Joelle Al Hage

Philippe Xu

Philippe Bonnifait

High Integrity Localization With Multi-Lane Camera Measurements

Localization with high integrity is crucial for highly autonomous vehicles. This requires that the localization system send a warning to a client application when it should not be used. The concept of integrity was firstly developed for aviation applications and recently became an active research area for autonomous vehicles. GNSS information merged with dead reckoning sensors is not sufficient for lane level localization in all navigation environments. Map-aided localization with vision sensors is essential to provide redundant and complementary information. In this work, a multi-sensor data fusion method that takes advantage of a high definition (HD) map is presented and the integrity of the obtained solution is quantified. A Fault Detection and Exclusion (FDE) step is added to exclude the faulty measurements from the fusion procedure. A second step is to bound the estimation errors in the Along Track (AT) and Cross Track (CT) directions through Protection Levels (PL). For this step, the usual Gaussian distribution is replaced by a Student's distribution with an adapted degree of freedom chosen according to the navigation environment. The performance of the approach is evaluated with an experimental vehicle equipped with a camera able to detect up to four lane markings simultaneously.

I. INTRODUCTION

Within the scope of this paper, autonomous driving includes levels 3 to 5 as defined by the Society of Automotive Engineers (SAE) in Standard J3016. That is when the vehicle can navigate autonomously either under certain conditions or without any constraint. Hence, high automation (levels 3 and 4) and full automation (level 5) require the ability of the car to give warning or to make a safe decision in case of a problem which is related to the integrity concept.

Reliable and accurate positions are an essential part for autonomous vehicles in order to navigate safely. Different sensors can be used like wheel speed sensors, gyros, GNSS, LiDARs and cameras. Low-cost GNSS receivers available currently in the market are not adapted for lane level localization especially in urban environments. Therefore, autonomous navigation cannot rely only on these kinds of sensors.

Localizing autonomous vehicles is generally associated with High Definition (HD) road maps that are used to retrieve accurate information for the navigation task [START_REF] Betaille | Creating enhanced maps for lane-level vehicle navigation[END_REF], [START_REF] Quddus | Current mapmatching algorithms for transport applications: State-of-the art and future research directions[END_REF]. Associating an estimated position obtained from GNSS or dead-reckoning with roads map is done via a map matching procedure [START_REF] White | Some map matching algorithms for personal navigation assistants[END_REF], [START_REF] Kasmi | Map matching and lanes number estimation with openstreetmap[END_REF]. Additional information from exteroceptive sensors (e.g. cameras) can be used to detect lane markings and enhance the positioning output [5], [6].

The authors are with Heudiasyc, UMR 7253, Université de Technologie de Compiègne, Sorbonne Universités, France. The authors are also members of SIVALab, a joint laboratory between Renault/Heudiasyc/CNRS/UTC.

Integrity is an emergent concern for road localization. Indeed, localization integrity was initially developed for aeronautical applications in order to provide a measure of confidence for GNSS-based navigation solutions [7]- [START_REF] Andres | Integrity monitoring applied to the reception of GNSS signals in urban environments[END_REF]. It is associated with a target integrity risk which represents the maximum probability that the error in position exceeds a given limit without warning the user. This limit is known as the Protection Level (PL). Aeronautical integrity methods cannot be applied to road vehicles for different reasons [START_REF] Hage | Bounding localization errors with student distribution for road vehicles[END_REF]. Indeed, they make the assumption of Gaussian distribution with zero mean and they suppose the occurrence of one fault at a time. These hypotheses verified in open sky environments cannot be generalized to urban applications because of multipath and non-line of sight signals which are very frequent in such environments [START_REF] Groves | Intelligent urban positioning using multi-constellation GNSS with 3d mapping and NLOS signal detection[END_REF], [START_REF] Ali | Measured performance of the application of EGNOS in the road traffic sector[END_REF]. Moreover, the PL obtained for aeronautical applications are too conservative to be applied to autonomous vehicles. Indeed, most airways are eight nautical miles (14 kilometers) wide while lane width for road vehicles is about 3.5 m.

In this paper, we propose a method for lane level localization for autonomous vehicles with integrity study. For this purpose dead-reckoning measurements are merged with GNSS measurements (GPS and GLONASS) obtained from a low-cost mono-frequency receiver. A smart camera returns measurements of detected lane markings that are used to enhance the localization procedure after associating with a prior HD map. The enhancement is especially done in the cross track direction.

Measurements from sensors are affected by errors where the origin is the sensors or the environmental noises. These errors have a direct effect on the integrity of the localization if they are not detected and excluded from the fusion procedure. Therefore, a Fault Detection and Exclusion (FDE) step is added in order to test the consistency of the GNSS and the camera measurements with the dead-reckoning measurements. A detected error on the camera measurements can get its origin from the camera or from the map. In a second step, the purpose is to bound the errors and to compute a PL after removing all the detected faults. For this step, the Gaussian distribution is replaced by a Student's t-distribution which is heavy tailed and thus permits to take into account the errors not considered when using a Gaussian distribution [START_REF] Hage | Bounding localization errors with student distribution for road vehicles[END_REF]. Likewise, with this distribution, we aim to avoid an undervaluation of the PL. Since we work with autonomous vehicles, it is interesting to express the PL in AT and CT directions; this allows to visualize the benefit of the camera in reducing the PL values in the CT direction.

This paper is organized as follows. Section 2 presents the system modeling and the multi-sensor data fusion. The integrity study divided into FDE and error bounding is presented in section 3. Section 4 shows the experimental study and the performance of the method on a recorded dataset in Rambouillet, France. Conclusion and future work are given in section 5.

II. SYSTEM MODELING

The system uses wheel speed sensors, a gyro, GNSS, an intelligent camera for lane markings detection and an HD map. The localization is defined relatively to a Cartesian ENU frame (East, North, Up) denoted R O (figure 1). The origin of this frame can be any position close to the navigation area. Other working frames are needed, and they are shown in figure 1. R B is the body frame located in the middle of the rear wheel axis of the vehicles where the vehicle pose (x, y, θ) is defined. The detected lanes are expressed in a virtual frame R M located at the center of the front bumper while the camera is located behind the windscreen. The GNSS fix is obtained relatively to the antenna (frame R G ) which is translated with a lever arm relatively to the body frame. The lanes in the map are represented as polylines with an absolute accuracy of the order of a few centimeters. A view of the map is given in figure 2.

