multi-sensor data fusion based on information theory
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Multi-sensor data fusion based on Information Theory.
Application to GNSS positionning and integrity
monitoring
Nourdine Aït Tmazirte, Maan E. El Najjar, Cherif Smaili and Denis Pomorski
LAGIS UMR 8219 CNRS/Université-Lille1
Avenue Paul Langevin 59655 Villeneuve d’Ascq, France
Maan.El-Badaoui-el-najjar@univ-lille1.fr
Abstract — Integrity monitoring is considered now as an
important part of a vehicle navigation system. Localisation
sensors faults due to systematic malfunctioning require integrity
reinforcement of multi-sensors fusion method through
systematic analysis and reconfiguration method in order to
exclude the erroneous information from the fusion procedure. In
this paper, we propose a method to detect faults of the GPS
signals by using a distributed information filter with a
probability test. In order to detect faults, consistency is
examined through a log likelihood ratio of the information
innovation of each satellite using mutual information concept.
Through GPS measurements and the application of the
autonomous integrity monitoring system, the current study
illustrates the performance of the proposed fault detection
algorithm and the pertinence of the reconfiguration of the multi-
sensors data fusion.
Keywords:data fusion, Fault detection, information theory,
localisation, GNSS, RAIM.
I. INTRODUCTION
Autonomous navigation system requires a safety positioning
system. When leading safety, positioning services not only need
to provide an estimate of the vehicle location, but also
uncertainty estimation. In practice, an upper bound on the
positioning error, representing an integrity risk, is required to
determine if a positioning system can be used for a given task.
At GNSS (Global Navigation Satellite System) receiver level,
standard approaches introduce a Fault Detection & Exclusion
(FDE) stage to monitor the integrity of the estimation of the
position. This is known as Receiver Autonomous Integrity
Monitoring (RAIM) [1] [2]. Numerous FD studies have been
conducted with real time sensors measurements. When we do
not dispose of a sensor model, which is always the case with
localisation sensors, FDE methods are approached in a statistical
manner. Classically, stochastic approaches are developed
through KALMAN filters [3], Particle Filter [4] or interval
analysis [5]. These procedures for FD introduce residual testing
through statistical approach [6]. In this residual test scheme,
generally, FD is conducted by examining the probabilistic
property of the estimated state itself [5], yet a precise reference
system unaffected by failures is required. The implementation of
this kind of architecture requires a thresholding process in order
to pick out measurements errors. This kind of process could
easily detect sporadic measurements errors but hardly detect the
gradually increasing error of measurements. In addition,
thresholds are fixed classically with a heuristic manner.
Recently, the information filter (IF), which is the informational
form of the KALMAN filter (KF), has proved to be attractive for
multi-sensors fusion, like in [7],[8] or [9]. The IF uses an
information matrix and an information vector to represent the
co-variance matrix and the state vector usually used in a KF.
This difference in representation makes the IF superior to the KF
concerning multiple sensor fusion, as computations are simpler
and no prior information of the system state is required [10]
[11]. In addition, consistency test using log likelihood ratio
(LLR), based on information matrix, permits to have a good
balance between the detection probability and the false alarm
probability.
With the help of information filtering, the main interest of this
paper is to ensure the reliability of the GNSS positioning in a
navigation system. In this paper, a new GNSS integrity
monitoring method is presented by applying a distributed IF and
a LLR test. Specifically, the presented (FDE) scheme takes
advantage of the highly LLR change between the information of
each satellite and the mutual information of a set of visible
satellites by a GNSS (GPS) receiver. The performance of the
distributed IF is notable after reconfiguration of the data fusion
procedure.
This paper is organized as follows: Section 2 briefly introduces
the general principle of distributed IF. In section 3, a detailed
GPS integrity monitoring algorithm that employs distributed IF
and ratio test is illustrated. A result using real GPS measurement
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data is presented in section 4, followed by the conclusion in
section 5.
II. INFORMATION THEORY AND FILTERING
A. Information filter
An Extended KF can easily be expressed in another form, called
Nonlinear Information Filter (NIF) or inverse covariance filter
[10]. Instead of working with the estimation of states 𝑥(𝑖|𝑗) and
the variance 𝑃(𝑖|𝑗) , the IF deals with the information state
vector 𝑦 (𝑖|𝑗) and information matrix 𝑌(𝑖|𝑗), where:
y 𝑖 𝑗 = P−1 𝑖 𝑗 . x 𝑖 𝑗 (1)
𝑌(𝑖|𝑗) = 𝑃−1(𝑖|𝑗) (2)
The information matrix is closely associated with the Fisher
information. The physical mean of the Fisher information is the
surface of a bounding region containing probability mass. It
measures the compactness of a density function.
