Elwan Héry

Philippe Xu

Philippe Bonnifait

Pose and covariance matrix propagation issues in cooperative localization with LiDAR perception

This work describes a cooperative pose estimation solution where several vehicles can perceive each other and share a geometrical model of their shape via wireless communication. We describe two formulations of the cooperation. In one case, a vehicle estimates its global pose from the one of a neighbor vehicle by localizing it in its body frame. In the other case, a vehicle uses its own pose and its perception to help localizing another one. An iterative minimization approach is used to compute the relative pose between the two vehicles by using a LiDAR-based perception method and a shared polygonal geometric model of the vehicles. This study shows how to obtain an observation of the pose of one vehicle given the perception and the pose communicated by another one without any filtering to properly characterize the cooperative problem independently of any other sensor. Accuracy and consistency of the proposed approaches are evaluated on real data from on-road experiments. It is shown that this kind of strategy for cooperative pose estimation can be accurate. We also analyze the advantages and drawbacks of the two approaches on a simple case study.

I. INTRODUCTION

Localization is a key feature for autonomous driving, in particular when vehicles cooperate in complex situations like lane merging on a motorway for instance [START_REF] Ph | System architecture of a driverless electric car in the grand cooperative driving challenge[END_REF]. The cooperation between vehicles can improve localization and this can be achieved through the use of new communication standards such as CAM (Cooperative Awareness Message) [START_REF] Etsi | Intelligent transport systems (ITS); Vehicular communications; Basic set of applications; Part 2: specification of cooperative awareness basic service[END_REF] to receive the pose of other vehicles and CPM (Cooperative Perception Message) to share perception information. Moreover, cooperative localization is a well-known issue in mobile robotics [START_REF] Martinelli | Multi-Robot Localization Using Relative Observations[END_REF], [START_REF] Roumeliotis | Distributed Multi-Robot Localization[END_REF], [START_REF] Roumeliotis | Propagation of Uncertainty in Cooperative Multirobot Localization: Analysis and Experimental Results[END_REF], [START_REF] Kurazume | Cooperative positioning with multiple robots[END_REF]. Perception is often described as bearing and range, sometimes with relative orientation. It is then essential to use multiple perceived robots to localize one robot of the fleet. It has also been shown that vehicles can improve their own pose estimates thanks to perception information and pose estimates of neighboring vehicles [START_REF] Karam | Localization of a Group of Communicating Vehicles by State Exchange[END_REF], [START_REF] Héry | Along-track localization for cooperative autonomous vehicles[END_REF]. However, these works often use simulated data, the perception problem is then not always perfectly represented.

A typical scenario of cooperative localization aided by perception is cooperative navigation through a local GNSS denied environment. For example, when a platoon of vehicles goes under a bridge, the vehicles of the platoon lose their GNSS signals one after the others, Fig. 1. The vehicles that are not under the bridge are able to help the vehicles which have lost their absolute position measurements.

France firstname.surname@hds.utc.fr In order for a vehicle to make use of the pose estimate of another one, it needs to be able to estimate the relative pose between them. Camera based vehicle detection has been extensively studied in the literature in the past decades. Recent deep learning based detectors reached relatively high accuracy for relative pose estimation in the image space [START_REF] Chateau | Deep MANTA: A Coarse-to-Fine Many-Task Network for Joint 2d and 3d Vehicle Analysis from Monocular Image[END_REF]. However, monocular vision based methods often do not provide accurate metric pose estimates. 3D point clouds given by stereovision or LiDAR can be used to obtain metric pose estimates. By fitting the points with a model, e.g., Lshape fitting [START_REF] Zhang | Efficient L-shape fitting for vehicle detection using laser scanners[END_REF] or more general 3D model fitting [START_REF] Coenen | Detection and 3d modelling of vehicles from terrestrial stereo image pairs[END_REF], the relative pose w.r.t. a detected vehicle can be estimated. To compute this relative pose, the Iterative Closest Point (ICP) method, often used for scan matching [START_REF] Censi | An ICP variant using a point-to-line metric[END_REF], can be applied [START_REF] Héry | LiDAR based relative pose and covariance estimation for communicating vehicles exchanging a polygonal model of their shape[END_REF].

For multiple cooperative vehicles to make the most of exchanged pose estimates within a data fusion framework, each vehicle not only needs to be able to provide a pose estimate but also needs to properly quantify the uncertainty associated with it. When using a classical Gaussian assumption, it is done through the estimation of the covariance matrix. Therefore, the ability to have a consistent estimation of the covariance matrix associated to the perception [START_REF] Bengtsson | Robot localization based on scanmatching-estimating the covariance matrix for the IDC algorithm[END_REF], [START_REF] Censi | An accurate closed-form estimate of ICP's covariance[END_REF], [START_REF] Prakhya | A closed-form estimate of 3d ICP covariance[END_REF] becomes of significant importance.