At instant k, the state vector is considered to be the position and the heading of the vehicle:

X = [ x y θ ] T (1)
The GNSS observation is given by:

Z 1 = x GNSS y GNSS = t x cos θ -t y sin θ + x t x sin θ + t y cosθ + y , (2)
where t x and t y are the translation of the antenna with respect to the body frame. A smart camera detects lane markings and returns the coefficients of a Taylor's expansion of a clothoid in the frame R M [START_REF] Tao | Road-centered map-aided localization for driverless cars using single-frequency GNSS receivers[END_REF]:

y = C 3 x 3 + C 2 x 2 + C 1 x + C 0 ( 3
)
The camera is able to detect up to two markings at a time on each side: right and left. In this paper, we consider only the use of the parameter C 0 (the lateral distance between the lane marking and the frame R M ) to enhance the localization procedure:

Z 2,j = C 0 (4)
where j could go from 1 to 4 representing C 0,right1 , C 0,right2 , C 0,left1 , C 0,left2 if they exist. The camera observation model was developed in [5], [START_REF] Tao | Road-centered map-aided localization for driverless cars using single-frequency GNSS receivers[END_REF] and can be written as:

C 0 = (P x sin θ + y -y A ).x AB -(P x cos θ + x -x A ).y AB x AB . cos θ + y AB . sin θ
where the lane marking is detected on the segment

[AB] at coordinates [x L , y L ] expressed in frame R O as: O L = O T B . B T M . M L (5)
O T B and B T M are the transformation matrices from R B to R O and from R M to R B , respectively. Therefore,

x L y L = P x cos θ + C 0 sin θ + x P x sin θ -C 0 cos θ + y (6)
where M L = [0, C 0 ] and P x is the distance between R B and R M (view figure 1).

For state estimation, an extended information filter is used. The information matrix and information vector are updated as follows [START_REF] Durrant-Whyte | Multisensor data fusion[END_REF]:

Y k/k = Y k/k-1 + n i=1 I i,k , (7)
y k/k = y k/k-1 + n i=1 i i,k , (8)
with n the number of observations at instant k and

I i,k = H T i,k R -1 i,k H i,k (9)
i i,k = H T i,k R -1 i,k Z i,k -Ẑi,k + H i,k X k/k-1 . ( 10
) (I i,k , i i,k
) are the information contributions of the observation i, that could be the GNSS measurements or the camera observations (left or right side).

H i is the observation matrix defined as a Jacobian. R i is the covariance associated to observation noise Z i .

It should be noted that the evolution model is obtained from the odometric model using the wheel speed sensors and the gyro [START_REF] Hage | Bounding localization errors with student distribution for road vehicles[END_REF]. The information matrix and the information vector are obtained as:

Y k+1/k = P -1 k+1/k (11)
y k+1/k = Y k+1/k X k+1/k (12)
with P k+1/k the covariance matrix.

The simple summations that appear in equations ( 7) and (8) are well adapted for multi-sensor data fusion as well as for multi-fault detection and exclusion.

III. MULTI-SENSOR DATA FUSION WITH INTEGRITY A. Fault detection and exclusion

The observations from the GNSS receiver or from the camera can be erroneous or faulty. The origin of the fault can come from the sensor, the map or the environmental conditions. Erroneous measurements, if not detected, affect directly the localization procedure. Hence, a FDE step is added where the Mahalanobis distance between the predicted state and the updated state is used as a residual:

r k = (X k/k -X k/k-1 ) T Y k/k (X k/k -X k/k-1 ) (13)
where X k/k and Y k/k are obtained from a filter that uses all available measurements at instant k. Please note that this residual acts in the state space and detects faults which have a direct influence on the position estimate (it is not an innovation gating as classically done). A fault is detected if r k exceeds a threshold value obtained from a Chi-squared distribution with 3-degrees of freedom according to a given false alarm probability. For the exclusion step, residuals that each use only one observation i (e.g. GNSS fix, C 0,right,1 , C 0,left,1 , C 0,right,2 , C 0,left,2 ) are generated:

r i,k = (X k/k,i -X k/k-1 ) T Y k/k,i (X k/k,i -X k/k-1 ) (14)
where X k/k,i and Y k/k,i are obtained from a filter that uses only the observation i in the update step. Using these residuals, it is possible to distinguish between GNSS and camera measurement errors (left and right).

The architecture of the multi-sensor data fusion with FDE step is given in figure 3 and a summary is given in the following.

At instant k, if observations are available (GNSS or camera), the residual for the fault detection can be computed using the informational contributions of the different available measurements (equation 13). If this residual exceeds the threshold value, a fault is detected and its origin has to be determined in order to proceed to its exclusion from the fusion procedure. Therefore, the set of residuals r i,k (i ∈ {GNSS, C 0,left,1 ,C 0,left,2 , C 0,right,1 , C 0,right,2 }) is generated by creating a bank of information filters (EIF i ) where each one uses one observation. Notice that the complete set of residuals is not always available since the sensors measurements are introduced in an asynchronous manner. Moreover, the availability of the camera measurements depends on the navigation environment. The camera gives an indication of the quality of the lane marking detection. Therefore, a lane marking with a quality below a given value is rejected.

A measurement i is excluded if the corresponding residual exceeds the threshold value. For example, if r gnss >threshold, the GNSS measurements are excluded from the fusion procedure by subtracting their information contributions from the main filter.

Given that the camera is able to detect up to two lane markings on the right or left side, the redundancy can be used to differentiate between camera and map faults which is useful to detect map errors. Let us take the following example to illustrate this. At instant k, suppose the camera observes simultaneously C 0,left,1 and C 0,left,2 . If r C0,left, 1 obtained from the first observation exceeds the threshold value, we can conclude that a problem appears on this measurement. However, we cannot specify if the error comes from the camera or from the map. Suppose now that the residual r C0,left, 2 is below the threshold. In this case, at the same instant and on the same side, the camera gives one good measurement and one bad measurement. In these conditions, it is likely that the error comes from the map since the second measurement, almost in the same environment, is error free and detected with high confidence. A flag can be stored in the map to indicate that the map-matched segment is potentially erroneous. Now, if r C0,left,2 is greater than the threshold, it cannot be concluded that there is or not an error in the map. Indeed, if the two observations on the same side are faulty, the error may get its origin from the environment or from the camera itself. Likewise, if only one measurement is available on a given side, the differentiation between the camera and the map is not possible. Notice that the origin of the error (camera or map) does not affect the exclusion procedure. In other words, if r C0,j indicates the presence of a fault, the measurement C 0,j will not be used in the update step independently from the origin of the camera measurement error.

Finally, if all the residuals r i indicate the presence of faults, the hypothesis of a filtering problem or deadreckoning problem is likely and an alarm has to be raised.

After determining the origin of the faults (GNSS, C 0,left,1 , ...), the faulty measurements are excluded from the fusion procedure by subtracting their information contributions from equations 7 and 8.

B. Protection level computation

After removing the faulty measurements, the objective is now to bound the errors. The aim is to avoid an underestimation of the PL that may be due to an undervaluation of the covariance matrix of the filter or from undetected biases. Therefore, when computing the PL, the multivariate Gaussian distribution is replaced by a multivariate Student's distribution which has a heavier tail compared to the latter [START_REF] Hage | Bounding localization errors with student distribution for road vehicles[END_REF], [START_REF] Dhital | A new approach for improving reliability of personal navigation devices under harsh GNSS signal conditions[END_REF]. For this purpose, the matrix P k/k will be considered as the covariance matrix of a multivariate Student's t-distribution with a fixed degree of freedom N . Notice that when N → +∞, the t-distribution converges to a Gaussian one. For more details about the properties of the t-distribution, the reader can refer to [START_REF] Hage | Bounding localization errors with student distribution for road vehicles[END_REF].