𝑥 𝑘 = 𝐹 𝑘 . 𝑥 𝑘 − 1 + 𝑤 𝑘 (3)
Where 𝑥(𝑘) is the state vector, 𝐹(𝑘) the state transition matrix
and 𝑤(𝑘) the process noise.
The observation is also modeled:
𝑧 𝑘 = 𝐻 𝑘 . 𝑥 𝑘 + 𝑣(𝑘) (4)
Where 𝑧(𝑘) is the observation vector, 𝐻 𝑘 the observation
matrix and 𝑣(𝑘) is a white noise.
The update stage of the NIF is written :
𝑦 𝑘|𝑘 = 𝑦 𝑘|𝑘 − 1 + 𝑖 𝑘 (5)
𝑌 𝑘|𝑘 = 𝑌 𝑘|𝑘 − 1 + 𝐼 𝑘 (6)
Where
𝑖 𝑘 = 𝐻𝑇 𝑘 .𝑅−1 𝑘 . 𝑧(𝑘) (7)
𝑖(𝑘) is a vector containing the contribution in term of
information from an observation 𝑧(𝑘).
𝐼 𝑘 = 𝐻𝑇 𝑘 .𝑅−1 𝑘 .𝐻𝑇 𝑘 (8)
𝐼(𝑘) is the associated information matrix. The couple
(𝑖 𝑘 , 𝐼(𝑘)) represents the Information Contribution (IC) of the
observation.
In multiple sensor problems:
zi k = Hi k . x k + vi k i = 1…N (9)
the estimate does not represent a simple linear combination of
contributions from individual sensors :
𝑥 𝑘 𝑘 ≠ 𝑥 𝑘 𝑘 − 1 + 𝑊𝑖𝑁𝑖=1 𝑘 . 𝑧𝑖 𝑘 −
𝐻𝑖 𝑘 . 𝑥 𝑘 𝑘 − 1
(10)
(with 𝑊𝑖(𝑘) independent gain matrices). The innovation
generated from each sensor is correlated. Indeed, they share
common information through the prediction 𝑥 (𝑘|𝑘 − 1).
In information form, estimates can be constructed from linear
combinations of observation information.
𝑦 𝑘 𝑘 = 𝑦 𝑘 𝑘 − 1 + 𝑖𝑖 𝑘 𝑁𝑖=1 (11)
It is because information terms 𝑖𝑖(𝑘) from each sensor are
assumed to be uncorrelated. Now it is straightforward to
evaluate the contribution of each sensor (unlike for a KF). Each
sensor node simply generates the information terms 𝑖𝑖(𝑘). These
are summed at the update step of the NIF to produce a global
information estimate.
B. Mutual Information
To evaluate the IC of each sensor, a well known concept of
Information theory can be used: the Mutual Information (MI).
The MI 𝐼(𝑥, 𝑧) is an a priori measure of the information which
will be gained by x with a set of observations z.
𝐼(𝑥, 𝑧) is defined in [12] as:
𝐼 𝑥, 𝑧 = −𝐸 ln𝑃 𝑥, 𝑧
𝑃 𝑥 .𝑃 𝑧 = −𝐸 ln
𝑃 𝑥 𝑧
𝑃 𝑥
= −𝐸 ln𝑃 𝑧 𝑥
𝑃 𝑧
(12)
In term of entropies, the MI is written:
𝐼 𝑥, 𝑧 = 𝐻 𝑥 + 𝐻 𝑧 − 𝐻 𝑥, 𝑧 = 𝐻 𝑥 − 𝐻 𝑥 𝑧 (13)
With 𝐻(𝑥), 𝐻(𝑧) the respective entropies of 𝑥 and 𝑧. 𝐻 𝑥 𝑧 and 𝐻 𝑧 𝑥 the conditional entropies.
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III. FAULTY DETECTION AND EXCLUSION : APPLICATION TO
GPS INTEGRITY MONITORING
GNSS integrity monitoring or Receiver Autonomous Integrity
Monitoring (RAIM) is a technology developed to control the
integrity of global navigation satellite system signals in a
receiver system. During a long time, integrity was exclusively
used in aviation or marine navigation. The integrity represents
the ability to associate to a position a reliable indication of trust.
With the development of different safety-critical urban
applications, the notion of integrity became an important field of
research, in particular in urban canyon, where satellites signals
are subject to hostile environment.
