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Journal of Global Positioning Systems (2012)
Vol.11, No.2 :89-99
DOI: 10.5081/jgps.11.2.89
Correlation Analysis for Fault Detection Statistics in Integrated
GNSS/INS Systems
Jinling Wang1, Ali Almagbile1, Youlong Wu1 and Toshiaki Tsujii2
1School of surveying and Geospatial Engineering, University of New South Wales, Sydney, NSW 2052, Australia.
2High-Precision Satellite Navigation Technology Section; Aviation Program Group, Japan Aerospace Exploration
Agency (JAXA), Tokyo, Japan
Abstract
Global Satellite Navigation Systems (GNSS) have been
widely used for positioning, navigation and timing
(PNT). Therefore, the integrity of the satellite based
navigation systems has been a major concern for many
liability critical applications, such as civil aviation, and
location-based services (LBS). Over the past two
decades, GNSS Receiver Autonomous Integrity
Monitoring (RAIM) procedures have been developed,
but the efficiency of such procedures is highly dependent
on measurement redundancy and geometric strength
within the GNSS positioning solutions. Reliability of a
PNT system can be measured by, not only the well-
known Minimal Detectable Biases (MDBs), but also the
recently derived Minimal Separable Biases (MSBs) for
the measurements. While the previous research has
shown that the MSBs are directly related to the
correlations between the faulty measurement detection
statistics, a comprehensive analysis for such correlations
between fault (or outlier) detection statistics is still
lacking, even for commonly used GNSS/INS integration
scenarios. In this research, we have demonstrated that
with the aid of inertial sensors, even with low-cost
MEMS sensors, the MDBs and correlation coefficients
between the measurement fault detection statistics can be
significantly reduced, thus improving the separability of
faults in GNSS measurements.
Keywords: GNSS, GNSS/INS Integration, Fault
Detection and Identification, Separability, Correlation
Analysis
1. Introduction
As more and more human activities are relying on the
use of satellite navigation technologies, the integrity of
satellite navigation solutions has become a major issue,
especially for the life-safety-critical and liability-critical
applications. Therefore, a reliable integrity monitoring
procedure must be used to eliminate hazardous and
misleading navigation information caused by failure(s)
within the navigation system and provide a timely
warning message to the user if the navigation
information is not good enough for certain applications
at a specific time. One effective approach to address the
satellite navigation integrity risks is the so-called
Receiver Autonomous Integrity Monitoring (RAIM).
The RAIM strategy is based on the consistency check
among satellite pseudo-range measurements used in a
navigation solution. If a faulty measurement/satellite
(failure) is detected, a procedure may be activated to
identify and exclude the failure from the navigation
solution, which will therefore remain fault-free and
reliable for use in the defined applications. Thus, a
RAIM procedure is self-contained and can be used as the
ultimate integrity monitor (e.g., Wang and Ober, 2009;
Wang and Kubo, 2000).
If measurements are contaminated by faults/outliers, the
user position solution may exceed the predefined
allowable accuracy range. If so, the role of RAIM comes
to provide a warning message (alarm) to the user within
a given period of time that the system must not be used
for navigation (Ochieng et al., 2002). If RAIM fails to
provide measurement check (mis-detection) when failure
occurs or falsely warn the user (false alarm) that the
system should not be used for navigation, the navigation
will be in risk. However, the performance of RAIM
depends on the number of visible satellites to the user,
the strength of geometry and the accuracy of the pseudo-
range measurements. For instance, when the number of
visible satellites is five, RAIM can provide a detection of
the fault, but the fault is inseparable due to the full
correlation between fault detection statistics (Hewitson
and Wang, 2007; Wang and Knight, 2012).
Although RAIM can provide an acceptable confidence
level for the user with the status of navigation solutions,
the design of RAIM is limited to detect one fault at a
Wang et al. Correlation Analysis for Fault Detection Statistics in Integrated GNSS/INS Systems 90
time. When the system encounter more than one failure
at a time, the capability of RAIM degrades or even stops
to provide any reliable integrity monitoring for the
system. As a result, several investigations (e.g., Wang
and Chen, 1999; Knight et al., 2010) have developed
methods for detecting and identifying multiple
simultaneous faults in either standalone navigation
systems or integrated navigation systems.
However, the reliability of RAIM procedure is highly
dependent upon sufficient satellite redundancy in the
navigation solutions, as well as the strong geometry of
visible satellites. But such conditions may not be
guaranteed under poor environments, such as in city
canyons, where majority of business activities are
located. This issue can be addressed through robust
integration between various GNSS satellite navigation
systems and between GNSS and other complementary
navigation sensors, such as pseudo-satellites (e.g., Wang,
2002) and/or Inertial Navigation System (INS) (e.g.,
Wang et al., 2003).
