Oscar De

La Cruz Fierro

Wael Bachta

Florian Gosselin

Guillaume Morel

A New Control Strategy for the Improvement of Contact Rendering with Encounter-type Haptic Displays

Keywords: Robot Control, Haptic Interface, Encounter-type Haptic Display (ETHD)

Encounter-type haptic interfaces are used to interact physically with virtual environments. They allow controlling the position of an avatar in the simulation while perceiving the forces applied on it when it interacts with the surrounding objects. Contrary to usual force feedback devices, the interface tracks the real user's finger without touching it when the user's finger avatar moves in free space. Only when a contact occurs in the virtual environment, the interface comes in contact with the user to display the mechanical properties of the encountered objects. This way, the device's behaviour is more natural as simulated contacts really occur in the real world. Existing control laws for such devices exhibit however limitations, especially when contacts occur at high speed. In such cases, the device tends to bounce against the user's finger, which decreases the realism of the interaction. In this paper, we propose a new control strategy where the interface is first stabilized against the obstacles before the user touches its end-effector. This way, contacts appear more natural, even at high speeds, as confirmed by preliminary user-tests made with an existing 2 DoF encounter type haptic interface at different speeds with the state of the art control law and the novel approach we propose here.

INTRODUCTION

Haptic interfaces allow motion interactions with virtual or remote environments with a reproduction of the sense of touch, using kinesthetic (force/position) and cutaneous (tactile) receptors [START_REF] Hannaford | Haptics[END_REF]. We can distinguish four methods for creating haptic sensations artificially: vibrotactile devices, force-feedback systems (discussed in this paper), surface displays and distributed tactile displays [START_REF] Hayward | Do it yourself haptics: part I[END_REF].

Force-feedback systems are robotic mechanisms capable to measure the user's movements and deliver a force signal to the operator's hand, usually through a pen-like interface, a knob or a thimble [START_REF] Campion | Literature Review[END_REF]. A non-exhaustive list of application cases are computer-aided design [START_REF] Nahvi | Haptic manipulation of virtual mechanisms from mechanical CAD designs[END_REF], maintenance and assembly tasks [START_REF] Mcneely | Six Degree-of-freedom Haptic Rendering Using Voxel Sampling[END_REF], games [START_REF] Martin | Characterisation of the Novint Falcon Haptic Device for Application as a Robot Manipulator[END_REF] and virtual reality task simulations [START_REF] Sagardia | VR-OOS: The DLR's virtual reality simulator for telerobotic on-orbit servicing with haptic feedback[END_REF] as well as teleoperation [START_REF] Gosselin | Design of a High Fidelity Haptic Device for Telesurgery[END_REF].

In an ideal force-feedback system, the user should be able to move in free space without feeling any force and the device should prevent him/her to move in the constraint direction if a stiff object is being touched. In the mentioned application contexts however, force-feedback interfaces usually require the user to be mechanically linked to them. This link has a non-negligible influence: the user experiences the friction, inertia and vibrations of the mechanical structure even when moving in free space, which reduces the realism of the interaction. In addition, the difference between free space and contact is less distinctively felt than in real world.

Encounter-Type Haptic Displays (ETHDs) propose, as a solution to this problem, to remove the mechanical link between the interface and the operator [START_REF] Mcneely | Robotic graphics: a new approach to force feedback for virtual reality[END_REF], [START_REF] Tachi | A construction method of virtual haptic space[END_REF]. This principle allows a perfect transparency in free space motion as the user touches the haptic device, usually with the fingertip, only when a contact occurs in the virtual/remote environment (see Figure 1). [START_REF] Yoshikawa | A touch and force display system for haptic interface[END_REF] and [START_REF] Yoshikawa | A three-dimensional touch/force display system for haptic interface[END_REF] use for example a set of optical glassfiber on-off sensors for measuring the position of the operator's finger, respectively in 2D space with a ring-like end-effector and in 3D space with a cap-like end-effector. The position of the finger is however only roughly estimated. In (Gonzalez et al., 2015a) a ring-like end-effector instrumented with infrared proximity sensors, mounted on a 2 Degrees of Freedom (DoF) interface, is used to reconstruct the shape of the finger and precisely estimate its position using distance measurements [START_REF] Gonzalez | A 2-D Infrared Instrumentation for Close-Range Finger Position Sensing[END_REF]. We find as well hand exoskeletons [START_REF] Nakagawara | An Encounter-Type Multi-Fingered Master Hand Using Circuitous Joints[END_REF], [START_REF] Fang | An exoskeleton master hand for controlling DLR/HIT hand[END_REF] where the position of a thin reflecting plate, pushed by the nail, is recorded thanks to an optical sensor. Force felt between the plate and the finger is negligible. Special attention should be given to the control law that governs ETHDs, particularly to the transitions between free space and contact modes. As shown in (Gonzalez et al., 2015a), control strategies usually implemented on ETHDs rely on an abrupt transition between these two modes, potentially generating vibrations and non-realistic impact forces at that moment. To cope with this issue, (Gonzalez et al., 2015a) proposes a smooth transition-based control. This solution was implemented on a 2DoF ETHD. It proves stable and more realistic, especially when finger interactions occur at low speeds (≈0.2 m/s). However at higher speeds the problem is not completely tackled and the sensation felt may be nonrealistic at the moment of contact.

In this paper, we propose a new control strategy aimed at allowing natural transitions between free space and contact modes, even at higher speeds (>0.2 m/s). It includes a bilateral damping allowing the stabilization of the robot's end-effector before application of force feedback. This comes at the price of a slight shift of the virtual wall, which remains however imperceptible for most users as proved by the results of our evaluations. The state of the art control law implemented in (Gonzalez et al., 2015a) is first briefly presented in section 2. The proposed upgraded control strategy is then explained in section 3. Section 4 presents the results of the experiments performed to validate the potential of our approach. Finally conclusions are given in section 5.

SMOOTH TRANSITION-BASED CONTROL

In free space, the ETHD should closely track the finger's position without touching it. When the user's avatar enters in contact with a virtual wall, the resulting contact force should be displayed to the user. A control law, which to our knowledge answers the most closely the aforementioned requirements, was proposed in (Gonzalez et al., 2015a) and implemented on a 2DoF ETHD. It will be briefly described in the following lines.

Control Algorithm for Finger Tracking in Free Space

We note here 𝝐 𝑿 = 𝑿 𝒇/𝟎 -𝑿 𝒓/𝟎 = 𝑿 𝒇/𝒓 the position error between the ring center and the finger center (see Figure 2). Being small as close tracking of the finger is desired, it can be expressed in joint space:

𝝐 𝒒 = 𝐉 -𝟏 (𝒒) • 𝝐 𝑿 (1)
where 𝒒 = [ q 1 q 2] T are the joints positions and 𝐉(𝒒) the robot's jacobian matrix expressed in its global reference frame 𝑹 𝟎 .

Figure 2: 2DoF ETHD with ring center 𝑋 𝑟/0 and finger center 𝑋 𝑓/0 (adapted from [START_REF] Gonzalez | Contributions au développement d'une interface haptique à contacts intermittents[END_REF]).

Error minimization is achieved with a Proportional Derivative controller, which provides the robot with a reference torque:

𝝉 𝒕 = 𝐙 𝐭 • 𝛜 𝒒 = (𝐊 𝐭 + 𝐁 𝐭 𝑠) • 𝛜 𝒒 (2)
where 𝐙 𝐭 is the equivalent impedance, 𝐊 𝐭 and 𝐁 𝐭 the proportional and derivative gains respectively. With this controller, a link equivalent to a spring and damper system is created between the centers of the user's finger and of the ring (see Figure 3).

Control Algorithm for Force Rendering at Contact

When a virtual object is encountered, the interface must render the corresponding interaction force 𝐅 𝐞 , which is defined as an unilateral constraint. A viscoelastic compliant virtual environment without tangential friction is assumed and a modified Kelvin-Voigt model [START_REF] Achhammer | Improvement of model-mediated teleoperation using a new hybrid environment estimation technique[END_REF] is used to calculate the resulting interaction forces (see Eq. 3).

Figure 4: At contact the influence of the tracking force 𝐅 𝐭 has been diminished by a factor 𝛽and a spring and damper coupling (𝐙 𝐞 = 𝐊 𝐞 + 𝐁 𝐞 𝑠) is created between the ring and the wall to render the interaction force 𝐅 𝐞 .

Let 𝑑 𝑟/𝑒 be the distance between the ring's avatar inner circumference and the closest point of a virtual object, 𝑥 𝑒 the position of a vertical wall (see Figure 4), 𝐧 a unitary vector normal to the surface of contact, 𝑋 ̇𝑟/0 the speed of the ring along 𝐧. The environment interaction force 𝐅 𝐞 can be expressed as follows: (3)

The transition between free space and contact is achieved by reducing the influence of the tracking force 𝐅 𝐭 by a factor 𝛽 nearby the obstacles, i.e. at a distance 𝑑 𝑓/𝑒 = 𝑑 𝛽 from the virtual object (VO) placed at position 𝑥 𝑒 . Equation 4shows how 𝛽 varies in function of 𝑑 𝑓/𝑒 , the distance between the finger's avatar and the VO (see Figure 3). Factor 𝛽 cannot be totally cancelled at the proximity of the wall as in this case the ring would not follow the finger when it moves away from it.

{ 𝑖𝑓 𝑑 𝑓/𝑒 > 𝑑 𝛽 𝛽 = 1 𝑖𝑓 𝑑 𝑓/𝑒 ∈ ]0; 𝑑 𝛽 ] 𝛽 = ( 1 -𝛽 𝑚𝑖𝑛 𝑑 𝛽 ) 𝑑 𝑓/𝑒 + 𝑖𝑓 𝑑 𝑓/𝑒 ≤ 0 𝛽 = 𝛽 𝑚𝑖𝑛 = 0.1 𝛽 𝑚𝑖𝑛 (4)
Here 𝑑 𝛽 = 𝑅 𝑟 -𝑅 𝑓,𝑚𝑎𝑥 ,with 𝑅 𝑓,𝑚𝑎𝑥 chosen so that ∀ 𝑅 𝑓 , 𝑅 𝑓,𝑚𝑎𝑥 > 𝑅 𝑓 . This way by the moment the user encounters the ring, the tracking effect is almost canceled and ‖𝐅 𝐞 ‖ ≠ 0. The updated Eq. ( 2) can now be expressed as:

𝝉 𝒕 = 𝛽𝐙 𝐭 𝛜 𝒒 (5)
While allowing a smooth transition between free space and contact, this algorithm presents in practice an undesired behaviour when impacting virtual objects at high speeds (> 0.2 m/s): oscillations appear when the ring encounters a virtual object and therefore an unnatural contact is perceived by the user when his/her finger encounters the ring. It may give the impression of touching a moving object instead of a static one as in the real life.