EIF C 0 , left,2 EIF C 0 , right,1 EIF C 0 , right,2 X k/k-1 , Y k/k-1 X k/k-1 , Y k/k-1
The horizontal protection level is expressed as follows [START_REF] Navarro Madrid | Device and method for computing an error bound of a kalman filter based gnss position solution[END_REF]:

P L k (α) = K(α, N ) √ N -2 max(eigenvalue(P k/k )), ( 15
)
where K is obtained from a multivariate t-distribution with degree of freedom N and according to a given confidence level α. The K(α, N ) value can be computed by solving numerically the following equation [START_REF] Navarro Madrid | Device and method for computing an error bound of a kalman filter based gnss position solution[END_REF]:

α = 2 B d 2 , N 2 ∞ K y d-1 (1 + y 2 ) N +d 2 dy. ( 16
)
where

B d 2 , N 2
is the Beta distribution and d is the dimension of the t-distribution; for horizontal P L, d = 2.

For road vehicles, we are interested by protection levels in the AT and CT directions. Therefore, the covariance matrix in the ENU frame is projected to the (AT, CT) frame using equation 17:

P k/k,proj = R proj P k/k(1,2),(1,2) R T proj (17) with R proj = cos θ sin θ -sin θ cos θ ( 18
)
where θ is the orientation of the vehicle. The AT and CT PLs are obtained as:

P L AT = K(α, N ). √ N -2 [eigenvalue(P k/k,proj )] 1 ( 19
) P L CT = K(α, N ). √ N -2 [eigenvalue(P k/k,proj )] 2 (
20) where the indexes 1 and 2 denote the first and second elements of the vector (eigenvalue(P k/k,proj )).

IV. EXPERIMENTAL RESULTS

For the evaluation of the approach, an experiment was carried out in the streets of Rambouillet, France (figure 4), using an experimental vehicle equipped with Ublox 8T which is a single frequency multi-constellation GPS/GLONASS receiver, a NovAtel SPAN-CPT with network RTK corrections used as a ground truth and a Mobileye camera for lane marking detections. The positioning errors in the AT and CT directions after the data fusion and after the FDE step are compared to the case when using only the GNSS as shown in figure 5. It can be seen that the improvement is more significant in the CT direction where the errors are most of the time very small. This shows the importance of lane marking detections in improving the localization in this direction. The sampling instant corresponds to the frequency of the dead-reckoning (0.02 s).

Figure 6 shows the errors before and after the FDE step. The effect of the undetected errors on the position estimates is noticeable. Indeed, the surges that appear before the FDE step (in blue) in the CT and AT directions disappear after the FDE step (in red).

Some statistical values of the error before and after the FDE are shown in tables I and II respectively. One can see the reduction in the maximum errors values in the AT and CT directions after the FDE step.

A small part of the trajectory is given in figure 7. The estimated trajectory is displayed with the ground truth and the map. Note that the map matching algorithm used in this paper is a point to curve matching [START_REF] White | Some map matching algorithms for personal navigation assistants[END_REF]. displayed in figure 9. Figure 9a shows the residual associated to the GNSS measurements, and figures 9b and 9c show the residuals associated to the camera measurements in the left and right sides, respectively. When two measurements are available simultaneously on the same side, the residual associated to the second measurement is displayed in red.

If a fault is detected on a camera measurements, a visualization of the possible origin of the fault is presented in figure 10. The blue value indicates that the error probably comes from the map, the red value indicates that the origin of the error cannot be specified for one of the two reasons presented before. Figure 11 shows two areas where possible faults have been detected on the map. These detected faults truly correspond to map errors. Indeed, the Mobileye provided two detections on the left side (the link border and the centerline) while only the link border is present on the map which led to an incorrect association.

Let us study the computation of the PL in the AT and CT directions. As we have mentioned before, the Student's t-distribution replaces the Gaussian distribution for this step. To adjust the degree of freedom of the t-distribution, data acquisitions with a ground truth system are needed in the target environment (urban, suburban, open sky, etc.) [START_REF] Hage | Bounding localization errors with student distribution for road vehicles[END_REF]. The experiment described in this paper has been done in a suburban area. Accordingly, the degree of freedom of t-distribution is fixed to N =6 to preserve a heavy tail necessary to obtain a consistent PL. The target integrity risk is fixed to 10 -3 since we cannot test with a smaller value given the available number of samples. The performance of the computed PL in the AT and CT directions are shown in figure 12. It can be seen that the perception sensor used for lane markings detection reduces the PL in the CT direction down to 1.05 m. The intervals where the PLs in the CT direction show an increase correspond to situations where the vehicle is on a roundabout area. Indeed, the camera cannot detect markings on these kinds of trajectories because of its limited field of view. Therefore, in these areas, the localization is done using only dead-reckoning and GNSS which results in a direct augmentation in the computed PL. The PL in the AT direction is about 2.5 m. To further reduce this value, the camera should be used to detect also information like pedestrian crossing or traffic signs which has not been done in this work.

It should be noted that the computation of the PLs was consistent since the measured frequency of errors that exceeded the PL was lower than the fixed integrity risk: P (error > P L) = 0 < 10 -3 in the AT, CT and horizontal directions. This is not the case if the Gaussian distribution is used where the measured frequencies of errors that exceed the PL are 4.32 × 10 -4 , 0.0035 and 0.0012 in AT, CT and horizontal directions, respectively.

V. CONCLUSION AND FUTURE WORK

In this work, we have proposed a localization method that exploits lane features for autonomous vehicles with integrity considerations. For this purpose, a FDE strategy has been proposed. It takes advantage of different types of sensors measurements present in the system: GNSS, deadreckoning, Mobileye for lane markings detection and an HD map. A bank of information filters was used to determine the origin of the faults after creating a set of residual tests. The obtained results confirm the importance of the FDE step as it improves significantly the localization accuracy. In a second step, we have proposed a method to bound in a consistent manner the errors by replacing the Gaussian distribution with a Student's distribution. The PLs are computed in the AT and CT directions in order to give a concrete meaning of PL for classical road vehicles navigation tasks like path following. The obtained PL is about 1.05 m in the CT direction (when data from the perception sensor are available) and about 2.5 m in the AT direction.

In future work, the use of other perception sensors like velodyne LiDAR will be investigated in order to improve the results, especially on roundabouts.

Fig. 1 .

 1 Fig. 1. Frames representation.

Fig. 2 .

 2 Fig. 2. Detailed view of an HD map. The lane markings are in green, the centerline is in purple, pedestrian crossing is the rectangle in green.

Fig. 3 .

 3 Fig. 3. Architecture of the method to compute an accurate localization with protection levels.

Fig. 4 .

 4 Fig. 4. Experimental environment (Rambouillet, France). The path of the experimental vehicle is displayed in white.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Errors after data fusion compared to GNSS (sampling period of 0.02s).

Fig. 7 .Fig. 8 .