In integrated GNSS/INS navigation systems, fault
detection and identification has been well documented in
the literature. Sturza (1989) presented a fault detection
algorithm based on hypothesis testing in parity space
while Teunissen (1990) investigated quality control of
integrated navigation systems using innovation and
recursive filtering. An innovation based autonomous
integrity monitoring extrapolation (AIME) was
presented by Diesel and King (1995). Brenner (1995)
used Kalman Filter (KF) to quantify the performance of
integrity monitoring. Nikiforve (2002) presented fault
detection and exclusion in multi-sensor integration based
on KF innovations. Hewitson and Wang (2010)
investigated the quality control for integrated GNSS and
inertial navigation systems.
It has been widely accepted that integration of
GNSS/INS offers great advantages because the
characteristics of both sensors complement each other.
One of these advantages is that through the integration,
the integrity and reliability of the navigation solution can
be improved. Integrity and reliability are two parameters
that are closely connected. While integrity refers to the
ability of the system to provide a warning message to the
user when the system must not be used for navigation,
the reliability is the ability of the system to detect the
faults (or outliers) in the measurements, hence evaluating
the impact of undetected faults on the positioning
It is obvious that detection and identication of single and
multiple faults has been given considerable attention by
many researchers. However, most of the investigations
that dealt with fault exclusion/separation in either
geodetic surveying or navigation applications are limited
to the single fault case. Förstner (1983) developed the
measurements from the contaminated one. The
separability measures are based on correlation
coefficients between faulty meaurement detection
statistics. The higher the correlation coefficients between
tests statistics, the less likely the system can separate the
fault. In other words, due to the high correlation
coefficients between fault detection statistics, several
meaurements may have a large value for their associated
detection statistics, even when there is only one fault,
increasing the linkelyhood to identify a wrong
measurement as a fault. The separability measures are
employed to evaluate the capability of the system to
separate any pair of the fault detection statistics. In
GNSS applications, an analysis of outlier separability
measures under different numbers of visible satellites
and various satellites geometries was investigated by
Hewitson and Wang (2006); Wang and Knight (2012),
while Almagbile and Wang (2011) and Almagbile et al.
(2011) evaluated outlier separability measures in
integrated GPS/INS systems. This evaluation considered
the factors such as the number of visible satellites,
satellites geometry and the number of system states
models that influences the correlation coefficients
between fault detection test statistics. However, such
analysis was based on a specific epoch (snapshot) results
for the case of single fault.
Using the same principles of single outlier separability
case, multiple faults separability measures were
discussed by Förstner (1983) and Li (1986). In this case
multi-dimensional correlation coefficients were
employed as an indicator of the capability of the system
to separate the true hypotyhesis from the false alternative
hypotheses in photogrammetric adjustment applications..
In this paper, we will present a comphrehensive
correlation analysis for the faulty measurement detection
statistics in integrated GNSS/INS systems. The structure
of the paper is as follows: Section 2 discusses the quality
control of integrated GNSS/INS systems including
statistical quality contreol procedures for detection and
identification for single and multiple faults. Section 3
presents simulated studies on the correlation analysis for
the fault detection statistics in various GNSS and
GNSS/INS integration scenarios, which will be followed
by the concluding remarks in Section 4.
2. Quality Control for GNSS/INS Integration
In the next generation GNSS receivers, measurements
from multiple satellite constellations are combined to
improve the integrity of navigation solutions. However,
to integrate the measurements from two or more satellite
constellations, the clock-offsets between the
constellations need to be treated properly. Options of
such treatments may include: a) adding GNSS system
Wang et al. Correlation Analysis for Fault Detection Statistics in Integrated GNSS/INS Systems 91
time-offsets as unknown parameters into the navigation
solution; or b) measuring/broadcasting precise time-
offset values which could be considered as errorless or
as pseudo-measurements (Wang et al., 2011). For
example, the GPS/QZSS system time offset has been
dealt with at the system level, and thus, users will not
estimate this time offset in their navigation solutions.
In a GNSS-only system, faulty measurement detection
and identification is normally carried out with a so-called
snapshot least squares framework. Such a procedure
should be implemented through a Kalman filter for
GNSS/INS integration where the dynamic information
captured by the INS sensors can be efficiently used. In
fact, the Kalman Filter can also be presented as a least-
squares procedure in each epoch (Sorenson, 1970;
Salzmann, 1993; Wang et al., 1997; 2008).
Mathematically, the discrete time of the system state and
measurement model of the Kalman filtering can be
written as follows:
111 kkkk wxx
(1)
kkkk vxHz
(2)
where
k
x
is the (
)1n
state vector,
k
is the
(
)nn
transition matrix,
k
z
is the (
)1r
measurments
vector,
k
H
is the
)( nr
measurement (or design) matrix.
The variables
k
w
and
k
v
are the uncorrelated white noise
errors with covariance matrices
k
Q
and
k
R
respectively.