OFFSET TRANSITION-BASED CONTROL

When the user's finger encounters the ring, he/she should feel as touching a static object. The offset transition-based control introduced in this paper proposes therefore to first apply a dissipating force when the ring's avatar penetrates into the VO in order to stop it before displaying the VO properties 𝐙 𝐞 . Further details will be given below.

Virtual Environment Force Estimation and Rendering

In free space, the user can move the interface freely, i.e. no interaction force exists. When the ring's inner periphery penetrates in a virtual object, we propose to completely stop it before displaying the VO properties. Therefore a dissipating force is applied on the ring until the interface is static (in practice until |𝑋 ̇𝑟/0 | < 𝑣 𝑡ℎ , with 𝑣 𝑡ℎ an experimentally tuned threshold introduced to cope with the speed signal's noise). When the mentioned condition is true, the VO properties (𝐊 𝐞 and 𝐁 𝐞 ) are rendered to the user. The new virtual wall position 𝑥 𝜀 is defined as the coordinate of the distal point on the inner periphery of the ring once it is static. This algorithm was implemented using a Finite State Machine (FSM) as shown in Figure 5. The initial state is called transparent: in this mode only the tracking force acts on the ring and 𝐅 𝐞 = 0. As soon as the interface approaches the VO and the inner periphery of the ring penetrates into it (i.e. 𝑑 𝑟/𝑒 < 0), the braking state becomes active. The applied bilateral force 𝐅 𝐝 exerted on the ring is shown in equation 6 with 𝐁 𝐝 the dissipative gain.

𝐅 𝐝 = -𝑋 ̇𝑟/0 𝐁 𝐝 n (6)

When |𝑋 ̇𝑟/0 | < 𝑣 𝑡ℎ the VO state becomes active. The speed threshold is fixed to 𝑣 𝑡ℎ ≈ 0.020 m/s, which in practice corresponds to a static interface with a finger inside the ring. In this state the VO properties are the same as implemented in the original control law.

Tracking Force

As previously explained, the influence of the tracking force must diminish when the ring touches the wall, however it must be strong enough for the interface to follow his/her finger when moving away from the VO. Also, it is important to have a continuous tracking force to ensure that the interface will behave correctly during transitions between free and contact modes. In the original control law 𝐅 𝐭 decreases by a factor 𝛽, which is function of 𝑑 𝑓/𝑒 , to ensure its continuity (see section 2.2).

The control algorithm presented in section 3.1 requires a different strategy to make sure that the tracking force remains continue. 𝛽 still varies in function of 𝑑 𝑓/𝑒 as in (4) but we make here 𝑅 𝑓 = 𝑅 𝑟 so that the minimum value of 𝐅 𝐭 when the ring's avatar penetrates in the reference wall is ensured. Indeed, because of the tracking error, the finger's center position is always in advance to that of the ring when it is in movement. This means that if we consider a finger avatar the size of the ring, it will always penetrate first in the reference (𝑥 𝑒 ) or offset (𝑥 𝜀 ) VO position. When the VO takes its new value at 𝑥 𝜀 , the augmented finger's avatar is already penetrating into it. At this moment 𝛽 varies in function of 𝑑 𝑓/𝜀 to ensure the continuity of 𝛽 and therefore that of 𝐅 𝐭 . When moving away from the VO, the augmented finger avatar can be far enough from it in order too fully reactivate 𝐅𝐭. When the ring comes back in transparent state, this is when 𝑑 𝑟/𝑒 > 𝑑 𝛽 , we do 𝛽 to vary in function of 𝑑 𝑓/𝑒 again. This way the continuity of the tracking force is ensured during all state transitions (see Figure 6). As we can observe in equation 8, the distance 𝑑 𝛽 is also valid when the VO state is active. Here 𝑑 𝑓/𝑒,𝜀 means that 𝛽 will be function either of 𝑑 𝑓/𝑒 or 𝑑 𝑓/𝜀 , according to the active state (see Figure 6). The control explained in sections 3.1 and 3.2 is resumed in the control block diagram from Figure 7.

EXPERIMENTS AND RESULTS

To validate the benefits of the new approach, called VO-B (Virtual Object B) in the following, we have compared its performances with that of the original control law, called VO-A (Virtual Object A).

Experimental Setup

The robot used for these experiments is an optimized version of a 2D substructure of a parallel 6DoF haptic interface developed at CEA, LIST for tele-surgery [START_REF] Gosselin | Design of a High Fidelity Haptic Device for Telesurgery[END_REF]. This 2 DoF robot is composed of two links 0.25 m long each (see Figure 8). Its workspace lies in a vertical plane. Actuation is provided by two Maxon RE-35 DC motors and cable capstan reducers, allowing a particularly transparent behaviour. 1000ppt encoders are used for position sensing and counterweights mounted on each axis allow gravity compensation. The 2D ring-shaped encounter-type end effector has an inner diameter of 24 mm, sufficient to track a finger at medium speeds. Sixteen Vishay VCNL4000 infrared proximity sensors distributed over the inner side of the ring make possible the estimation of a finger's center position at a rate of 300 Hz with a ±0.3 mm precision. ATMega328P microcontrollers retrieve and send proximity sensor measurements to the haptic interface controller trough a fast serial bus at a rate of 400 kbps. Estimation of the finger's location is computed as the center of the polygon obtained from the measurement. The controller is composed of a PC104 computer running Xenomai realtime operating system and a servo-drive controlling both motors. A telerobotics library [START_REF] Gicquel | Tao: 2000: A generic control architecture for advanced computer aided teleoperation systems[END_REF] acquires the state of the robot, computes the finger's position and sends the reference torques to the servodrive at a rate of 1 KHz. Rate mismatch between the control loop (1 KHz) and the estimation of the finger's center (300 Hz) is handled by a Kalman filter committed to extrapolating the finger's position.

Practical Comparison between Smooth Transition-based and Offset Transition-based Control

In order to compare algorithms VO-A and VO-B, we made a typical encounter with a vertical virtual wall. We asked therefore a participant to move the ETHD horizontally from the right to the left with his index finger until he taps on a vertical wall located a few centimetres on the left of the initial position. In order to ensure an as natural as possible contact, we asked the user to keep his index finger straight and perpendicular to the working plane of the robot, with the pulp oriented to the left. These experiments were made both at low speed, where the existing control law is assumed to work properly, and at higher speeds, for which the device's behaviour becomes unnatural. The participant had a sufficient time to understand these instructions and familiarize with the interface. We checked both visually during the experiments and during data post-processing that the gesture was performed properly. It is worth noting that in practice, due to the limited dynamics of the robot, the user enters in contact with the ring before he encounters the wall for speeds higher than 0.5 m/s. We chose therefore 0.4 m/s as high speed. A value of 0.2 m/s was chosen for the low speed so as to remain significantly lower than the high speed. The robot's gains were defined experimentally, so as to be the highest possible while remaining stable. Their values in free space and during contacts are given in equations 9 and 10 respectively. We use the same gains 𝐊 𝒕 , 𝐁 𝒕 and 𝐊 𝐞 , 𝐁 𝐞 in both conditions VO-A and VO-B. The dissipative gain 𝐁 𝐝 is defined in equation 11. A high value is chosen in order to stop the interface as fast as possible (in practice in less than 20ms, see Table 1). With these values, no instabilities were observed in practice. In both cases, we denote 𝑡 𝑟/𝑣𝑜 the instant of time at which the ring's inner periphery encounters the object's reference constraint, 𝑡𝑜 𝑓/𝑟 the instant of time when the user's finger contacts the ring, 𝑑 𝑚𝑎𝑥 the amplitude of the very first rebound of the end-effector at contact and 𝑥 𝑒 as the reference constraint. In condition VO-B, 𝑥 𝜀 represent the modified constraint which is actually 𝑥 𝑒 plus the offset 𝜀. Regarding speeds, we denote 𝑣 𝑟 (𝑡 𝑟/𝑣𝑜 ) the speed of the ring's center when it encounters the reference wall 𝑥 𝜀 , 𝑣 𝑟 (𝑡𝑜 𝑓/𝑟 ) its speed when it encounters the user's finger and 𝑣 𝑟 𝑚𝑖𝑛 (> 𝑡𝑜 𝑓/𝑟 ) its minimal speed just after the finger encountered the ring.

Low Speed Case

The results obtained at low speed are given in Figures 9 and10. No significant discrepancy can be observed between VO-A and VO-B. Both 𝑑 𝑚𝑎𝑥 and 𝑣 𝑟 (𝑡𝑜 𝑓/𝑟 ) look similar. Only the speed 𝑣 𝑟 𝑚𝑖𝑛 (> 𝑡𝑜 𝑓/𝑟 ), i.e. the rebound speed, is slightly smaller in condition VO-B. As a whole, the behaviour of both control laws is very similar. This is not a surprise as VO-A has already good performances at low speeds.

High Speed Case

A clear difference between VO-A and VO-B can be observed in this case (see Figure 11 and Figure 12), considering both 𝑑 𝑚𝑎𝑥 , which is smaller for VO-B (observable oscillations appear in VO-A) and 𝑣 𝑟 (𝑡𝑜 𝑓/𝑟 ), which is closer to zero in VO-B, indicating that in this case the ring is quasi-static, as expected (the observed oscillations prove on the contrary that this is not the case in condition VO-A).

Rebound speed 𝑣 𝑟 𝑚𝑖𝑛 (> 𝑡𝑜 𝑓/𝑟 ) is also higher in VO-A than in VO-B. These observations lead us to make the hypothesis that with the new algorithm proposed in this article the sensation felt by the user will be more realistic since 𝑣 𝑟 (𝑡𝑜 𝑓/𝑟 ) and 𝑣 𝑟 𝑚𝑖𝑛 (> 𝑡𝑜 𝑓/𝑟 ) tend to be smaller at high speeds than in condition VO-A.

