Zhongkai Zhang
email: zhongkai.zhang@inria.fr.2

Jeremie Dequidt
email: jeremie.dequidt@inria.fr.2

Junghwan Back
email: junghwan.back@kcl.ac.uk.

Hongbin Liu
email: hongbin.liu@kcl.ac.uk.

Christian Duriez
email: chris-tian.duriez@inria.fr.2

Motion Control of Cable-Driven Continuum Catheter Robot through Contacts

Catheter-based intervention plays an important role in minimally invasive surgery. For the closed-loop control of catheter robot through contacts, the loss of contact sensing along the entire catheter might result in task failure. To deal with this problem, we propose a decoupled motion control strategy which allows to control insertion and bending independently. We model the catheter robot and the contacts using the Finite Element Method. Then, we combine the simulated system and the real system for the closed-loop motion control. The control inputs are computed by solving a quadratic programming (QP) problem with a linear complementarity problem (LCP). A simplified method is proposed to solve this optimization problem by converting it into a standard QP problem. Using the proposed strategy, not only the control inputs but also the contact forces along the entire catheter can be computed without using force sensors. Finally, we validate the proposed methods using both simulation and experiments on a cabledriven continuum catheter robot for the real-time motion control through contacts.

I. INTRODUCTION

Robotic catheter systems eliminate radiation exposure to surgeons by allowing to be operated remotely. In the past decade, both manual and automatic steerable systems have been designed to increase the safety and the procedural effectiveness. However, the precise steering using a more accurate, safer, and more reliable approach still remains to be one of the main challenges for catheter insertion.

Unlike rigid robots, catheter robots are soft with infinite degrees of freedom so that the accurate modeling of catheter robots is difficult, especially in a constrained environment. In the literature, several methods have been proposed to model catheter robots. One approach employs the pseudorigid-body (PRB) model which consists of a set of rigid links connected with rotation joints [START_REF] Khoshnam | A pseudo-rigid-body 3r model for a steerable ablation catheter[END_REF], [START_REF] Greigarn | Task-space motion planning of mriactuated catheters for catheter ablation of atrial fibrillation[END_REF]. In order to capture more accurate deformation, Cosserat rod theory is used to model the 3-D static deflection of catheters [START_REF] Tang | A stable and real-time nonlinear elastic approach to simulating guidewire and catheter insertions based on cosserat rod[END_REF], [START_REF] Kratchman | Guiding elastic rods with a robot-manipulated magnet for medical applications[END_REF]. Another method is proposed to model the catheter with the assumption of constant curvature [START_REF] Ganji | Robot-assisted catheter manipulation for intracardiac navigation[END_REF]. Catheters can also be modeled as multiple beams and simulated using Finite Element Method (FEM) [START_REF] Duriez | New approaches to catheter navigation for interventional radiology simulation[END_REF]. FEM is an effective strategy to simulate the 3D deformation of structures with external forces. Using the realtime FEM technology, computational cost is dramatically reduced (The model can be updated at 30 to 50 Hz.) for the interactive simulation.

Catheter robots can be controlled using a model based on the assumption of constant curvature [START_REF] Bailly | Modeling and control of a continuum style microrobot for endovascular surgery[END_REF]. Most position controllers are designed based on the inverse kinematic model so that the computation of inverse Jacobian matrix is indispensable. Jacobian matrix can be computed based on either the analytical model [START_REF] Bailly | Modeling and control of a continuum style microrobot for endovascular surgery[END_REF], [START_REF] Loschak | Algorithms for automatically pointing ultrasound imaging catheters[END_REF] or the numerical model [START_REF] Zhang | Kinematic modeling and observer based control of soft robot using real-time finite element method[END_REF], [START_REF] Zhang | Visual servoing control of soft robots based on finite element model[END_REF]. These methods are proposed for the position control of catheters or soft robots without considering the interaction with the environment. The loss of contact information might result in imprecise steering and even the failure of the task because of the unsensed Jacobian matrix rotation [START_REF] Yip | Model-less feedback control of continuum manipulators in constrained environments[END_REF]. The position of catheter tip can also be controlled by the modelless [START_REF] Yip | Autonomous control of continuum robot manipulators for complex cardiac ablation tasks[END_REF] (Jacobian matrix is estimated empirically using the input and output data) or the model-free [START_REF] Back | Model-free position control for cardiac ablation catheter steering using electromagnetic position tracking and tension feedback[END_REF] control strategies. Although they are robust to the environment, contact forces along the entire robot are difficult to be captured.