Then equations (1) and (2) can be integrated as follows:
kx
zk
k
k
k
k
k
kv
v
v
E
H
A
x
z
l,,
(3)
i.e.,
kkkk xAvl
(4)
where
E
is the
)( mm
identity matrix and
k
x
is the
vector of the predicted states at epoch k. The variance-
covariance matrix
k
l
C
, which is derived from both the
measurement noise covariance matrix
k
R
and the
covariance matrix
k
P
of the predicted states within the
Kalman filtering, can be written as (Wang et al., 1997):
1
0
0
P
P
R
Ck
k
lk
(5)
where
P
is the weight matrix with the priori variance
factor being assigned as one.. The optimal estimates of
the state parameters
k
x
and their covariance matrix
k
x
Q
can be written as:
k
T
kk
T
kk PlAPAAx 1
)(
(6)
1
)(
k
T
k
xPAAQ k
(7)
The KF residuals
k
v
and the associated cofactor
k
v
Q
can
be written as:
kkk
kx
zk
klxA
v
v
v
(8)
T
k
x
klv AQACQ k
kk
(9)
where the symbol k is used as a notation for the current
epoch and it will be ignored throughout this paper for
simplification. By performing the KF as least squares,
many existing faulty measurement detection and
identification procedures for GNSS can also be used in
the integrated GNSS/INS systems.
2.1. Faulty measurement detection statistics
For the case of single faulty measurement, the w-test can
then be used to identify the corresponding measurement,
where the test statistic is (e.g., Baarda, 1968; Teunissen,
1990; Wang and Chen, 1999)
iv
T
i
T
i
iPePQe
Pve
w
(10)
where
i
e
is a vector in which the ith element is equal to
one and all other elements are equal to zero. Under the
null hypothesis,
i
w
has a standard normal distribution
and under the alternative,
i
w
has the following non-
centrality
iv
T
iii PePQeS
(11)
where
i
S
is the size of the fault in the ith measurement.
The critical value for the test is
)1,0(
2/1
N
, where is
the significance level.
The Minimal Detectable Bias (MDB) specifies the lower
bound for detectable fault (outlier) with a certain
probability and confidence level. The MDB is
determined as (Baarda, 1968; Teunissen, 1990; Wang
and Chen, 1999)
iv
T
i
iPePQe
S0
0
(12)
where
0
is the non-centrality parameter, which depends
on the chosen power of the test (1- β) and significance
level or false alarm rate ().
For the cases of multiple faulty measuremnts, regardless
the true number of faults (outliers) that exist in the
Wang et al. Correlation Analysis for Fault Detection Statistics in Integrated GNSS/INS Systems 92
measurements, the faulty measurement detection statisitc
can be written as follows (e.g., Teunissen, 1990; Wang
and Chen, 1999; Knight et al, 2010):
2
0
1
2)(
PlPQGPGPQGPGPQl
Tv
T
iiv
T
iiv
T
i
(13)
where
2
0
the priori variance factor assigned as one in
this paper;
i
G
is an
n
by
matrix, with rank
,
containing zeros with a in each column
corresponding to the faults. The statistic
2
i
T
has a Chi-
square distribution with
as the degrees of freedom.
Fault identification based on Equation (13) can be
applied for different number of outliers, such as two or
three outliers. In this study, we consider the case that two
outliers exist in the measurements. In this case
is
equal to 2 and the
i
G
matrix takes the following form:
T
i
G
01.000
001.00
2
(14)
All the possible
2
i
T
may be presented as the
)(
n
matrix corresponding to the
i
G
matrix.
Simialr to the single fault case, the minimal detectable
bais can also be derived for the case of multiple faults,
but with very complicated formulae as such MDBs are
multiple dimensional, see the details in, e.g., Wang and
Chen, (1999), Knight et al. (2010).
2.2. Correlation coefficients between the fault
detection statistics
In the fault detection process, the largest fault detection
statistic (in absloute value) is associated with the most
likely faulty measurment. However, due to the
correlation between this statistic and any other fault
detection statistics, one fault may result in many fault
statistics being close each other. Therefore, such
correlation coeeficients are closely related to the fault
separability (e.g., Forstner, 1983; Li, 1986; Wang and
Knight, 2012). In order to ensure any two alternatives
are separable, such correlation coeeficeints should be
small.
The correlation coefficient between two single fault
detection statistics is given below (e.g., Förstner, 1983;
Hewitson and Wang, 2007)
jv
T
jiv
T
i
jv
T
i
ij PePQePePQe
PePQe
.
(15)
In this paper, the absloute value of
ij
is considered.
The bigger the correlation between two test statistics, the
more difficult they are to separate.
For the multiple faults case, the fault detection statistics
2
i
T
and
2
j
T
are for two groups of faulty measurements,
e.g., Group i and Group j. In the two fault case, these
two groups are actually two pairs of measurements, for
example, measurement pairs (2, 4) and (3,5).
Similarly, multiple fault separability may also be related
to the correlation between two fault vectors associated
with the two groups of measurements to be tested. For
convenience, the maximum correlation and the global
correlation are derived as follows (Förstner, 1983; Li,
1986):
)()( maxmax ijij MQ
(16)
).()).(.()( 11 jijjijijij ppppM
(17)
jviij PGPQGp
(18)
where
)(( max ij
M
is the maximum eigenvalue of
matrix
)( ij
M
. The rank of this matrix is the same as the
number of faults to be detected. And
i
G
and
j
G
correspond to the faults tests (i and j), respectively.