Usually, RAIM algorithms use a comparison of the result of a
main equation system and subsystems results [4], we propose to
use the information theory to quantify the information
contribution of each satellite (Figure 1). In other words, we
propose to detect and to exclude faulty satellite upstream,
meaning before the equation system resolution.
Figure 1. Satelllite Information contribution for GNSS Monitoring
A. State space representaion for GNSS Positionning
GNSS positioning with pseudo-range is a Time of Arrival
method [5]. Pseudo-ranges are the distances between visible
satellites and the receiver plus the unknown difference between
the receiver clock and the GNSS time. Thus, GNSS positioning
is a four dimensional problem: the 3D coordinates (𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 ) of
the user and the clock offset 𝑑𝑡𝑖 are unknown.
As well detailed in [13], the process model depends on the
dynamical characteristics of the vehicle. The system is assumed
to evolve according to the equation:
𝑥 𝑘 + 1 = 𝐴. 𝑥 𝑘 + 𝑤 𝑘 + 1
(14)
Where the state vector is composed of the eight following
variables:
𝑥 𝑘 =
𝑋 𝑘 ,𝑋 𝑘 ,𝑌 𝑘 ,𝑌 𝑘 ,𝑍 𝑘 ,𝑍 𝑘 , 𝑐𝜕𝑡 𝑘 , 𝑐𝜕𝑡 𝑘 (15)
[𝑋,𝑌,𝑍] representing the receiver position,[𝑋 ,𝑌 ,𝑍 ]the velocity
in Earth Centered Earth Fixed (ECEF) frame, 𝜕𝑡 the clock range
and 𝜕𝑡 the clock drift.
The state transition matrix is given by:
𝐴 =
1 𝑇 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 𝑇 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 𝑇 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 𝑇0 0 0 0 0 0 0 1
(16)
Where 𝑇 is the sample period (in our case 1 second). The
measurement noise 𝑤 . is assumed to be time invariant:
𝑤 . =
𝑇2𝜍𝑥
2
2,𝑇𝜍𝑥
2,𝑇2𝜍𝑦
2
2,𝑇𝜍𝑦
2,𝑇2𝜍𝑧
2
2,𝑇𝜍𝑧
2, 𝑐𝜍𝑓2𝑇 +
𝑐𝜍𝑔2𝑇2
2, 𝑐𝜍𝑔
2𝑇
(17)
Where 𝜍𝑥2 ,𝜍𝑦
2 ,𝜍𝑧2 represent the process noise variances related
to the states 𝑥, 𝑦 and 𝑧 , 𝜍𝑓 and 𝜍𝑔 are the variances related to
clock offset and velocity, and 𝑐 the speed of light .
Making the assumption that 𝑤~𝑁 0,𝑄 means that 𝑤 is assumed to be a white Gaussian noise with covariance 𝑄 :
𝑄 =
𝛼𝜍𝑥2
𝛽𝜍𝑥2
000000
𝛽𝜍𝑥2
𝛾𝜍𝑥2
000000
00
𝛼𝜍𝑥2
𝛽𝜍𝑥2
0000
00
𝛽𝜍𝑥2
𝛾𝜍𝑥2
0000
0000
𝛼𝜍𝑥2
𝛽𝜍𝑥2
00
0000
𝛽𝜍𝑥2
𝛾𝜍𝑥2
00
000000
𝛾𝜍𝑓2 + 𝛼𝜍𝑔
2
𝛽𝜍𝑔2
000000
𝛽𝜍𝑔2
𝛾𝜍𝑔2
(18)
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With
𝛼 =
𝑐2𝑇3
3
(19)
𝛽 =
𝑐2𝑇2
2
(20)
𝛾 = 𝑇
(21)
The observation [𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖] of each satellite 𝑖 , is calculated