 78 Fig. 7. Zoom in a part of the trajectory: estimated trajectory (blue), ground truth (magenta), map (node and shape point) (red), matched points on the map (green and cyan). Scales in meters.The residual used for the fault detection is shown in figure8. When r indicated the presence of faults, the residuals used for the fault exclusion are generated (if available) and are

Fig. 9 .Fig. 10 .

 910 Fig. 9. Residuals for fault exclusion, in green the thresholds.

Fig. 11 .

 11 Fig. 11. Areas where map faults have been detected.

Fig. 12 .

 12 Fig. 12. Errors and PLs in the AT and CT directions.

TABLE I STATISTICAL

 I VALUES OF THE ERRORS BEFORE THE FDE STEP.

	error (m)	Horizontal	CT	AT
	Root mean square	0.72
	Median		-0.014	0.144
	Min (absolute value)		2.42×10 -5	2.38×10 -5
	Max (absolute value)		4.34	3.54
		TABLE II
	STATISTICAL VALUES OF THE ERRORS AFTER THE FDE STEP.
	error (m)	Horizontal	CT	AT
	Root mean square	0.585
	Median		-0.016	0.091
	Min (absolute value)		0.31×10 -5	1.81×10 -5
	Max (absolute value)		1.76	1.80

ACKNOWLEDGMENTS

This work was carried out in the framework of the Equipex ROBOTEX (ANR-10-EQPX-44-01) and SIVALab, a shared laboratory between Renault and Heudiasyc UMR UTC/CNRS. The authors thank Corentin Sanchez, Anthony Welte, Emmanuel Stawarsky, Stéphane Bonnet and Clément Zinoune for their support in the experiments.


Joelle Al Hage

Philippe Xu

Philippe Bonnifait

Student's t Information Filter with Adaptive Degree of Freedom for Multi-Sensor Fusion

Safety-critical applications such as autonomous driving require a high-integrity localization system that bounds the errors of the estimation process. In this paper, the classical Kalman filter used for multi-sensor data fusion, which is unable to consistently bound estimation errors with a low probability risk, is replaced by a Student's t filter. The degree of freedom of the t distribution offers a way of shaping the heavy tail of the distribution that makes the estimation process more robust in the presence of non-detectable bias and results in a more consistent confidence interval computation. We make use of the heavy-tailed property of the t distribution by introducing a novel real-time adaptive computation of the degree of freedom. The filtering process is formalized through an informational form, since this makes it easier to include a fault detection and exclusion step where a bank of filters is generated. The performance of the proposed approach is evaluated through a localization problem using data acquired from an experimental vehicle equipped with multiple sensors: a GNSS receiver, wheelspeed sensors, a yaw rate gyro and a smart camera that can detect several lane markings, together with high-definition maps.

I. INTRODUCTION

The social acceptability of autonomous vehicles depends on their ability to operate seamlessly in today's road networks. Accurate and reliable localization is therefore vital. Localizing autonomous vehicles generally relies on sensors such as GNSS (Global Navigation Satellite System) receivers, perception sensors (camera, LiDAR) and deadreckoning sensors. It also makes use of High Definition (HD) road maps [START_REF] Betaille | Creating enhanced maps for lanelevel vehicle navigation[END_REF], [START_REF] Quddus | Current mapmatching algorithms for transport applications: State-of-the art and future research directions[END_REF] used to perform a map matching by associating the estimated position (obtained via GNSS or dead-reckoning) with the map [START_REF] White | Some map matching algorithms for personal navigation assistants[END_REF], [START_REF] Kasmi | Map matching and lanes number estimation with Openstreetmap[END_REF]. Additional information from exteroceptive sensors such as cameras can be used to detect lane markings and to improve positioning.

One of the most commonly used filters for combining data from different sensors is the Kalman Filter (KF). This filter is optimal in terms of minimum mean square error for a linear model with Gaussian noise. However, for autonomous vehicles, the assumption of a Gaussian distribution is often not justified, especially in urban environments where the GNSS signals are affected by multi-path and non-line of sight. Large errors with small probabilities can appear in these environments, generating heavy-tailed distributions.

In this work, we are looking for a state estimator that provides consistency in terms of confidence intervals. Consistency is here a more important criterion than optimality. The Student's t Filter (StF) can be seen as a modification of KF for managing process and measurement noise that has a heavy-tailed distribution. This filter is known to be more robust to large errors and outliers [START_REF] Roth | Kalman filters for nonlinear systems and heavy-tailed noise[END_REF], [START_REF] Huang | Robust student's t based nonlinear filter and smoother[END_REF]. In this paper we develop an informational form of the StF to take advantage of its update step. This form is better suited to multi-sensor data fusion and to decentralized architectures. The development of the Student's t Information Filter (StIF) is done in a similar way to the development of the Information Filter (IF) [START_REF] Durrant-Whyte | Multisensor data fusion[END_REF]. Although the Student's t filter is quite robust to large errors, they still affect the localization procedure. For this reason we add a Fault Detection and Exclusion stage (FDE) to exclude erroneous measurements that have a direct effect on the estimation procedure. The Student's t distribution is characterized by a degree of freedom (dof) that shapes the tail of the distribution. It directly affects the performance of the filter and the computed confidence interval. In order to avoid setting a fixed degree of freedom, which is difficult to do when the environment changes, the residual used in the FDE stage is used to adapt the dof of the StIF in realtime. The literature has little to say about the determination of the appropriate dof of a Student's t filter, apart from the work done in [START_REF] Li | Robust selection of the degrees of freedom in the student's distribution through multiple model adaptive estimation[END_REF], where the authors propose using a multiple model adaptive estimation.

We apply our proposed robust StIF with adaptive degree of freedom to the localization of an autonomous vehicle. Dead-reckoning measurements are merged with GNSS measurements (GPS and GLONASS) obtained from a low-cost mono-frequency receiver and with measurements from a smart camera. The camera returns measurements of detected lane markings that are used to enhance the localization procedure after solving the data association with the HD map.

A second step involves computing a confidence interval that bounds the position errors with a given risk. Compared to a classical Gaussian assumption, the Student's t distribution is well adapted to this kind of computation. The heavy tail of the t distribution takes into account errors that have a small probability. Since we are dealing with autonomous vehicles, it is interesting to express the confidence interval in the along-track (AT) and cross-track (CT) directions. This shows the contribution made by the camera in reducing the confidence interval in the CT direction.

This paper is organized as follows. After an introduction on Student's t distributions, section 2 presents the main contributions of the research, namely the Student's t information filter (StIF) and the robust StIF with adaptive degree of freedom. In this section the FDE stage is also described in detail. Section 3 presents the localization of a selfdriving car using the proposed robust filter. The computation of confidence intervals applied to the localization is also presented in this section. Section 4 reports an experimental study and the performance of the method on a recorded dataset in Rambouillet, France. Our conclusion and prospects for future work are given in section 5.