The examples of them are given by Equation (14)
The global correlation coefficients can also be computed
as follows (e.g., Li, 1986):
ji
ij
Globalij rr
Mtr
Q.
)(
)(
(19)
where
i
r
and
j
r
are the degrees of freedom for
2
i
T
and
2
j
T
, respectively.
3. Simulated Data Sets for GNSS and INS
Trajectory
In this research, GNSS and INS data simulations as well
as the tight GNSS/INS integration data processing were
based on GPSoft® Satellite Navigation, Inertial
Navigation System, Navigation System Integration and
Kalman Filter Toolbox. This software has been modified
to include reliability and correlation analsysis
functionalities for use in this study. The simulation
process is divided into two steps:
Wang et al. Correlation Analysis for Fault Detection Statistics in Integrated GNSS/INS Systems 93
Firstly, as shown in Fig. 1, a reference trajectory was
created through a moving reciever by using initial
position, initial velocity and initial attitude.
151.15 151.2 151.25 151.3
-33.95
-33.94
-33.93
-33.92
-33.91
-33.9
-33.89
-33.88
-33.87
-33.86
-33.85
longitude (deg)
latitude (deg)
Ground truth trajectory
Figure 1: Simulated vehicle trajectory
Then the INS delta velocity and delta theta were created
for a selected site in Sydney. The geodetic coordinates of
the selected location are 33 55
an elevation of 87m above the sea level. The MEMS,
Tactical and Navigation grade INS data sets were
simulated with an output rate of 200Hz along with
simulated initial alignment, velocity error and gyros and
acclerometer biases. The details for IMU sensor noise
parameters used in the simulations are listed in Table 1.
Table 1: IMU Noise Parameters for Simulations
IMU
Grades
Accelerometers
(m/s2)
Gyros
(deg/ )
MEMS
0.002
3
Tactical
0.0006
0.09
Navigation
0.0001
0.0015
The ground truth of the navigation solutions was created
from the error-free INS data.
Secondly, the satellite positions for each GNSS system
including GPS, GLONASS, GALILEO, COMPASS and
QZSS were calculated through Keplerian elements with
a 10-degree masking angle. (It should be stated here that
such simulated constellations are not related to
operational GNSS constellation scenarios). All the
GNSS pseudo-ranges were simulated with the standard
deviation of one meter.
The examples of simulated GNSS systems were: GPS,
GPS/QZSS, BeiDou, Glonass and Galileo. These data
sets were first processed individually, and then, in the
GNSS/INS intergration scenarios. The Kalman filter
(KF) states included three states for each of position,
velocity, attitude, gyros, accelerometer plus two states
for reciever clock error and drift. The KF states were
updated every second with GNSS pseudo-range
measurements.
The analysis of the MDBs for the signle fault as well as
the correlation analysis for the fault detection statisics in
various typical GNSS, and GNSS/INS integration
systems are presnted below.
4. Numerical Experiments and Analysis
4.1 Single fault scenarios
In order to evaluate the capability of various GNSS
systems to detect a faulty pseudo-range, the MDBs
defined by Equation (11) were calculated and presented
in Figs. (2)-(6).
0200 400 600 800 1000 1200 1400 1600
5
10
15
20
25
30
35 GPS only MDB
Time (seconds)
MDBs (m)
SV 1
SV 2
SV 3
SV 4
SV 5
Figure 2: MDBs for GPS Measurements
0200 400 600 800 1000 1200 1400 1600
5
6
7
8
9
10
11
12
13
14
15 GPS/QZSS only MDB
Time (seconds)
MDBs (m)
SV 1
SV 2
SV 3
SV 4
SV 5
SV 6
Figure 3: MDBs for GPS/QZSS Measurements
Wang et al. Correlation Analysis for Fault Detection Statistics in Integrated GNSS/INS Systems 94
0200 400 600 800 1000 1200 1400 1600
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10 BeiDou only MDB
Time (seconds)
MDBs (m)
SV 1
SV 2
SV 3
SV 4
SV 5
SV 6
SV 7
SV 8
Figure 4: MDBs for BeiDou Measurements
0200 400 600 800 1000 1200 1400 1600
4
5
6
7
8
9
10
11
12 GALILEO only MDB
Time (seconds)
MDBs (m)
GALILEO 1
GALILEO 2
GALILEO 3
GALILEO 4
GALILEO 5
GALILEO 6
GALILEO 7
Figure 5: MDBs for Galileo Measurements
0200 400 600 800 1000 1200 1400 1600
5
6
7
8
9
10
11 GLONASS only MDB
Time (seconds)
MDBs (m)
SV 1
SV 2
SV 3
SV 4
SV 5
SV 6
Figure 6: MDBs for Glonass Measurements
From the above figures, it is noted that in the various
GNSS constellations simulated here, the MDBs vary
from about 5 meters to 30 meters, even with the
simultaed noise level of 1 meter. These results
demonstarted the reality of how difficult it is to detect a
fault in the measurments. At the worst situations in the
simulated GPS only positioning, a fault could only be
detected when the its size became large as 30 meters.
The MDBs are highly denpendent on the number of the
satellites tracted and their geometric distributions.