The available force feedback during catheter insertion is an important strategy to avoid damaging the blood vessel by applying excessive force. The contact forces that the catheter tip applies on the blood vessel can be measured by an embedded force sensor [START_REF] Polygerinos | Mri-compatible intensity-modulated force sensor for cardiac catheterization procedures[END_REF], [START_REF] Do | Design and control of a mechatronic tracheostomy tube for automated tracheal suctioning[END_REF] or by an indirect force measurements [START_REF] Wei | Modeling, force sensing, and control of flexible cannulas for microstent delivery[END_REF], [START_REF] Khoshnam | Modeling and estimation of tip contact force for steerable ablation catheters[END_REF], [START_REF] Back | Three dimensional force estimation for steerable catheters through bi-point tracking[END_REF] (like the image-based force sensing). However, none of the available strategies is capable of providing the catheter-tissue interaction forces along the entire catheter. There is, therefore, a need to report sufficient information of contact forces to the surgeon by visual or haptic feedback to increase the safety of catheter insertion. Given the known environment and catheter model, the location of contact forces is detected by collision detection algorithm and the intensity along the entire catheter can be computed by the FEM [START_REF] Duriez | New approaches to catheter navigation for interventional radiology simulation[END_REF]. This strategy is widely used for medical simulation where the contact is modeled based on the Signorini's law [START_REF] Duriez | Signorini's contact model for deformable objects in haptic simulations[END_REF], [START_REF] Courtecuisse | Real-time simulation of contact and cutting of heterogeneous soft-tissues[END_REF].

To achieve the closed-loop control, the catheter robot and the contacts are simulated in a real-time simulator and are combined with the real system. The simulated system provides the Jacobian matrix and detects the collision between the robot and its environment. The contact is usually modeled as a linear complementarity problem (LCP). Then, the motion control with contacts can be converted as a quadratic programming (QP) problem with a LCP. Treating the contact forces as optimization variables is employed for both the rigid robot [START_REF] Posa | A direct method for trajectory optimization of rigid bodies through contact[END_REF] and the soft manipulator [START_REF] Coevoet | Optimization-based inverse model of soft robots with contact handling[END_REF]. The computation of the QP problem with LCP is difficult and it is necessary to explore a simpler strategy to solve this optimization problem.

This work aims to provide a possible strategy to increase the safety of catheter robots by independent control of insertion and bending as well as visualization of contracts forces along the catheter. Three main contributions are shown in this paper: (1) A closed-loop decoupled control strategy is proposed for catheter robot. It allows the surgeon to remain in direct control of the catheter insertion; (2) The contact forces are computed along the entire robot by solving a QP problem with LCP; (3) We proposed a simplified strategy to solve the problem of QP with LCP where the LCP holds strictly for a reachable trajectory.

This paper is organized as follows. Section II introduces the approach of quasi-static modeling of catheter robots with contacts based on real-time FEM. The problem statement and the overview of the proposed strategy is shown in Section III. The closed-loop decoupled control is proposed in Section IV where the bending and insertion are decoupled. The simulation-based validation and experiments are presented in Section V. Finally, conclusion and future work are shown in Section VI.

II. QUASI-STATIC MODELING OF CATHETER ROBOTS WITH

CONTACTS BASED ON REAL-TIME FEM The quasi-static equilibrium equation of catheter robots is deduced based on the real-time FEM without considering the dynamic feature (like the inertia force).

f ext + F (x) + H T a λ a + H T c λ c = 0 (1)
where x ∈ R n (n is the total degrees of freedom of all nodes) is the position vector of all nodes and is updated according to the configuration of the catheter robots at each sampling step. λ a ∈ R a (a is the number of actuators) and λ c ∈ R c (c is the number of contact points) are the constraint force of actuators and contacts, respectively. The directions of the forces on the nodes exerted by actuators and contacts are provided by H T a ∈ R n×a and H T c ∈ R n×c . Therefore, H T a λ a and H T c λ c represent respectively the contributions of the actuators and the contacts to the position of FEM nodes. f ext ∈ R n represents the external loads (like the gravity) and F (x) ∈ R n gathers the internal stiffness forces.

The internal stiffness forces F (x) can be expressed by its first-order Taylor expansion:

F (x + dx) ≈ F (x) + K (x) dx (2)
where K (x) ∈ R n×n is the tangent stiffness matrix which is highly sparse and depends on the actual position of the nodes x. dx is the displacement vector of FEM nodes between consecutive configurations. Substituting (2) into (1), the linearization of (1) per simulation step can be obtained:

K (x) dx = f ext + F (x + dx) + H T a λ a + H T c λ c (3)
We define δ e ∈ R 3e (e is the number of effectors), δ a ∈ R a and δ c ∈ R c as respectively the positions of effectors (the tip of the catheter robot), the displacement of actuators (the change of cables' length), and the interpenetration gap of contacts.

