Ali Hamdan

Reine Talj

Véronique Cherfaoui

A Fuzzy Logic Shared Steering Control Approach For Semi-Autonomous Vehicle

Keywords: semi-autonomous vehicle, human-machine cooperative control, cooperative driving, shared lateral control, Fuzzy logic decision-making

This paper deals with the shared lateral control between the driver and the automated system for the lane keeping purpose. The objective of this shared control is to ensure road safety and enhance driving performance. The control authority is realized using the blended shared control that permits the fusion of two inputs via a fusion parameter. To do that, a driver model is detailed to represent the human in the loop. The vehicle longitudinal and lateral movement are performed by an autonomous system based on Super Twisting Sliding Mode (STSM) control approach. A Fuzzy Logic Controller followed by a situation-based analysis block for the decision-making process are developed for the fusion parameter determination. Finally, the proposed shared control is validated on Matlab/Simulink for two case studies with a complete nonlinear model of the vehicle validated on "SCANeR Studio" (OKtal) professional simulator.

I. INTRODUCTION

Safety is an important criterion that should be realized while driving on the road. Road accidents are caused by the human's errors in most cases. According to National Highway Traffic Safety Administration (NHTSA) statistics, these errors lead to 90% of road accidents [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF]. In a critical situation, when the driver is tired or distracted, the integration of an Advanced Driving Assistance System (ADAS) helps him to prevent this dangerous situation. There are many examples of ADAS systems, like ESP, ABS, and lane keeping assist systems. The latter can help the driver to stay within the lane and warn him when he is crossing the boundaries. These systems ensure safety and performance of driving tasks. However, these systems are dedicated for specific tasks, in some precise driving situations, and their functionalities are limited. Hence, full autonomous vehicles are taking an important attention in the researches and automotive industry where several tasks of ADAS systems are done by an autonomous system itself. But, the realization and commercialization of these autonomous systems need more time for many reasons: high cost of hardware and software, safety on the road, robustness of the system, test and validation of the system for any possible scenario on the public road, etc. The autonomous driving system is the result of different tasks: Environment perception, motion planning and vehicle control. However, many problems are caused by these modules, and will need more time to be fixed. Thus, a shared cooperative control is necessary to 1 Université de Technologie de Compiègne, CNRS, Heudiasyc, CS 60 319, 60 203 Compiègne, France. Email: ali.hamdan@hds.utc.fr compensate the gap between the manual driving (level 0/1 and 2) and full autonomous driving (level 5) [START_REF] Favarò | Examining accident reports involving autonomous vehicles in california[END_REF]. This new aspect of control in the automotive field aims to reduce the workload of the driver. It helps and assists him to prevent a critical situation when he is distracted or tired. In addition, interaction between the human and the autonomous system that keeps the driver in the loop, should compensate the limits and problems of autonomous systems and improve driving safety. Therefore, if a conflict occurs during driving, it will be reduced by this shared strategy of control.

In this work, we will focus on the shared lateral control for the lane keeping purpose. The vehicle is asked to realize a path following. Two case studies of driving are considered later to address this control strategy under different driving conditions. Many advanced studies have been presented in the literature to investigate shared control between driver and an automated system. In fact, these studies differ in the way how to manage the control authority between the two agents. A shared lateral control is presented in [START_REF] Sentouh | Toward a shared lateral control between driver and steering assist controller[END_REF] to illustrate the authority of driving between the driver and the automatic system. A decision making algorithm using a weighting function related to the lateral error was used to manage this authority and switch between them. Similar approach is presented in [START_REF] Borroni | A weighting approach to the shared-control of lateral vehicle dynamics[END_REF] to realize the control authority through a blending parameter varying between 0 and 1, adjusted manually or automatically depending on the driver's attention and situation. A first order filter applied to a coordination variable is used in [START_REF] Sentouh | Cooperative steering assist control system[END_REF], to determine the control authority depending on the product of the driver and assistance system torque. The authors in [START_REF] Nguyen | Online adaptation of the authority level for shared lateral control of driver steering assist system using dynamic output feedback controller[END_REF], [START_REF] Nguyen | Driver-automation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF] have been developing a shared steering control for lane keeping and obstacle avoidance. The control input of assistance system is computed considering the driver's activity which reflects his actions and intentions. The authors in [START_REF] Iwano | A study on shared control between the driver and an active steering control system in emergency obstacle avoidance situations[END_REF] suppose that the steering torque provides the driver some degree of freedom to control his vehicle. For that, a shared control has been developed for the emergency obstacle avoidance, where the assistance torque is multiplied by a weight coefficient ω varying between 0 and 1 to do the control allocation. However, the conflict between the driver and the steering assistance increases with the increasing of parameter ω. An assistance steer-by-wire system is presented in [START_REF] Perozzi | On nonlinear control for lane keeping assist system in steer-by-wire road wheeled vehicles[END_REF], where the fusion of two steering inputs is performed considering the availability of the driver via monitoring system. Another lane keeping assistance system is used in [START_REF] Soualmi | Fuzzy takagi-sugeno lq controller for a shared control of vehicle[END_REF] based on the fuzzy Takagi-Sugeno control method. Finally, many factors related to safety and driver (avoidance of lane departure, prevention of high acceleration and excessive steering) are taken into account when calculating the fusion parameter to deal with the tire blowout on the road [START_REF] Li | Shared steering control of tire blowout for ground vehicles[END_REF]. Other studies consider the driver's steering in the trajectory planning, where the desired trajectory of the system is adjusted to meet the driver's intention [START_REF] Benloucif | A new scheme for haptic shared lateral control in highway driving using trajectory planning[END_REF].

All these interesting studies have motivated us to develop a shared control for the lane keeping objective using steer-by-wire system. However, this method can be adapted to consider the tracking of a local trajectory with more complex maneuvers (overtaking, collision avoidance...). Thus, in our present work, a new shared lateral control is developed to manage the control authority between driver and autonomous system in order to reduce the conflict and promote road safety. The novelty of this work is illustrated in the computation of the fusion parameter. A Fuzzy Logic Controller (FLC) associated to a situation-based Analysis Block are used to determine the fusion parameter according to many dynamics criteria such as: the lateral deviation error, the conflict of the automated steering angle and the driver's confidence. Then, the proposed approach is tested for two case studies to demonstrate the effectiveness of this method to ensure safety and improve driving performance.

The paper structure is as follows: in Section II a driver model reformulation is provided to represent the driver in the loop. Then, a longitudinal and lateral controller for autonomous system are exposed based on the the Supertwisting sliding mode (STSM) control approach, to follow the desired trajectory at the desired velocity. A full description of the proposed shared lateral control is introduced in Section III. The shared lateral control authority is detailed to blend the two inputs. To do that, A fuzzy logic controller and a situation-based analysis block are developed to determine the fusion parameter considering the two case studies. In Section IV the two case studies, simulation results and analysis are described. The validation of the proposed shared control is done on Matlab/Simulink with a complete nonlinear model of the vehicle, validated on "SCANeR Studio" (OKtal) simulator. Finally, the conclusions and the perspectives for future work are given in Section V.

II. SHARED STEERING ARCHITECTURE

The block diagram of the shared control is given in the Fig. 1. Its main components are: the Driver Model, the Autonomous System (detailed in the following), and the shared lateral control (detailed in Section III).