Figure 11: Typical encounter with a vertical virtual wall at a high speed in condition VO-A.

Figure 12: Typical encounter with a vertical virtual wall at a high speed in condition VO-B.

User Tests

According to [START_REF] Samur | Performance Metrics for Haptic Interfaces[END_REF], many factor studies can be used to assess the benefits of haptic feedback on sensory-motor tasks: peg-in-hole, tapping, targeting, etc. In the present work, we are more particularly interested in comparing how much the interface stabilizes when contacting a virtual object and how natural the contact is perceived by the user. As a consequence, we chose a tapping test to qualify the behaviour of the interface at both low and high speeds using VO-A and VO-B. The metrics introduced in section 4.2 will guide our analysis. The perception of the interaction was also evaluated through a survey.

Methodology

Four right-handed volunteers (3 men, 1 woman), aged 23-31, were invited to perform the tapping test. A printed document describing the experiment was given to each participant and he/she was asked to sign a letter of consent. The experiment was performed in an isolated room. The participant was standing, facing a screen and wearing an anti-noise helmet. The haptic interface was placed on a table so that his/her right index finger can be placed comfortably inside the end effector and so that a horizontal movement to the left can be performed easily.

A devoted graphical user interface (GUI) was developed for this experiment. It displays a 2D virtual environment in which the free space appears as a black vertical rectangle, surrounded by a thick green contour representing four virtual walls. The user is asked to tap on the left wall at low and high speeds, as previously defined in section 4.2, in conditions VO-A and VO-B. A white circle represents his/her finger and a vertical line indicates where to start before each tap (see Figure 13). In both conditions the finger's avatar is stopped against the wall, even if the robot, hidden by a vertical barrier, goes further, in order to avoid the influence of visual cues. It is worth noting that it is of crucial importance that the user keeps the index finger straight and perpendicular to the working plane of the robot throughout the experiments, with the pulp oriented to the left, so that it can make a full and proper contact with the wall. The non-respect of this gesture may impact the user's perception and the quality of the recorded data. To avoid this, the participants had a sufficient time to familiarize with the interface and practice taping at low and high speeds. Also, data was recorded for each single tap and its exploitability was verified in situ (as for the results given in section 4.2, we checked both visually during the experiments and during data post-processing performed just after each tap that the gesture was performed properly). Each participant was asked to perform taps until we get at least three valid taps (correct speed and absence of contact with the ring before the obstacle).

Results

Between 10 and 15 taps were necessary in each case to obtain three valid taps (one of the subject being unable to perform taps at high speeds, his data were discarded for the analysis).

Figure 14 illustrates the contact speed in each case (the median value appears in red, the box represents the first and third quartiles and the lower and higher bars the extremal values over the 9 trials, i.e. 3 users with 3 taps each). We can observe from these results that our data set is close to the low and high speeds defined in section 4.2 (i.e. 0.2 and 0.4 m/s). Figures 15, 16 and 17 illustrate respectively the speed of the ring center when the finger touches it (it should be as low as possible to realistically simulate the wall, which is fixed), the amplitude of the first rebound of the ring against the wall and the highest value of the speed of the ring after contact with the virtual wall (both should be as small as possible).

These results confirm that the behaviour of the interface in conditions VO-A and VO-B is very similar at low speeds. It differs only for high speeds. In this case, the speed of the ring when it encounters the user's finger tends to be closer to zero in condition VO-B (see Figure 15). At least 25% of the samples show positive speed, which means that at contact the ring and the finger were moving in the same direction. This is preferable than having a ring moving against the finger at contact. A clear difference can also be observed if we take into account the rebound 𝑑 𝑚𝑎𝑥 and the speed 𝑣 𝑟 𝑚𝑖𝑛 (> 𝑡𝑜 𝑓/𝑟 ). Figure 16 shows that conditions VO-A produces a very important rebound compared to VO-B. The same tendency is observed for speeds in Figure 17, the absolute value of the speed in VO-A being much higher than in the VO-B case. The rebound amplitude and speed in VO-B shows a considerable reduction, confirming that conditions VO-B allows a more realistic simulation of a fixed wall, even at high speeds. Further details on the behaviour of the interface in condition VO-B are given in Table 1. Results show that in average the ring stabilizes in about 3 ms at low speed and 15 ms at high speed, the constraint offset remaining below 3 mm. We can expect that a human operator wouldn't realize these differences when performing a tap (according to [START_REF] Knorlein | Influence of Visual and Haptic Delays on Stiffness Perception in Augmented Reality[END_REF] and [START_REF] Vogels | Detection of temporal delays in visual-haptic interfaces[END_REF], a visuo-haptic delay is imperceptible if it is lower than 45 ms). The participants were asked to answer three questions after the completion of the tests in each case. Q1 asks if the user perceived the contact before (score 1 or 2), just when (3) or after (4 or 5) the finger's avatar touched the virtual wall. It provides information on the perception of the visuo-haptic delay (ideal result is 3). Q2 asks if at contact the touched wall was perceived as moving to the left (score 1 or 2), being static (3) or moving to the right (4 or 5). It tell us if the user was perceiving the rebound (ideal result is also 3). Finally, Q3 asks if the sensation at contact was felt very natural (1), natural (2), neutral (3), unnatural (4) or very unnatural (5). Results are given in Figure 18 (mean scores for the three participants). They show almost no difference at low speed, as expected. At high speed however, conditions VO-B gives better results. The ring appears static while in VO-A it appears slightly moving. Also, the contact is perceived as being more natural in condition VO-B.

CONCLUSIONS

In this paper, we introduced a new control law intended to improved contact rendering with encounter type haptic displays. The results of our experiments show that this offset transition-based control allows to reduce the speed of the end effector before the user's finger encounters it, as well as its rebound amplitude against the obstacles. As a consequence, the contact is perceived as more natural.

Figure 1 :

 1 Figure 1: Encounter-type haptic display principle.

Figure 3 :

 3 Figure 3: Spring damper coupling between the center of the finger 𝑋 𝑓/0 and the center of the ring 𝑋 𝑟/0 in free space.

{

  𝑖𝑓 𝑑 𝑟/𝑒 < 0 & 𝑋 ̇𝑟/0 > 0 𝐅 𝐞 = 𝑑 𝑟/𝑒 (𝐊 𝐞 + 𝐁 𝐞 𝑠)𝐧 𝑖𝑓 𝑑 𝑟/𝑒 < 0 & 𝑋 ̇𝑟/0 < 0 𝐅 𝐞 = 𝑑 𝑟/𝑒 𝐊 𝐞 𝐧 𝑒𝑙𝑠𝑒 𝐅 𝐞 = 0

Figure 5 :

 5 Figure 5: Finite State Machine governing the proposed control law. Equation 7 defines 𝐅 𝐞 in reference to the new wall at position 𝑥 𝜀 , where 𝑑 𝑟/𝜀 is the distance between the inner ring periphery and it.

Figure 6 :

 6 Figure 6: Variation of 𝛽for a typical encounter and related active state at each stage. εrepresents the wall offset.

Figure 7 :

 7 Figure 7: Offset transition-based control. KVM:interaction force. 𝑓(•): estimation of the robot's end-effector speed 𝑋 ̇𝑟/0 with negative sign and of the distance 𝑑 𝑓/𝜀 .

Figure 8 :

 8 Figure 8: 2DoF ETHD.

Figure 9 :

 9 Figure 9: Typical encounter with a vertical virtual wall at a low speed in condition VO-A.

Figure 10 :

 10 Figure 10: Typical encounter with a vertical virtual wall at a low speed in condition VO-B.

Figure 13 :

 13 Figure 13: Setup of the experiment. Low speed tests were always performed first, the order of presentation of each case being alternative, i.e. either training with VO-A low and then perform tests in condition VO-B low then VO-A low or training with VO-B low and then perform cases VO-A low then VO-B low. The same principle was used in high speed conditions.It is worth noting that it is of crucial importance that the user keeps the index finger straight and perpendicular to the working plane of the robot throughout the experiments, with the pulp oriented to the left, so that it can make a full and proper contact with the wall. The non-respect of this gesture may

Figure 14 :

 14 Figure 14: Speed of the ring center at time 𝑡 𝑟/𝑣𝑜 .

Figure 15 :

 15 Figure 15: Speed of the ring center at time 𝑡𝑜 𝑓/𝑟 .

Figure 16 :

 16 Figure 16: Rebound amplitude 𝑑 𝑚𝑎𝑥 .

Figure 17 :

 17 Figure 17: Speed of the ring center at a time > 𝑡𝑜 𝑓/𝑟 .

Figure 18 :

 18 Figure 18: Survey scores.

Table 1 :

 1 Mean and standard deviation for ∆sand ε.

	mean(std)	∆𝒔 (ms)	𝜺 (mm)
	Low speed	3.111(2.804)	0.388(0.251)
	High speed	14.556(1.424)	2.434(0.357)


Maira Possas-Abreu

Lionel Rousseau

Farbod Ghassemi

Gaelle Lissorgues

Massiel Habchi

Emmanuel Scorsone

Khasim Cal

Krishna Persaud

BIOMIMETIC DIAMOND MEMS SENSORS BASED ON ODORANT-BINDING PROTEINS: SENSORS VALIDATION THROUGH AN AUTONOMOUS ELECTRONIC SYSTEM

Keywords: Odorant Binding Proteins, Diamond micro-cantilever, Sensor read-out, Sensor Array, MEMS

Odorant binding proteins (OBPs) have been considered one of the best candidates for conceiving a biosensor to detect biochemical molecules and for odor sensor applications. In the same way, MEMS sensors for chemical and biological applications proved to be a good option for applications requiring high sensitivity. In this context, micro-cantilever sensor is one of the most promising micromechanical sensor. In this paper a grafting method, compatible with MEMS technology, was used for grafting porcine OBPs on polycrystalline diamond cantilevers. These sensors have been tested, for the first time, in combination with an autonomous dedicated electronic system developed to automatically detect sensors response signals. Electronic detection of binding 2-isobutyl-3-methoxypyrazine (IBMP) to OBP was demonstrated.

