
bw









(3d)
In this equation,
n is the number of the ANs,
[]
T
MM
xy
is the nominal point, and
i
r is the
calculated range using the nominal point as
22
()()
iM iM
xx yy
. The nominal point error
can be estimated as follows (Mendel, 1995):
1
ˆ
()
TT
X
HH HR


(4)
The location of the MN can be updated as
ˆ
ˆ
ˆ
M
M
M
M
ILS
x
x
X
y
y







(5)
This process is iterated until
ˆˆ
T
XX
is smaller than
the threshold that is set previously. In this process,
two main issues have to be considered: the large
initial error of the nominal point may cause local
minimum problem; the non-Gaussian error as well
as the Gaussian noise cannot be taken into account.
The former problem can be solved using the
particular algorithm. However, it is difficult to
overcome the latter problem and this causes
unavoidable localization errors.
2.2 Direct Solution
In the DS method, the measurement errors in the
equation (1) are ignored, then the both sides of (1)
are squared to remove the square root. The DS
method yields a closed-form solution as follows
(Biton et al., 1998):
2
ˆ
(1) 4
ˆ
2
i
i
M
i
M
DS
x
bbac
LA B
a
y


 

 




(6)
where {1, 2 }i

,
1
()
TT
L
GG G


(7a)
11
22
22
nn
x
y
G
x
y












(7b)
22
11 1
22
nn n
rx y
A
rx y
















(7c)
[1 1]
T
B


(7d)
()
T
aLBLB
(7e)
(2 ) 1
T
bLALB


(7f)
()
T
cLALA
(7g)
The DS method has two candidate solutions, and one
of them is selected based on the measurement
residual calculated as


2
22
1
ˆˆ
() ( ) ( )
n
ii
jjM jM
j
ei r x x y y




(8)
The reduction of the computational burden is the
merit of the DS method in comparison to the ILS
method. There is the red sea zone (RSZ) problem,
however, caused according to the relations among
the locations of the ANs and MN (Cho and Kim,
2013). Also, the neglect of the measurement errors is
the same as the ILS method.
2.3 Difference of Squared Ranging
Measurements
The DSRM method, on the other hand, does not
ignore the measurement errors. This method makes
ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics
406

the squared ranging measurements equations by
squaring the both sides of the equation (1). One of
the ANs is selected as a common node. Then, the
DSRM equation is formulated by subtracting each
squared ranging measurement equation of ANs from
that of the common node as follows (Cho and Kim,
2013):
1, 1 1
1, 1 1
CC C
M
M
nC C n C n
M
M
xx yy
x
V
y
xx yy
x
ZG V
y




















 



(9)
where V contains the measurement errors.
The location of the MN can be calculated as
11 1
ˆ
()
ˆ
M
TT
M
DSRM
x
GQ G GQ Z
y
 




(10)
where
[]
T
Z
EV V
can be calculated by assuming
the non-Gaussian error is Gaussian noise.
The important thing in this method can be
confirmed in the process of calculating
,
j
C

as
22
,
22
()/2
()/2
jC j C
jC
rr
rr BCV



 

(11)
where
2222
CCj j
x
yxy  
(12a)
22
()/2
jj CC j C
Brb rb b b 
(12b)
j
jCC
Cbwbw
(12c)
22
()/2
jj CC j C
Vrwrw ww 
(12d)
The last three terms in this equation are related to
the measurement errors, and there are several
considerations: the size of the Gaussian noise is
smaller than that of the non-Gaussian error; the non-
Gaussian error is always positive numbers; and the
last term V is considered in the equation (9) by Q.
Based on the first consideration, it can be guessed
that C is too small. Also B is analysed as a relatively
small number due to the second consideration.
Therefore, the DSRM method can yield more
accurate solutions than ILS and DS methods even in
the case of the measurement error that is not
Gaussian.
2.4 Cubature Kalman Filter
The measurement equation the equation (1) is
nonlinear. So, nonlinear Kalman filters such as
extended Kalman filter, unscented Kalman filter
(UKF), CKF, etc. can be used in the wireless
localization. For the system model of the Kalman
filter, a constant velocity (CV) model or constant
acceleration model can be selected in the light of the
dynamics of the MN. In this paper, the localization
filter is designed using the CKF with a CV model.
CKF is the cubature rule-based approximate
Bayesian filter, and the performance of the 3
rd
-
degreee CKF is similar to that of UKF (Arasaratnam
and Haykin, 2009; Jia et al., 2013). If the CV model
is defined in the 2-D coordinate frame, 2N cubature
points (
,1,2,,2
i
iN