A. Driver Model

The proposed human driver model for the lane keeping purpose in [START_REF] Saleh | Human-like cybernetic driver model for lane keeping[END_REF], [START_REF] Mars | Modeling the visual and motor control of steering with an eye to shared-control automation[END_REF] is used to represent the human's behavior in the loop. This model is based on the hypothesis that the driver uses the visual information to anticipate with the upcoming road changing, and the vehicle state's information (velocity, position, heading...) to position the vehicle on the road (Anticipatory and Compensatory action). The structure of the adopted driver model, used in this work, is given in Fig. 2. The calculation of angles' block that gives θ near and θ f ar angles is based on the definition in [START_REF] Mars | Modeling the visual and motor control of steering with an eye to shared-control automation[END_REF]. θ near is calculated at a distance l p in front of the vehicle to maintain the vehicle's lateral position. It is function of the vehicle's heading error e ψ (e ψ = ψ-ψ * ) and the lateral error e y (e y = y-y * ), where ψ * and y * are the desired heading angle and the lateral coordinate of the road respectively. e ψ and e y are calculated by a Map Matching block (see Fig. 1) that aims to localize the vehicle position on the reference map extracted from a "SCANeR Studio" scenario. θ f ar is calculated as a tangent point (Fig. 3) at a far distance D f ar that depends on the curvature of the road. The far distance becomes constant when the driver drives on a curve road (usually between 10 and 20 m according to the radius of road curvature). It depends on the vehicle's velocity V x and the heading velocity of the road ψ * . θ near and θ f ar (see Fig. 3) are given as:

θ near = e y l p -e ψ , (1)
θ f ar = D f ar V x * ψ * . (2)
The visual information is composed of two parts: A compensatory part G comp that depends on θ near to maintain a centerline position of the vehicle, and anticipatory part G anti which acts upon θ f ar to consider the upcoming road curvature. G anti and G comp are given as:

G anti = k p , (3)
G comp = k c V x T L s + 1 T I s + 1 , (4)
where k p is a positive constant parameter and k c represents the driver's cautiousness when he drives close to the lanes markers. T L and T I define the compensation frequency band and the compensation rate respectively. s is the Laplace transformation variable. Finally, the visual processing delay module is represented by e -τ p s . An approximation for the visual processing delay is given as:

e -τ p s = 1 -0.5τ p s 1 + 0.5τ p s , (5)
where τ p represents the time delay. Thus, the output of the adopted driver model is the steering wheel angle δ sw . Noting that the relation between the steering wheel angle δ sw and the real angle of rotation of the front wheels δ d is given as a proportional function:

δ d = δ sw 14.04 (6)
The values of the different parameters are given in Table I [START_REF] Saleh | Human-like cybernetic driver model for lane keeping[END_REF].

B. Design of Longitudinal and Lateral Controllers for Autonomous System

The autonomous system consists of longitudinal and lateral controllers based on the super-twisting sliding mode (STSM) algorithm, validated separately to realize a path following at the desired velocity. The main idea of the STSM is to define a sliding surface, representing the desired behavior of the system, where the dynamic states are forced to reach this surface during a finite time and remain on it. Consider the second order system given as:

Ẋ = f (X,t) + g(X,t)u(t) (7)
where X = [x, ẋ] T ∈ ℜ 2 is the state vector, u is the control input, and f , g are continuous functions. X * is the desired state of X with X * = [x * , ẋ * ] T ∈ ℜ 2 . The error vector is given by E = X -X * = [e, ė] T ∈ ℜ 2 where e = xx * and ė = ẋẋ * . Therefore, a sliding variable s with relative degree r = 1 w.r.t the control input, is defined as:

s = ė + k e. (8)
The second order derivative of s is:

s(s,t) = Φ(s,t) + ξ (s,t) u(t) (9)
where Φ(s,t) and ξ (s,t) are unknown bounded functions.

The goal of the Super-Twisting algorithm is to enforce the sliding variable s to converge to zero (s = 0) in finite time.

Assume that there exist positive constants S 0 , b min , b max , C 0 , U max verifying for all X ∈ ℜ n and |s(X,t)| < S 0 :

   |u(t)| ≤ U max |Φ(s,t)| < C 0 0 < b min ≤ |ξ (s,t)| ≤ b max (10)
Thus, the control input based on the Super-Twisting Sliding Mode algorithm [START_REF] Rivera | Supertwisting sliding mode in motion control systems[END_REF] is given as:

u(t) = u 1 + u 2 u 1 = -α 1 |s| τ sign(s), τ ∈]0, 0.5] u2 = -α 2 sign(s) (11)
α 1 and α 2 are positive gains. The following conditions guarantee the finite time convergence:

   α 1 ≥ 4C 0 (b max α 2 +C 0 ) b 2 min (b min α 2 -C 0 ) α 2 > C 0 b min (12)
The convergence analysis is shown in [START_REF] Utkin | On convergence time and disturbance rejection of supertwisting control[END_REF].

The controller synthesis is based on a robotic formalism model presented in [START_REF] Chebly | Coupled longitudinal/lateral controllers for autonomous vehicles navigation, with experimental validation[END_REF], [START_REF] Chebly | Trajectory planning and tracking for autonomous vehicles navigation[END_REF] (more details in APPENDIX), that represent the coupling between the longitudinal and lateral dynamics. Based on this model, we choose the two sliding variables for the longitudinal and lateral controllers as follows:

s 1 = e V x + λ x e V x , λ x > 0 s 2 = ėy + λ y e y , λ y > 0 ( 13
)
where λ x and λ y are positive constants, and, e V x (e V x = V x -V * x ) and e y are the vehicle longitudinal speed error and the lateral error respectively. The sliding variables s 1 and s 2 have a relative degree equal to one w.r.t the inputs respectively, the driving/braking torque Γ c for the longitudinal dynamics and the steering angle δ c for the lateral dynamics. Thus, in order to converge these variables to zero and the controlled states follow the desired ones, and based on the above discussion, the torque and the steering angle control applied to the vehicle, are given by:

Γ c = -α Γ c ,1 |s 1 | τ Γc sign(s 1 ) -α Γ c ,2 t 0 sign(s 1 )dτ, δ c = u 1 + u 2 + δ *    u 1 = -α δ ,1 |s 2 | τ δ sign(s 2 ), u 2 = -α δ ,2 t 0 sign(s 2 )dτ, δ * is the equivalent control input, (14)
where α δ ,i and α Γ c ,i with i = [1, 2], are positive constants satisfying the conditions in [START_REF] Benloucif | A new scheme for haptic shared lateral control in highway driving using trajectory planning[END_REF]. τ Γ c and τ δ are constants in ]0, 0.5]. The controller gains are given in [START_REF] Chebly | Coupled longitudinal/lateral controllers for autonomous vehicles navigation, with experimental validation[END_REF]. Finally, δ * is the equivalent control input, corresponding to the steering wheels angle at the equilibrium when ṡ2 = 0.

III. DESIGN OF SHARED LATERAL CONTROL

This section provides a full description of the proposed shared lateral control (see Fig. 1). The fusion approach based on the fuzzy logic controller and the situation-based analysis block are developed in the following for decision-making.

A. Shared Lateral Control Authority

The shared control authority between the driver and the autonomous system is performed by using the blended shared control [START_REF] Borroni | A weighting approach to the shared-control of lateral vehicle dynamics[END_REF] which allows the fusion of two inputs from each agent. This type of shared control is used for the systems where there is no mechanic connection between both inputs such as the steer-by-wire system. The familiar form of blending shared control is the blending using weight parameter (see. [START_REF] Li | Shared steering control of tire blowout for ground vehicles[END_REF], [START_REF] Li | Shared control driver assistance system based on driving intention and situation assessment[END_REF]). The total blending control input is given as:

δ total = α * δ d + (1 -α) * δ c , (15)
where δ total , δ d and δ c are the total control input, inputs of human driver and automated system respectively. α is the fusion parameter representing the influence proportion of each agent on the total input. α is bounded in [0,1]. The blending shared control permits a direct interaction between human driver and autonomous system. The fusion parameter α is calculated based on Fuzzy Logic Controller and a situation-based analysis block (Fig. 1) detailed later.