INTRODUCTION

The OBPs are soluble proteins of transportation of odorant molecules. Many studies defend that in vertebrates, they contribute to the transport of odorous molecules through the mucus to the cilia of the olfactory neurons [START_REF] Breer | Odor recognition and second messenger signaling in olfactory receptor neurons[END_REF][START_REF] Meillour | Function of the odorant-binding proteins (OBP) in the olfactory transduction[END_REF]. Odorant Binding Proteins are able to bind diverse odorants and present high conformational stability which are attractive characteristics for the design of label free biosensors in particular for the detection of small volatile organic compounds [START_REF] Persaud | Sensor array techniques for mimicking the mammalian olfactory system[END_REF]4]. The most part of existing biosensors have been developed combining bio-receptors with different transducers like quartz crystal microbalance (QCM), surface acoustic wave (SAW) sensors, electrochemical sensors, etc. Microcantilever based sensors offer many advantages for odor detection, since they can be miniaturized (and are suitable for parallelization into arrays) and can achieve high sensitivity [START_REF] Raiteri | Micromechanical cantilever-based biosensors[END_REF][START_REF] Boisen | Cantilever-like micromechanical sensors[END_REF]. However, the most important challenge on using microcantilevers as bio-sensors transducers for odor detection (enose applications) is related to the need of reliable actuation and readout techniques.

Optical techniques are the most commonly-used method for measure cantilever bending or dynamic movement. They are conventionally used in atomic force microscopy (AFM). The main challenges of optical readout technologies are high cost, surface preparation, and optical alignment and adjustment requirements. Optical external read-out is also susceptible of interfere with bio-molecules in cantilever surface. In addition, integrated optical readout makes cantilever fabrication more complex, requiring waveguide light integration and alignment. In contrast, piezoresistive strainsensing techniques provide benefits over optical in readout size and the ability to work in light-sensitive environments or use opaque materials. With regard to sensor material for bio-interfaces, synthetic diamond has been recognized for many years for being an interesting bio-sensing interface because of the wide range of possibilities to covalently immobilize bio-receptors onto its carbon based surface and for the long term stability of carbon sp3 [START_REF] Hébert | Chapt.9 Diamond Biosensors, Carbon for Sensing Devices[END_REF]. Previous works [START_REF] Manai | Grafting odorant binding proteins on diamond bio-MEMS[END_REF] have shown the feasibility of grafting OBPs on polycrystalline diamond cantilevers. However, the use of OBP-grafted micro-cantilevers together with a dedicated electronic read-out has never been demonstrated. Indeed, such a device would require the integration of the read-out component, which is, generally, not compatible with grafting techniques. In this paper, we investigate a new approach consisting of grafting chemically the OBPs onto synthetic diamond microcantilevers at low temperatures, allowing the integration of electronic components in the transducer. Electronic detection of binding 2-isobutyl-3-methoxypyrazine (IBMP) to OBP is demonstrated in order to validate our approach.

MATERIALS AND METHODS

Diamond Cantilever Fabrication

Micro-cantilevers have been conceived to work on dynamic mode. In this mode, the analyte causes a shift in the cantilever's resonance frequency due to the increased mass loading. The response of the sensors is monitored by integrated piezoresistors.

Diamond micro-cantilevers were fabricated according to a process described in previous works [START_REF] Possas | Fabrication and micromechanical characterization of polycrystalline diamond microcantilevers[END_REF][START_REF] Possas | Comparing silicon and diamond microcantilevers based sensors for detection of added mass and stiffness changes[END_REF]. Cantilever dimensions were 310 µm × 140 µm (Figure 1). The cantilever geometry led to a resonance frequency of typically f=85000 Hz and a Q-factor of 600 in air. Diamond micro-cantilevers are cleaned before grafting the OBPs.

Figure 1: SEM image of a diamond micro-cantilever

OBPs Immobilization

All chemicals used for OBP grafting were purchased from Sigma-Aldrich, France. Here the method consisted of the activation of the diazonium salt [START_REF] Viel | Covalent grafting onto self-adhesive surfaces based on aryldiazonium salt seed layers[END_REF]. This activation is done by forming the diazonium function N2+ from an aromatic primary amine (phenylenediaine: NH2-φ-NH2 in this case), then to create a radical cycle φ which can react with the surface of the nanodiamonds in the presence of the C-C covalent bonds. We first proceed to the in situ formation of nitrous acid (HNO2) by adding sodium nitrite NaNO2 in an acid medium (HCl (0,5 M)). Subsequently, the nitrous acid solution will react with phenylenediamine (NH2-φ-NH2) for the formation of the hydrophilic function diazonium N2+, which can be removed in gaseous form, causing the radical phenyl to form. In our case, 4-aminobenzoic acid (NH2-φ-COOH) is used in order to complete the grafting by the immobilization of OBPs on the surface of the polycrystalline diamonds. OBPs solution (1.0 mg/mL in 20 mM PBS, pH=8.00) are used as received (preparation is described in [START_REF] Manai | Grafting odorant binding proteins on diamond bio-MEMS[END_REF]). Cantilevers are exposed to it for 2 hours. Again the substrates were rinsed thoroughly with deionized water and stored in PBS solution. The main difference between this OBP-grafting technique in diamond surface and other techniques presented on the literature [START_REF] Manai | Grafting odorant binding proteins on diamond bio-MEMS[END_REF] is the fact that hydrogenation steps at high temperatures are not needed. Indeed, it requires only low temperatures (<4°C) for its preparation. For this reason, piezoresistive gauges integrated in diamond cantilever can be preserved, and electronic detection is possible.

Gas cell and electronic system

The sensor chips were placed in a customized gas analysis cell, which consists on stainless steel chamber containing eight sensor cavities with a gas inlet/outlet (Figure 2). The sensors are organized radially on a holding part (represented in green) below which lies a piezoelectric cell used to bring each micro cantilever to its main resonance frequency. A low noise and highly reconfigurable electronic system has been designed as well as a dedicated human-machine interface. The main board is composed of a micro-controller, a signal generation circuit to excite the sensors around their resonance frequencies and an analog processing chain to extract the amplitude of each sensor response. An image of the complete system is showed on Figure 3.

Cantilever

Fluorescein has the ability to bind the OBP thus offering the possibility to visualize under UV excitation the presence of OBPs on the diamond surface.

Grafting validation through electronic system measurements

In order to verify the presence of OBPs grafted on the sensors, the 2-isobutyl-3-methoxypyrazine (IBMP) was used as reference gas due to its excellent affinity with porcine OBP [13].

CONCLUSION

We reported on the grafting of porcine OBP onto diamond micro-cantilevers using a different chemical approach enabling the use of integrated components and offering the possibility to use a dedicated electronic system. Surface immobilization of the proteins was validated by fluorescence imaging. These sensors have been tested, for the first time, in combination with an autonomous dedicated electronic system developed to automatically detect sensor response signals. This is highly promising for further general investigations of ligand binding properties of OBPs and for the development of reliable biosensors that may be used in vapor sensing applications as well as electronic nose applications.

Figure 2 :

 2 Figure 2: A photography of the gas cell and eight sensors placed inside

Figure 3 :

 3 Figure 3: Gas analysis cell and electronic system 3. RESULTS AND DISCUSSION 3.1. Fluorescence imaging First confirmation of protein immobilization was obtained by Fluorescence imaging. After the last step of OBPs grafting, a first sample was immersed in a fluorescein solution then rinsed vigorously with deionized water.

Figure 4 :

 4 Figure 4: Fluorescence microscopy picture observed at λexc=550 nm and λem=570nm of diamond cantilever functionalized with OBP immobilization after immersion in a fluorescein solution.

Figure 5 :

 5 Figure 5: Response of two OBP-grafted diamond microcantilevers and two references to water vapor and IBMP Results of Figure 5 presents the response of two OBP grafted sensors (red and green) compared with two nonfunctionalized cantilevers (blue). Here nitrogen is used as reference gas. The response of our sensors saturated IBMP atmosphere is around 40 Hz. Considering only the mass effect hypothesis of this detection, we can estimate the presence of 10 8 IBMP molecules bound to immobilized OBPs.


Adrien Chan-Hon-Tong
email: adrien.chan-hon-tong@cea.fr

Catherine Achard
email: catherine.achard@upmc.fr

Laurent Lucat
email: laurent.lucat@cea.fr

Deeply Optimized Hough Transform: Application to Action Segmentation

Keywords: Hough Transform, Learning, Action Segmentation

Hough-like methods like Implicit Shape Model (ISM) and Hough forest have been successfully applied in multiple computer vision fields like object detection, tracking, skeleton extraction or human action detection. However, these methods are known to generate false positives. To handle this issue, several works like Max-Margin Hough Transform (MMHT ) or Implicit Shape Kernel (ISK ) have reported significant performance improvements by adding discriminative parameters to the generative ones introduced by ISM. In this paper, we offer to use only discriminative parameters that are globally optimized according to all the variables of the Hough transform. To this end, we abstract the common vote process of all Hough methods into linear equations, leading to a training formulation that can be solved using linear programming solvers. Our new Hough Transform significantly outperforms the previous ones on HoneyBee and TUM datasets, two public databases of action and behaviour segmentation.

Introduction

The Hough Transform [START_REF] Hough | Method and means for recognizing complex patterns[END_REF] has first been introduced to detect lines in picture. The main idea of this method is to perform the detection not directly in the picture space but in the line parameter space (Hough space) where each line in the image is mapped into a single point. This method has subsequently been extended to parametric objects [START_REF] Ballard | Generalizing the Hough transform to detect arbitrary shapes[END_REF], and non-parametric objects [START_REF] Leibe | Combined object categorization and segmentation with an implicit shape model[END_REF] (e.g. car, pedestrian, sport activities, ...). For non parametric objects, the Hough Transform first learns a probabilistic-like parametrization of the objects on a training database, and, then performs the detections as a local problem in the corresponding Hough space.

Due to this property of local detection, Hough Transform is a very fast process both in theory (time complexity theory) and practice. For this reason, it has been applied in context of real-time systems like [START_REF] Girshick | Efficient regression of general-activity human poses from depth images[END_REF] for skeleton extraction and more generally in multiple computer vision fields like tracking [START_REF] Gall | Hough forests for object detection, tracking, and action recognition[END_REF], object detection [START_REF] Gall | Class-specific Hough forests for object detection[END_REF], human action detection [START_REF] Yao | A Hough transform-based voting framework for action recognition[END_REF]. Actually, this method can be used to perform multi-class segmentation in multi-dimensional data e.g. pixel segmentation in image [START_REF] Sreedevi | An efficient image segmentation using Hough transformation[END_REF], temporal human action segmentation [START_REF] Yao | Does human action recognition benefit from pose estimation?[END_REF].