In the literature, cooperative localization problems using data fusion often describe the state of one vehicle with the pose and the velocity of this vehicle and of its neighbor and use the relative measurements as observations. In this work, we show how a vehicle can propagate the pose estimate and the associated uncertainty of a neighboring vehicle by using the LiDAR based perception approach combined with the exchange of geometrical models of the vehicles proposed in [START_REF] Héry | LiDAR based relative pose and covariance estimation for communicating vehicles exchanging a polygonal model of their shape[END_REF]. This relative localization method can be used even when only the back of the vehicle is in the field of view of the LiDAR. Other more classical methods such as L-shape fitting need two sides of the vehicle. The contribution of this paper is to show the accuracy and consistency of the cooperative pose estimation using real LiDAR perception, without any filtering, to properly characterized this raw observation computed from one epoch only without the use of other sensors, e.g. GNSS or odometry observations. The results of the proposed methods can be seen as observations which can be used in the update of any filtering algorithm, e.g., Extended Kalman Filter or covariance intersection filter. This approach has the advantage of only requiring another vehicle to estimate the position and the orientation. In the case of multiple surrounding cars, each vehicle can be seen as a different observation.

We consider two formulations of the cooperative localization problem. In the first one, the ego-vehicle perceives a cooperating vehicle and uses the pose estimate of the latter to estimate its own pose. In the second formulation, we consider that it is the ego-vehicle that is perceived by the cooperating vehicle which can directly help estimate the pose of the egovehicle. The relative pose between the vehicles is estimated by minimizing the distances between LiDAR scans and a polygonal model of the shape of the vehicles.

Both accuracy and consistency of the localization resulting from the two formulations are evaluated and compared using real data recorded from two communicating experimental vehicles in the city of Compiègne, France (see Fig. 2).

The cooperative localization problem is first introduced in section II, where we present the pose and the covariance propagation. The relative localization process based on an iterative minimization method is explained in section III. Experimental results are studied and analyzed in section IV. We finally analyze the advantages and drawbacks of the two approaches on a simple case study.

II. PROBLEM STATEMENT

As stated in the previous section, the aim of this work is to estimate the pose of a given vehicle from the localization information of another communicating vehicle via perception information. Throughout this work, we consider that the vehicles are equipped with a LiDAR capable of detecting surrounding vehicles. In order to be able to estimate an accurate relative pose between two vehicles, in particular relative orientation, we also suppose that they can exchange a model of their geometrical shape as illustrated in Fig. [START_REF] Censi | An accurate closed-form estimate of ICP's covariance[END_REF].

Using a 2D formulation is often sufficient in vehicle localization problems as many road environments are locally planar. Therefore, the pose of a vehicle is defined by its 2D coordinates and its heading : q = [x y θ] T . The data coming from the LiDAR sensor is considered as a set of twodimensional points P = {p i } i=1,...,N . We suppose that this set of points corresponds to a detected vehicle. The detection task itself is out of the scope of this paper, many LiDAR clustering algorithms can be found in the literature [START_REF] Douillard | On the segmentation of 3d LIDAR point clouds[END_REF]. The shape of the vehicle is also given in 2D by a polygonal model representing the outer hull from a bird's-eye view. A shape model is represented as a set of consecutive points M = (m 1 , . . . , m K ).

The considered 2D geometrical shapes are typically defined by the bumpers of the vehicles which are situated at the most outer place. For safety reasons in case of collision, the height of the bumpers is almost the same for the majority of vehicles. Therefore, placing a LiDAR layer at this same height provides a perception that is consistent with the considered models. This assumption may not be respected in some cases, e.g., for trucks or non-planar roads. In this situation, a multi-layer 3D LiDAR may be used to enhance the detection of the 2D hull of a detected vehicle. The method proposed in this paper can be generalized to this 3D case.

Let us consider a first approach in which the ego-vehicle aims at estimating its own pose q e using the pose q c of a cooperating vehicle as a deported antenna with a lever arm corresponding to their relative pose e q c (expressed in the mobile frame of the ego-vehicle). This relative pose is estimated using perception. Here, the goal is to find an estimate qe in a fixed working frame along with its associated covariance matrix Σ e using the estimates qc , Σ c , e qc and e Σ c .

Let R(θ) be the rotation matrix defined as

R(θ) =   cos(θ) -sin(θ) 0 sin(θ) cos(θ) 0 0 0 1   . (1)
Suppose that the relative pose is estimated from the egovehicle point of view. In this first formulation, the pose of the ego-vehicle is computed with the following equation:

q e = q c -R (θ e ) e q c =   x c -e x c • cos(θ e ) + e y c • sin(θ e ) y c -e x c • sin(θ e ) -e y c • cos(θ e ) θ c -e θ c   (2) =   x c -e x c • cos(θ c -e θ c ) + e y c • sin(θ c -e θ c ) y c -e x c • sin(θ c -e θ c ) -e y c • cos(θ c -e θ c ) θ c -e θ c   .
Since the relation q e = f 1 (q c , e q c ) is not linear, a first order approximation is used to estimate Σ e from Σ c and e Σ c , as presented by [START_REF] Durrant-Whyte | Uncertain geometry in robotics[END_REF], [START_REF] Smith | Estimating Uncertain Spatial Relationships in Robotics[END_REF]:

Σ e ≈ ∂f 1 ∂q c Σ c ∂f 1 ∂q c T + ∂f 1 ∂ e q c e Σ c ∂f 1 ∂ e q c T , (3)
where the Jacobian matrices of f 1 , ∂f1 ∂qc and ∂f1 ∂ e qc are defined as follows:

∂f 1 ∂q c =   1 0 e x c s 1 + e y c c 1 0 1 -e x c c 1 + e y c s 1 0 0 1   , (4)
∂f 1 ∂ e q c =   -c 1 s 1 -e x c s 1 -e y c c 1 -s 1 -c 1 e x c c 1 -e y c s 1 0 0 -1   , (5)
with s 1 = sin (θ c -e θ c ) and c 1 = cos (θ c -e θ c ).

