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

743

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.

744

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.

747

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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