In this paper, we focus on temporal action segmentation in video: for each frame, the goal is to determine automatically which action is performed by an actor among a set of actions. The Hough Transform, used in this context, is composed of three main steps:

1: feature extraction and quantization to form codewords. 2: each of these codewords ω extracted at frame t votes for the action l at frame t+Δ t with a specific weight θ (ω, l, Δ t ). During this step, the same codeword ω votes for all actions l and all relative time displacements Δ t . The weights have been learned previously, during a learning step. 3: All votes, for all codewords and all frames, are agglomerated to build the Hough score which is the basis for segmentation decisions.

More formally, the Hough Transform (step 2-3) is based on the function θ() (traditionally positive) that links codewords, time displacements and actions to vote weights. Thus, a codeword w extracted at time t votes with a weight θ (w, l, Δ t ) for the hypothesis that an action l is present at time t + Δ t (this weight does not depend on the time t but only on l, Δ t and w). Hence, given a set of localized codewords W = {w, t} (w for codewords, t for the extraction time), the Hough score H for the action l at the time t is:

H t, l = (w,t)∈W θ w, l, t -t (1)
and, the decision about the action at time t is given by:

l t = arg max l H t, l (2)
Thus an action is decided for each frame of the video. Hence, all the purpose of the training is to select values for θ (w, l, Δ t ) that will provide correct decisions with respect to the equations (1) (2) at testing time. Several works, recalled in section 2, offer to improve the generative votes introduced by the ISM method by introducing a partial discriminative optimization process during the vote estimation step. In section 3, we offer to extend these methods by optimizing globally all the votes in a discriminative way. With this new learning process, our Hough method significantly outperforms previous ones on two public datasets of action segmentation (the Honeybee dataset [START_REF] Oh | Learning and inferring motion patterns using parametric segmental switching linear dynamic systems[END_REF] and the TUM dataset [START_REF] Tenorth | The tum kitchen data set of everyday manipulation activities for motion tracking and action recognition[END_REF]) as reported in section 4, before the conclusion in section 5.

State of the Art

In this section, we present the different published methods to select the vote weight during the training step of Hough Transform.

Implicit Shape Model

In the ISM [START_REF] Leibe | Combined object categorization and segmentation with an implicit shape model[END_REF], the Hough Transform (the values of θ ()) is based on generative weights. Let P (l, Δ t |w) be the probability that the action at time t + Δ t is l, knowing that a codeword w has been extracted at time t. This probability is estimated with statistics on the training dataset and is supposed to be independent of t (it just depends on l, Δ t and w). Then, the weights are given by:

θ ISM (w, l, Δ t ) = P (l, Δ t |w) ( 3 )
In practice, the probability P (l, Δ t |w) is estimated by:

P (l, Δ t |w) ≈ N (l, Δ t , w) N (w) (4)
where N (l, Δ t , w) is the number of occurrences of an action l observed with a displacement Δ t from a codeword w and N (w) is the number of occurrences of the codeword w. These ISM -based weights have several advantages (e.g. parameter-free training, robustness to over-training), but they suffer from several drawbacks. In particular, all codewords and training examples have the same importance and are considered independently from each other. Two methods, MMHT [START_REF] Maji | Object detection using a max-margin Hough transform[END_REF] and ISK [START_REF] Zhang | Implicit shape kernel for discriminative learning of the Hough transform detector[END_REF] have been introduced to address these drawbacks.

Max-Margin Hough Transform

In MMHT [START_REF] Maji | Object detection using a max-margin Hough transform[END_REF], a coefficient is introduced for each codeword to ponderate the ISM values, resulting in:

θ MMHT (w, l, Δ t ) = λ w × θ ISM (w, l, Δ t ) = λ w × P (l, Δ t |w) ( 5)
The weights λ w give more or less importance to the different codewords w according to their discriminative power. They are learnt simultaneously in a discriminative way through an optimisation process similar to support vector machine (SVM ) training [START_REF] Cortes | Support-vector networks[END_REF].

Implicit Shape Kernel

In ISK [START_REF] Zhang | Implicit shape kernel for discriminative learning of the Hough transform detector[END_REF], the votes are also based on the ISM generative ones, but some coefficients are introduced to weight the different training examples. Hence, ISK training leads to:

θ ISK (w, l, Δ t ) = i λ i × P i (l, Δ t |w) ( 6
)
where P i (l, Δ t |w) is an estimation of the probability P (l, Δ t |w) based only on the training example i. The weights λ i are learnt simultaneously in a discriminative way using a specific kernel-SVM training [START_REF] Zhang | Implicit shape kernel for discriminative learning of the Hough transform detector[END_REF].

MMHT and ISK report experimental improvements over ISM by adding discriminative parameters. This trend is also supported by [START_REF] Wohlhart | Discriminative Hough forests for object detection[END_REF].

ISM + SVM

In [START_REF] Wohlhart | Discriminative Hough forests for object detection[END_REF], it is highlighted that learning directly the Hough map H t, l with a SVM is equivalent to the ISM with the introduction of a weighting coefficient for each displacement. This results in:

θ ISM+SV M (w, l, Δ t ) = λ Δt × P (l, Δ t |w) = λ Δt × θ ISM (w, l, Δ t ) (7)

Hough Forest

To our knowledge ISM [START_REF] Leibe | Combined object categorization and segmentation with an implicit shape model[END_REF] and the presented extensions [START_REF] Maji | Object detection using a max-margin Hough transform[END_REF][START_REF] Wohlhart | Discriminative Hough forests for object detection[END_REF][START_REF] Zhang | Implicit shape kernel for discriminative learning of the Hough transform detector[END_REF] are the only published methods to estimate the weights of Hough Transform. More precisely, these methods define links between codewords and votes. There are, of course, various ways to select the features and the codewords, like, the Hough forest [START_REF] Gall | Class-specific Hough forests for object detection[END_REF] methods which are major methods of the state of the art. Hough forests use ISM votes, but the mapping between features (usually data patches) and codewords (a leaf in a weak binary classifier tree) is constructed such that all training features associated with the same codeword are expected to come from training examples with a same label. Several works, like [START_REF] Gall | Class-specific Hough forests for object detection[END_REF][START_REF] Yao | Does human action recognition benefit from pose estimation?[END_REF], report that this automatic feature mapping process associated with ISM votes leads to significant experimental improvements against codewords obtained without learning, by Kmeans algorithm for example. However, in this paper, we focus on the optimisation of the weights used during the vote process and so to the link between codewords and votes which is generic whatever the features and codewords used. Thus, the offered method can be employed in the Hough forest context by substituting the weights estimated by ISM by the weights optimized by our offered method.

The common point between MMHT, ISK and ISM+SVM is that they add discriminative parameters to the generative ones introduced by the ISM. In this paper, we offer to use only discriminative votes strongly optimized. We call this method Deeply Optimized Hough Transform (DOHT ).

Deeply Optimized Hough Transform

The goal of the training process is to establish a mapping between codewords and weights. While ISM method only uses generative weights, MMHT, ISK and ISM+SVM introduce discriminative parameters optimized according to codewords, training examples or displacements. We offer in this paper to optimize all these weights in a global way, according to all parameters of θ (w, l, Δ t ) in multi-class context. Hence, our set of variables is indexed by codewords w (as in HHMT ), displacements Δ t (as in ISM+SVM ) and also by actions l. These differences are summarized in table 1. In this way, we do not use ISM values and the method becomes deeply discriminative.

The goal is to define a function θ() such that for all training examples W and all times t, the predicted action l is the right one l * (known on training data).

Considering the definition of the predicted action l (eq. ( 2)), our problem formulation is equivalent to:

∀W, t, l = l * t H t, l < H t, l * t (8)
By dividing θ() by the minimal gap in eq. ( 8), this is equivalent to

H t, l + 1 ≤ H t, l * t (9)
and, using equation ( 1), to ∀W,

t, l = l * t ⎛ ⎝ (w,t)∈W θ w, l, t -t ⎞ ⎠ + 1 ≤ ⎛ ⎝ (w,t)∈W θ w, l * t , t -t ⎞ ⎠ (10)
In addition, as a codeword extracted at time t should not provide more information about the time t ± 1 than about the time t, we constraint θ to be decreasing with the absolute value of Δ t .

However, it is not sure that a function can satisfy these constraints. Hence, we introduce a soft margin as in [START_REF] Cortes | Support-vector networks[END_REF]: some variables ξ are introduced in eq. ( 10) leading to:

∀t, l = l * t , W (w,t)∈W θ w, l, t -t + 1 -ξ t ≤ (w,t)∈W θ w, l * t , t -t (11)
and these variables are minimized to reduce the number of not-satisfied constraints, leading to the objective function: min θ≥0,ξ≥0 t ξ t .

To prevent over-fitting, a regularity term in added to the objective function as in [START_REF] Chakrabartty | Gini support vector machine: Quadratic entropy based robust multi-class probability regression[END_REF]. A coefficient Υ regulates the trade-off between the attachment to data and the regularity as in [START_REF] Cortes | Support-vector networks[END_REF][START_REF] Chakrabartty | Gini support vector machine: Quadratic entropy based robust multi-class probability regression[END_REF].

Finally, the training problem is formulated as:

min θ≥0,ξ≥0 (w,l,Δt) θ(w, l, Δ t ) + Υ t ξ t under constraints: ∀W, t, l = l * t , (w,t)∈W θ w, l * t , t -t -θ w, l, t -t + ξ t ≥ 1 and: ∀w, l, Δ t , θ (w, a, Δ t + sign (Δ t )) ≤ θ (w, a, Δ t ) ( 12
)
where sign is the sign function. These formulation is a linear program which which is a well studied problem in literature (e.g. [START_REF] Karmarkar | A new polynomial-time algorithm for linear programming[END_REF]), and, which can be solved efficiency (for example using the solver CPLEX1 , freely available for academic purposes).