Based on the Signorini's law, for every contact point, there is a complementarity relation between the interpenetration gap δ c and the contact force λ c among the normal direction. The complementarity relation can be written as [START_REF] Duriez | Signorini's contact model for deformable objects in haptic simulations[END_REF]:

0 ≤ δ c ⊥ λ c ≥ 0 (4)
The complementarity condition (4) can also be formulated as δ c ≥ 0, λ c ≥ 0 and δ T c λ c = 0. The contact positions can be detected using a collision detection algorithm. Then, the contact forces are computed by solving [START_REF] Kratchman | Guiding elastic rods with a robot-manipulated magnet for medical applications[END_REF].

The computation of (3) consists of two steps [START_REF] Coevoet | Software toolkit for modeling, simulation, and control of soft robots[END_REF]: (1) the computation of a free configuration x f ree considering that there is no actuation and no contact; (2) the computation of λ a and λ c by solving an optimization problem.

Based on (3), the optimization in the motion space is timeconsuming because the size of matrix K is usually very large. By projecting the equilibrium equation (3) into the constraint space 1 , the dimension of the matrices can be dramatically reduced so that the control inputs can be computed in realtime [START_REF] Coevoet | Software toolkit for modeling, simulation, and control of soft robots[END_REF]. The projected equations can be written as:

δ e = W e λ + δ f ree e (5) δ a = W a λ + δ f ree a (6) δ c = W c λ + δ f ree c (7)
where W i = W ia (x) W ic (x) with i standing for the subscripts e (effector), a (actuator) and c (contact), respectively. W e ∈ R e×(a+c) , W a ∈ R a×(a+c) and W c ∈ R c×(a+c) are compliance matrices which show the mechanical coupling between the effectors, actuators and contacts. The method to compute the compliance matrices is introduced in [START_REF] Coevoet | Software toolkit for modeling, simulation, and control of soft robots[END_REF]. The external forces applied on the FEM can be written as λ = λ a λ c T ∈ R a+c . The free motion vectors

δ f ree e ∈ R 3e , δ f ree a ∈ R a and δ f ree c ∈ R c are computed by δ f ree i = δ i (x) + H i dx f ree (i = a, b, c) with the assumption of λ = 0.
At the next step, the configurations of both the catheter robot and the contacts are updated by

dx = K -1 H T a H T c • λ + dx f ree (8)
Remark 1. Compliance matrices W e and W c depend on the configuration of catheter robot so that they change at each iteration and can not be pre-computed. If the actuators and contacts act on different FEM nodes or with different directions, the matrices W e has a full row rank (e ≤ (a + c)) or a full column rank (e ≥ (a + c)), and W c has a full row rank.

III. PROBLEM STATEMENT AND OVERVIEW OF PROPOSED CONTROL STRATEGY

In this section, we first introduce the problems for the control design without considering the contacts. Then, we give an overview of the decoupled motion control strategy for a cable-actuated catheter robot through contacts.

A. Problems for control design without considering contact

As shown in Fig. 1 which is first introduced in [START_REF] Yip | Model-less feedback control of continuum manipulators in constrained environments[END_REF], there are two canonical problems (artificial singularity and inverted mapping) for the cable-driven continuum robot if contacts exist on the robot but are not considered for the closed-loop control design. In the worst case, unsensed contacts would result in a positive-feedback loop. Both the artificial singularity and the inverted mapping are generated by the coupling between the insertion and bending. Therefore, to avoid the two canonical problems (shown in Fig. 1), we propose a decoupled control strategy by decoupling the insertion and the bending. Besides, the decoupled implementation of insertion and bending provides a possible solution to increase the safety for catheter insertion and to allows the surgeon to remain in direct control of the catheter insertion.

B. Decoupled closed-loop bending control

The proposed closed-loop strategy (see Fig. 2) needs to combine both the catheter robot and its FE model which is simulated in real-time. The FE model is employed to compute the Jacobian matrix and to detect collisions between the robot and the environment. The same insertion input δ in is sent to both the catheter robot and its FE model.

The bending control strategy is implemented based on the equations in 2D space converted by the projection module. Both the catheter robot and its FE model are controlled to follow the same predefined trajectories (δ d and δ FEM d respectively). For the actuation of FE model, the control inputs and the contact forces are computed by solving a quadratic programming (QP) problem with a linear complementarity constraint. Using the decoupled strategy, the robot controller is designed without taking contacts into consideration, which aims to simplify the control design and to avoid the impacts of imprecise contact model.

IV. DECOUPLED CONTROL

In this section, we introduce the design of the three components (Projection, FEM Controller and Robot Controller) needed for the decoupled control of catheter robots.