Another possible solution is to have the relative pose computed from the cooperating vehicle ( c q e ). The pose of the ego-vehicle is then estimated as follows:

q e = q c + R (θ c ) c q e (6) =   x c + c x e • cos(θ c ) + c y e • sin(θ c ) y c + c x e • sin(θ c ) -c y e • cos(θ c ) θ c + c θ e   .
Notice that using this formulation q e = f 2 (q c , c q e ), the position coordinates (x e , y e ) become independent of the relative orientation c θ e . This implies that the position estimation becomes more robust to inaccurate estimation of the relative orientation. Similarly to the previous case, the covariance matrix is estimated using Jacobian matrices

Σ e ≈ ∂f 2 ∂q c Σ c ∂f 2 ∂q c T + ∂f 2 ∂ c q e c Σ e ∂f 2 ∂ c q e T , (7)
where

∂f 2 ∂q c =   1 0 0 0 1 0 0 0 1   , (8)
∂f 2 ∂ c q e =   c 2 -s 2 -c x e s 2 -c y e c 2 s 2 c 2 c x e c 2 -c y e s 2 0 0 1   , (9)
with s 2 = sin (θ c ) and c 2 = cos (θ c ).

III. RELATIVE LOCALIZATION A. LiDAR based perception

The Iterative Closest Point (ICP) algorithm is often used to match two LiDAR scans in order to obtain the relative pose between them. In our case, we aim at matching two sets of points but with one that corresponds to a LiDAR scan and another that represents the polygonal shape model of a vehicle. However, these two sets do not come from the same sensor as in classical ICP resulting in a disparity in terms of the number of points. Therefore, we use a variant of the ICP algorithm that uses line segments to represent the model points, [START_REF] Héry | LiDAR based relative pose and covariance estimation for communicating vehicles exchanging a polygonal model of their shape[END_REF], [START_REF] Censi | An ICP variant using a point-to-line metric[END_REF], [START_REF] Low | Linear Least-Squares Optimization for Point-to-Plane ICP Surface Registration[END_REF]. The principle of the ICP is to have an iterative process that alternates a matching step with a minimization one. This relative localization problem using LiDAR measurements and a polygonal model has been described and compared to other approaches in [START_REF] Héry | LiDAR based relative pose and covariance estimation for communicating vehicles exchanging a polygonal model of their shape[END_REF]. In practice, only one cluster of points of the scan is matched with the model. This clustering step and the matching of the different clusters with the models of different vehicles in the case of a multiple vehicles scenario is not studied in this paper. A consistency test between the computed relative pose using this algorithm and the initial relative pose can be computed to check if the matching between the cluster and the vehicle model is correct.

This method provides an estimate of the relative pose which minimizes the distances between the LiDAR points, in red, and the model, in blue in Fig. 3. These distances are shown in orange in Fig. 3.

Once each LiDAR point is matched with one segment of the polygonal model, the minimization problem can be written as follows:

min q E (q; P, M ) = min q N i=1 d(q; p i , M ) 2 , ( 10
)
where d(q; p i , M ) is the distance between the point p i and the model M when the transformation associated to the pose q is applied to the LiDAR scan P . The authors of [START_REF] Low | Linear Least-Squares Optimization for Point-to-Plane ICP Surface Registration[END_REF] showed that this minimization problem can be simplified as a quadratic convex problem with the following form:

E(∆q; P, M ) = A∆q -b 2 . ( 11
)
This problem can be solved using the pseudo-inverse of the matrix A:

∆q = pinv (A) b. ( 12
)
Once the pose is found, the covariance matrix can be computed, as described in [START_REF] Bengtsson | Robot localization based on scanmatching-estimating the covariance matrix for the IDC algorithm[END_REF]:

∆ Σ = 2 E (∆q) N -D ∂ 2 E ∂q 2 (∆q) -1 = E (∆q) N -D A T A -1 , (13)
where D is the dimension of ∆q, i.e., D = 3.

It is then possible to estimate the relative pose e qc,k and its associated covariance matrix for this iteration k using the previous iteration e qc,k-1 and the pose ∆q k obtain with the minimization:

) ∆ Σ k R ( e θ c,k ) T , (15)
where R ( e θ c,k ) is the rotation matrix with the angle e θ c,k .

Once the ICP has converged, the pose e qc,k and the covariance matrix e Σc,k of the last iteration gives a good estimate of the relative pose and of its covariance matrix.

B. Propagation accuracy and consistency

1) LiDAR detections: In real driving scenarios the LiDAR perception does not always provide accurate relative localization. When only the back of the vehicle is in the field of view of the LiDAR a C-shape perception can be seen, Fig. 4a. This shape creates a rotation invariance during the execution of the ICP algorithm which leads to larger relative heading and lateral errors. When the back and one side of the vehicle are in the field of view of the LiDAR, an L-shape is visible, Fig. 4b. The angle ambiguity disappears and the relative orientation becomes accurate.