In the next section, we evaluate ISM, HHMT, ISM+SVM and DOHT in action segmentation or behavior segmentation contexts. As ISK is only intended [START_REF] Leibe | Combined object categorization and segmentation with an implicit shape model[END_REF] θISM (w, l, Δt) = P (l, Δt|w) -MMHT [START_REF] Maji | Object detection using a max-margin Hough transform[END_REF] θMMHT (w, l, Δt) = λw × P (l, Δt|w) λw ISK [START_REF] Zhang | Implicit shape kernel for discriminative learning of the Hough transform detector[END_REF] θISK (w, l, Δt) = i (λi × Pi (l, Δt|w)) λi ISM+SVM [START_REF] Wohlhart | Discriminative Hough forests for object detection[END_REF] θISM+SV M (w, l, Δt) = λΔ t × P (l, Δt|w) λΔ t DOHT (our) θDOHT (w, l, Δt) = λ w,l,Δt λ w,l,Δt P (l, Δt|w) is the probability that the action at time t + Δt is l knowing that a codeword w has been extracted at time t. Pi (l, Δt|w) is the same probability estimated using only the training example i.

for detection and can not be straightforwardly extended to segmentation, we can not compare it to the others methods.

Experimental Results

Experiments have been conducted on the TUM [START_REF] Tenorth | The tum kitchen data set of everyday manipulation activities for motion tracking and action recognition[END_REF] and Honeybee [START_REF] Oh | Learning and inferring motion patterns using parametric segmental switching linear dynamic systems[END_REF] datasets. These datasets are well designed for segmentation as each frame is associated to an action.

Application to Human Action Segmentation

TUM is a multi-sensor dataset and in particular it contains skeleton streams (fig. 1). It is composed of 19 sequences about 2 minutes each, containing 9 kinds of actions like Lowering an object, Opening a drawer, performed by 5 subjects.

To provide results comparable to [START_REF] Yao | Does human action recognition benefit from pose estimation?[END_REF], the dataset is separated in training and testing sequences, all algorithms decide an action for each frame of each testing sequence and the performance is measured by the ratio of correctly decided actions.

example of action: Lowering an object Provided skeleton Fig. 1. TUM dataset [START_REF] Tenorth | The tum kitchen data set of everyday manipulation activities for motion tracking and action recognition[END_REF] As [START_REF] Yao | Does human action recognition benefit from pose estimation?[END_REF] reports better performances using skeleton features (than visual or visual plus skeleton ones), we decide to consider only skeleton based features. Hence, the input signal of our algorithm is the 3D positions of each articulation at each time.

We use the same preprocessing (features and codewords) than the bag-ofgestures from [START_REF] Chan-Hon-Tong | Skeleton point trajectories for human daily activity recognition[END_REF] which achieves the best published performance on this dataset (with a manual segmentation). First, the positions are normalized (positions are expressed in a system of coordinates linked to the subject in order to be invariant to camera point of view, global body position, rotation and size). Then, we consider short temporal series of 3D positions of each articulation as features: let the vector (p 1 , ..., p T ) be the normalized trajectory of one articulation, then, we consider the vector (p t-τ , ..., p t+τ ) as a feature extracted at time t. Similar features are also considered in [START_REF] Sun | Hierarchical spatio-temporal context modeling for action recognition[END_REF][START_REF] Yao | Does human action recognition benefit from pose estimation?[END_REF][START_REF] Wang | Action Recognition by Dense Trajectories[END_REF] which emphasize the efficiency of interest points trajectories for human action recognition. Finally, all these features are clustered by K-means. The cluster centers define the codebook and features are mapped to their nearest codeword. We consider the 8 main articulations: feet, hands, knees, elbows. The quantization with K-means is performed independently for each articulation.

The parameters of this set of experiments are τ , K, Υ (for the learning process). On this dataset, the maximal performances achieved by ISM, MMHT and ISM+SVM when empirically varying the parameters values are less than the mean performances of DOHT. A typical run is obtained with τ = 6, K = 10, and C = 1. Results of this experiment are presented in table 2. In this set of experiments, DOHT significantly outperforms ISM, MMHT and ISM+SVM and achieves equivalent performance than a SVM based on the same features and codeword applied on the optimal segmentation (obtained from the ground truth) from [START_REF] Chan-Hon-Tong | Skeleton point trajectories for human daily activity recognition[END_REF]. Hence, for this dataset, we achieve equivalent performance than the best published (82.6% against 84.3%) without using any manual segmentation.

Application to Behaviour Segmentation

Experiments have also been conducted on the Honeybee dataset [START_REF] Oh | Learning and inferring motion patterns using parametric segmental switching linear dynamic systems[END_REF]. The Honeybee dataset provides tracking output of honey bees having 3 kinds of behaviour correlated with their trajectories (figure 2). It composed of 6 large sequences. To provide results comparable to [START_REF] Oh | Learning and inferring motion patterns using parametric segmental switching linear dynamic systems[END_REF], all algorithms decided an action for each frame in a leave-one-out cross validation setting and the performance is measured by the ratio of correctly decided actions.

The input signals in this dataset are the sequences of bee 2D positions and orientations (x t , y t , α t ). As in the previous experiment, normalized short temporal series of (2D here) positions are considered as features. Let us call R (β) the matrix of the 2D rotation of angle β and p (t) = (x t , y t ), then we consider the vector (R (-α t ) (p t-τ -p t ) , ..., R (-α t ) (p t+τ -p t )) as the feature extracted at time t. All these features are clustered using K-means. The cluster centers define the codebook and features are mapped to their nearest codeword. K-means is performed independently for each τ . The parameters of this set of experiments are τ , K, Υ . On this dataset, the maximal performances achieved by ISM, MMHT and ISM+SVM when empirically varying the parameters values are less than the mean performances of DOHT. A typical run is obtained with τ ∈ {1, 3, 6}, K = 10, and C = 1. Results of this experiment are presented in table 2.

In this set of experiment, DOHT significantly outperforms ISM, MMHT and ISM+SVM. In addition, DOHT achieves equivalent performances than the latter best published results [START_REF] Hoai | Joint segmentation and classification of human actions in video[END_REF]. In [START_REF] Hoai | Joint segmentation and classification of human actions in video[END_REF], a multi-class SVM is applied on each temporal window (with similar kind of features and codewords). Then, segmentation is computed using dynamic programming. As scores are computed on each temporal window, this method is quadratic with respect to the maximal length of an activity while our is linear. This quadratic property is a common drawback caused by performing scoring as a global problem. Hence, for this dataset, we achieve equivalent performances (86.5% against 89.3%) than the best published results while being significantly faster.

Table 2. Global results on TUM [START_REF] Tenorth | The tum kitchen data set of everyday manipulation activities for motion tracking and action recognition[END_REF] and Honeybee [START_REF] Oh | Learning and inferring motion patterns using parametric segmental switching linear dynamic systems[END_REF]

Method

Accuracy on TUM Accuracy mean on Honeybee ISM [START_REF] Leibe | Combined object categorization and segmentation with an implicit shape model[END_REF] 58.4 71.9 MMHT [START_REF] Maji | Object detection using a max-margin Hough transform[END_REF] 69.6 78.8 ISM+SVM [START_REF] Wohlhart | Discriminative Hough forests for object detection[END_REF] 68.5 77.5 DOHT (our) 82.6 86.5

Conclusion

In this paper, we offer to use Hough transform to segment temporal series. In a non parametric context, the training of Hough transform consists in properly selecting the weights used in the voting process. The simple way (Implicit Shape Model) consists in computing some probabilities on the training database, leading to a generative model. Some methods (Max-Margin Hough Transform, Implicit Shape Kernel) offer to add some parameters optimized on a training database in a discriminative way. In this article, we offer to skip the first step based on a generative model, and, to globally learn all the parameters of the Hough transform on the training database, resulting a deeply discriminative model. This required to reformulate the voting process to express it in a linear form in order to use linear programming solvers. We performed several experiments on public datasets where the Hough transform trained with our method significantly outperforms other Hough transform methods and provides equivalent results than best published results for these datasets while avoiding some limitations of the corresponding algorithms (e.g. manual segmentation).

In future works, we will adapt and apply our method on other contexts e.g. object segmentation in image, video spatio-temporal segmentation, automatic speech segmentation, sign language segmentation.

  tracking output Green correspond to waggle, magenta to right turn and blue to left turn.

Fig. 2 .

 2 Fig. 2. Honeybee dataset[START_REF] Oh | Learning and inferring motion patterns using parametric segmental switching linear dynamic systems[END_REF]

Table 1 .

 1 The different learning methods of the Hough Transform

	methods	θ	variables
	ISM

www-01.ibm.com/software/websphere/products/ optimization/academic-initiative/


Manuel Vermillac

Francois Peters

Courtney Kucera

T W Hawkins

John Ballato

Wilfried Blanc

Drawing Nanocomposite Optical Fibers: Shaping Particles and Bubbles

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Nikodem Dymski
email: nikodem.dymski@inria.fr

Paola Goatin
email: paola.goatin@inria.fr

Massimiliano D Rosini
email: mrosini@umcs.lublin.pl

Modeling moving bottlenecks on road networks

In this paper we generalize the Lighthill-Witham-Richards model for vehicular traffic coupled with moving bottlenecks to the case of road networks. Such models can be applied to study the traffic evolution in the presence of a slow-moving vehicle, like a bus. At last, a numerical experiment is shown.

Introduction

In this paper we study the Lighthill-Witham-Richards (LWR) model for vehicular traffic coupled with moving bottlenecks on road networks. We recall that the LWR model was introduced in [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF] and gave rise to macroscopic modelling of traffic flow. A moving bottleneck models the presence of a slow vehicle, like a bus or a truck, which causes the reduction of the road capacity at its position and thus generates a moving constraint for the traffic flow. From the analytical point of view our model is the natural generalization to the case of a network of the LWR model with moving constraint on a single road developed in [START_REF] Delle Monache | Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result[END_REF], which in turn can be considered as a generalization of the fixed in space point constraint on the flow theory, see [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF][START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF], [START_REF] Rosini | Macroscopic models for vehicular flows and crowd dynamics: theory and applications[END_REF]Chapter 6]. For completeness we mention that a 2 × 2 system of conservation laws coupled with a fixed in space point constraint on the flow has been studied in [START_REF] Andreianov | A second-order model for vehicular traffics with local point constraints on the flow[END_REF][START_REF] Dymski | Existence of BV solutions for a non-conservative constrained Aw-Rascle-Zhang model for vehicular traffic[END_REF][START_REF] Garavello | The Aw-Rascle traffic model with locally constrained flow[END_REF] in the case of a single road, while the case of a phase transition model coupled with a fixed in space point constraint on the flow has been studied in [START_REF] Santo | General phase transition models for vehicular traffic with point constraints on the flow[END_REF].