A. Projection design

The projection component is used to decouple the insertion and the bending for the catheter robot. As shown in Fig. 3, the movement of catheter robot is modeled in 3D space defined by the coordinate system OXY Z. To achieve the projection, the local coordinate system Pxyz is computed at each step based on the 3D tip position and the predefined trajectory so that it moves with the insertion of catheter robot. After the projection process, the movement of catheter tip in 3D space is converted to the insertion along the tangent axis x and the bending movement in 2D plane defined by the normal axis z and the projection axis y. We project the 3D equations ( 5), ( 6) and ( 7) on a moving 2D plane orthogonal to the tip direction. This projection provides equations for the control of the bending movement 2 . The algorithm for the projection component design is summarized in Algorithm 1. During the bending control design, the desired position of catheter tip is the nearest point P on the predefined trajectory. Then the 3D position error δ 3D e P is projected to a 2D vector in the plan Pyz and its norm is minimized with the bending controller.

Algorithm 1 Projection algorithm Initialization: Define the trajectory for the catheter tip; For each step do 1. Local coordinate system computation Compute the 3D tip position δ 3D e in OXY Z Compute point P and vector Px Compute vector Pz and vector Py 2. Project the 3D position error δ 3D e P to the plan Pyz 3. Project equations ( 5), ( 6) and [START_REF] Bailly | Modeling and control of a continuum style microrobot for endovascular surgery[END_REF] to the plan Pyz end For the implementation of the bending control, both the FEM controller and the Robot controller are introduced in the following subsections.

B. FEM controller design

The object of this section is to design a closed-loop bending controller for the FE model so that the tip of the FE model δ e converges to its desired value δ d . 3 We employ the optimization-based approach to compute the control inputs. The optimization problem can be defined as λ = arg minΓ (λ ) where Γ (λ ) is the objective function:

Γ (λ ) = 1 2 W e λ + δ f ree e -δ d 2 (9)
For the safe interaction with the environment, the contact force is constrained as 0 ≤ λ c ≤ λ c,max . Besides, the constraints of the control inputs should be also considered: λ a,min ≤ λ a ≤ λ a,max where λ a,min and λ a,max are chosen based on the physical limitation.

In order to solve the constrained optimization, the numerical optimization algorithm is necessary to be employed. In order to match the standard quadratic programming (QP) formulation for the usage of software packages [START_REF] Ferreau | qpoases: A parametric active-set algorithm for quadratic programming[END_REF], (9) can be converted to min

λ k 1 2 λ T Q 1 λ + c T 1 λ (10)
s.t. 0 ≤ λ ≤ λ max (11)
and

0 ≤ δ c ⊥ λ c ≥ 0 (12)
where

λ max = λ a,max λ c,max T , Q 1 = W T e W e and c 1 = W T e δ f ree e -δ d .
For the optimization, both the control inputs and the contact forces are treated as optimization variables. The linear complementarity problem (LCP) [START_REF] Yip | Autonomous control of continuum robot manipulators for complex cardiac ablation tasks[END_REF] is employed to capture the contact constraint.

In the following, we show a simplified method to solve the QP problem with LCP.

Finding a solution to the LCP ( 12) is associated with minimizing the quadratic function

f (λ c ) = λ T W T c λ c + δ f ree c T λ c (13)
subject to the constraints W c λ + δ f ree c ≥ 0 and λ c ≥ 0. 3 The variables in this subsection represent the variables for the FE model. However, we use the same variables for the real robot in Section IV.C.

Considering that λ c = 0 c×a I c×c λ , the quadratic function ( 13) can be rewritten as

f (λ ) = λ T Q lcc λ + c T lcc λ (14)
where

Q lcc = 0 (a+c)×a W T c and c lcc = 0 1×a δ f ree c T .
Then, the QP problem with a LCP (10) can be separated into two standard QP problems:

QP1 : min λ 1 2 λ T Q 1 λ + c T 1 λ (15)
s.t. 0 ≤ λ ≤ λ max and QP2 : min

λ 1 2 λ T Q 2 λ + c T 2 λ (16) s.t. W c λ + δ f ree c ≥ 0 and λ ≥ 0
where

Q 2 = Q lcc + Q T lcc (Because λ T Q lcc λ is a scalar, we have λ T Q lcc λ T = λ T Q T lcc λ = λ T Q lcc λ .
) and c 2 = c lcc . The QP problem ( 16) corresponds to the LCP.

Combining both ( 10) and ( 16), the QP problem with the LCP (10)can be converted into a standard QP problem without the LCP: converted QP : min For general problems, the parameter ε should be chosen as a large value to make sure that the LCP holds during the optimization. Unlike former solvers [START_REF] Chen | An active-set method for mathematical programs with linear complementarity constraints[END_REF], [START_REF] Bai | On convex quadratic programs with linear complementarity constraints[END_REF], the proposed strategy has easier background, faster computation and can employ standard QP solvers to solve the problem of QP with the LCP. However, we make the assumption that the defined trajectory is always reachable. In practice, with assumption of static environment, we plan it off-line and make sure that it is always reachable by our robot. For the case with a reachable trajectory, the solution of ( 17) is equal to the solution of [START_REF] Zhang | Visual servoing control of soft robots based on finite element model[END_REF] if ε > 0 (The proof is presented as the appendix.).