B. Fuzzy Logic Controller (FLC)

An intermediate fusion parameter α is determined by a fuzzy logic controller (FLC) as in Fig. 1. The situation-based analysis block analyzes the driving situation and gives the final value of fusion parameter α in order to promote driving safety and avoid dangerous situations. The fuzzy logic structure is given by the Fig. 4. The lateral error e y , the conflict of the automated steering angle δ con f lict and the driver's confidence are applied to the fuzzy logic controller (FLC) as inputs, and the fuzzy parameter α is the output. Two scenarios are discussed and later addressed to demonstrate the effectiveness of this approach against different driving conditions. Using the FLC for the decision-making process for the cooperative control, makes the determination of the intermediate fusion parameter α simple, depending on the lateral error, the conflict of the automated steering angle δ con f lict and the driver's confidence. Note that the conflict of the automated steering angle δ con f lict is the difference between the steering wheel angle applied by the autonomous system (δ c ) and the reference steering angle (δ c-re f ) on the desired trajectory given as:

δ con f lict = |δ c-re f -δ c |, (16)
On the other hand, the driver's confidence can be determined in different ways depending on: driver's eyes observation, environment conditions, trajectory prediction, etc. However, the calculation of driver's confidence is not in the scope of this paper and it is considered as an input to the FLC fusion block. Three fuzzy sets are defined for the lateral Two fuzzy sets are defined for the conflict automated steering angle δ con f lict : {S(Small), B(Big))}. Note that δ con f lict and the autonomous system's confidence are directly dependent. A small δ con f lict corresponds to a high autonomous system's confidence and vice-versa. Three fuzzy sets are defined for the driver's confidence: {S (Small), M(Medium), B(Big)}. And finally, three fuzzy sets are defined for the output parameter α : {S(Small), M(Medium), H(High)}.

The normalized Membership Functions (MFs) of fuzzification of the controller inputs and defuzzification of the controller output are respectively given in Figures. 5a, 5b, 5c and 5d. The values and the types of these fuzzy sets are chosen depending on the feedback and observation of simulation for different scenarios and a lot of experiments. To determine the fuzzy controller output α for the given fuzzy controller inputs e y , δ con f lict and the driver's confidence, the decision matrix of the linguistic control rules is used. There are 18 rules (in total) where 15 are determined in the FLC to give the value α , and the 3 remaining rules are detailed in the analysis block (see Fig. 1). So, the analysis block receives α and modifies it if necessary, depending on the driving situation. The final output of the analysis block is α. Noting that α is equal to the output of FLC α if no check needed. The goal of using the analysis block with the FLC is to cover all the possibilities of driving situations that can occur while driving on the road. The rules of FLC are designed and presented in the Table II and Table III for δ con f lict small or big respectively. These fuzzy sets, membership functions, and the linguistic rules are usually determined based on an expert knowledge of the system by performing several simulations for different driving conditions. Finally, to defuzzify the result/output, the "Mamdani centroid fuzzy inference method" is used [START_REF] Reznik | Fuzzy controllers handbook: how to design them, how they work[END_REF].

C. Situation-Based Analysis Block

In order to check the remaining 3 cases, an algorithm (given below) is developed in the analysis block to cover the possible driving situations that cannot be resolved appropriately by the FLC approach.

Algorithm 1 checked case in the analysis block if (lateral error is S) and (driver's confidence is S) and (δ con f lict is B) then retain the previous input else if (lateral error is B) and (driver's confidence is B) and (δ con f lict is S) then α = 1 and brake the vehicle else if (lateral error is B) and (driver's confidence is S) and (δ con f lict is B) then Emergency mode until driver will be attentive else Keep the fusion parameter α from FLC end if

IV. VALIDATION OF THE APPROACH

In this section, the developed shared lateral control is validated on Matlab/Simulink with a complete nonlinear model of the vehicle, validated on "SCANeR Studio" (OKtal) simulator. The vehicle is asked to follow the desired trajectory given in the Fig. 6, of a track extracted from a "SCANeR Studio" scenario. A control authority decision is generated by the FLC and the analysis block in order to determine the fusion parameter and then the leader of the vehicle for each case. The results are presented to show the adaptation and functionality of FLC and analysis block in each case.

A. Case Study Definition

In this subsection, the case studies are detailed to test and validate the proposed shared lateral control for different driving situations. Indeed, two case studies are presented for the same scenario to show the adaptation and functionality of the proposed approach in each case:

• Case 1: This case is defined to show functionalities managed by the FLC. The developed controller aims to follow the desired trajectory by reducing the lateral error, at a desired velocity. It is addressed to assist the driver and help him in a shared control mode (an example of Advanced driving assistance system ADAS).

A human driver (presented by the driver model in the simulation) is kept in the loop to supervise the scene and interact if necessary. A sudden failures occur in the autonomous system's behavior at 9s and 40s respectively (see Fig. 8). The vehicle deviates from the centerline of the road and the driving situation is considered dangerous. A driving mode is defined in a way to give the total authority of control to the driver at 9s even when the driver's confidence is Medium (0.5)(Fig. 9). In addition, at 40s, the driver takes the control action to compensate the second failure of the automated system and retain the driving stability. Noting that the driver is aware and attentive about the scene at 40s to the rest of trajectory (full Driver's confidence=1, see Fig. 9). The importance of this case is to illustrate the effectiveness of our method in terms of smooth switching between the two agents and failure's detection on the autonomous system. So, the control authority is determined through the fuzzy logic controller to manage the driving situation. • Case 2: This case is defined to show the functionality of the analysis block with the FLC. Two functionalities of the analysis block are tested in this case: An emergency mode (EM) situation where the confidence in the driver is lost (Fig. 13) and a failure occurs on the autonomous system at a defined time (Fig. 12). The vehicle decelerates to stop. The ON/OFF conditions of emergency mode are given in the Fig. 7. Note that e y min and V x min are the lower limits of the lateral error and the longitudinal speed respectively, that define a no critical driving situation. δ threshold is a threshold from which we consider that is an error on the autonomous system and the conflict on the automated steering angle is big that means the autonomous system's confidence is low. The second mode keeps the previous input that realizes a small lateral error when there is no driver's confidence (driver's confidence=0) at the same time of autonomous system's failure (Fig. 12). The driver's confidence varies between [0;1] (Fig. 13).

Results will be addressed in the next subsection, including the fusion parameter α to show the shared control authority between the two agents for each case given below.