We describe the evolution of the traffic in presence of a slow-moving vehicle by the strongly coupled PDE-ODE system (1) introduced in [START_REF] Delle Monache | Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result[END_REF][START_REF] Giorgi | Prise en compte des transports en commun de surface dans la modélisation macroscopique de lécoulement du trafic[END_REF], where the PDE (1a) consists of a scalar conservation law which models the evolution of traffic, while the ODE (1b) describes the trajectory of the slow-moving vehicle. The study of coupled PDE-ODE systems is not new in the conservation laws framework, we refer the reader to [START_REF] Borsche | On the coupling of systems of hyperbolic conservation laws with ordinary differential equations[END_REF][START_REF] Delle Monache | A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow[END_REF][START_REF] Delle Monache | Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result[END_REF][START_REF] Delle Monache | A numerical scheme for moving bottlenecks in traffic flow[END_REF][START_REF] Dymski | Existence of BV solutions for a non-conservative constrained Aw-Rascle-Zhang model for vehicular traffic[END_REF][START_REF] Lattanzio | Moving bottlenecks in car traffic flow: a PDE-ODE coupled model[END_REF]. The paper is organized as follows. In the next section we consider the case of a single unidirectional road. The case of a network is then considered in Section 3. Finally, in Section 4, we compare the solutions of the standard model with that with moving constraint for the same Riemann problem.

A single unidirectional road

We consider a single road parametrized by the coordinate x ∈ R and assume that the vehicles move in the direction of increasing x with maximal speed V > 0. Let y = y(t) ∈ R be the position of the bus and ρ = ρ(t, x) ∈ [0, 1] be the mean (normalized) density of cars at time t ≥ 0 and position x ∈ R. The resulting model is expressed by the following system

∂ t ρ + ∂ x f (ρ) = 0 (t, x) ∈ R + × R, (1a)
ẏ(t) = ω ρ(t, y(t)+) t ∈ R + , (1b)
f ρ(t, y(t)) -ẏ(t)ρ(t, y(t)) ≤ α 4V V -ẏ(t) 2 t ∈ R + . (1c)
Above, the flux f ≥ 0 is defined by f (ρ) := ρv(ρ), where v(ρ) is the mean velocity of the cars. We let v : [0, 1] → R + be the strictly decreasing function defined by v(ρ)

:= V (1 -ρ). Clearly f : [0, 1] → R + is a strictly concave function such that f (0) = f (1) = 0, f (1/2) = max ρ∈[0,1] f (ρ) and sign(ρ -1/2)f (ρ) < 0 ∀ρ ∈ [0, 1] \ {1/2}.
We stress that v(1) = 0 and v(0) = V . If the bus has maximal speed V b ∈ [0, V [, then it moves with velocity

ω(ρ) = min{v(ρ), V b }.
As a result, the trajectory of the bus is given by the function y : R + → R satisfying (1b). Notice that if the road is sufficiently congested, then v(ρ) < V b and the speed of the bus coincides with the speed of the cars. Condition (1c) can be derived as follows. By setting X = x-y(t) we obtain the bus reference frame, where the velocity of the bus is zero and the conservation law (1a) becomes

∂ t ρ + ∂ X f (ρ) -ẏρ = 0.
The presence of the bus hinders the maximum flow at X = 0 according to the rule

f (ρ) -ẏρ ≤ α 4V V -ẏ 2 ,
where the constant coefficient α ∈ ]0, 1[ is the reduction rate of the road capacity due to the presence of the bus. Notice that a higher velocity of the bus ẏ corresponds to a lower capacity of the road at its position and that

α 4V V -ẏ 2 ∈ α 4V V -V b 2 , αV 4
.

We augment system (1) with an initial datum for the density of the form of a Heaviside function with a jump at the initial bus position, which is assumed to be x = 0, that is

ρ(0, x) = ρ if x < 0, ρ r if x ≥ 0, (2a)
y(0) = 0, ( 2b
)
where ρ , ρ r ∈ [0, 1] are fixed constants. We consider solutions of the problem ( 1), ( 2) that are self-similar, hence the bus velocity ẏ(t) is assumed to be constant.

Let RS : [0, 1] 2 → L 1 loc (R; [0, 1]
) be the standard Riemann solver for (1a), (2a), which is described for instance in [START_REF] Bressan | Hyperbolic systems of conservation laws[END_REF]. We point out that the associated self-similar weak solution (t, x) → RS(ρ , ρ r )(x/t) does not always satisfy the constraint condition (1c). For this reason we define below the constrained Riemann solver

RS α : [0, 1] 2 → L 1 loc (R; [0, 1]
) for the Riemann problem ( 1), ( 2), see [START_REF] Delle Monache | Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result[END_REF]. First, see Figure 1, we need to introduce the density values ρα and ρα defined by

ρα = min{ρ ∈ [0, 1] : f (ρ) = ρV b + F α }, ρα = max{ρ ∈ [0, 1] : f (ρ) = ρV b + F α },
where

F α := α 4V V -V b 2 .
α : [0, 1] 2 → L 1 loc (R; [0, 1]) is defined as follows: 1. If f (RS(ρ , ρ r )(V b ) ≤ V b RS(ρ , ρ r )(V b ) + F α , then RS α (ρ , ρ r )(x/t) = RS(ρ , ρ r )(x/t) and y(t) = ω(ρ r )t. 2. If f (RS(ρ , ρ r )(V b ) > V b RS(ρ , ρ r )(V b ) + F α , then RS α (ρ , ρ r )(x/t) = RS(ρ , ρα )(x/t) if x < V b t, RS(ρ α , ρ r )(x/t) if x ≥ V b t, and
y(t) = V b t.
Notice that if constraint condition (1c) is not satisfied by the standard weak solution (t, x) → RS(ρ , ρ r )(x/t), then the weak solution (t, x) → RS α (ρ , ρ r )(x/t) has a single undercompressive shock (ρ α , ρα ) moving with speed of propagation equal to V b , according to the Rankine-Hugoniot condition.

Networks

In this section we introduce the LWR model with moving constraint on road networks. As in [START_REF] Garavello | Traffic flow on networks[END_REF], we define a network as a directed graph (I, J ), that is a pair consisting of a finite set I of unidirectional roads and a finite set J of junctions. For the rest of the work, if it is not stated differently, a junction is placed at x = 0. Below we consider a node J ∈ J having n incoming roads

I i = ] -∞, 0[ ∈ I for i ∈ I := {1, .
. . , n} and m outgoing roads

I j = ]0, ∞[ ∈ I for j ∈ J := {n + 1, . . . , n + m}. Let f h (ρ) := V h ρ(1 -ρ)
be the flux corresponding to the road I h , for h ∈ H = I ∪ J. Assume that at time t = 0 the bus is at the junction, that is y(0) = 0. Let I k , k ∈ J, be the road corresponding to the route of the bus. A constrained Riemann problem at the node J is the following system of scalar conservation laws with constant initial datum on every road, augmented by the ODE for the bus trajectory and the constraint inequality:

∂ t ρ i + ∂ x f i (ρ i ) = 0 ρ i (0, x) = ρ 0 i (t, x) ∈ R + × I i , i ∈ I, ∂ t ρ j + ∂ x f j (ρ j ) = 0 ρ j (0, x) = ρ 0 j (t, x) ∈ R + × I j , j ∈ J, y(t) = ω(ρ k (t, y(t)+)), y(0) = 0, f k ρ k (t, y(t)) -ẏ(t)ρ k (t, y(t)) ≤ α k 4V k V k -ẏ 2 t ∈ R + , (3)
for some α k depending on the k-th road characteristics. Before stating the constrained Riemann solver at the junction for (3), we define the admissible solutions. Definition 3.1 An admissible constrained Riemann solver at the junction J ∈ J for (3) is a map RS J α k : [0, 1] n+m → [0, 1] n+m such that for any (ρ 0 1 , . . . , ρ 0 n+m ) ∈ [0, 1] n+m we have that (ρ 1 , . . . , ρn+m ) := RS J α k (ρ 0 1 , . . . , ρ 0 n+m ) satisfies the following properties:

• For every i ∈ I, RS i (ρ 0 i , ρi ) has only waves with negative speed. • For every j ∈ J \ {k}, RS j (ρ j , ρ 0 j ) and RS α k (ρ k , ρ 0 k ) have only waves with positive speed.

• The mass through the junction is conserved, that is :

n i=1 f i (ρ i ) = n+m j=n+1 f j (ρ j ).
• RS J α k is consistent, that is:

RS J α k (ρ 1 , . . . , ρn+m ) = (ρ 1 , . . . , ρn+m ).
The last condition above says that the vector of traces at junction of an admissible solution is a fixed point for RS J α k . We propose below the possible traces and their maximal fluxes. To reach this goal, we need first to define the following function.

Definition 3.2 For any h ∈ H, the function τ

h : [0, 1] → [0, 1] is such that • f h (τ h (ρ)) = f h (ρ) for every ρ ∈ [0, 1]; • τ h (ρ) = ρ for every ρ ∈ [0, 1] \ {1/2}.
The function τ h is well defined, continuous and satisfies

0 ≤ ρ ≤ 1/2 ⇐⇒ 1/2 ≤ τ h (ρ) ≤ 1, 1/2 ≤ ρ ≤ 1 ⇐⇒ 0 ≤ τ h (ρ) ≤ 1/2.
In next propositions, we show the range of admissible fluxes for a given initial datum.

Proposition 1 Let i ∈ I and ρ 0 i be the initial datum on the incoming road I i . The set of reachable fluxes f i ( ρi ) is

Ω i (ρ 0 i ) = [0, f i (ρ 0 i )] if ρ 0 i ∈ [0, 1/2], [0, f i (1/2)] if ρ 0 i ∈ ]1/2, 1]
. Proof 1 Since the constraint does not affect an incoming road, we can apply the construction done in [START_REF] Garavello | Traffic flow on networks[END_REF]Proposition 4.3.3]. For completeness, we consider the case ρ 0 i ∈ [0, 1/2]; the case ρ 0 i ∈ ]1/2, 1] is analogous. We stress that RS i (ρ 0 i , ρi ) must have only waves with negative speed.