λ 1 2 λ T Qλ + c T λ (17)

C. Robot controller design

The quasi-static equations ( 5) and ( 6) can be converted as the discrete-time kinematic equations [START_REF] Zhang | Kinematic modeling and observer based control of soft robot using real-time finite element method[END_REF] (k is the timestep index):

δ e,k+1 = δ e,k + W ea,k λ a,k (18)
δ a,k+1 = δ a,k + W aa,k λ a,k (19)
Combining ( 18) and ( 19), we have

δ e,k+1 = δ e,k + J a,k δ a,k (20)
where J a,k = W ea,k W aa,k -1 is the Jacobian matrix of the catheter robot.

As shown in Section III, it is not necessary to consider the contact information for the the controller design. Therefore, the robot controller is designed for the bending control of catheter robot based on the kinematic equation [START_REF] Courtecuisse | Real-time simulation of contact and cutting of heterogeneous soft-tissues[END_REF]. It allows to control the position of catheter robot by changing the displacements of actuators.

By defining a new control vector U k = J a,k δ a,k , the kinematic model can be rewritten as δ e,k+1 = δ e,k +U k . The tracking error is defined as e k = δ e,kδ d,k . In the task space, the control vector U k can be designed as:

U k = -K P e k -K D (e k -e k-1 ) (21)
where K P and K D are constant parameters for the proportional gain and differential gain, respectively. Then, the Pseudo-inverse based control allocation is employed to obtain a unique solution:

u = J + a,k U k (22)
which is the minimum 2-norm solution of U k = J a,k δ a,k and u = δ a,k is the control input of the robot. Considering that the matrix J a,k has full row rank (There are more actuators than effectors in our experiment), the Pseudo-inverse matrix

J + a,k can be computed by J + a,k = J T a,k J a,k J T a,k -1
. Substituting [START_REF] Posa | A direct method for trajectory optimization of rigid bodies through contact[END_REF] into the [START_REF] Coevoet | Optimization-based inverse model of soft robots with contact handling[END_REF], the control law can be written as:

u = -J + a,k [K P e k + K D (e k -e k-1 )] (23)
Remark 2. Considering the contacts, the kinematic equation is written as δ e = J a λ a + J c λ c where J c is defined as the contact Jacobian matrix and the Jacobian matrix J a only depends on the robot shape. The kinematic equation can also be written as δ e = (J a + ω) λ a = J λ a where ω λ a = J c λ c . In the bending space, J is not inverted by the contacts. Therefore, based on the assumption that we have an accurate J a (accurate FE model of the robot shape), we can design the bending controller [START_REF] Courtecuisse | Real-time simulation of contact and cutting of heterogeneous soft-tissues[END_REF]. If the FE model has a lower accuracy, the condition for a stable loop using [START_REF] Coevoet | Software toolkit for modeling, simulation, and control of soft robots[END_REF] is that the error of J a does not generate an inverted mapping of J a . Then, we can increase the robustness of the system by choosing a suitable control parameters [START_REF] Zhang | Visual servoing control of soft robots based on finite element model[END_REF].

V. EXPERIMENT RESULTS USING DECOUPLED CONTROL

In this section, we first introduce the experimental setup for the validation of the proposed decoupled control strategy. Then, we show the experiment results of trajectory tracking and positioning for a cable-driven continuum catheter robot. Besides, we validate the collision detection and contact forces estimation by experiment using a soft rod. Finally, we test respectively the decoupled control with Jacobian matrix rotation and the trajectory tracking through a soft tube by simulation.

A. Experimental setup

The experimental setup for the position control of a cabledriven catheter robot through contacts is shown in Fig. 4. Three components are needed for the implementation of the experiments: a position tracking system (a), an obstacle to generate the contacts between catheter robot and its environment (b), and a continuum catheter robot (c).

The commercial electromagnetic tracking system is employed (Aurora by NDI company) with the sampling frequency 20Hz and the sub-millimetric accuracy. A rigid tube (diameter 28mm) is fixed with respect to the base of the catheter robot. For the calibration of coordinate systems between the Aurorat and the catheter robot, we choose two corresponding sets (seven points for each set) of keypoints defined for the rigid tube. The positions of the keypoints are measured respectively from the Aurora system and the FE model. Then, the transformation between two sets of points can be solved robustly using the Singular Value Decomposition (SVD) method.