 , N is the system dimension,
that is 4) are generated, and then time-propagated as
follows:
,1 ,
,1 ,
,1 ,
,1 ,
ˆ
ˆ
(1) (3)
ˆˆ
(2) (4)
ˆˆ
(3) (3)
ˆ
ˆ
(4) (4)
ik ik
ik ik
ik ik
ik ik
dt
dt


















(13)
where dt is the time interval of the measurement
acquisition, and the indices 1, 2, 3, and 4 denote
location and velocity of x and y axes, respectively.
The time-propagated state vector and covariance
matrix are calculated as follows:
2
,
1
1
ˆ
ˆ
2
N
kik
i
x
N






(14)
2
,,
1
1
ˆˆ
ˆ
ˆ
()()
2
N
T
kikkikk
i
PxxQ
N






(15)
where Q is the process noise covariance matrix.
Then, measurement-update of the state vector
and error covariance matrix is performed as
ˆ
ˆ
ˆ
()
kk kkk
x
xKyz


 

(16)
T
kk kxz
P
PKP


(17)
where
2
,
1
1
ˆ
2
N
kik
i
zy
N






(18a)
, 1,, ,,
ˆ
ˆ
[r]
T
ik ik nik
yr
 



(18b)
and the other parameters can be obtained in
(Arasaratnam and Haykin, 2009).
Cubature Kalman Filter-based Performance Enhancement of Wireless Indoor Localization using Ultra-wideband
407

New cubature points are generated as
,
ˆ
[1]
ik k i k
SN x  
(19)
where S
k
can be calculated using the Cholesky
factorization as
T
kkk
PSS
.
The Kalman filter estimates the state variables
using the systems equations as well as the
measurements. That is, the model-based localization
filtering can yield more accurate location solutions
than the model-free localization methods when the
model reflects the movement of the MN as it is. So,
the solution of the Kalman filter can have good
features of a low-pass filter. Also, the effect of the
non-Gaussian measurement errors can be
diminished.
3 SIMULATION ANALYSIS
To analyse the performance of the several model-
free and model-based localization methods, some
simulations are performed. In these simulations, it is
assumed that the wireless communication infra used
for localization is the UWB, so the noise of the
ranging measurements is set to
2
(0,(0.3 ) )Nm
. In
addition, the non-Gaussian error denoted in (1) is
defined as
2
|(0,(1.5))|Nm
. The size of the test area
is set to 20
m ൈ 15 m , and four ANs are installed in
the area for the first simulation.
Figure 1 shows the comparative results of the
localization methods. In this figure, four circles in
the corners of the test area denote the ANs. Based on
the error statistics, 1000 ranging measurements are
generated each in the 24 fixed reference locations.
The location of the MN is calculated using the
individual localization method and, then, the
location error is calculated. In this figure, the sized
of the circles denote the comparative mean values of
the location errors.
From the outcome of this simulation results, it
can be stated that (i) the performance of the DS
method may be degraded according to the test
location due to the RSZ problem as can be seen in
the top right of figure 1(b); (ii) among the model-
free methods, the DSRM method has better
localization performance than the DS and ILS
methods because the non-Gaussian errors can be
somewhat diminished in the DSRM method; and (iii)
the location solution of the CKF is more accurate
than the model-free methods because it uses the
dynamic model of a MN as well as the ranging
measurement. The location errors at each test
location are summarized in Table 1.
(a) ILS method
(b) DS method
(c) DSRM method
(d) CKF
Figure 1: Simulation 1 – comparative localization errors
according to the localization methods.
ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics
408