In Figs. (7) to (10), the correlation coefficients between
the fault detection statisitcs for the measurement from
SV1 and the rest of the measuremnts are illudsrated for
various GNSS positioning secnarios.
In the simulated GPS constelation, there were only 5
SVs. Therefore, there was only one redundant
measurement in the positioning solution. As exptected, it
turned out that all the correlation coefficients between
any two fault detection statistics were exactly one! This
means that no matter which measuremnt is the faulty
one, all the fault detection statistics will have the same
value, which makes the fault identification impossible.
Thus, the faults in such a systen are insparable according
to both the new separability test (Wang and Knight,
2012) or the multiple hypothesis test (Forstner, 1983; Li,
1986).
The results shown in Fig. 7 demonstrate that, with even
one more QZSS satellite combined with GPS, the
correlation coefficients can be significantly reduced, but,
one of the correlation coefficients was still very close to
one over the first 5 minutes.
In the Glonass only positioning scenario, the some
correlation coefficients shown in Fig. (10) were very
close to one over a long period of time which will result
in an extremely weak fault separability.
The correlation coefficents shown in Figs. (8)(9) also
indicate that even with bigger number of SVs tracked,
such as 7 SVs for the BeiDou case, and 6 SVs for the
Galileo case, the correlation coefficients could reach 0.7-
0.8. Such weak geometries for poor fault separability are
not rare in satellite navigation. For example, Wang and
Knight (2012) demonstarted an even much worse
scenario where there were 8 SVs in a positioning
solution, but one of the correlation coefficients was as
high as 0.9999.
0200 400 600 800 1000 1200 1400 1600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1GPS/QZSS-only: correlation coefficients between SV1 and other SVs
Time (seconds)
Correlation Coefficient
Figure 7: Correlation Coefficients between Fault
Detection Statistics in GPS/QZSS Positioning
Wang et al. Correlation Analysis for Fault Detection Statistics in Integrated GNSS/INS Systems 95
0200 400 600 800 1000 1200 1400 1600
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 BeiDou-only: correlation coefficients between SV1 and other SVs
Time (seconds)
Correlation Coefficients
Figure 8: Correlation Coefficients between Fault
Detection Statistics in BeiDou Positioning
0200 400 600 800 1000 1200 1400 1600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 GALILEO-only: correlation coefficients between SV1 and other SVs
Time (seconds)
Correlation Coefficients
Figure 9: Correlation Coefficients between Fault
Detection Statistics in Galileo Positioning
Figure 10: Correlation Coefficients between Fault
Detection Statistics in Glonass Positioning
In this research it has been found that with the aiding of
INS, through a tight integration of GNSS/INS, both the
MDBs for GNSS measurements and the correlation
coefficients between faulty measurement detection
statistics can be remarkedly reduced. The details of these
results are illustrated in Figs. (11)-(18).
Figure 11: MDBs for GPS Measurements in GPS/INS
(Tactical Grade) Integration
0200 400 600 800 1000 1200 1400 1600
4
5
6
7
8
9
10
11
12
13
14 GPS/QZSS/INS
Time (seconds)
MDBs (m)
SV 1
SV 2
SV 3
SV 4
SV 5
SV 6
Figure 12: MDBs for GPS/QZSS Measurements in
GPS/QZSS/INS (Tactical Grade) Integration
0200 400 600 800 1000 1200 1400 1600
4
5
6
7
8
9
10 BeiDou/INS
Time (seconds)
MDBs (m)
SV 1
SV 2
SV 3
SV 4
SV 5
SV 6
SV 7
SV 8
Figure 13: MDBs for BeiDou Measurements in
BeiDou/INS (Tactical Grade) Integration
Wang et al. Correlation Analysis for Fault Detection Statistics in Integrated GNSS/INS Systems 96
Figure 14: MDBs for Galileo Measurements in
Galileo/INS (Tactical Grade) Integration
0200 400 600 800 1000 1200 1400 1600
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9GLONASS/INS
Time (seconds)
MDBs (m)
SV 1
SV 2
SV 3
SV 4
SV 5
SV 6
Figure 15: MDBs for Glonass Measurements in
Glonass/INS (Tactical Grade) Integration
The MDB results shown in Figs. (11)-(15) have
demonstrated the INS sensor can aid GNSS to reduce the
MDBs significantly. It is noted that for all the SVs, the
MDBs were coverged to about 4-5 meters. In Fig.16,
similar trends are noted for the maximum correlation
coefficients, which were reduced to the range below 0.3.
0200 400 600 800 1000 1200 1400 1600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Maximum correlation coefficients in GNSS/INS syst ems
Maximum corrlation coefficients
Time (seconds)
GPS/INS
GLONASS/INS
GPS/QZSS/INS
GALILEO/INS
BeiDou/INS
Figure 16: Correlations between Fault Detection
Statistics in GNSS/INS (Tactical Grade) Integration
In order to find out the impacts of the IMU sensor noise
levels on the MDBs and the correlation coefficients, the
GPS/INS integration was carried with simulated MEMS,
Tactical and Navigation IMU sensors, respectively. The
results illustrated in Figs. (17)-(18) for this simulated
trajectory indicate that IMU noise levels do not have a
significant impact on the MDBs and correlation
coefficients.