Moreover, the position obtained by the first formulation depends on the relative orientation which is not the case for the second formulation. This is visible when the relative heading is badly estimated in Fig. 5b when a C-shape is visible and when the first formulation is used. In Fig. 5d, when a C-shape is detected and when the second formulation is used, even if the relative heading is not accurate, the obtained position is not impacted.

2) Non linearity issues during covariance matrix propagation: The inconsistency of the first formulation is therefore only due to the highly non-linear mathematical expressions in Eq. ( 2). Fig. 6 illustrates through Monte Carlo simulations the empirical distribution of the estimated pose of the egovehicle when the communicated pose (in black) and the relative pose have Gaussian distributions. One can see that the results are actually following banana-shaped distributions [START_REF] Wheeler | Relative Navigation: A Keyframe-Based Approach for Observable GPS-Degraded Navigation[END_REF], [START_REF] Thrun | A real-time algorithm for mobile robot mapping with applications to multi-robot and 3d mapping[END_REF] which are badly approximated by Gaussian distributions when they are large. The first formulation (in blue) gives a distribution less Gaussian than the second one (in red) because of the dependency of the relative orientation error, see Eq. ( 3). Both formulations are dependent on the orientation error of the communicated pose, as shown by Eq. ( 3) and [START_REF] Douillard | On the segmentation of 3d LIDAR point clouds[END_REF]. Therefore, both formulations give bananashaped distributions. The propagation of the uncertainty through the non-linear transformation is an approximation.

With the second formulation, the cooperating vehicle computes the pose of the ego-vehicle for it with a good un-certainty estimate. Therefore, the ego-vehicle pose estimate is entirely dependent on the cooperating vehicle. This is acceptable if the cars are in a trustworthy network but it can lead to trust issues in other real case scenarios. The first formulation does not make this assumption. Indeed, the relative pose is computed with the perception sensor of the ego-vehicle. Misleading information sent by the cooperating vehicle such as erroneous pose estimate or wrong mode can be detected by using the error computed from the minimization step as a trust indicator. Indeed, if the cooperating vehicle does not send an accurate pose or model, it will not be coherent with the estimated pose of the ego-vehicle. This can be detected by its perception leading to a large value of the cost function.

IV. EXPERIMENTAL RESULTS

A. Experimental platform

The dataset used for this experimentation was recorded on a platooning scenario where two Renault Zoe cars were driving on a test road composed of two roundabouts and a straight lane between them, as pictured in Fig. 2. Nine laps were completed in this test track for a total of 2.82 km in 10 minutes and 7,489 LiDAR scans. The inter-distance between the poses of the two vehicles was between 6.2 m and 18.8 m. This inter-distance was around 8 m in the roundabouts and 13 m in the straight lane. Each vehicle was equipped with a Novatel's SPAN-CPT, an IMU with a GNSS receiver using RTK corrections for the ground truth. No tracking or data fusion was implemented in this work. The frequency of the ground truth being high compared to the speed of the vehicle, the temporal error was supposed to be negligible compared to the spatial error. The following vehicle was also equipped with a four-layer SICK LDMRS LiDAR, but only one layer was used to compute the relative pose. The 2D geometric model of the Renault Zoe car has been manually built from a blueprint. The ground truth having an uncertainty in the order of one decimeter, its covariance matrix was used to compute the consistency. Since the error of the ground truth is not negligible compared to the LiDAR measurement, small errors should be interpreted carefully.

To evaluate the consistency, we test the following condition:

(q e -q ref ) T Σe + Σ ref -1 (q e -q ref ) ≤ χ 2 3;0.05 , (16)
where χ 2 3;0.05 = 7.815 is the quantile of a χ 2 distribution with 3 degrees of freedom and a confidence level of 95%. q ref and Σ ref are the pose and the covariance of the ground truth. In practice, it means that Eq. ( 16) has to be verified for at least for 95% of the samples.

Table I reports the results of the experiments. We decomposed the horizontal error e h into a longitudinal e x and lateral e y components in the mobile frame of the ground truth for a better analysis.

B. First formulation

The ego-vehicle computes its pose from its own perception sensor given the pose and the model received by the cooperating vehicle (see Fig. 5 (a-b)). This first formulation gives an accuracy of 1.24 m for the position and 6.92°for the orientation. This is not accurate enough for autonomous driving. The consistency rate is 91.5% and so the pose estimate is not consistent.

Two different perceptions of the cooperating vehicle exist in this platooning scenario: an L-shape, when the back and one side of the vehicle are in the field of view of the LiDAR and a C-shape, when only the back is in the field of view.

1) L-shape: L-shape can be seen by the LiDAR when both vehicles are driving on the roundabout, Fig. 5-a. This perception gives a good accuracy, 0.27 m for the position and 4.20°for the orientation. The covariance also becomes a bit pessimistic with a 99.2% consistency rate.