Table 1: Summary of Simulation 1 (1000 Samples).
Errors
Localization Methods
DS ILS DSRM CKF
Mean [m] 2.559 1.248 1.071 0.723
Std. [m] 1.035 0.794 0.609 0.274
(a) an example of the localization (2 m, 2 m)
(b) localization errors (2 m, 2 m)
(c) an example of the localization (5 m, 2.5 m)
(d) localization errors (5 m, 2.5 m)
Figure 2: Simulation 2 – comparative results according to
the localization methods in the small area.
Another simulation is performed and the results
are shown in Figure 2. In this simulation, the small
test area is set to 8 m

3 m and three ANs are
installed in the test area denoted in Figure 2(a) and
2(c). Figure 2(a) and 2(b) are the results of the
localization of the MN located in (2 m, 2 m), where
the RSZ problem does not occur (Case 1). Figure 2(c)
and 2(d) are the results of the localization of the MN
located in (5 m, 2.5 m) where the RSZ problem
occurs (Case 2).
In the Case 1, the results of the DS and ILS
methods are similar, and that of the DSRM is
improved. The CKF yields more accurate and stable
solutions irrespective of the measurement error as
well as noise. In the Case 2, the performance of the
DS method is degraded due to the RSZ problem. In
this case, the results of the DS and ILS methods may
be out of the test area. The simulation results are
summarized in Table 2.
Table 2: Summary of Simulation 2 (1000 Samples).
Test
Loc.
Localization Methods
DS ILS DSRM CKF
Mean Value of the Location Errors [m]
Standard Deviation of the Location Errors [m]
2, 2
0.771 0.814 0.586 0.463
0.436 0.510 0.358 0.069
5, 2.5
1.451 0.858 0.636 0.527
0.597 0.498 0.417 0.189
4 CONCLUSIONS
In this paper, several wireless localization methods
using the ranging measurements are reviewed when
the measurements include non-Gaussian errors as
well as Gaussian noise. The measurement errors are
considered as always positive. The localization
methods analysed in this paper are ILS, DS, and
DSRM methods for model-free methods, and CKF
for model-based Kalman filtering. First, the
characteristics of each method are analysed based on
the expansion of the localization equations. Then,
some simulations are carried out to verify the
performance of the localization methods under the
measurement error occurrence. The simulation
results show that the relative location errors of the
DSRM method compared with the DS and ILS
methods are 41.8% and 85.8%, respectively. Also,
the relative location errors of the CKF compared
with the DS, ILS and DSRM methods are 28.2%,
57.9% and 67.5%, respectively. Consequently, it can
be concluded that the DSRM method can yield
0 1 2 3 4 5 6 7 8
0
0.5
1
1.5
2
2.5
3
3.5
4
x [m]
y [m]
AN 1
AN 2
AN 3
True
DS
ILS
DSRM
CKF
0 100 200 300 400 500 600 700 800 900 1000
0
0.5
1
1.5
2
2.5
3
3.5
samples
localization error [m]
DS
ILS
DSRM
CKF
0 1 2 3 4 5 6 7 8
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x
[
m
]
y [m]
AN 1
AN 2
AN 3
True
DS
ILS
DSRM
CKF
0 100 200 300 400 500 600 700 800 900 1000
0
0.5
1
1.5
2
2.5
3
3.5
samples
localization error [m]
DS
ILS
DSRM
CKF
Cubature Kalman Filter-based Performance Enhancement of Wireless Indoor Localization using Ultra-wideband
409

comparatively more accurate location solution
among the model-free localization methods when
the ranging measurements contain non-Gaussian
errors with positive numbers In addition, the model-
based Kalman filtering can enhance the localization
performance compared with the model-free
methods.
ACKNOWLEDGEMENTS
This research was supported by Basic Science
Research Program through the National Research
Foundation of Korea (NRF) funded by the Minist ry
of Education (NRF-2015R1D1A1A01059606).
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