0200 400 600 800 1000 1200 1400 1600
4
6
8
10
12
14
16
18
20
22
24 Maximum MDBs in GPS /INS Integration
Time (seconds)
MDBs (m)
NAVIGATION GRADE IMU
TACTICAL GRADE IMU
MEMS GRADE IMU
Figure 17: Maximum MDBs in GPS/INS (MEMS,
Tactical and Navigation Grades) Integration
0200 400 600 800 1000 1200 1400 1600
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Maximum correlation coefficients in GPS/INS sys tem
Maximum corrlation coefficients
Time (seconds)
NAVIGATION GRADE IMU
TACTICAL GRADE IMU
MEMS GRADE IMU
Figure 18: Maximum Correlation between Fault
Detection Statistics in GPS/INS (MEMS, Tactical and
Navigation Grades) Integration
4.2 Two fault scenarios
Given the
PPQv
matrix and the dimension of faults,
which is equal to two in this case, the maximum and
global correlation coefficients between the multiple fault
detection statistics can be calculated. The numerical
results of such correlation coefficients in GPS/INS and
GPS/QZSS/INS integration are given below.
A numerical example for equation (18) at epoch 12 in
the GPS/INS inetegration is given below:
Wang et al. Correlation Analysis for Fault Detection Statistics in Integrated GNSS/INS Systems 97
22
00
00
00
10
01
0.0780.0100.020-0.0090.018-
0.0100.0530.020-0.012-0.008
0.020-0.020-0.0690.0130.010
0.0090.012-0.0130.0590.018-
0.018-0.0080.0100.018-0.081
00
10
00
01
00
j
v
iG
PPQ
G
ij
p
(20)
where
i
G
and
j
G
correspond to multiple fault test
2
i
T
and test
2
j
T
for measurement pairs (i) and (j).
One can note that the matrix
M
for the measurement
pair (2, 4) in
2
i
T
and (2, 4) in
2
j
T
has a diagonal of ones
and zeros. In fact, this is true by the definition because it
express the correlation coefficients between the
measurement pair and itself. Hence the correlation
coefficients will inevitably equal to one. The computed
correlation coefficients
Maxij
Q)(
, and
Globalij
Q)(
,
between the multiple fault detection statistics for
measurement pair (2, 4) in
2
i
T
and all the measurement
pairs in
2
j
T
are shown in Table 2 below and the similar
results are listed in Table 3 for GPS/QZSS/INS
integration.
Table 2 Multiple fault detection statistics and correlation
coefficients between measurement pair (2, 4) in
i
G
and
all the pairs in
j
G
in GPS/INS integration
j
G
Maxij
Q)(
Globalij
Q)(
2
k
T
(1,2)
1
0.709
19.80
(1,3)
0.421
0.305
3.29
(1,4)
1
0.726
17.42
(1,5)
0.266
0.245
0.82
(2,3)
1
0.726
25.80
(2,4)
1
1
44.58
(2,5)
1
0.72
20.70
(3,4)
1
0.716
17.82
(3,5)
0.310
0.271
5.08
(4,5)
1
0.717
19.23
It can be clearly seen in Tables 2 and 3 that the
correlation coefficients for multiple fault detection cases
can be categorized into three groups, i.e., the correlation
coefficients between the fault detection statistics for:
(a) the measurements pairs that share one same
measurement, e.g., (2, 4) and each of (1, 2), (1, 4), (2, 3)
(2, 5), (3, 4), (4, 5);
(b) the measurements pairs that do not have any common
measurements, e.g., measurement pair (2, 4) and (1, 3),
(1, 5) and (3, 5);
(c) the measurement pairs that share the two same
measurements, e.g, (2,4) and (2,4).
In the case of the maximum correlation method, one can
note that the correlation coefficients between the fault
detection statictics for any two measurement pairs in the
category (a) are equal to one.
In the case of global correlation approach, however, the
correlation coefficients between the fault detection
statictics for any two measurement pairs in the category
(a) are round 0.7.
In both correlation evaluation methods, the correlation
coefficients between the fault detection statictics for
those measurements pairs in the category (b) are tiny. In
addition, with both approaches the correlation
coefficients between the fault detection statictics for the
measurement pairs in the category (c) are equal to one
because it demonstrates the correlation between the fault
detection statictics for the measurement pair and itself.
In addition, the maximum correlation coefficients are
slightly higher than the associated global correlation
coefficients.
In order to show the capability of the system to separate
multiple faults in the integrated GNSS/INS systems, two
faults of 15m and 13m were added into measurements 2
and 4 respectively, for both GPS/INS, and
GPS/QZSS/INS integration. The multiple fault detection
statistics as well as the correlation coefficients between
measurement pair (2, 4) in
i
G
and all the pairs in
j
G
are
shown in Tables 2 and 3.