B. Simulation Results

For the case 1, errors are injected between 9s and 23s and between 40s and 70s respectively on the automated system (see Fig. 8). Remember that the human (presented by the driver model) is the supervisor of the scene who compensates for the errors that occurred in the behavior of the autonomous system. The Fig. 8 shows the lateral error of: autonomous system with failure, manual driving mode and shared mode. As we can see, the shared mode diminishes the lateral error (1.5 m) caused by the autonomous system failure on the curvy road (see Fig. 11), by penalizing the last one and gives the authority of control to the driver. So, the driver takes the action of control at the beginning of the failure 9s, that is the decision of the FLC, until 23s, even though his confidence is medium (Fig. 9). The curve of lateral error in the shared mode is covered with the manual curve between [9s;23s](Fig. 8). Similar to the second failure, the driver acts on the vehicle at 40s to 70s in order to diminish the lateral error of autonomous system. A smooth switching is done between the two agents that makes the system stable. The shared control is done thanks to the fusion parameter α (Fig. 9) delivered by the FLC and analysis block modules. As shown in the Fig. 9, the value of α is equal to 0 expect the two intervals of time: [9s;23s] and [40s;70s]. For the region where α = 0, the autonomous system is able to control the vehicle. However, for two intervals of time: [9s;23s] and [40s;70s], an unexpected error occurred on the autonomous system, illustrated by a conflict in the Fig. 10. At this time, the driver interacts by taking the action of driving to reduce the conflict of autonomous system and α increases from 0 to 1. The STSM steering angle, the driver steering angle, the shared steering angle calculated by the fusion system and δ con f lict are given in the Fig. 10. And finally, Fig. 11 shows the longitudinal speed converging to the desired one through the STSM controller, the road curvature of the desired trajectory and finally, the actual lateral acceleration in the shared mode that demonstrate a stable and comfortable driving. Concerning case 2 the failure of autonomous system is injected at 40s until 70s (see Fig. 12) where there is no driver's confidence between 40s and 50s (driver's confidence= 0; see Fig. 13). An emergency mode is activated corresponding to α=1 (Manual steering) while braking the vehicle that means the driver takes the driving task in the emergency mode situation until the vehicle is stopped (Fig. 13). Then, the driver is more attentive and careful to take the action of control between 50s until 60s (driver's confidence= 1; see Fig. 13). He is still acting on the vehicle's control between 60s and 70s even though his confidence decreased again to 0 (Fig. 13). Therefore, this mode is defined in the algorithm above which retains the previous input if it leads to a small lateral error, even if the confidence is small on this input. Fig. 12 shows the different lateral errors. The different values of fusion parameter α are given in Fig. 13, and Fig. 14 shows the different steering angles. The longitudinal speed, the road curvature of the desired trajectory, the driving/braking torque and the longitudinal and lateral acceleration of the shared mode are given in Fig. 15. As we can see in Fig. 15, when the emergency mode is activated, the vehicle starts the emergency mode by decreasing its velocity until the driver will be more attentive and will be able to drive the vehicle. And finally, the longitudinal speed starts to increase at 50s, ending the emergency mode. Noting that driving is comfortable for the overall trajectory including the emergency situation that ensure a stable driving situation. Moreover, many different tests are done for several driving situations to validate the proposed approach based on the decision making of the FLC and the different functionalities of block analysis. To conclude, in this paper a shared lateral control has been developed to improve the system's performance and enhance driving safety. This cooperative control is done by using blended shared control, where a fuzzy logic controller and a situation-based analysis block are presented to determine the decision making authority. The proposed shared control is validated in Matlab/Simulink with a complete nonlinear model of the vehicle, validated on "SCANeR Studio" (OKtal) simulator. The validation of this shared control under different possible critical driving situations, is done in this work. In addition, the results show the effectiveness of the proposed approach to compensate the errors of autonomous system and prevent an undesirable driving situation. In the future work, we will consider other criteria to prove the effectiveness of the decision making process, and validate the approach on the "SCANeR Studio" simulator, connected to a hardware-inthe-loop steering system.

APPENDIX

The vehicle model is based on a robotic formalism model presented in [START_REF] Chebly | Coupled longitudinal/lateral controllers for autonomous vehicles navigation, with experimental validation[END_REF], [START_REF] Chebly | Trajectory planning and tracking for autonomous vehicles navigation[END_REF]. However, to simplify the controller design, some assumptions are given in [START_REF] Chebly | Coupled longitudinal/lateral controllers for autonomous vehicles navigation, with experimental validation[END_REF]. With all assumptions, the reduced vehicle model used to establish control laws, can be written as:

m e ẍ -m ẏ ψ + L 3 ψ2 + F aero + δ (2C α f δ -2C α f ẋ( ẏ + L f ψ) ẋ2 -( E 2 ψ) 2 ) = g 1 m ÿ + m ẋ ψ -L 3 ψ + 2C α f ẋ( ẏ + L f ψ) ẋ2 -( E 2 ψ) 2 + 2C α r ẋ( ẏ -L r ψ) ẋ2 -( E 2 ψ) 2 = g 2 I 3 ψ + 2L f C α f ẋ( ẏ + L f ψ) ẋ2 -( E 2 ψ) 2 -2L r C α r ẋ( ẏ -L r ψ)) ẋ2 -( E 2 ψ) 2 -L 3 ( ÿ + ẋ ψ) = g 3 ( 17
)
where x and y are the longitudinal and the lateral positions of the vehicle at its center of gravity (COG). ψ is the yaw angle. F aero is the longitudinal aerodynamic force. m e , g 1 , g 2 and g 3 are given by:

m e = m + 4 I w R 2 e f f
,

g 1 = τ w R e f f , g 2 = (2C α f -2 I w R 2 e f f ẍ)δ , g 3 = L f g 2 + (-E 2 C α f E ψ( ẏ+L f ψ) ẋ2 -( E

Fig. 2 :

 2 Fig. 2: Structure of the driver steering model.

Fig. 3 :

 3 Fig.3: θ near and θ f ar angles[START_REF] Mars | Modeling the visual and motor control of steering with an eye to shared-control automation[END_REF].

Fig. 4 :Fig. 5 :

 45 Fig. 4: The structure of fuzzy logic controller.

Fig. 6 :

 6 Fig. 6: Scaner Studio: Map of the test track

Fig. 7 :

 7 Fig. 7: Emergency mode ON/OFF

Fig. 8 :Fig. 9 :Fig. 10 :

 8910 Fig. 8: The lateral error -case 1

Fig. 11 :Fig. 12 :Fig. 13 :Fig. 14 :Fig. 15 :

 1112131415 Fig. 11: Vehicle dynamic variables -case 1

TABLE I :

 I Driver model Parameters[START_REF] Saleh | Human-like cybernetic driver model for lane keeping[END_REF]

	Parameters	Values
	l p ; D f ar	2; 20
	k c ; k p	20; 2.5
	T L : T I	2; 0.5
	τ p	0.04

TABLE II :

 II Rules of FLC when δ con f lict is S

	α		S	M	e y	B
		S	S	S		M
	Driver's confidence	M	S	S		M
		B	S	H	check case

TABLE III :

 III Rules of FLC when δ con f lict is B

	α		S	e y M	B
		S	check case	M	check case
	Driver's confidence	M	H	H	H
		B	H	H	H

ψ)[START_REF] Favarò | Examining accident reports involving autonomous vehicles in california[END_REF] )δ . The control inputs to the vehicle are the steering wheel angle, δ , and the Driving/Braking wheels torque τ w . More details about the vehicle parameters are given in[START_REF] Chebly | Coupled longitudinal/lateral controllers for autonomous vehicles navigation, with experimental validation[END_REF].


domain: spi phys

text:
Thomas Pierre
email: thomas.pierre@univ-ubs.fr

Jean-Claude Krapez

Helcio R B Orlande

Christophe Rodiet

Dylan Le Maux

Mickaël Courtois

Philippe Le Masson

Bernard Lamien

Philippe Le Masson

Simultaneous estimation of

come

Introduction

This article aims at the estimation of temperature and emissivity of metals above their melting points. It is a part of a global work dealing with the estimation of the physical properties of metals in solid and liquid states (density [START_REF] Le | Density measurement of liquid 22MnB5 by aerodynamic levitation[END_REF], viscosity, surface tension, diffusivity [START_REF] Lamien | A Bayesian Approach for the Simultaneous Estimation of the Thermal Diffusivity and Thermal Conductivity of Aerodynamically Levitated Solid Metals at High Temperatures -Theoretical Study[END_REF]). An apparatus, specifically developed, allows to perform aerodynamic levitation of samples of millimeter size. Once in levitation, the samples are heated with a laser up to and above their melting temperature. Therefore, temperature measurements with thermocouple are not possible and non-contact measurements are necessary, such as pyrometry. Among all the instrumentations around the apparatus, a six-wavelength pyrometer collects the radiative flux in six narrow spectral bands.

The simultaneous estimation of temperature and emissivity of the sample is possible with bispectral or multispectral pyrometry. This technique is largely used in many fields, but presents several drawbacks. The major one is that the inversion problem is ill-posed in the sense that it is underdetermined; indeed in multiwavelength pyrometry there is always one unknown parameter more than there are equations [START_REF] Krapez | Radiative measurements of temperature in Thermal Measurements and Inverse Techniques[END_REF]: for example in bispectral pyrometry,

for two measurements at wavelengths 𝜆 𝑖 and 𝜆 𝑗 , there are three unknowns, the temperature and the two spectral emissivities 𝜀 𝑖 and 𝜀 𝑗 . Nevertheless, some possibilities exist to solve the problem; however they all require some assumptions.