2) C-shape: C-shape can be seen by the LiDAR when both vehicles are driving on the straight lane, Fig. 5-b. The back of the car being curved, a rotation invariance appears. The LiDAR points are free to move along the back of the model without changing the error. From Eq. ( 2), one can see that the estimated position of the ego-vehicle is dependent on the relative orientation. We obtain an accuracy of 1.82 m for the position and 8.55°for the orientation. The covariance is less consistent for this kind of perception (86.9%).

As one might expect, the lateral error, in the frame of the ground truth, is the largest part of the error, as shown by the uncertainty ellipse in Fig. 5b. This error is of 1.72 m compared to 0.35 m for the longitudinal error.

C. Second formulation

The cooperating vehicle computes the pose of the egovehicle with its pose and its perception sensor, Fig. 5 (cd). This second formulation of the cooperative localization problem gives a better accuracy, with 0.39 m for the position and 6.92°for the orientation. It also has a very good consistency rate of 98.0%. Indeed from Eq. ( 6), one can see that the estimated position of the ego-vehicle does not depend on the relative orientation anymore. One can also see that the orientation error does not change from one formulation to the other.

1) L-shape: As for the first formulation, the pose found for this perception is very accurate with a mean error of 0.29 m for the position and 4.20°for the orientation and also a bit too pessimistic (consistency rate of 99.3%).

2) C-shape: This perception is much more accurate and provides consistent information for this formulation. Indeed, the position is independent of the relative orientation. The mean error is 0.44 m for the position and 8.55°for the orientation and the consistency is 97.2% which is excellent.

D. Analysis

Based on these results, we can conclude that it is more accurate and more consistent to use the perception of the vehicle sending its pose than computing the pose of the ego-vehicle from its own perception. The accuracy of the second formulation is better because the position estimation is independent of the relative orientation uncertainty. The inconsistency of the global estimate possibly comes from an underestimation of the covariance matrix of the relative pose e Σ c . However, as the second formulation is consistent, it is reasonable to assume that e Σ c is actually consistent.

In conclusion, if two cooperative vehicles are driving together the vehicle with the most accurate localization should drive behind the other. Nevertheless, even if the second formulation gives better results, both formulations can be necessary in some scenarios, for example when only one vehicle is able to perceive the other.

V. CONCLUSION

This work has presented two formulations of a cooperative localization problem using the pose and shape of another vehicle. To compute the relative pose from the LiDAR points, an iterative minimization problem has been applied with points to lines matching. This led to a relatively consistent estimation of the relative pose. Two formulations have been tested experimentally and the accuracy and consistency have been evaluated in different situations. On curved roads where L-shapes can be perceived, both formulations gave good results with an accuracy that is of the order of magnitude of the decimeter. However, on straight lanes where the perception leads to C-shapes, the second formulation is more consistent and more accurate.

In future work, this cooperative localization method will be used with data fusion to improve the accuracy while keeping a good integrity. Different challenging scenarios will be tested, e.g., multiple vehicles in long tunnels or having only one vehicle accurately localized in a platoon. The cooperative method proposed in this paper can also be extended when two vehicles are perceiving a same feature of the environment. One vehicle can communicate its observation and the other one can localize itself using this shared observation.

Fig. 1 .

 1 Fig. 1. Platoon of two vehicles losing the GNSS signals while driving under a bridge.

Fig. 2 .

 2 Fig. 2. Two experimental vehicles equipped with GNSS receivers and LiDAR sensors on a test road at Compiègne, France. The LiDAR points are shown in red and the polygonal model is in blue. This figure shows the relative pose e qc of the mobile frame Rc of the cooperating vehicle (white car) in the frame Re of the ego-vehicle (gray car). R 0 is a local working frame in which the vehicles need to be localized for their navigation tasks.

Fig. 3 .

 3 Fig. 3. Distances to minimize in orange between the LiDAR point in red and the model in blue. The initial pose communicated by the cooperating vehicle is shown in blue and the final pose is in green.

Fig. 4 .

 4 Fig. 4. LiDAR detections in a straight line and a curved line.

  (a) 1st formulation, L-shape. (b) 1st formulation, C-shape. (c) 2nd formulation, L-shape. (d) 2nd formulation, C-shape.

Fig. 5 .

 5 Fig. 5. Estimated pose of the ego-vehicle, in green, computed from the pose of the cooperating vehicle, in blue, and the LiDAR points of the cluster, in red. This pose is compared to the ground truth in black. The gray polyline is the center of the road. For each pose, the model and the uncertainty ellipse and the uncertainty cone are shown in the same color.

Fig. 6 .

 6 Fig.6.Banana shape distribution of the estimated pose and its 95% confidence ellipse approximation, with the first formulation in blue and with the second formulation in red, from the communicated pose in black.

TABLE I MEAN

 I ERROR [ex ey e θ ] T IN THE FRAME OF THE GROUND TRUTH, MEAN HORIZONTAL ERROR e h AND CONSISTENCY RATE C OF THE TWO FORMULATIONS FOR THE COMPLETE SCENARIO, FOR THE L-SHAPE PERCEPTIONS ONLY AND FOR THE C-SHAPE PERCEPTIONS ONLY.