Table 3 Multiple fault detection statistics and correlation
coefficients between measurement pair (2, 4) in
i
G
and
all the pairs in
j
G
in GPS/QZSS/INS integration
j
G
Maxij
Q)(
Globalij
Q)(
2
k
T
(1,2)
1
0.707
18.07
(1,3)
0.373
0.264
0.55
(1,4)
1
0.735
18.87
(1,5)
0.296
0.216
2.56
(1,6)
0.411
0.348
11.65
(2,3)
1
0.707
16.49
(2,4)
1
1
40.50
(2,5)
1
0.710
17.49
(2,6)
1
0.762
25.75
(3,4)
1
0.726
19.63
(3,5)
0.314
0.228
3.81
(3,6)
0.415
0.337
11.36
(4,5)
1
0.714
20.39
(4,6)
1
0.714
22.21
(5,6)
0.434
0.316
14.45
Wang et al. Correlation Analysis for Fault Detection Statistics in Integrated GNSS/INS Systems 98
The characteristics of the multiple fault detection
statistics in association with the correlation coefficients
can be categorized into three groups as follows:
The first group consists of measurement pair (2, 4)
only. Because the faults were injected in those
measurement pair, their fault detection statistic value
is the highest among other groups.
The second group are the measurements pairs that
include either 2 or 4. Due to the high correlation
coefficients between this group and the first group,
their fault detection statistic values are quite high.
The third group are the measurements pairs that
include neither 2 nor 4. Their correlation coefficients
with the first group is relatively low comparing with
the second group, therefore fault detection statistic
values are low too.
One can note that the second group is almost fully
correlated with the first group especially in the
maximum correlation approach case; nonetheless, their
fault detection statistic values are not the same as those
of the first group. In other words, the rates of change of
the fault detection statistic values are not constant. This
means that the correlation coefficients between these
groups of statistics are not linked with the fault detction
statistics proportionally. In the GPS/INS integration (see
Table 2), for instance, the correlation coefficients
between the measurement pairs (2, 4) and (1, 2) are
equal to one. However, the fault detection statistic value
of the first pair is 44.58, which is the largest one, while
the fault detection statistic for the second pair is only
19.80. Similar situations are noted in Table 3 for the
GPS/QZSS/INS integration. But, with a given
significance level of 0.1%, and thus, the critical value of
the fault test as 13.82, the two simulated fault
measurements were detected and identified correctly.
5 Concluding Remarks
Fault detection and identification is an important
procedure for use in modern positioning and navigation.
The ability of a positioning and navigation system to
detect and identify one or muliple faults is highly
dependent on the geometric strength of the system.
With the use of the minimal detectable bias (MDB) and
the correlation coefficients between the fault detection
statistics, the reliability of various GNSS constellations
and GNSS/INS integration scenarios have been
analysed. Through the simulated data sets, this research
has demonstrated some critical reliability issues within a
GNSS positioning system with the pseudo-ranges based
single point positioning (SPP) mode, particualry at the
times when the number of the tracked SVs is low, and
the MDBs and correlation coefficients are high. Such
critical issues can be well addressed with the use of INS
sensors through the tight integration of GNSS/INS, even
with low-cost MEMS sensors.
For multiple fault scenarios, the maximum and global
correlation coefficients between the fault detection
statistics have been used to analyse the mutiple fault
separability performances in GPS/INS integration. It has
been shown that, for the case of two faults, when the
measurement pairs share at least one common
measurement, the maximum correlation coefficients
between those measurements pairs become one.
However, not like the situation of single facut, although
the maximum correlation coefficients are exatly one, the
successful identification of multiple faults is still
possible. This is due to the fact that the rate change of
multiple fault detection statistic values is not
proportionally related with the maximum or global
correlation coefficients. Therefore, such a relationship
should be further investigated before the multiple fault
separability can be properly related to these correlation
coefficients.
References
Baarda W. (1968) A testing procedure for sse in
geodetic networks, Netherlands Geodetic
Commission, New Series, 2(4).
Almagbile A. and Wang J. (2011) Analysis of outlier
separability in integrated GPS/INS systems. IGNSS
symp. Sydney Australia 15-17 November.
Almagbile A., Wang J., Ding W. & Knight N. (2011)
Sensitivity analysis of multiple fault test and
reliability measures in integrated GPS/INS systems.
Archives of Photogrammetry, Cartography and
Remote Sensing (APCRS), 22, 25-37
Brenner M. (1995) Integrated GPS/Inertial Fault
Detection Availability, 9th International Technical
Meeting of the Satellite Division of the Institute of
Navigation, Kansas City, USA, 1949-1958
Diesel J. and King J. (1995) Integration of Navigation
Systems for Fault Detection, Exclusion and
Integrity Determination – Without WAAS, ION
National Technical Meeting, California, USA, 683-
692.
Förstner W. (1983) Reliability and discernability of
extended Gauss-Markov models, Deutsche
Geodätische Kommission (DGK), Report A, 98, 79-
103.
Hewitson S. and Wang J. (2006) GNSS Receiver
Autonomous Integrity Monitoring (RAIM)
performance analysis, GPS Solutions, 10(3), 155-
170.