II. MULTI-SENSOR DATA FUSION USING THE INFORMATIONAL FORM OF STUDENT'S t FILTER

A. Student's t distribution

A random variable X is said to follow a multivariate t distribution with mean vector µ, scale matrix P and degree of freedom (dof) ν, if it can be written in the form [START_REF] Kotz | Multivariate T-Distributions and Their Applications[END_REF]:

X = µ + ω -1 2 y ( 1
)
where ω is distributed according to a Gamma distribution Γ ν 2 , ν 2 and y has a zero-mean Gaussian distribution with covariance P . It is denoted as X ∼ St(µ, P, ν).

The degree of freedom of the Student's t distribution has a direct effect on the tails of the distribution and if ν → ∞ the t distribution converges to a Gaussian distribution. The covariance of the t distribution is defined only if ν > 2, and in this case it can be written as Σ = ν ν-2 P . The Student's t distribution can also be viewed as a mixture of normal density [START_REF] Nadarajah | Mathematical properties of the multivariate t distribution[END_REF].

The Student's t distribution shares some properties with the Gaussian distribution [START_REF] Kotz | Multivariate T-Distributions and Their Applications[END_REF]. For example, for any a and any non-singular matrix B, the random variable z = BX + a follows a t distribution in the form: St(Bµ + a, BP B T , ν). This property is used in the derivation of Student's t Kalman Filter (StF).

B. System modeling

Consider the following evolution and observation models used in a multi-sensor data fusion problem:

X k+1 = f (X k , u k ) + v k (2) Z k = h(X k ) + w k (3)
X is the state vector with scale matrix P u k is the input vector (for instance speed or acceleration) Z is the observation vector (typically exteroceptive sensors measurements)

v k and w k are the process and measurements noises. Suppose that the initial state and the noises follow Student's t distributions [START_REF] Roth | Robust bayesian filtering and smoothing using student's t distribution[END_REF], [START_REF] Roth | A student's t filter for heavy tailed process and measurement noise[END_REF], [START_REF] Huang | Robust student's t based nonlinear filter and smoother[END_REF]:

p(X 0 ) = St(X 0 ; X0 , P 0 , ν 0 ) (4) p(v k ) = St(v k ; 0, Q k , γ k ) (5) p(w k ) = St(w k ; 0, R k , δ k ) (6)
where X0 , P 0 and ν 0 correspond to the initial guess. A scale matrix Q u is associated with the input vector u k to manage the input noise.

C. Student's t Filter

Similarly to the Kalman Filter, the StF includes two steps: time update and measurement update.

At time k, suppose that p(X k |Z 1:k ) follows a t distribution St(X k , Xk/k , P k/k , ν k ) and suppose that p(v k ) follows equation 5. In order to derive the StF, these two distributions have to share a common degree of freedom denoted ν k [START_REF] Roth | Robust bayesian filtering and smoothing using student's t distribution[END_REF], [START_REF] Roth | A student's t filter for heavy tailed process and measurement noise[END_REF]:

p(X k , v k |Z 1/k ) = St X k v k ; Xk/k 0 , P k/k 0 0 Q k , ν k (7)
X k and v k are assumed to be uncorrelated.

In order to preserve the heaviest tails between X k and v k , ν k can be chosen to be the minimum value between (ν k , γ k ) as proposed in [START_REF] Roth | Robust bayesian filtering and smoothing using student's t distribution[END_REF]. Then, P k/k and Q k are the adjusted values of P k/k and Q k given the common degree of freedom ν k . Different methods can be used to adjust the matrices, such as moment matching or minimizing the Kullback-Leibler divergence [START_REF] Roth | Robust bayesian filtering and smoothing using student's t distribution[END_REF].

Suppose that the uncertainty propagation is well modeled by a first-order Taylor approximation. Then, since linear transformations preserve the degree of freedom of a Student's t distribution (see section II-A), the predicted density can be approximated as

p(X k+1 |Z 1:k ) = St(X k+1; Xk+1/k , P k+1/k , ν k ) (8)
The predicted estimators Xk+1/k and P k+1/k are obtained as in the case of an Extended KF, using Jacobian matrices (F k and B k ):

Xk+1/k = f ( Xk/k , u k ) (9)
P k+1/k = F k P k/k F T k + B k Q u B T k + Q (10)
Notice that P k+1/k represents the scale matrix and not the covariance matrix. The latter can be easily retrieved as a function of the scale matrix:

Σ k+1/k = ν k ν k -2 P k+1/k .
Similarly to the time update, the measurement update requires an assumption regarding the degree of freedom. Hence, the joint distribution between the predicted state and the measurement noise can be written as follows:

p(X k , w k |Z 1:k-1 ) = St X k w k ; Xk/k-1 0 , P k/k-1 0 0 R k , ν k (11)
where X k and w k are assumed to be uncorrelated and P k/k-1 and R k are the adjusted matrices corresponding to the joint degree of freedom

ν k = min(ν k-1 , δ k ).
The joint distribution of the state and measurement can be written as

p(X k , Z k |Z 1:k-1 ) = St X k Z k ; Xk/k-1 Ẑk , P k/k-1 P k/k-1 H T k H k P k/k-1 S k , ν k (12)
where

S k = H k P k/k-1 H T k + R
k and H k is the Jacobian matrix corresponding to the measurement model.

The updated degree of freedom, state vector and scale matrix are [START_REF] Roth | Robust bayesian filtering and smoothing using student's t distribution[END_REF]

ν k = ν k + d Z (13) Xk/k = Xk/k-1 + P k/k-1 H T k S -1 k (Z k -h( Xk/k-1 )) (14)
P k/k = ν k + ∆ 2 k ν k + d Z (P k/k-1 -P k/k-1 H T k S -1 k H k P k/k-1 ) (15)
= ν k + ∆ 2 k ν k + d Z (I -W k H k )P k/k-1 (16)
where d Z is the dimension of the observation vector,

∆ 2 k = (Z k -h( Xk/k-1 )) T S -1 k (Z k -h( Xk/k-1 )) and W k = P k/k-1 H T k S -1
k . The P k/k matrix is the same as that obtained in the case of KF, but scaled by a term that depends on the quality of the observations.

From equation 13 and after the first iteration, the dof increases to ν k . In the absence of a limitation of this dof such as that presented in the prediction step (ν k = min(ν k , γ k )), it will tend to infinity and the StF will converge to a KF.

D. Informational form of the Student's t Filter

A derivation of the informational form of the StF can be given similarly to the informational form of the KF, known as the Information Filter (IF) [START_REF] Durrant-Whyte | Multisensor data fusion[END_REF]. The IF is well adapted to decentralized multi-sensor data fusion and to multiple Fault Detection and Exclusion (FDE) thanks to its update step that is modeled as a simple summation of the information contributions of the different observations [START_REF] Hage | Multi-sensor fusion approach with fault detection and exclusion based on the Kullback-Leibler divergence: Application on collaborative multi-robot system[END_REF]. Likewise, the IF does not need to inverse a matrix that has the dimension of the observation vector, meaning that it has a shorter execution time than the KF in the case of multi-sensor fusion. In this section we derive the Student's t Information Filter (StIF) and examine its advantages in relation to an StF.