		1st formulation	2nd formulation
	shape	all	C	L	all	C	L
	ex (m)	0.27	0.35	0.13	0.11	0.11 0.09
	ey (m)	1.16	1.72	0.22	0.36	0.41 0.27
	e h (m)	1.24	1.82	0.27	0.39	0.44 0.29
	e θ (°)	6.92	8.55	4.20	6.92	8.55 4.20
	C (%)	91.5	86.9	99.2	98.0	97.2 99.3

The authors are with Sorbonne universités, Université de technologie de Compiègne, CNRS, Heudiasyc UMR 7253, 60203 Compiègne Cedex,

e qc,k = e qc,k-1 -R ( e θ c,k-1 ) ∆q k ,(14)

ACKNOWLEDGMENT

This work was carried out in the framework of the Equipex ROBO-TEX (ANR-10-EQPX-44-01) and Labex MS2T (ANR-11-IDEX-0004-02) projects. It was also carried out within SIVALab, a shared laboratory between Renault and Heudiasyc UMR UTC/CNRS.


S Idlahcen
email: idlahcen@coria.fr

C Rozé

L Méès

T Girasole

J B Blaisot

Spatio-temporal analysis of a liquid jet using a sub-picosecond optical gate

Spatio-temporal analysis of a liquid jet using a sub-picosecond optical gate

Introduction

Optimal use of liquid fuel needs a perfect understanding of the liquid injection inside the combustion chamber and of its break-up into small droplets. High pressure injection leads to complex structures in the near field of the nozzle, where the liquid column is stripped of droplets, liquid filaments, etc. This highly non-homogeneous medium makes tough imagery due to the almost random propagation of light. Time resolved imaging techniques have been proposed [1,2,3,4,5] to visualise the liquid core in spite of the surrounding scattering droplets cloud. In first approximation, the light transmission through a strongly scattering medium consists of two components, a ballistic and a scattered one. The first peak corresponds to the ballistic photons which have not interacted with the scatterers and constitute the first detected light. Scattered photons undergo multiple interactions, their trajectories are elongated and they are time delayed by a few picoseconds (second peak). In time resolved imaging techniques (ballistic imaging), an appropriate temporal selection of the ballistic photons reduces the influence of the scattered photons. A shadowgraph of the liquid core is then obtained by limiting the scrambling effect of multiple scattering.

The efficiency of ballistic imaging has been demonstrated through a 3.5 mm-thick human breast tissue for a 8 ps time gate [6] and through a 5 mm spray with a 2 ps time gate [4]. The relation between the spray dimension and the appropriated time gate has not been clearly established from now, probably because it depends strongly on many parameters. However, scattered photons time delay clearly depends on the actual length of scattering medium. So, we can state that reducing the time resolution in ballistic imaging technique is suitable to investigate smaller sprays. Furthermore, a shorter time gate provides additional information on the liquid core, based on the separation between refractive light and ballistic light. Such a separation between scattering modes (reflection/ diffraction, refraction, etc.) have been investigated experimentally [7] and numerically [8] in single scattering configurations and in a multiple scattering configuration [9].

Near the injector nozzle, the liquid has not yet completely exploded into a cloud of droplets. At this stage, the liquid keeps memory of the injector hole and is nearly cylindrical. Nevertheless, hydrodynamic surface instabilities appear in this domain and will lead to jet atomisation further from the nozzle. Some photons experience refractions through the liquid jet and may be delayed by a typical time  t =  n d c , where  n is the difference between the refractive index of the liquid and the surrounding gas, d is the diameter of the jet and c is the speed of light in free space. Knowing that in automotive internal combustion engines, diameter of injection hole is about 200 µm, a temporal detection with a resolution of about 300 fs is sufficient to discriminate refracted photons from ballistic ones. To reach this aim, an ultra-short light pulse (duration of 100 fs) illuminates the liquid jet and a time-resolved image is recorded thanks to an ultra-fast time gate and a single CCD camera.

No electronic device can achieve such an ultrafast detection and an optical gate must be used. Two kinds of optical gates may be found in the literature based on (i) an optical Kerr effect, or (ii) a second harmonic generation (SHG) in a non linear crystal. In the referenced papers, optical Kerr gate is obtained by inducing a temporary birefringence in a Kerr active liquid between two crossed polarisers [1,2,3,4,5]. The temporal resolution of the gate is fixed by molecular reorientation time of the Kerr liquid and is of the order of 1 ps or greater. The second kind of optical gate has been used to obtain ballistic images and detect hidden objects in a turbid medium [10,11,12]. The aim of this paper is to study the potential of SHG crystal time gate to extract liquid shape properties of the liquid in the near field of an injector. In the next section, temporal and spatial resolutions are presented. Glass fibre and spray images are presented in the last section.

Experimental setup

The principle of optical gating is to divide the laser pulse of wavelength 800 nm into two pulses: the imaging pulse crosses the spray, while the gating pulse is used to open the time gate and may be shifted in time by means of a delay line. The time gate consists of a beta barium-borate (BBO) crystal, which has a quadratic nonlinearity. The imaging pulse and the gating pulse are incident on the crystal with an angle θ. First, collinear second harmonic generation (SHG) occurs, so that light at wavelength 400 nm is generated along the direction of each pulse. When the two pulses overlap spatially and temporally on the crystal, non-collinear second harmonic is generated in direction  k 2=  k 1   k 2  , which is the intermediate direction between the two collinear second-harmonic pulses. The crystal orientation is adjusted by a rotation around a vertical axis, in order to obtain the best conversion efficiency. The SHG is quasi-instantaneous, so that the gate duration is only limited by the gating pulse duration and angle between the pulses.