thanks to broadcasted ephemeris. The atmospheric errors are
also modeled 𝛿𝜌𝑖 𝐼𝑜𝑛𝑜𝑆 , 𝛿𝜌𝑖 𝑇𝑟𝑜𝑝𝑜
𝑆 . The pseudo-range can be
modeled as follow:
𝑅𝑖𝑠 = 𝑥𝑖 − 𝑥𝑆 2 + 𝑦𝑖 − 𝑦𝑆 2 + 𝑧𝑖 − 𝑧𝑆 2 +
𝑐. 𝑑𝑡𝑖 − 𝑑𝑡𝑠 + 𝛿𝜌𝑖 𝐼𝑜𝑛𝑜𝑆 + 𝛿𝜌𝑖 𝑇𝑟𝑜𝑝𝑜
𝑆
(22)
Being non-linear, the observation model is linearised around a
predicted state [𝑋𝑝𝑟 ] to obtain an observation matrix:
𝐻 =
∇𝑥1
⋮
∇𝑥𝑗
⋮∇𝑥
𝑛
0⋮0⋮0
∇𝑦1
⋮
∇𝑦𝑗
⋮∇𝑦
𝑛
0⋮0⋮0
∇𝑧1
⋮
∇𝑧𝑗
⋮∇𝑧
𝑛
0⋮0⋮0
1⋮1⋮1
0⋮0⋮0
(23)
With
∇𝑥𝑗
=𝜕𝑅𝑗
𝑠
𝜕𝑥= −(𝑥𝑆 − 𝑥𝑝𝑟 ) 𝑅𝑝𝑟
𝑠
(24)
∇𝑦𝑗
=𝜕𝑅𝑗
𝑠
𝜕𝑦= −(𝑦𝑆 − 𝑦𝑝𝑟 ) 𝑅𝑝𝑟
𝑠
(25)
∇𝑧𝑗
=𝜕𝑅𝑗
𝑠
𝜕𝑧= −(𝑧𝑆 − 𝑧𝑝𝑟 ) 𝑅𝑝𝑟
𝑠
(26)
And
𝑅𝑝𝑟𝑠 = 𝑥𝑆 − 𝑥𝑝𝑟
2+ 𝑦𝑆 − 𝑦𝑝𝑟
2+ 𝑧𝑆 − 𝑧𝑝𝑟
2
(27)
Each satellite noise is assumed to be uncorrelated with all others.
So, the noise described as follow 𝑣~𝑁 0,𝑅 is simply
represented by its diagonal time-invariant matrix R:
𝑅(. ) =
𝜍1
⋮
0
⋱
…
𝜍𝑗
…
⋱
0
⋮
𝜍𝑛
(28)
B. Mutual Information for FDE
As proved in [14] the entropy, for a multivariate Gaussian
distribution (which is assumed to be the case in Eq. 14 for our
study), goes as the log-determinant of the covariance. Precisely,
the differential entropy of a d-dimensional random vector 𝑋
drawn from the Gaussian 𝑁(𝜇, 휀) is:
𝐻 𝑋 = 𝑁 𝑋 . ln 𝑁 𝑋 .𝑑𝑋
=𝑑
2+ 𝑑.
ln(2𝜋)
2+ ln
휀
2
(29)
From Eq.13 & Eq. 29, the MI in our case is defined:
𝐼 𝑋,𝑍 =𝑑
2+ 𝑑.
ln 2𝜋
2+ ln
𝑃(𝑋)
2− (
𝑑
2+ 𝑑.
ln 2𝜋
2
+ ln 𝑃(𝑋|𝑍)
2)
It becomes a basic Log Likelihood Ratio :
(30)
𝐼 𝑋,𝑍 =1
2ln
𝑃(𝑋)
𝑃(𝑋|𝑍)
(31)
A Global Information Mutual Information (GOMI) can be
defined. 𝑃(𝑋) is the prediction covariance and 𝑃(𝑋|𝑍) is the
updated covariance. The GOMI becomes, in information form:
𝐼 𝑋,𝑍 =1
2ln
𝑌(𝑘|𝑘)
𝑌(𝑘|𝑘 − 1) (32)
𝐼 𝑋,𝑍 =1
2ln
𝑌 𝑘 𝑘 − 1 + 𝐼𝑖(𝑘)𝑁𝑖=1
𝑌(𝑘|𝑘 − 1) (33)
We also define the Partial Observation Mutual Information
(POMI) which will take in account only one information term of
sensor 𝑖.
𝐼𝑖 𝑋,𝑍 =1
2ln
𝑌 𝑘 𝑘 − 1 + 𝐼𝑖(𝑘)
𝑌(𝑘|𝑘 − 1)
(34)
The GOMI will prove to be an adapted residual for fault
detection, whereas the POMI will isolate the faulty observation.
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C. Principle of the FDE method
The structure of the proposed GNSS integrity monitoring
method is illustrated in figure 2 which correspond to a classical
prediction/correction filtering using IF but including a FDE
stage.
Figure 2. Classical GNSS Integrity Monitoring schema
In the proposed FDE approach, described in the following
flowchart, in figure 3, only five measurements are needed for the
integrity monitoring.