The movement of catheter robot consists of bending and insertion. The bending is controlled by four cables using four step motors and the insertion is actuated using one step motor. The robot is a modified version in [START_REF] Back | Three dimensional force estimation for steerable catheters through bi-point tracking[END_REF]. The structure of the catheter robot (diameter 7mm) is made of polymer material which is MR-compatible and is manufactured using the rapid prototyping technique. Both the robot and the rigid tube are simulated in SOFA framework (a real-time FEM simulation software). The frequency of the simulation can reach to 50Hz in our experiment (100 mesh nodes). The detail strategy for FEM modeling can be seen in [START_REF] Coevoet | Software toolkit for modeling, simulation, and control of soft robots[END_REF].

B. Trajectory Tracking Results

The central line of the tube is defined as the desired trajectory for the robot tip. The tracking performances are illustrated in Fig. 5 where both the desired trajectory and the catheter tip trajectory are projected to the XZ plane and XY plane. In our experiments, the desired trajectory consists of twice bending along the insertion direction. Three different speeds of insertion (For each case, the insertion speed is uniform.) are tested in the experiments. We choose the same control parameters for all cases, K P = 0.2 and K D = 0.002. The tracking errors are defined as the distance between the catheter tip and its desired position. We compute the tracking error based on the positions on the bending plane. As shown in Fig. 6, the average tracking error is around 0.3mm when the desired trajectory is a straight line. However, the higher insertion speed deteriorates the tracking performances when there is a bending along the desired trajectory. The largest error increases to be 2mm, 3.2mm and 4.8mm respectively for the insertion speeds 0.6mm/s, 1.2mm/s and 2.4mm/s. The error increases with the increment of the insertion speed because the lower insertion speed allows longer time for the convergence of the tip position. For the application, the insertion is controlled manually and its speed should be lower when the catheter robot starts to bend.

C. Positioning Results

The positioning results are shown in Fig. 7 which includes two cases with different catheter length (60mm and 166mm). The tracking error converges faster when the length of catheter is shorter. For both cases, the positioning errors converge to a small bound (around 0.2mm). Usually a longer length of catheter robot generates a larger convergence bound if there is no interaction with the environment. However, in our experiments, the contacts limit the deflection and therefore reduce the convergence bound.

D. Validation of contact forces estimation

As shown in Fig. 8, three experiments are designed to validate respectively the collision detection, the accuracy of FE model and the contact force computation. The collision is detected based on the relative positions between the contact and the soft rod. The contact positions are very close for both the FE model and the real system (see (a) and (b) in Fig. 8). The FE model is tested using a 3D tube (see (c) and (d) in Fig. 8). Ten static positions are measured with different insertions and compared between the real system and the FE model. The positions are drawn in Fig. 9 and the average position error is 3.35mm in our experiment. The accuracy of the FE model ensures the success to capture the relation between the insertion and the tip position.

The computation of contact force is validated using the setup (e) and ( f ) in Fig. 8. The length of the soft rod is 210mm and its Young's Modulus is 1.4GPa. Nine groups of forces are measured and computed with different target tip positions and different contact points. The measured forces using a force sensor and the computed forces using the FE method are listed in Tab. I. The accuracy of contact force computation can reach to 97.1%. The contact forces computed based on the FE model can be regarded as the approximation of the contact forces for the catheter robot. However, our contribution here is based on the assumption that the boundary conditions are fully and perfectly known. To simplify the experiment, we validate our method in the situation of single contact.

E. Simulation-based validation

The simulation-based validation allow to focus on the control algorithms themselves and can get rid of the sensor noise, imprecise modeling of robot and contacts. In this subsection, we first validate the decoupled control strategy to deal with the problem of the Jacobian matrix rotation. Then, we show the simulation result when the catheter robot interacts with a soft tube.

1) Control strategy to deal with the Rotation of Jacobian Matrix: As shown in Fig. 10, the catheter robot can be controlled from the initial configuration (a) to the finial configuration (b) with the decoupled control strategy. If the insertion and bending are coupled between each other, the Jacobian matrix is inverted during the movement. However, the decoupled control strategy can successfully avoid the problem of the Jacobian matrix rotation when the robot interacts with the contacts.

2) decoupled control with soft tube: The trajectory tracking can also be achieved when the catheter robot interacts with soft contacts. The decoupled control strategy is em-

VI. CONCLUSION

This work contributes to the position control of catheter robot in the constrained space. With the aim to provide safe insertion for the catheter robot, in this paper, a decoupled controller is designed and allows to control the bending and the insertion independently. The proposed strategy can achieve a stable position control and can avoid the problem of Jacobian matrix rotation. Besides, the decoupled control allows the surgeon to remain in direct control of the catheter insertion. As a benefit of the proposed method, the contact forces along the entire catheter robot can be computed without using force sensors. So, during the insertion, it is possible to provide to the surgeon a feedback on the force intensity applied on the entire catheter. A simplified method to compute quadratic programming problem with a linear complementarity problem is proposed in our work. The proposed strategy is general and the application is not limited for the cable-driven catheter robot. We perform some experiments and some simulations to test our methods. In the following, we address the main limitations of this work, possible solutions and future works.