The StIF features an information matrix and an information vector obtained from the scale matrix (and not the covariance matrix) and the state vector:

Y k = P -1 k (17) y k = Y k X k (18)
The time update remains unchanged relatively to the IF. For the measurement update, equation 15 can be expressed as

P k/k = ν k + ∆ 2 k ν k + d Z (P -1 k/k-1 + H T k R -1 k H k ) -1 ( 19
)
This equation is obtained in a similar way to the computation of the information matrix in the case of the IF.

By setting

c k = ν k +∆ 2 k
ν" k +d Z , the information matrix can be written as

Y k/k = c -1 k (Y k/k-1 + H T k R -1 k H k ) (20)
The derivation of the information vector is expressed in the same way as for IF and using the following equations [START_REF] Durrant-Whyte | Multisensor data fusion[END_REF]:

(I -W k H k ) = (P k/k-1 -W k S k W T k )P -1 k/k-1 = c -1 k P k/k P -1 k/k-1 (21)
W k = P k/k-1 H T k (H k P k/k-1 H T k + R k ) -1 W k (H k P k/k-1 H T k + R k ) = P k/k-1 H T k W k R k = (I -W k H k )P k/k-1 H T k W k = (I -W k H k )P k/k-1 H T k R -1 k (22) = c -1 k P k/k H T k R -1 k (23)
Therefore, using equation 21 and 23, in the linear case, X k/k can be written as

Xk/k = (I -W k H k ) Xk/k-1 + W k Z k (24) Xk/k = c -1 k P k/k [Y k/k-1 Xk/k-1 + H T k R -1 k Z k ] (25)
and the information vector is

ŷk/k = c -1 k ŷk/k-1 + c -1 k H T k R -1 k Z k (26)
In the case of multi-sensor data fusion and in a similar way to the IF, the equations of the StIF are obtained as

Y k/k = c -1 k Y k/k-1 + N i=1 I i,k (27)
ŷk/k = c -1 k ŷk/k-1 + N i=1 i i,k (28)
where N is the number of observation vectors, assumed to be uncorrelated with each other,

I i,k = H T i,k R -1 i,k H i,k and i i,k = H T i,k R -1 i,k Z i,k are the information contributions of the observation Z i,k .
Note that equations 27 and 28 distribute the computation of I i,k and i i,k thanks to the summation part. However, the quantity c -1 is present in both terms and in its present form it cannot be computed in a distributed manner since it needs to determine the value of ∆ 2 k , and this involves inversing a matrix S that has the dimension of the observation vector. Note that S contains correlation terms. A new expression of ∆ 2 k can be determined using the matrix inversion lemma:

S -1 = (H k P k/k-1 H T k + R k ) -1 (29) = R -1 k -R -1 k H k (Y k/k-1 + H T k R -1 k H k ) -1 H T k R -1 k = R -1 k -R -1 k H k (Y k/k-1 + N i=1 H T i,k R -1 i,k H i,k ) -1 H T k R -1 k The term (Y k/k-1 + N i=1 H T i,k R -1 i,k H i,k
) has already been computed and the new expression of S -1 avoids the inversion of a non-diagonal matrix with the dimension of the observation vector. The matrices R k and H k present in this expression are obtained by concatenation.

Algorithm 1 StIF with fixed degree of freedom.

Input: Xk-1/k-1 , Y k-1/k-1 , ν k-1 ,
# We assume that γ = δ (no approximation of the dof is needed when going from time update to measurement update) ν k-1 = min(ν k-1 , γ) = γ Moment matching to adjust the matrices Q and P k-1/k-1 : # We suppose that the matrices Q and R are the same as in the Gaussian case

P k-1/k-1 = ν k-1 -2 ν k-1 ν k-1 ν k-1 -2 P k-1/k-1 (31)
Q = ν k-1 -2 ν k-1 Q (32) Time update X k/k-1 = f (X k-1/k-1 , u k-1 ) + v k P k/k-1 = F k-1 P k-1/k-1 F T k-1 + B k-1 Q u,k-1 B T k-1 + Q k-1 # F k and B k are the Jacobian matrices Measurement update R = ν k-1 -2 ν k-1 R c k = ν k-1 + ∆ 2 k ν k-1 + d Z Y k/k = c -1 k (Y k/k-1 + N i=1 I i,k ) ŷk/k = c -1 k y k/k-1 + N i=1 i i,k ν k = ν k-1 + d Z Covariance matrix: Σ k/k = ν k ν k -2 Y -1 k/k X k/k = Y -1 k/k y k/k Output: Xk/k , Y k/k , ν k
For non-linear systems, i i,k in equation 28 becomes

i i,k = H T i,k R -1 i,k Z i,k -Ẑi,k + H i,k Xk/k-1 (30)
The algorithm for the StIF with a fixed degree of freedom (γ = δ) is given in algorithm 1. Another algorithm will be given in section II-E after developing a fault detection and exclusion stage.

E. Robust StIF with adaptive degree of freedom

Although Student's t Filter is designed to be more robust to faults than KF insofar as larger errors are taken into account via the heavier tail of the density function, these faults still influence the quality of the filter estimation. Therefore we propose adding a fault detection and exclusion stage in order to exclude these errors from the fusion procedure. Note that in the StF, when a fault is presented in the system, ∆ 2 increases, leading to a significant increase in the value of the covariance matrix.

The first step of an FDE is the generation of the residual. In this work, a residual based on the Mahalanobis distance between the prediction (obtained from dead-reckoning) and the update estimation (obtained after merging the measurements Z) is constructed as

r k = ( Xk/k -Xk/k-1 ) T Y k/k ( Xk/k -Xk/k-1 ) (33)
where Xk/k and Y k/k are obtained from a filter that uses all available measurements at instant k.

Similarly to the use of the Chi-squared distribution in the Gaussian case for threshold computation, an F-distribution is employed in the Student's t case [START_REF] Roth | Kalman filters for nonlinear systems and heavy-tailed noise[END_REF]. The variable r k d X follows an F-distribution (F d X ,ν k ) with parameter d X (dimension of the state vector) and ν k = ν k + d Z , and therefore

r k ∼ d X F d X ,ν k (34)
The threshold value for the statistical test over r k corresponding to a given false alarm probability P F a is given as

T h P F a = d X F -1 1-P F a (F d X ,ν k ) = d X F -1 P 1-F a (F d X ,ν k +d Z ) (35
) where F -1 is the inverse cumulative distribution.

If the residual r k indicates the presence of faults by exceeding the threshold value, residuals used for the exclusion stage are generated by creating a bank of StIF where each one uses only one observation. These residuals are computed as

r i,k = ( Xi,k/k -Xk/k-1 ) T Y i,k/k ( Xi,k/k -Xk/k-1 ) (36)
where Xi,k/k and Y i,k/k are obtained from a filter that uses only the observation i in the update step.