The classical bispectral assumption is to consider a constant emissivity 𝜀 𝑖 ≈ 𝜀 𝑗 for two measurements at wavelengths 𝜆 𝑖 and 𝜆 𝑗 not to far from each other, and nevertheless not to close to not increase the temperature uncertainty [START_REF] Pierre | Micro-scale temperature by multi-spectral and statistic method in the UV-visible wavelengths[END_REF]. In the case of the multiwavelength pyrometry, that is, for more than two wavelengths, it is common to use a mathematical or physical spectral model for the emissivity, on the condition that the number of unknowns is smaller than the number of wavelengths [START_REF] Duvaut | Multiwavelength infrared pyrometry: optimization and computer simulations[END_REF]. The model choice depends on the spectral range width of the pyrometer, and a low-order polynomial (order 1 or 2) form can be sufficient. However, a common result is that the identification error is highly sensitive to the matching of the chosen model with the real emissivity spectrum, which, by principle, is a difficult task since the real emissivity spectrum is not known beforehand [START_REF] Krapez | Radiative measurements of temperature in Thermal Measurements and Inverse Techniques[END_REF], [START_REF] Coates | Multiwavelength pyrometry[END_REF].

To solve the estimation problem three techniques are proposed here. The first one is based on bispectral pyrometry and ordinary least squares (OLS) coupled with a Levenberg-Marquardt algorithm using two out of the six measurements with different possibilities of combinations [START_REF] Gardner | Computer modeling of a multiwavelength pyrometer for measuring true surface temperature[END_REF]. The second technique is based on multispectral pyrometry and regularization by OLS estimation using a low-order model and some criteria allowing to not amplify the standard deviation on the estimated temperature [START_REF] Rodiet | Optimisation of wavelength selection used for the multi-spectral temperature measurement by ordinary least squares method of surfaces exhibiting non-uniform emissivity[END_REF].The third method is based on Bayesian inference, a stochastic technique, which allows to take into account a priori information about the parameters to be estimated [START_REF] Heasler | Nonlinear Bayesian Algorithms for Gas Plume Detection and Estimation from Hyper-Spectral Thermal Image Data Sensors[END_REF]. The present work is developed as follows. First the apparatus and the six-wavelength pyrometer are presented in section 2. Section 3 is dedicated to theoretical considerations regarding the apparatus. The parameter estimation methodologies are presented in section 4, which are finally used in the last section 5 with the experimental data obtained from measurements with two metallic samples: niobium with known melting temperature (1 750 K) [START_REF] Seifter | Microsecond Laser Polarimetry for Emissivity Measurements on Liquid Metals at High Temperatures-Application to Niobium[END_REF], and a steel sample, which has not been characterized, but according to ASTM, its melting temperature is about 1 697 K [14].

The levitation apparatus and the pyrometers

Figure 1 presents the apparatus dedicated to the levitation of spherical sample with a diameter of 3 mm. The levitation is assured by a controlled argon flow. Once in levitation the sample is heated above its melting point by a laser (300 W, 1.070 µm). Figure 1 shows also infrared and high-speed cameras that are not dealt in this study. The six-wavelength pyrometer is aimed at the sample top surface. A commercial pyrometer, which does not appear in Figure 1, is also associated to the sixwavelength one. It is bispectral with wavelengths 0.900 µm and 1.050 µm. Therefore, as the laser wavelength is within its spectral band during the sample heating, the measurements of the pyrometer can only be used during the cooling of the sample, meaning when the laser is stopped. The parameter of the bispectral pyrometer to set is an emissivity ratio.

For both niobium and steel samples, the heating procedure is as follows: 140 W during 1 s, then 30 W for another five seconds.

Theoretical considerations

The flux Φ 𝑖 𝑡ℎ (𝑇) collected by the i th sensor is expressed theoretically by the following expression:

Φ 𝑖 𝑡ℎ (𝑇) = 𝐻 𝑖 ∫ 𝑓 𝑖 (𝜆)𝜀 𝑖 (𝜆, 𝑇)𝑀 0 (𝜆, 𝑇)𝑑𝜆 ∞ 0 ( 1
)
where T is the temperature (in kelvin),  is the wavelength (in µm), H i is an amplitude correction coefficient estimated by calibration, f i () is the spectral transfer function of the i th optical path (OP#i) gathering all the optical manufacturing data (e.g. spectral transmittivity),  i (,T) is the sample surface emissivity, and M 0 (,T) is the Planck's law given by equation [START_REF] Lamien | A Bayesian Approach for the Simultaneous Estimation of the Thermal Diffusivity and Thermal Conductivity of Aerodynamically Levitated Solid Metals at High Temperatures -Theoretical Study[END_REF]. This equation can be rewritten by using the Wien's approximation as equation (3) which is valid for T less than 3 000 µm•K, which is mostly the case here [START_REF] Krapez | Radiative measurements of temperature in Thermal Measurements and Inverse Techniques[END_REF]. In the case of a blackbody at temperature T, the theoretical flux collected is Φ 𝑖 0,𝑡ℎ (𝑇). 𝜆𝑇 -1

(2)

𝑀 0 (𝜆, 𝑇) = 𝐶 1 𝜆 -5 𝑒 - 𝐶 2 𝜆𝑇 (3)
with C 1 = 3.741 × 10 8 W•µm 4 •m -2 and C 2 = 14 388 µm•K [START_REF] Siegel | Thermal radiation heat transfer[END_REF]. It is assumed that the influence of both the environment and the atmosphere inside the apparatus are negligible.

Whatever the relation used to express the theoretical flux, (2) or (3), the signal is experimentally contaminated by an additive random noise e i . Thus, let's define the observable Y i as following:

𝑌 𝑖 = 𝜀 𝑖 (𝑇)Φ 𝑖 0,𝑡ℎ (𝑇) + 𝑒 𝑖 (4)
Section 4.3 will present the temperature-emissivity estimation through the Bayesian technique using a linear approximation. Therefore, assuming Wien's approximation (3), and introducing mean values 𝜆 𝑖 , 𝜀 𝑖 and f i for the wavelength, the emissivity and the transfer function, respectively, in each spectral band i. The flux in the band i can thus be simplified to:

Φ 𝑖 𝑡ℎ (𝑇) = 𝐻 𝑖 𝑓 𝑖 𝜀 𝑖 𝐶 1 𝜆 𝑖 -5 𝑒 - 𝐶 2 𝜆 𝑖 𝑇 (5)
The mean wavelengths 𝜆 𝑖 are gathered in

The parameters of interest to estimate are the temperature and the emissivity. And one of the error sources comes from the spectral transfer functions f i (), that is why the H i coefficients have been introduced and estimated by calibration against a blackbody. As an example Figure 3 presents the evolution of both the commercial bispectral pyrometer temperature and the OP#5 flux versus time during the cooling of a blackbody, that is, after the laser heating is stopped. Note that this blackbody has been placed in the levitation apparatus instead of the levitation nozzle (Figure 1). The estimation of H i has been performed by solving equation ( 8), considering the temperature given by the commercial bispectral pyrometer as being the real one and the emissivity of the blackbody simulator being equal to one.

Φ 𝑖 0,𝑡ℎ (𝐻 𝑖 ) = 𝑌 𝑖 (8)
The evolution of H 5 is plotted in Figure 4. Except during the very first period of time when the blackbody temperature is not uniform, the estimation of H 5 gives rather constant values;

Figure 5 presents the histogram of the constant H 5 from which, assuming a Gaussian distribution, the mean and the standard deviation are calculated. The mean values of the amplitude correction coefficients and the associated standard deviations are information of prime importance. Firstly, they allow to discuss the credibility of the estimated parameters. Secondly, they will be used for the Bayesian inference as a priori information [START_REF] Heasler | Nonlinear Bayesian Algorithms for Gas Plume Detection and Estimation from Hyper-Spectral Thermal Image Data Sensors[END_REF], so that their uncertainties are taken into account as regards the estimation of the parameters of interest, that is, the temperature and the emissivity of the sample (see section 4.3).