To estimate the contact forces, we need the assumption that we have an accurate model of the contact and the catheter robot. However, it is very difficult to always fulfill this assumption in complex situation of real applications. When the contact objects are not modeled well, the registration method for deformation structures [START_REF] Peterlík | Fast elastic registration of soft tissues under large deformations[END_REF] could be employed to improve the accuracy of FE model.

Besides, to make the tracking error as small as possible, the insertion speed should be limited in a small value. When the catheter robot starts to bend, the tracking error increases and needs relatively longer time to converge. Instead of limiting the insertion speed in a quite small range, we could use adaptive control parameters for the control law to increase the convergence speed.

In the future, we would also like to explore the hybrid force and position control for the catheter robot and improve the control system by registration of both contacts and catheter robot with their FE models. Finally, the proposed techniques should be validated in vivo.

APPENDIX

It is assumed that F 1 , F 2 , F 3 , and F 4 are the minimum value of objective functions for [START_REF] Zhang | Visual servoing control of soft robots based on finite element model[END_REF], ( 15), ( 16) and [START_REF] Khoshnam | Modeling and estimation of tip contact force for steerable ablation catheters[END_REF], respectively. λ 1 , λ 2 , λ 3 and λ 4 are respectively the solution of QP problems [START_REF] Zhang | Visual servoing control of soft robots based on finite element model[END_REF], ( 15), ( 16) and [START_REF] Khoshnam | Modeling and estimation of tip contact force for steerable ablation catheters[END_REF], respectively. Then, a corollary about the proposed approach is provided as follows:

Corollary: If the desired position is reachable (the solution of ( 9) is zero), the solution of ( 17) is equal to the solution of [START_REF] Zhang | Visual servoing control of soft robots based on finite element model[END_REF].

proof: The proof consistsof two steps:

Step 1: λ 1 is the solution of the QP problem [START_REF] Khoshnam | Modeling and estimation of tip contact force for steerable ablation catheters[END_REF].

For the QP problem [START_REF] Zhang | Visual servoing control of soft robots based on finite element model[END_REF] with the solution λ 1 , we have F 1 = -1 2 δ f ree e -δ d 2 and the LCP (12) holds so that, for QP problem [START_REF] Wei | Modeling, force sensing, and control of flexible cannulas for microstent delivery[END_REF] with λ 3 = λ 1 , we have F 3 = 0. Then, for the QP problem [START_REF] Do | Design and control of a mechatronic tracheostomy tube for automated tracheal suctioning[END_REF] with λ 2 = λ 1 , we have F 2 = -1 2 δ f ree e δ d 2 . Therefore, the objective functions [START_REF] Khoshnam | Modeling and estimation of tip contact force for steerable ablation catheters[END_REF] has the same minimum value F 4 = -1 2 δ f ree e δ d 2 and λ 4 = λ 1 is the solution of the QP problem [START_REF] Khoshnam | Modeling and estimation of tip contact force for steerable ablation catheters[END_REF].

Step 2: λ 4 is the solution of the QP problem [START_REF] Zhang | Visual servoing control of soft robots based on finite element model[END_REF].

Because the objective function [START_REF] Khoshnam | Modeling and estimation of tip contact force for steerable ablation catheters[END_REF] is the sum of ( 15) and ( 16) and have a more strict constraint, we have the conclusion that F 4 ≥ F 2 + F 3 ≥ - 2 with λ 2 = λ 4 , and F 3 = 0 with λ 3 = λ 4 . Therefore, λ 4 is a solution to make [START_REF] Zhang | Visual servoing control of soft robots based on finite element model[END_REF] reach the minimum value and make [START_REF] Yip | Autonomous control of continuum robot manipulators for complex cardiac ablation tasks[END_REF] hold. Then, we can conclude that λ 1 = λ 4 is the solution of the QP problem [START_REF] Zhang | Visual servoing control of soft robots based on finite element model[END_REF].

Fig. 1 .

 1 Fig. 1. Two canonical problems for the closed-loop control design [11]. (a) artificial singularity and (b) inverted mapping result in the deterioration and even the positive-feedback loop.

Fig. 2 .

 2 Fig.2. Framework of decoupled closed-loop bending control for catheter robot through contacts. For the implementation, the FE model of catheter robot and its environment is simulated in real-time. The insertion and bending are decoupled by the projection block which is introduced in Section IV.A. Two independent controllers are employed for the control of catheter robot and its FE model respectively. δ d and δ be are respectively the desired position and the bending control inputs for the catheter robot, while δ FEM d

Fig. 3 .