A measurement i is excluded if

r i > T h i = d X F -1 1-P F a F d X ,ν k +d Z,i (37)
Then, the measurement i is excluded from the fusion procedure by subtracting its information contribution from the main filter and by updating the c term. This exclusion step is more complicated to do when the KF (or StF) is used.

Regarding the degree of freedom of the Student's t filter, instead of fixing it as proposed in section II-C, it can be adapted in real-time by taking into account the value of the residual. This residual can provide an indication about the quality of the measurements on this particular sample. Intuitively, when the residual is large, one may wish to be more cautious in the estimation process and when computing the confidence interval, and for this purpose a heavy-tailed distribution may be preferred. Inversely, when the residual is small, a Gaussian distribution may be more reasonable. Starting from large values of the residual, the degree of freedom increases slowly up to a given limit when the residual decreases (e.g dof=20), and if r k is greater than a given value the dof should be limited to 2.1. We propose an expression of the degree of freedom according to the residual as follows:

dof k (r k ) = a. exp(b.r k ) if r k < d dof k (r k ) = 2.1 if r k > d (38)
Algorithm 2 Multi-sensor data fusion with FDE

Input: Xk-1/k-1 , Y k-1/k-1 , ν k-1 , γ k = δ k if k = 0 then ν k-1 = ν k-1 = γ k = δ k else γ k = δ k = ν k-1 , (ν k-
ν k = ν k-1 + d z ( 39
)
Compute the corresponding residual r k if r k > T h then compute the set of residuals r i,k using a bank of StIF exclude the measurements that have r i,k > T h i , compute the new value of c k , Y k/k and ŷk/k Xk/k = Y -1 k/k ŷk/k if measurements available then compute the new value of ν k that will be used in the next iteration, as in equation 38.

else ν k = ν k = ν k-1 Output: Xk/k , Y k/k , ν k , ν k
We note that in the case study given in section III, a , b and d were tuned with the following values: a =20.1137, b = -0.0565 and d = 40. With this tuning, the degree of freedom tends to 20 if r k tends to 0 and to 2.1 if the residual is greater than d.

Algorithm 1 is then modified to algorithm 2 in order to take into account the FDE stage and the computation of the adaptive degree of freedom.

III. CASE STUDY

A. Localization for self-driving vehicles

Let consider the localization problem of self-driving vehicle equipped with wheel-speed sensors, a gyro, a lowcost GNSS receiver, an intelligent camera for lane-marking detection and an HD map. The localization is defined relatively to a Cartesian ENU frame (East, North, Up) denoted R O . The vehicle pose is defined at the body frame located at the middle of the rear wheel axis (frame R B ) (see figure 1).

At instant k, the state vector is considered to be the position and the heading of the vehicle:

X = [ x y θ ] T (40)
The GNSS observation is where [t x t y ] is the translation of the antenna (located at frame R G ) with respect to the body frame.

Z 1 =   x GNSS y GNSS θ GN SS   =   t x cos θ -t y sin θ + x t x sin θ + t y cos θ + y θ   , (41)
A smart camera detects the lane markings and returns the coefficients of a Taylor's expansion of a clothoid in the frame R M [START_REF] Tao | Road-centered map-aided localization for driverless cars using singlefrequency GNSS receivers[END_REF]:

y = C 3 x 3 + C 2 x 2 + C 1 x + C 0 ( 42
)
The camera is able to detect up to two markings at a time on each side (right and left) of the car. In this paper, we consider only the use of the lateral distance to the lane markings (the C 0 parameter):

Z 2,j = C 0 ( 43
)
where j ∈ {1, 2, 3, 4} is an index representing the detected marking: right 1 , right 2 , left 1 or left 2 if they exist. A camera observation model was developed in [START_REF] Tao | Mapping and localization using GPS, lane markings and proprioceptive sensors[END_REF], [START_REF] Tao | Road-centered map-aided localization for driverless cars using singlefrequency GNSS receivers[END_REF] and is given by

C 0 = (P x sin θ + y -y A )x AB -(P x cos θ + x -x A )y AB x AB • cos θ + y AB • sin θ
where the lane marking is detected on the segment

[AB] at coordinates [x L , y L ] expressed in frame R O as O L = O T B . B T M . M L (44)
O T B and B T M are the transformation matrices between (R B and R O ) and (R M and R B ) respectively. It results that

x L y L = P x cos θ + C 0 sin θ + x P x sin θ -C 0 cos θ + y ( 45
)
where M L = [0, C 0 ] and P x is the distance between R B and R M (figure 1). For state estimation, the robust StIF developed in algorithm 2 is used. Therefore, the residual r k defined in equation 33 is computed in order to detect the presence of faults. Note that the prediction is obtained from the wheel speed sensors and the gyro while the update is done using the GNSS fix and the Mobileye measurements. Mobileye measurements are not always available, since the camera cannot operate in highly curved roads. For the fault exclusion, a set of residual r i,k is computed (equation 36) for each measurement i, which can be the GNSS measurement or one of the four Mobileye measurements. Let us choose the example of the residual corresponding to the GNSS measurement (i = gnss, d Z|gnss = 3) as illustration. Xi,k/k and Y i,k/k are obtained from the following equations:

S -1 = R -1 i -R -1 i H i (Y k/k-1 + H T i,k R -1 i,k H i,k ) -1 H T i R -1 i c i,k = ν k + ∆ 2 k ν k + 3 Y i,k/k = c -1 i,k (Y k/k-1 + H T i,k R -1 i,k H i,k ) ŷi,k/k = c -1 i,k ŷk/k-1 + i i,k Xi,k/k = Y -1 i,k/k ŷi,k/k If r gnss >T h gnss = d X F -1 1-P F a (F d X ,ν k +3
), the GNSS measurements are excluded from the fusion procedure by subtracting their information contributions from the main filter and by updating the c k term.

B. External integrity and computation of protection level

External integrity is an emergent concern for road localization. External localization integrity was initially developed in aeronautical applications in order to provide a measure of confidence for GNSS navigation solutions [START_REF] Zhu | GNSS position integrity in urban environments: A review of literature[END_REF]. It is associated with a target integrity risk that represents the maximum probability that the error in position exceeds a limit without warning the user. This limit is known as the Protection Level (PL). In this paper we are seeking to bound the errors by computing a consistent PL that makes use of Student's t distributions. For this computation, the matrix P k/k obtained in the output of algorithm 2 is used. The heavy tail property of Student's t distribution is essential for computing a consistent PL without underestimating its value, which may happen in the Gaussian case [START_REF] Hage | Bounding localization errors with Student distribution for road vehicles[END_REF]. For this purpose, the matrix P k/k is adjusted according to the minimum degree of freedom between ν k and ν k (denoted M in equations 48 and 49). If this minimum value is greater than a limit value fixed according to the application and to the environment, the degree of freedom for the PL computation is taken to be this limit value. For example, in urban environments the degree of freedom for the PL computation should not go above M = 5. After updating the degree of freedom, the matrix P k/k is adjusted using the moment matching. The adjusted matrix is denoted P k and is only used for the PL computation. The reason for choosing the degree of freedom in this way is to preserve a heavy tail, which is necessary in order to obtain a consistent protection level (for more details the reader can refer to [START_REF] Hage | Bounding localization errors with Student distribution for road vehicles[END_REF]).