Hewitson S. and Wang J. (2007) GNSS Receiver
Autonomous Integrity Monitoring (RAIM) with a
Wang et al. Correlation Analysis for Fault Detection Statistics in Integrated GNSS/INS Systems 99
dynamic model, Journal of Navigation, 60(2), 247-
263.
Hewitson S. and Wang J. (2010) Extended Receiver
Autonomous Integrity Monitoring (eRAIM) for
GNSS/INS Integration. Journal of Surveying
Engineering, 136(1), 13-22.
Knight N., Wang, J. and Rizos C. (2010) Generalized
measures of reliability for multiple outliers, J Geod,
84(4), 625-635.
Kok J. (1984) On data snooping and multiple outlier
testing. NOAA Technical Report, NOS
NGS.30,USA. Department of Commerece
Rockvillle, Maryland
Li D. (1986) Trennbarkeit und Zuverlässigkeit bei zwei
verschiedenen Alternativhypothesen im Gauß-
Markov-Mödell. Z.f.Verm,Wesen 111, 114-128
Nikiforov I. (2002) Integrity Monitoring for Multi-
Sensor Integrated Navigation Systems, ION GPS
2002, 24-27 September, Portland, OR.
Ochieng W.Y., Sheridan K.F., Sauer K., Han X., Cross
P.A., Lannelongue S., Ammour N. & Petit K. (2002)
An Assessment of the RAIM Performance of a
Combined Galileo/GPS Navigation System Using
the Marginally Detectable Errors (MDE)
Algorithm, GPS Solutions, 5(3), 42-51.
Availability and Reliabilty
Advantages of GPS/Galileo Integration, ION GPS
2001, Salt Lake City, Utah, September 11-14, 1-10
Salzmann M. (1993) Least squares Filtering and
Testing for Geodetic Navigation Applications,
Netherlands Geodetic Commission, publications on
Geodesy, New Series, Delft, The Netherlands, No.
37, 209 pp.
Sturza M. A. (1989) Navigation System Integrity
Monitoring Using Redundant Measurements,
Navigation: Journal of the Institute of Navigation,
35(4), 69-87.
Sorenson H. W. (1970) Least Squares Estimation: from
Gauss to Kalman, IEEE Spectrum 7, 63-68
Teunissen P. J. G. (1990) Quality control in integrated
navigation systems, Proc IEEE PLANS’90, Las
Vegas, U.S.A. 158-165
Wang J. (2002) Applications of pseudolites in
positioning and navigation: Progress and problems.
Journal of Global Positioning Systems, 1(1), 48-56.
Wang J. and Chen Y. (1999) Outlier detection and
reliability measures for singular adjustment models.
Geomat Res Aust. 71, 57-72
Wang J., Knight N. and Lu X. (2011) Impact of the
GNSS time offsets on Positioning Reliability.
Journal of Global Positioning Systems, 10(2), 165-
172.
Wang J. and Knight N. (2012) New Outlier Separability
Test and Its Application in GNSS Positioning.
Journal of Global Positioning Systems, 11(1), 46-57.
Wang J. and Kubo Y. (2010) GNSS Receiver
Autonomous Integrity Monitoring, In: (Eds.
Sugimoto S & R. Shibasaki): GPS Handbook,
Asakura, Tokyo, 197-207.
Wang J., Lee H.K., Hewitson S. and Lee H.-K. (2003)
Influence of Dynamics and Trajectory on
Integrated GPS/INS Navigation Performance.
Journal of Global Positioning Systems, 2(2), 109-
116.
Wang J. and Ober P.B. (2009) On the Availability of
Fault Detection and Exclusion in GNSS Receiver
Autonomous Integrity Monitoring. Journal of
Navigation, 62(2), 251-261.
Wang J., Stewart M. and Tsakiri M. (1997) On quality
control in hydrographic GPS surveying, Third
Australian Hydrographic Symposium, 30 November-
3December, Fremantle, Westren Australia, 1-10.
Wang J., Xu C. and Wang J.A. (2008) Applications of
robust Kalman filtering schemes in GNSS
navigation. Int. Symp. on GPS/GNSS, Yokohama,
Japan, 25-28 November, 308-316.
Biography
Jinling Wang is an Associate Professor in the School of
Surveying and Geospatial Engineering, University of
New South Wales (UNSW). His major research interests
are in the areas of navigation and geospatial mapping
with multi-sensors, such as GNSS, INS, cameras. He has
published over 200 papers in journals and conference
proceedings as well as two commercial software
packages (www.gmat.unsw.edu.au/wang). He is a
Fellow of the Royal Institute of Navigation (RIN), UK, a
Fellow of the International Association of Geodesy
(IAG), and is a member of the Editorial Board for the
international journal GPS Solutions, Journal of
Navigation, International Journal of Navigation and
Observations, and President of IAG Sub-Commission
4.2 (2011-2015) on Geodesy in Geospatial Mapping and
Engineering. He was elected 2004 President of the
International Association of Chinese Professionals in
Global Positioning Systems (CPGPS), and the Founding
Editor-in-Chief for the Journal of Global Positioning
Systems (2002-2007).