Simultaneous estimation of Temperature and Emissivity

The bispectral method

Fundamentally, the bispectral method consists in estimating the temperature by using two radiative fluxes measured at wavelengths i  and j  . Regarding the spectral emissivity values, one of them is estimated in the same time as temperature, say i  at i  , whereas a functional relationship is assumed between j  (at j  ) and i  . One possible functional relationship is

ji i j    
where ji  is a predetermined constant. In the classical "greybody" approximation the constant ji  is set to one, which, to be satisfactory, requires that the wavelengths i  and j  be not too far from each other.

Consider two radiative fluxes Φ 𝑖 𝑒𝑥𝑝 (𝑇) ≡ 𝑌 𝑖 and Φ 𝑗 𝑒𝑥𝑝 (𝑇) ≡ 𝑌 𝑗 measured at wavelengths  i and  j . The method lays on the simultaneous temperature emissivity estimation by solving the system ( 9)-( 10):

where the coefficient 𝜉 𝑗𝑖 is the assumed emissivity ratio 𝜀 𝑗 /𝜀 𝑖 , which is 1 in the classical "greybody" approximation.

The multispectral method

In the case of multispectral pyrometry, that is to say using three and more wavelengths, the emissivity can be considered in terms of a low order polynomial form [START_REF] Ash | Temperature-emissivity separation for LWIR sensing using MCMC[END_REF] even if some physical models exist (e.g. Drude, Hagen-Rubens [START_REF] Duvaut | Multiwavelength infrared pyrometry: optimization and computer simulations[END_REF]).

𝜀(𝜆, 𝑇) = ∑ 𝑎 𝑘 𝜆 𝑘 𝑚 𝑘=0 (11)
The proposed method here implies to correct the experimental flux 𝑌 𝑖 by 𝑌 𝑖 𝑐𝑜𝑟 = 𝑌 𝑖 /𝐻 𝑖 𝑓 𝑖 according to the linearization of the Planck's law [START_REF] Le | Density measurement of liquid 22MnB5 by aerodynamic levitation[END_REF]. In order to not amplify the standard deviation on the estimated temperature and emissivity by multispectral methods, a parsimonious model is preferable, that is, the parametrization of a spectral emissivity of order 2 maximum, m = 0, 1, or 2 in [START_REF] Ash | Temperature-emissivity separation for LWIR sensing using MCMC[END_REF], and a minimum distance between two consecutive wavelengths [START_REF] Pierre | Micro-scale temperature by multi-spectral and statistic method in the UV-visible wavelengths[END_REF][8] [16][17] expressed by relation [START_REF] Krapez | Measurements without contact in heat transfer. Part A: radiative thermometry: principles, implementation and pitfalls[END_REF]. The objective is to minimize the cost function [START_REF] Seifter | Microsecond Laser Polarimetry for Emissivity Measurements on Liquid Metals at High Temperatures-Application to Niobium[END_REF], where W is the number of considered wavelengths, Φ 𝑖 𝑡ℎ,𝑐𝑜𝑟 is the corrected

𝛷 𝑖 𝑡ℎ (𝜀 𝑖 , 𝑇 𝑖𝑗 ) = 𝑌 𝑖 (9)
𝜉 𝑗𝑖 𝛷 𝑗 𝑡ℎ (𝜀 𝑖 , 𝑇 𝑖𝑗 ) = 𝑌 𝑗 [START_REF] Berrett | A Bayesian Nonparametric Model for Temperature-Emissivity Separation of Long-Wave Hyperspectral Images[END_REF] theoretical flux using the Planck's law (without 𝐻 𝑖 𝑓 𝑖 (𝜆)), and 𝐚 = [𝑎 0 , … 𝑎 𝑊 ] 𝑇 is the parameter vector of the emissivity model [START_REF] Ash | Temperature-emissivity separation for LWIR sensing using MCMC[END_REF].

Δλ 𝑖𝑗,𝑚𝑖𝑛 = 𝜆 𝑖 2 𝑇 𝐶 2 | 𝜆 𝑖 >𝜆 𝑗 (12) ∑[Φ 𝑖 𝑡ℎ,𝑐𝑜𝑟 (𝑇, 𝐚) -𝑌 𝑖 𝑐𝑜𝑟𝑟 ] 2 𝑊 𝑖=1 (13)

Estimation by Bayesian inference

The Bayesian theorem is given in reference [START_REF] Krapez | Measurements without contact in heat transfer. Part A: radiative thermometry: principles, implementation and pitfalls[END_REF]:

𝜋(𝛃|𝐘) = 𝜋(𝐘|𝛃)𝜋(𝛃) 𝜋(𝐘) (14)
𝛃 = [𝜺, 𝑇] 𝑇 ( 15
)
where  is the vector of the following parameters: the temperature and the emissivities 𝑇 , where m represents the number of wavelengths. 𝜋(𝐘|𝛃) is the likelihood function or the probability density of the measurements with the parameters  given, 𝜋(𝛃) is the a priori density of the unknown or uncertain parameters, and 𝜋(𝐘) is the marginal probability density of the measurements, which plays the role of a normalization constant. As per equation ( 6), measurement uncertainties are supposed Gaussian with known zero mean and a covariance matrix, additive and independent from the unknown parameters. Thus, the likelihood function can be written:

𝜺 = [𝜀 1 , … 𝜀 𝑚 ]
𝜋(𝐘|𝛃) ∝ 𝑒𝑥𝑝 {- 1 2 [𝐘 -𝛆⨂𝚽 𝟎,𝒕𝒉 (𝑇)] 𝑇 𝛀 -1 [𝐘 -𝛆⨂𝚽 𝟎,𝒕𝒉 (𝑇)]} ( 16
)
where  is the measurement error covariance matrix and  denotes the element-wise product. Regarding the density of an a priori distribution of the parameters, it was assumed independent and given by [START_REF] Rodiet | Influence of measurement noise and number of wavelengths on the temperature measurement of opaque surface with variable emissivity by a multispectral method based on the flow ratio in the infrared-ultraviolet range[END_REF], where 𝜋(𝛆) and 𝜋(T) are the probability densities of the emissivity and the temperature, respectively:

𝜋(𝛃) = 𝜋(𝛆)𝜋(𝑇) (17)

Multiwavelength pyrometer in linear approximation

In the case of the linear approximation, let's consider the system ( 5)- [START_REF] Gardner | Computer modeling of a multiwavelength pyrometer for measuring true surface temperature[END_REF]. Equation ( 6) can be rewritten as:

𝐘 = 𝐗𝛃 + 𝐞′ (18) 𝐗 = [𝐈 𝐦𝐦 -𝛍 𝒎𝟏 ] (19)
𝛍 = [𝜇 1 𝜇 2 … 𝜇 𝑚 ] 𝑇 (20)
where 𝐈 𝐦𝐦 is the identity matrix of size m × m. Thus the relation ( 16) becomes, taking into account equation [START_REF] Rodiet | Influence of measurement noise and number of wavelengths on the temperature measurement of opaque surface with variable emissivity by a multispectral method based on the flow ratio in the infrared-ultraviolet range[END_REF] and the linear hypothesis:

𝜋(𝛃|𝐘) ∝ 𝑒𝑥𝑝 {- 1 2 [(𝐘 -𝐗𝛃) 𝑇 𝛀 -1 (𝐘 -𝐗𝛃) + (𝛃 -𝛃 𝒑𝒓𝒊𝒐𝒓 ) 𝑇 𝐖 -1 (𝛃 -𝛃 𝒑𝒓𝒊𝒐𝒓 )]} (21)
The maximum a posteriori (MAP) estimator is then obtained when its derivative is null with respect to the parameter vector. Introducing the MAP estimator and the posterior covariance matrix, relation [START_REF] Cezairliyan | High-speed (subsecond) measurement of heat capacity, electrical resistivity, and thermal radiative properties of molybdenium in the range 1900 to 2800 K[END_REF] becomes:

𝜋(𝛃|𝐘) ∝ {- 1 2 [(𝛃 -𝛃 ̂𝑀𝐴𝑃 ) 𝑇 𝚪 𝛃|𝐘 -𝟏 (𝛃 -𝛃 ̂𝑀𝐴𝑃 )]} (22)
In relation ( 22), 𝛽 ̂𝑀𝐴𝑃 is the MAP estimator (23) and 𝚪 𝛃|𝐘 is the posterior covariance matrix (24). They are given by [START_REF] Siegel | Thermal radiation heat transfer[END_REF]:

𝛃 ̂𝑀𝐴𝑃 = 𝛃 𝐩𝐫𝐢𝐨𝐫 + 𝐖𝐗 𝐓 (𝐗𝐖𝐗 𝐓 + 𝛀) -𝟏 (𝐘 -𝐗𝛃 𝐩𝐫𝐢𝐨𝐫 ) (23)
𝚪 𝛃|𝐘 = 𝐖 -𝐖𝐗 𝐓 (𝐗𝐖𝐗 𝐓 + 𝛀) -1 𝐗𝐖 (24)

Multiwavelength pyrometry in non-linear estimation

In the non-linear case, the theoretical fluxes are calculated from relations (1), [START_REF] Lamien | A Bayesian Approach for the Simultaneous Estimation of the Thermal Diffusivity and Thermal Conductivity of Aerodynamically Levitated Solid Metals at High Temperatures -Theoretical Study[END_REF], and (4), that is, with Planck's law. The  vector of the parameter is the same as in eq. [START_REF] Rodiet | Influence of measurement noise and number of wavelengths on the temperature measurement of opaque surface with variable emissivity by a multispectral method based on the flow ratio in the infrared-ultraviolet range[END_REF].

A sampling method based on the Markov Chain Monte Carlo (MCMC) method is used in this work. The Metropolis-Hastings algorithm was applied to generate samples of the posterior distribution based on the likelihood given by equation ( 16). This algorithm begins with the selection of a proposal distribution 𝑝(𝛃 * , 𝛃 𝑡-1 ), which is used to define a new candidate *,

given the current state 𝛃 𝑡-1 of the Markov Chain. Once the proposal distribution is defined, the Metropolis-Hastings sampling algorithm can be implemented by repeating the following steps:

1-Sample a candidate * from the proposal distribution 𝑝(𝛃 * , 𝛃 𝑡-1 ).

2-Calculate the acceptance:

𝛼 = 𝑚𝑖𝑛 [1, 𝜋(𝛃 * |𝐘), 𝑝(𝛃 * , 𝛃 𝐭-𝟏 ) 𝜋(𝛃 𝑡-1 |𝐘), 𝑝(𝛃 𝑡-1 , 𝛃) ] (25)
3-Generate a random value U uniformly distributed between 0 and 1.

4-If U ≤ 𝛼, set 𝛃 𝑡 = 𝛃 * ; otherwise, set 𝛃 𝑡 = 𝛃 𝑡-1 .

5-Return to step 1 to generate the sequence {𝛃 1 , 𝛃 2 … , 𝛃 𝑛 }.

In this manner, a sequence is generated to represent the posterior distribution, and inference on this posterior distribution is obtained from inference on the samples {𝛽 1 , 𝛽 2 … , 𝛽 𝑛 }. Note that the values 𝛽 𝑖 must be rejected as long as the chain has not converged. The two state changes (fusion, solidification), the large temperature variation, and the broad spectral range are three reasons that make the emissivity change highly likely. However, the niobium melting temperature is known (2 750 K) and the literature mentions values of emissivity around 0.35 at 0.684 5 µm in the liquid phase [START_REF] Seifter | Microsecond Laser Polarimetry for Emissivity Measurements on Liquid Metals at High Temperatures-Application to Niobium[END_REF] and 0.35 at 0.650 µm in the solid phase [START_REF] Y S Touloukian | Thermal radiative properties[END_REF], respectively. All these values are convenient to verify the estimations at the solidification plateau mainly, and useful as a priori data. We first analyze the solution of the system ( 9)-( 10) by considering 𝜀 𝑖 = 𝜀 𝑗 , for two bispectral combinations and different initial guesses for temperature and emissivity, by comparison with literature values of the emissivity at the melting temperature. As the results are not satisfying, the greybody assumption should be relieved, which means that one should solve the system ( 9)-( 10) by introducing an emissivity ratio 𝜉 𝑗𝑖 . The estimation of this parameter is only possible using a known data, such as the solidification plateau.

Results and discussions

Estimation with niobium

This estimation is based on the experimental data, obtained during the cooling period, when the bispectral commercial pyrometer measures temperature T bis . This minimization is based on a high confidence in the temperature given by the commercial bispectral pyrometer T bis .

Indeed, the measured temperature during the solidification plateau has been estimated at (2 731 ± 12) K (i.e. 0.5 %), which made it coincides with the literature [START_REF] F P Incropera | Fundamentals of Heat and Mass Transfer[END_REF], and its emissivity ratio has been set to 1.025 (consistent with 𝜉 43 = 1.087, see further).

Results of two bispectral combinations are presented: 3/4 where  34 = 170 nm; 4/5 where  45 = 90 nm. Then the principle is to minimize the mean of the residuals (26) between the estimated temperature T ij and T bis during a short time interval (N = 50 between 5.9 s and 6.9 s).

1 𝑁 ∑[𝑇 𝑖𝑗 (𝑡 𝑘 ) -𝑇 𝑏𝑖𝑠 (𝑡 𝑘 )] 2 𝑁 𝑘=1 (26)
Figure 7 plots the evolution of the mean of residuals in eq. ( 23) versus  43 . Consequently,  43 = 1.087 and the results of emissivity and temperature, obtained for this value are presented in Figures 8 and9. Even if the emissivity ratio has been evaluated during the short time interval of solidification, the obtained value has been applied to the whole duration of the experiment. Even if literature mentions a rather constant emissivity in the liquid state [START_REF] Seifter | Microsecond Laser Polarimetry for Emissivity Measurements on Liquid Metals at High Temperatures-Application to Niobium[END_REF],

this is a major assumption. Nevertheless, we can observe that in Figures 9 and10 the estimated temperatures T 34 and T 45 are very close. Measurements on known materials, such as pure niobium, are convenient, since there is a possibility to have an a priori value such as the melting temperature as the point of reference.

But in the case of unknown materials, such as alloys, the solidification plateau would not be that constant but rather shows a slight decrease. This will be discussed in section 5.2. on the basis of the steel sample.

In the multispectral case (section 4.2), tests are performed considering emissivity as a polynomial of orders 0 (m = 0) and 1 (m = 1); the order 2 is not conceivable due to the spectral range being too narrow in the case of this study. Considering the six wavelengths, only four are kept to respect the minimal spectral criteria given by relation [START_REF] Krapez | Measurements without contact in heat transfer. Part A: radiative thermometry: principles, implementation and pitfalls[END_REF] with T = 2 750 K:  2 = 0.530 µm,  3 = 0.680 µm,  5 = 0.940 µm, and  6 = 1.055 µm. The estimation has been performed by ordinary least squares coupled with the Levenberg-Marquardt algorithm; estimated temperatures are presented in Figures 11 and12. Figure 12 shows a difference of estimation between the two tested emissivity models. Nevertheless, estimated temperatures with the order 1 polynomial are better, since the temperature at the solidification plateau is estimated at (2 753 ± 20) K (0.8 %) (Figure 12). The order 0 polynomial overestimates the temperatures. This result is predictable, since the emissivity is considered constant along the spectral range [0.530 µm -1.055 µm], which seems improbable with such a spectral range. Now parameter estimation is performed according to the Bayesian method in two ways: by a linear approximation (section 4.3.1) and in non-linear case, using MCMC (section 4.3.2).