Figure 1. Schematic diagram of experiment

The integration of the gate inside the complete experimental setup is shown on Fig. 1. A Ti:Sa amplified laser system (800 nm, 1 kHz, 1 mJ) generates constant 100-femtosecond pulses. The beam profile is approximately Gaussian with a 5 mm full width at half maximum (FWHM). The laser pulse is separated in two parts by means of a beam splitter (BSP), the imaging pulse and the gating pulse. The gating pulse passes through a delay line, a half wave plate (λ/2) and a neutral density filter (ND), in order to adjust time delay, polarization direction and intensity. The imaging pulse passes through a Pockels cell (PC) in order to decrease the repetition rate of the pulses down to 1 Hz (the spray repetition rate). A set of neutral density filters (NDs) is then used to adjust the lighting power. The imaging pulse next illuminates the spray injected in a chamber, equipped with glass windows. Imaging pulse and gating pulse are incident on a BBO crystal with the angle θ between them. Careful optical alignments are conducted to overlap the two beams in both spatial and temporal domains on the crystal. Non-collinear second harmonic is generated at half angle θ/2 between the directions of the two pulses direction when and where they overlap. A time gated second harmonic image of the spray is then formed by means of a lens (L1, f=150 mm). A vertical slit may be placed in the corresponding image plane, the use of which will be explained later. A second lens (L2, f=100 mm) forms a second image of the spray on the BBO crystal. A third lens is used to record the non collinear SHG on a CCD camera (Hamamatsu C9100). Light at fundamental wavelength (800 nm) is scattered the camera. This noise source is eliminated by a spectral filter placed between the BBO crystal and the CCD camera, increasing notably the signal-to-noise ratio.

In order to measure the temporal profile of the gate, the power of the SHG pulse at angle θ/2 is measured for various delays between the imaging pulse and the gating pulse, without liquid jet. The maximum corresponds to a temporal and spatial synchronisation between the two pulses in the BBO crystal, whereas the tails of the curve are obtained for nonzero delays between the pulses. The duration of the gate is defined as the full width at half maximum (FWHM) of the curve and is equal to 270 fs. It depends strongly on the angle θ between gating pulse and imaging pulse and finally, θ ≈ 8° seems to be a good compromise.

To evaluate the spatial resolution of the overall system, the image of an U.S. Air Force Chart is recorded on the camera. It allows of evaluate the modulation transfer function (MTF) of the gate for a range of spatial frequencies. This procedure has been used to assess the resolution of the imaging system to 10 µm.

Spatio-temporal analysis of the transmitted light

Images of a glass fibre

In order to investigate the potential of the BBO crystal based time gate, preliminary images have been recorded using a glass fibre as the object, without any other scattering medium. Fig. 2(a) shows the non-gated image of a fibre (diameter = 240 µm). The two strips located at each side of the image is the background light, whereas the thin line at the centre corresponds to refracted light into the fibre. When the optical gate is operating, the ballistic light may be isolated (Fig. 2(b)). By adjusting the delay line, refractive light is isolated (Fig. 2(c

Images of the liquid jet near the injector

The same set-up has been applied to a Diesel jet visualisation. The injection pressure is 400 bars, the diameter of the nozzle hole is 200 µm and the liquid is injected at atmospheric pressure and ambient temperature in a chamber. In the near field of the injector, the liquid roughly keeps the cylindrical shape of the injector hole. Observation of refractive light is expected at this location. Time gated images are shown on Fig. 3, for two different delays between imaging and gating pulses. In both cases, the two pulses overlapped only partially, as a consequence of the angle θ, as schematically illustrated in the figure bottom. In figure 3(a), the delay line is adjusted in such a way that the gating pulse and the ballistic component of the imaging pulse overlap in the left side of the image. The right side of the image, where the two pulses do not overlap remains dark. In the middle part of the image, ballistic component does not overlap either temporally with the gating pulse. But refractive light, which is time delayed due to the jet crossing, overlap with the gating pulse and is clearly visible on the image. Conversely, when the delay is adjusted to overlap the gating pulse and ballistic light at the right side of the image, refracted light is no more detected and the liquid core appears to be completely dark (Fig. 3(b)). It must be noticed that if the image and gating pulses are symbolised by rectangles on the schemas of Fig. 3, their temporal and spatial shapes are approximately Gaussian, so that the tails of the pulses may be used to detect light at time delays smaller than the 270 fs duration, depending on the sensitivity of the detector. The time delay between image 3(a) and 3(b) is  =500 fs. Knowing the refractive index of the injected liquid equal to 1.46, we can deduce an estimation of the jet diameter of the jet diameter to about 325 µm.