Figure 3. Information Theory based proposed method
With n measurements, n Information Contribution (IC1 … ICn)
(Eq. 7 & Eq.8) are computed. A GOMI (Eq. 33) is then
computed using the n IC. The GOMI permits to have an auto-
thresholding stage in order to detect the inconsistency of one or
several satellites.
In case of fault detection, a POMI (Eq. 34) is computed for each
satellite. These different POMIs are used to isolate the faulty
satellites measurements in order to be excluded from the
correction step of the NIF algorithm.
IV. EXPERIMENTAL RESULTS
In order to test the performance of the RAIM developed
approach, real data acquisition has been carried out with CyCab
vehicle produced by Robosoft (www.robosoft.fr/) with several
embedded sensors. In this work, measurements of GPS RTK
Thales Sagitta 02 system and open GPS Septentrio Polarx2e@
(Figure 4) are used.
Figure 4. Experimental vehicle.
The data acquisition has been carried out around LORIA Lab.
During these experiments we remarked, as shown in figure 5,
that when the positionning system uses all visible satellite
without a FDE stage, one satellite introduces a bias in the
positionning process. It seems that this satellite was in a bad
geometric configuration in respect to the building in front (Bat C
in the figure 4, 5 and 6). To compare, the test trajectory
reference with centimetric accuracy (GPS RTK) is plotted in
green in figure 6. This trajectory is about eighty meters length.
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Figure 5. Trajectory after fusion without FDE
Figure 6. Reference trajectory given by RTK GPS
In figure 7 and 8, the results of the detection and exclusion
process are presented for the set of visible satellites during the
test trajectory. GOMI concept, introduced in section 2 helps to
auto-threshold information contribution of all satellites. After
few steps of initialization, the contribution of each satellite goes
to ―stabilize‖ around a value. Indeed, a satellite, if still visible,
should not bring a contribution different to the information gives
by the steps before.
Figure 7, one can easily see the different surges of information
contribution. That because a pseudo-range is generally
overestimated when the corresponding satellite waves is subject
to multi-path. Thus, the IC of the faulty satellite is also
overvalued. This IC becomes inconsistent, and need to be
excluded.
In this test, we detected only one satellite real errors, but the
proposed approach can be generalized for multiple faulty
satellites.
Figure 7. Global Observation Mutual Information for Detection
The fault exclusion step is realised using the POMI computed
for each satellite. In fact, the POMI permits to identify the
inconsistency of the faulty satellite like discribed in figure 8.
One can see the surges of POMI of satellite 6 represented in
blue.
Figure 8. Partials Observation Mutual Information for Exclusion
In figure 9, is shown the performance of the fusion with FDE
after exclusion of satellite 6 from the fusion procedure. We note
that the trajectory in figure 9 is close to the GPS RTK trajectory
showed in figure 6.
748
Figure 9. Trajectory after fusion with FDE
Finally, the figure 10 shows performance of the proposed
approach with faulty satellite exclusion in blue. This figure
shows the position computation in the ECEF frame for each axis
(X,Y,Z). In red, are the position computation on ECEF frame
without faulty satellite exclusion.
Figure 10. Exclusion of a faulty satellite in GNSS positioning
V. CONCLUSION AND FUTURES WORKS
This paper proposes a reliable multi sensor data fusion approach
with integrity monitoring. This approach integrates Fault
Detection stage by using distributed NIF and LLR test using
mutual information. The proposed method makes it possible to
detect fault without heuristic thresholing step. It permits an
automatic reconfiguration of the fusion procedure in order to
exclude the faulty measure. The proposed approach is applied
for a GPS integrity monitoring. Measurements were pseudo-
ranges from the GPS satellites. The basic concept is to use the
Information Filter in a distributed architecture. The likelihood
function is established and examined by integrating state
estimate from distributed Information Filters. Thus, the LLR test
is used in fault detection, which compares the information
introduced by each satellite observation compared to the
expected information.
The evaluation of FDE is conducted through simulation using
real GPS measurements data. In the presented test trajectory, the
measured data from GPS contain the well known wave’s multi-
path error illustrated with an abnormal bias. Based on real data,
results demonstrate performance of the proposed approach in
detecting GPS measurements faults and fusion method
reconfiguration. In future work, our objective is to integrate dead
reckoning sensors in order to develop a tightly coupled data
fusion method. These sensors are known to introduce gradually
increasing errors (drift). The LLR is well adapted to detect this
kind of errors.
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[13] S. Cooper and H. F. Durrant-Whyte. ―A Kalman filter model for GPS navigation of land vehicles‖. In: IEEE/RSJ/GI International Conference on Intelligent Robots and Systems, p. 157—163, 1994.
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