If ρi ∈ {ρ 0 i } ∪ ]τ i (ρ 0 i ), 1] then RS i (ρ 0 i , ρi
) is either constant or has a single shock with negative speed. On the other hand, if ρi ∈ [0, τ i (ρ 0 i )] \ {ρ 0 i } then RS i (ρ 0 i , ρi ) is either a rarefaction or a single shock, but in both cases with non negative speed, which concludes the proof.

A direct consequence of the above proposition is the following

Corollary 1 The maximal flow of the incoming road I i at the junction J is

γ max i (ρ 0 i ) = f i (ρ 0 i ) if ρ 0 i ∈ [0, 1/2], f i (1/2) if ρ 0 i ∈ ]1/2, 1].
Additionally, there exists a unique ρi ∈ [0, 1] such that the admissible solution of the Riemann problem with initial datum (ρ 0 i , ρi ) consists of waves with only negative speed and the condition

f i (ρ i ) = γ max i (ρ 0 i ) holds.
Proposition 2 Let j ∈ J and ρ 0 j be the initial datum on the outgoing road I j . The set of reachable fluxes f j (ρ j ) is

Ω j (ρ 0 j ) =          [0, f j (1/2)] if ρ 0 j ∈ [0, 1/2] and j = k, [0, f j (ρ 0 j )] if ρ 0 j ∈ ]1/2, 1] and j = k, [0, f k (ρ α k )] if ρ 0 k ∈ [0, ρα k ], [0, f k (ρ 0 k )] if ρ 0 k ∈ ]ρ α k , 1].
Proof 2 The proof for j = k is analogous to proof of Proposition 1. The only difference is that RS j (ρ j , ρ 0 j ) must have only waves with positive speed. Let j = k and ρ 0 k ∈ [0, ρα k ]. We observe that ρk ∈ [0, ρα k ] can be connected with ρ 0 k by a classical waves.

For ρk ∈ ]ρ α k , τ k (ρ α k )[ ∪ {ρ α k } the RS k (ρ k , ρ 0 k
) consists of a possibly null shock (ρ k , ρα k ), followed by a non-classical shock (ρ α k , ρα k ) and a shock (ρ α k , ρ 0 k ). Notice that ρk ∈ [τ (ρ α k ), ρα k [ cannot be joined with ρα k by a wave with positive speed. The case j = k and ρ 0 k ∈ ]ρ α k , 1] is analogous to the situation when j = k and ρ 0 j ∈ ]1/2, 1].

Corollary 2

The maximal flow of the outgoing road I j at the junction J is

γ max j (ρ 0 j ) =          f j (1/2) if ρ 0 j ∈ [0, 1/2] and j = k, f j (ρ 0 j ) if ρ 0 j ∈ ]1/2, 1] and j = k, f k (ρ α k ) if ρ 0 k ∈ [0, ρα k ], f k (ρ 0 k ) if ρ 0 k ∈ ]ρ α k , 1].
Additionally, there exists a unique ρj ∈ [0, 1] such that the admissible solution of the Riemann problem with initial datum (ρ j , ρ 0 j ) consists of waves with only positive speed and the condition f j (ρ j ) = γ max j (ρ 0 j ) holds.

For each junction we consider a traffic distribution matrix, i.e. a matrix representing the distribution of cars among the roads.

Definition 3.3 A distribution matrix A m×n for the junction J ∈ J is given by

A m×n =    α n+1,1 • • • α n+1,n . . . . . . . . . α n+m,1 • • • α n+m,n    ,
where α j,i ≥ 0 for every i, j and n+m j=n+1 α j,i = 1 for every i. A distribution matrix A m×n gives the percentage of cars from each incoming road I i choosing the outgoing road I j . In other words, if C is the amount of cars coming from road I i , then Cα j,i is the amount of cars moving towards road I j from I i .

The construction of the admissible solution at the junction J corresponding to the initial datum (ρ 0 1 , . . . , ρ 0 n+m ) ∈ [0, 1] n+m is the following:

1. Fix a distribution matrix A m×n by choosing m × n non-negative constants α j,i such that n+m j=n+1 α j,i = 1 for every i ∈ I. 2. Define the closed, convex and non-empty sets of admissible fluxes

Ω = {(γ 1 , . . . , γ n ) ∈ Ω 1 ×. . .×Ω n : A•(γ n+1 , . . . γ n+m ) T ∈ Ω n+1 ×. . .×Ω n+m },
where Ω i (ρ 0 i ) = [0, γ max i (ρ 0 i )] and Ω j (ρ 0 j ) = [0, γ max j (ρ 0 j )] are respectively defined in propositions 1 and 2, see also corollaries 1 and 2.

Then by Corollary 1 there exists unique ρi ∈ [0, 1] such that f i (ρ i ) = γi . Then by Corollary 2 there exists unique ρj ∈ [0, 1] such that f j (ρ j ) = γj .

5. Finally, let RS J α k (ρ 0 1 , . . . , ρ 0 n+m ) = (ρ 1 , . . . , ρn+m ).

Remark 1

The maximization problem (4) may admit more than one solution. Additional assumptions are in general required to get uniqueness of the Riemann solver. This can be obtained either imposing further conditions on the distribution matrix A, see [13, Section 5.1], or introducing a priority vector as in [START_REF] Delle Monache | Priority-based Riemann solver for traffic flow on networks[END_REF].

A case study

We consider a junction with two incoming (n = 2) and two outgoing (m = 2) roads. Let V h = 4, namely f h (ρ) = 4ρ(1 -ρ), h ∈ {1, . . . , 4}. Fix constant initial density (ρ 0 1 , . . . , ρ 0 4 ) ∈ [0, 1] 4 , see Figure 2, center, such that

0 < ρ 0 1 < 1/2, 1/2 < ρ 0 2 < 1, 1/2 < ρ 0 3 < 1, 1/2 < ρ 0 4 < 1, f (ρ 0 1 ) = 1/2, f (ρ 0 2 ) = 2/5, f (ρ 0 3 ) = 7/10, f (ρ 0 4 ) = 1/2,
The parameter α 3 is suitably chosen to obtain f (ρ α3 ) = 7/20 and we take the distribution matrix We consider two cases: the case a slow vehicle with maximal velocity V b = 1/6 enters road I 3 and the case there is no slow vehicle at the junction. According to propositions 1 and 2, the sets of admissible fluxes at the junction are

A = 1/2 1/3 1/2 2/3 .
Ω 1 = Ω 4 = [0, 1/2], Ω 2 = [0, 1], Ω b 3 = [0, 7/20], Ω 3 = [0, 7/10],
where Ω b 3 and Ω 3 are the sets of admissible fluxes on I 3 in case the bus is present or not, respectively. In the case without the bus we let

Ω = {(γ 1 , γ 2 ) ∈ Ω 1 × Ω 2 : A • (γ 1 , γ 2 ) T ∈ Ω 3 × Ω 4 },
and find that the maximal admissible flow through junction max (γ1,γ2)∈Ω (γ 1 + γ 2 ) is reached at the point Q = (1/2, 3/8), see Figure 2, left, hence the solution for the fluxes of this problem is (γ 1 , . . . , γ4 ) = (1/2, 3/8, 3/8, 1/2). In the case with the bus, we let The solution of the Riemann problem at the junction is completely determined.

Ω b = {(γ 1 , γ 2 ) ∈ Ω 1 × Ω 2 : A • (γ 1 , γ 2 ) T ∈ Ω b 3 × Ω 4 },
For better understanding the solution behavior, we display in Figure 3 the two solutions at time t = 1/5. The blue line describes the density profile without the bus, while the red line represents the solution in the presence of the bus. We notice that a shock wave arises on road I 1 , on road I 2 we observe a rarefaction wave instead of a shock wave, on road I 3 the undercompressive shock is visible in the situation with the bus. Only the solution on road I 4 is the same in both cases.

Road I1 Road I2

Road I3 Road I4

Figure 1 :

 1 Figure 1: Fundamental diagram with constraint. Left: Fixed reference frame. Right: Bus reference frame.

3 .

 3 Compute a vector (γ 1 , . . . , γn ) ∈ Ω such that

4 .

 4 Compute the vector (γ n+1 , . . . , γn+m ) such that γj = n i=1 α j,i γi .

Figure 2 :

 2 Figure 2: Left: the set Ω. Center: the fundamental diagram with initial datum. Right: the set Ω b .

  and find that the maximal admissible flow through the junction max (γ1,γ2)∈ Ωb (γ 1 + γ 2 ) is reached at the point Q b = (2/5, 9/20), see Figure2, right, therefore1/2 < ρ1 < 1, 1/2 < ρ2 < 1, ρ3 = ρα3 , ρ4 = ρ 4,0 , f (ρ 1 ) = 2/5, f (ρ 2 ) = 9/20, f (ρ 3 ) = 7/20, f (ρ 4 ) = 1/2.

Figure 3 :

 3 Figure 3: Blue line indicates the situation without bus and red with the bus.

Acknowledgement ND acknowledges the support of the French Government Scholarship (BGF) program for joint PhD thesis of the French Embassy in Poland. MDR acknowledges the support of the National Science Centre, Poland, Project "Mathematics of multi-scale approaches in life and social sciences" No. 2017/25/B/ST1/00051


est destinée au dépôt

Fiber

  optic refractive index sensors through spectral detection of Rayleigh backscattering in a chemically etched MgO-based nanoparticle-doped fiber Marzhan Sypabekova, Sanzhar Korganbayev, Wilfried Blanc, Takhmina Ayupova, Aliya Bekmurzayeva, Madina Shaimerdenova, Kanat Dukenbayev, Carlo Molardi, Daniele Tosi To cite this version: Marzhan Sypabekova, Sanzhar Korganbayev, Wilfried Blanc, Takhmina Ayupova, Aliya Bekmurzayeva, et al.. Fiber optic refractive index sensors through spectral detection of Rayleigh backscattering in a chemically etched MgO-based nanoparticle-doped fiber. Optics Letters, 2018, 43 (24), pp.5945. hal-01985843