Spatio-temporal diagrams

The qualitative description of the preceding images leads to the conclusion that a spatio-temporal study of the light after interaction with an object is possible by changing the delay   between the gating pulse and the image pulse. Indeed, the image on the camera is the result of the spatio-temporal correlation of the gating pulse with the image pulse along the width of the BBO crystal. The complete model is complex, because of interdependent phenomena: the spatio-temporal profile of each pulse, the angle between beams, the slight translation of one pulse relatively to the other inside the crystal due to their different propagation directions, etc. However, a rough interpretation of the image may be proposed by making some assumptions : (i) the image pulse and gating pulse are so short enough to be modelled by a time Dirac, (ii) the magnification of the optical system is large, so that the image pulse may be considered uniform in space before interaction with any object, (iii) the thickness of the BBO crystal is small, so that the shift of the gating pulse relatively to the image pulse is neglected, (iv) SHG light rays from the BBO are parallel.

With these assumptions, when the pump beam crosses the image beam somewhere in the crystal with angle θ between their propagation directions, a third pulse is produced, due to the SHG, at half angle. This intersection (point A on Fig. 4) produced a bright zone on the image recorded by the camera.

Spatio-temporal diagrams of the spray

The same process may be applied on the spray far from the nozzle. Fig. 6 is an example of a recorded image at 5 mm from the nozzle, showing many bright spots corresponding to scattered light, which do not exist in the case of the fibre. When the delay   is changed, the texture of the image changes, leading to the conclusion that the source of this scattering may not be the same.

Again, images are recorded by varying the delay the gating pulse and the image pulse. 100 images are recorded for each delay and the same image processing than for the fibre is applied. Spatio-temporal diagrams I  -  x , x are shown on Fig. 7 in two cases: (a) without the spray, in order to determine coefficient  , and (b) with the spray. Cases (c) and (d) show corrected diagrams by using a normalisation. One can observe that the optical gate is able to select the ballistic light. Nevertheless, due to the image averaging, and because of the limited dynamical range of the camera, neither scattered light nor ballistic one is visible on the spatio-temporal diagrams in the centre of the spray. Increasing the intensity of the imaging pulse is not a solution: a more intense imaging pulse would damage the camera. That is why a vertical slit has been added in an intermediate image plane, in order to eliminate the sides of the image (Fig. 1). The imaging pulse may then be more intense, without the risk of a camera damage. In this case, the intensity of the imaging pulse may be increased until getting a detectable level of the transmitted light through the spray.

We have used this method at 200 µm from the nozzle, in a more dense part of the spray. Images are again recorded by changing delay   between gating pulse and imaging pulse. An average is made over 100 images for each  , and spatio-temporal diagrams are computed and shown on Fig. 8. Transmitted light at the centre of the spray is now observable. A time shift is visible between the reference diagram without spray (a) and the diagram with the spray (b). A delay of ~ 0.4 ps between the ballistic light and the transmitted light may be estimated by comparing these diagrams. It corresponds to an average jet liquid depth of about 300 µm, which is realistic at such distance of the nozzle.

Conclusion

The use of an optical gate based on SHG in a BBO crystal allows to reduce the temporal resolution in ballistic imaging, which is now compatible with the dimension of the spray resulting from a Diesel injector. This duration opens the way to a new analysis of the liquid jet by studying the transmitted light as a function of time. In the near field of the nozzle, the time delay at which the refraction is observed is a measurement of the jet diameter. The refracted light pattern would be a signature of the surface state of the liquid. The first instabilities appearing in the liquid may then be quantified and compared for example with direct numerical simulations (DNS) [13] to evaluate if correct destabilisation mechanisms are taken into account in the simulations.

Spatio-temporal diagrams of transmitted light open perspectives to investigate liquid jet atomisation, including in high speed and high pressure conditions. The characterisation of the structure of the spray would be possible by the analysis of the image texture produced by the scattered light.

Figure 2 .a

 2 Figure 2. a Image of a fibre without optical gate b, c for two different delays of the optical gate

Figure 3 .

 3 Figure 3. Images of the liquid jet at the exit from injector hole for two different delays between image and gating pulses

Figure 4 .Figure 5 .

 45 Figure 4. Relation between the maximum of intensity recorded on camera and the delay. The propagation directions of both pulses are in the horizontal plane Now, if the gating pulse is affected by a delay  , the intersection point A is shifted to point B. Then a shift  X appears on the camera screen. Following the schema of Fig. 4, the relation between   and  X leads to  =  X , with =2 sin n /2 c / n , where c is the speed of light in vacuum,  n is the angle between the two beams inside the crystal, and n the refraction index of BBO crystal. The preceding considerations may then be applied to the fibre. Images are recorded by varying the delay   between gating pulse and image pulse. For each value  , 20 images of size 1000x1000 pixels are recorded from the CCD camera and an average image I   , x , y  is computed. Then, intensity I   , x  is obtained by averaging the pixels of each column y. The spatio-temporal diagrams I   , x  are shown on Fig. 5, in two cases: (a) without the fibre and (b) with the fibre. In Fig. 5(a), the diagram is composed of ballistic photons. The bright area is oriented in accordance with the slope coefficient  previously introduced. The diagram of Fig. 5(b)shows a hole in the ballistic light, corresponding to the shadow of the fibre. As expected, refracted light undergoes a delay due to the fibre crossing. It shows that with a suitable processing, the interaction of light with an object may be followed in time with the time gate based on a BBO crystal.

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. Image of the spray far from the nozzle, for some delays between gating pulse and image pulse