For road vehicles, we would like to be able to bound the errors in the along track (AT) and cross track (CT) directions. Therefore, the shape matrix in the ENU frame is projected onto the (AT, CT) frame:

P k/k,proj = R proj P k/k(1,2),(1,2) R T proj (46) with R proj = cos θ sin θ -sin θ cos θ ( 47
)
where θ is the orientation of the vehicle. The protection levels are obtained as

P L AT = K(α, N ). √ M .[eigenvalue(P k/k,proj )] 1 (48) P L CT = K(α, N ). √ M .[eigenvalue(P k/k,proj )] 2 (49)
where the indexes 1 and 2 denote the first and second elements of the vector (eigenvalue(P k/k,proj )) and K(α, M ) is obtained from Student's t distribution with degree of freedom M according to a given confidence level α [START_REF] Navarro Madrid | Device and method for computing an error bound of a Kalman filter based gnss position solution[END_REF]. In order to evaluate the performance of the proposed approach, results were processed using sensor data recorded by an experimental vehicle equipped with wheel-speed sensors and the yaw rate gyro of the trajectory stability system of the car, a Ublox 8T which is a single frequency GPS/GLONASS constellation receiver, a Mobileye camera for lane-marking detection and a NovAtel SPAN-CPT with network RTK corrections used as a ground truth. Experiments were carried out in Rambouillet, France (figure 2). The results shown were obtained in a challenging environment where the GNSS measurements were affected by significant bias throughout the trajectory. The purpose of choosing this kind of trajectory was to study the performance of our proposed approach in this kind of challenging environment and to compare the StIF to the IF. The processing of the data was handled asynchronously. The wheel-speed sensor and the gyro data were available at a frequency of 50 Hz, the GNSS at 2 Hz and the frequency of the data from Mobileye camera was down-sampled to 3.3 Hz. The sampling period of the filter is the same as for the dead-reckoning data (0.02s).

IV. EXPERIMENTAL RESULTS

Figure 3 shows the position errors obtained using an IF and an StIF without an FDE stage. To provide a better understanding of the results, simulated biases on GNSS measurements were injected between the sampling instants 5000 to 5010 and 10000 to 10010. It can be remarked that after the error source is removed and without an FDE stage, the Student's t filter (StIF) convergence is faster than a Gaussian filter (IF). Figure 4 shows the residual used for the fault detection with the associated threshold value. This threshold varies with the degree of freedom computed in real-time (figure 4b). When r indicates the presence of a fault, the set of residuals used for the fault exclusion are generated (if available) and are shown in figure 5. Figure 5a shows the residual corresponding to the GNSS measurements and figures 5b and 5c show the residuals corresponding to the camera measurements on the left and right-hand sides respectively.

The errors in the CT and AT directions are shown in figure 6 with the corresponding PL value. The target integrity risk was fixed to 10 -3 . The contribution made by the Mobileye camera in reducing the error and the PL values in the CT direction (highlighted zone) can be seen. When there is no camera measurement, the non-observable biases from GNSS lead to inconsistent PL despite the use of Student's t distribution. Moreover, this sensor does not have any effect on the AT direction, since it is not used to detect markings in this direction. Therefore, the PL in the AT direction is not consistent. The limit value of the dof for PL computation was fixed to M = 5 (see section III-B). Note that if this limit were fixed to M = 3, the PLs value would be consistent in the AT and CT directions.

Table I shows the errors and the percentile values before and after the FDE stage for both the StIF and the IF.

For this experiment where a bias was present throughout the trajectory, even though the StIF and the FDE stage do not significantly improve the results, the position estimation using StIF was more accurate and more consistent than when using IF. For this particular case, the IF after the FDE stage results in a larger error than before FDE. The reason is that the algorithm excludes many camera measurements; using these measurements would give better results. For this trajectory, StIF with dof=3 was slightly more accurate than StIF with adaptive dof. However, on other trajectories, the StIF with dof=3 may give the poorest results, according to the error model. This shows that the choice of a fixed dof may not be straightforward in real applications. In this experiment, according to the percentile values, it can be remarked that the largest errors that occur are smaller when using StIF with FDE than when using an IF (a reduction from 4.15m to 3.8m for the 97.5 percentile).

From a consistency point of view, table I also shows the measured integrity risk obtained from the experiment in the AT and CT directions for a target integrity risk of 10 -3 . Given the number of available samples in the experiment, we cannot perform tests for a target integrity risk below 10 -3 . However, the Student's t distribution and the Student's t filter are more advantageous and more consistent for small integrity risk in comparison to the Gaussian cases, especially when using the moment matching to adjust distributions. Regarding the measured IR, the StIF is more consistent than the IF, and the StIF with adaptive dof leads to better results since the dof is modified according to the residual, and heavier tails are used when the environment is too noisy.

V. CONCLUSION AND FUTURE WORK

In this paper a new robust Student's t information filter was proposed. The StIF, which is the informational form of the StF, was developed. Using this formulation, a fault detection and exclusion stage was added in order to exclude erroneous measurements that have a direct influence on the state estimation. The computed residual gives an indication regarding the environmental conditions and the quality of the measurements in real-time. This residual is used to adapt the degree of freedom and to avoid setting it a priori. Experimental results using a vehicle equipped with different sensors showed that the StIF with an adaptive degree of freedom improves performance, particularly in terms of integrity.

In future works, additional studies on the variation of the degree of freedom in real-time will be done and the approach will be tested with trajectories in different environments. Likewise, the results using the moment matching will be compared with other approaches, such as minimizing the Kullback-Leibler divergence.

1

 1 obtained in the measurement update) Moment matching: Compute P k-1/k-1 and Q as in algorithm 1 Time update: same as algorithm 1 Measurement update: Compute R , c k ,Y k/k and y k/k as in algorithm 1.

Fig. 1 :

 1 Fig. 1: Frames representation.

Fig. 2 :Fig. 3 :

 23 Fig. 2: Test trajectory of 3.5 km.

Fig. 4 :

 4 Fig. 4: (a) Residual for the fault detection (in blue), threshold (in green), (b) the associated degree of freedom.

  Although the StIF behaves better than the IF in the presence of errors, large errors still affect its behavior. It should be noted that the StIF used for this figure is the one presented in algorithm 2 without an FDE stage. The degree of freedom is adapted in real-time according to the residual value. The values chosen for a , b and d in equation 38 were a =20.1137, b = 0.0565, and d = 40.

Fig. 5 :

 5 Fig. 5: Residuals for fault exclusion (without unit), in green the thresholds.

Fig. 6 :

 6 Fig. 6: The error (in blue) and the PL (in red) in the AT and CT directions (case of StIF with adaptive dof). Highlighted zones correspond to straight line where the Mobileye detect well the markings.

TABLE I :

 I Statistical values of the horizontal positioning errors.

The authors are also members of the SIVALab, joint laboratory between Renault/