Only bispectral estimations are presented. To start with the linear approximation, the theoretical fluxes are linearized, as presented in relations (3),( 5)-( 7): Figure 13 presents them for the measurements of OP#3 and OP#4. Estimations are performed during the first part of the cooling, mainly during the solidification plateau.

Three parameters are considered: the temperature and the two emissivities. Tests are performed with different levels of confidence, with priors of good or bad quality. In the first stage we concentrated on the solidification plateau (Figure 14). Since the phase-change temperature is well known, we introduced a temperature prior of small uncertainty, 𝑇 𝑝𝑟𝑖𝑜𝑟 = (2 750 ± 30) K whereas the emissivity priors were given a high uncertainty, namely 𝜀 3 𝑝𝑟𝑖𝑜𝑟 = 𝜀 4 𝑝𝑟𝑖𝑜𝑟 = 0.30 ± 0.15. The results of this second identification phase are described in Figures 17 and18. The MAP estimator of the emissivity 𝜀 3 and 𝜀 4 are quite stable in the liquid phase. On the contrary, in the solid phase, 𝜀 3 shows a slight increase whereas 𝜀 4 shows a slight decrease. The mean standard-deviation of the estimator of temperature is between 7 and 32 K during the period [1s, 9s] of the experiment.

For the case of non-linear approximation, still with the measurements of OP#3 and OP#4, the estimation has been performed during the whole experiment and only two parameters are considered: the temperature and the emissivity 𝜀 34 . The Markov chains were started with parameters with the same priors as previously (of good quality for the temperature and bad quality for the emissivity). The chains were run with 30 000 states and the 15 000 first states were neglected for the computation of the posterior statistics (burn-in period).

Estimation with steel sample

This last part concerns some results for the 100c6 steel, also referenced as AISI 52100 by ASTM [14]. Only Bayesian results are presented for this material with the linear approximation and the non-linear case. Figure 23

Conclusion

The article presented the first steps of the simultaneous temperature and emissivity estimation of molten metals by multispectral pyrometry using both deterministic and Bayesian techniques. Estimations were focused on bispectral pyrometry and one case by the multispectral method. The materials, niobium and steel samples, were aerodynamically levitated, then heated by laser above their melting point then cooled until room temperature.

Radiative flux measurements have been performed with the help of a multispectral pyrometer, whereas a second commercial bispectral pyrometer measured the temperature as a supplementary information. The estimation results have shown that all techniques presented similar temperature and emissivity values consistent with the literature.

Nevertheless, whatever the considered multispectral pyrometry, both temperature and emissivity are not easy to estimate since, firstly, they are correlated parameters, and, secondly, the problem is ill-posed since the number of unknown is always one more than the number of information. In the case of the bispectral pyrometry and the multispectral pyrometry with m = 0 (constant emissivity), the current approach to assume a greybody hypothesis, considering two measurements at two wavelengths, was not successful in this study. On the contrary, bispectral pyrometry with  ji ≠ 1 and multispectral pyrometry in the case of a linear emissivity (m = 1) gave better temperature values. Concerning the Bayesian technique, it has been necessary to take into account prior information, namely the emissivity, encountered in the literature and the temperature measured by the commercial bispectral pyrometer. However, it must be clear that the latter information was used only for initial calibration and for validation. Its systematic use is of course not needed. Nevertheless, with this information, the three tested methods give satisfying results, but need to be improved. The detection and the rigorous quantification of the uncertainties are of prime importance; it concerns for instance the calibration with a blackbody, which should be improved (e.g. blackbody ratio length/diameter [START_REF] Cezairliyan | High-speed (subsecond) measurement of heat capacity, electrical resistivity, and thermal radiative properties of molybdenium in the range 1900 to 2800 K[END_REF]). Even if they are necessary in the Bayesian inference (𝜋(𝛃)), the major point to solve concerns the fact that we must use external information (priors) to commit estimation, due to the strong correlation between temperature and emissivity. Since the Bayesian techniques seem relevant to circumvent the ill-posed problem, it would be appropriate to couple this methodology with temperature-emissivity separation considerations [3][12]. It is all the more appropriate, since we have to consider emissivity variation due to phase changes, for instance oxidation.

Figure 1 .

 1 Figure 1. Sketch of the aerodynamic levitation apparatus

Figure 2 .

 2 Figure 2. Sketch of the six-wavelength pyrometer.

Figure 3 .

 3 Figure 3. Evolution of the commercial bispectral pyrometer temperature (solid line) and of

Figure 4 .

 4 Figure 4. Example of estimating the H 5 coefficient during calibration.

Figure 5 .

 5 Figure 5. Gaussian representation of H 5 .

Figure 6

 6 Figure 6 presents all the fluxes and temperature measurements during the experiment. The six

Figure 6 .

 6 Figure 6. Niobium sample fluxes and commercial bispectral pyrometer temperature as a

Figure 8 .

 8 Figure 8. Estimated emissivity  3 ( 3 = 0.68 µm) versus temperature and comparison with

Figure 7 .

 7 Figure 7. Residuals mean between the estimated temperature T 34 and T bis versus the

Figure 8

 8 Figure 8 plots the evolution of the estimated emissivity  3 versus the estimated temperature

Figure 9 .

 9 Figure 9. Bispectral temperature emissivity estimation for the combination 3/4.

Figure 10 .

 10 Figure 10. Bispectral temperature emissivity estimation for the combination 4/5.

Figure 11 .

 11 Figure 11. Estimated temperatures by multispectral method.

Figure 12 .

 12 Figure 12. Zoom of the estimated temperatures at the solidification plateau.

Figure 13 .

 13 Figure 13. Modified observables for the combination 3/4 with Bayesian inference,

Figure 14 .

 14 Figure 14. Estimated temperature around the melting point and comparison with T bis .

Figure 15 .

 15 Figure 15. A priori and estimated emissivities  4 and  5 at the solidification plateau.

Figure 16 .

 16 Figure 16. A priori and estimated temperature at the solidification plateau.

Figure 17 .

 17 Figure 17. MAP estimator of temperature (in blue) with plus or minus one standard-

Figure 18 .

 18 Figure 18. MAP estimator of emissivity 𝜀 3 (in orange) and 𝜀 4 (in violet) and plus or minus

Figures 15 and 16

 16 Figures 15 and 16 show the normalized probability densities for the three parameters. On the

  figures, for the verification of the estimated fluxes and temperature versus the experimental

Figure 19 .

 19 Figure 19. Estimated and experimental fluxes.

Figure 20 .

 20 Figure 20. Estimated and experimental temperatures.

Figure 21 .

 21 Figure 21. Markov chain states for the temperature estimation.

Figure 22 .

 22 Figure 22. Markov chain states for the emissivity estimation.

  presents the experimental fluxes measured with the multispectral pyrometer and the temperature T bis with an emissivities ratio adjusted at 1.025. The fluxes measured by means of OP#1 and OP#2 are too weak to be used for the parameter estimation. As before, bispectral estimations are presented from the measurements of OP#3 and OP#4.

Figure 23 .

 23 Figure 23. 100c6 steel sample fluxes.

Figure 24 .

 24 Figure 24. Estimated temperature through Bayesian inference with linear approximation.

Figure 25 .

 25 Figure 25. Estimated emissivities through Bayesian inference with linear approximation.

Figure 26 .

 26 Figure 26. Estimated temperature versus Markov chain states with non-linear Bayesian.

Figure 27 .

 27 Figure 27. Estimated emissivity versus Markov chains states with non-linear Bayesian.

Table 1

 1

	. The relation (5) can be linearized by taking

Table 1 .

 1 Table 1 gathers all results for the six optical paths. Means values and standard deviations of the H i coefficients.

	OP#i	1	2	3	4	5	6
	𝝀 𝒊 (µm)	0.483	0.532	0.680	0.851	0.940	1.554
	𝑯 ̅ 𝒊 × 10 9	32.68	2.23	3.82	2.46	3.22	3.63
	𝝈 𝑯 𝒊 × 10 11	91.31	2.64	2.19	1.10	1.25	2.86