 3 Fig.3. Projection strategy for the catheter robot. The 3D position of the catheter tip in the global coordinate system OXY Z is projected to be the 2D position in the local coordinate system Pxyz. P is the nearest point on the desired trajectory to the catheter tip δ 3D e . Px is the tangent vector for the trajectory at the point P. Py and Pz are respectively the projection vector and the normal vector.

  s.t. 0 ≤ λ ≤ λ max and W c λ + δ f ree c ≥ 0 where Q = Q 1 +εQ 2 and c = c 1 +εc 2 with ε being the weight of the LCP.

Fig. 4 .

 4 Fig. 4. Experimental setup for the motion control of a catheter robot through contacts. (a) NDI electromagnetic tracking system: Aurora; (b) a rigid tube for the experiment. The white points are the feature points which are used for the calibration; (c) the cable-driven continuum catheter robot which has four step motors to actuate the bending and one step motor to control the insertion.

Fig. 5 .

 5 Fig. 5. Trajectories of catheter tip in XZ plane and in XY plane with different speeds of insertion (0.6mm/s, 1.2mm/s and 2.4mm/s).

Fig. 6 .

 6 Fig. 6. Trajectory errors of catheter tip with different speeds of insertion.

Fig. 7 .

 7 Fig. 7. Convergence of positioning error with different catheter length (60mm and 166mm).

Fig. 8 .

 8 Fig. 8. Experiment setup for the validation of the contact detection ((a) and (b)), the accuracy of simulation ((c) and (d)) and the contact force computation ((e) and ( f )). In (e) and ( f ), A1-A3 and B1-B3 are respectively the contact locations and the desired positions of the tip.

Fig. 9 .

 9 Fig. 9. Comparison of tip positions for the experiment in Fig. 8 ((c) and (d)).

Fig. 10 .

 10 Fig. 10. Simulation-based control validation for catheter robot through contacts. (a) and (b) are respectively the initial configuration and the finial configuration of the catheter robot.

Fig. 11 .

 11 Fig. 11. Simulation-based validation of decoupled control of catheter robot through a soft tube. (a) The initial state of catheter robot and the soft tube; (b) The simulated deformation of the soft tube and the robot.

1 2 δ f ree e -δ d 2 .

 2 If the desired position is reachable, there is solution λ 4 for (17) so that its objective function has the minimum value: F 4 = -

The constraint space is defined by the effectors, actuators and contacts (they are treated as constraint for the FE model). The details are shown in our former work[START_REF] Coevoet | Software toolkit for modeling, simulation, and control of soft robots[END_REF].

To simplify the notation, the equations (5), (6) and[START_REF] Bailly | Modeling and control of a continuum style microrobot for endovascular surgery[END_REF] in the following of this paper are used to capture the motion in 2D space.

This work was supported by the Region Hauts-de-France, the project COMOROS (ERDF funds), and EPSRC UK-grant No: EP/R013977/1.

1.


Y Brechet

J D Embury

Architectured materials: Expanding materials space

Keywords: Expanding materials space Preface, Architectured materials, Viewpoint set

published or not. The documents may come   L'archive ouverte pluridisciplinaire

The historical evolution in the use and development of materials reflects a progression from "materials by chance" to "materials by design". In early civilizations, the strategy with respect to materials was "using the materials available on site" (such as flint obsidian, wood, or even native metals). This is evident in a variety of tools and weapons in archaeological collections. Later, the strategy progressively evolved from optimization of specific classes of materials, first by empiricism, then by science-based approaches. This evolution was the essential contribution of metallurgy and later polymer science. Then came the stage of "hyperchoice of materials", which triggered the development of tools for comparing and selecting materials from different classes which were already optimized in terms of their engineering potential. Finally, and more recently, environmental as well as economical concerns have generated a tendency toward multifunctionality of materials, a decrease of over-dimensioning using parameters such as safety factors, and in general a search for materials capable of fulfilling conflicting needs. This is the most recent step in the evolution: the development of "materials by design strategies". This general evolution from "materials by chance" to "materials by design" is a long term-and durable trend, and comes with an increased importance of modelling, and an increased demand for multifunctionality. The present viewpoint set aims at illustrating one possible innovative route to meet this challenge, the development of "architectured materials" [1,2]. This approach blends, in addition to microstructural design, combinations of materials and the optimization of geometry, to obtain "materials" with unusual combinations of properties Figure 1.

The classical strategies to meet requirements for a given component can be usefully pictured in terms of scales. Metallurgists and polymer scientists have thoroughly explored the possibility of "microstructure by design" (grain size, precipitation, polymer chain design and interchain bonding, state of crystallization). These strategies operate at scales between 1 nm and 10 lm. The ultimate illustration of this strategy is to be found in the rapid recent development of a variety of nanomaterials. The general aim is to obtain, when possible, microstructures which are much smaller that the overall component size, and are homogeneous in their spatial distribution At the opposite end of the scale, engineers in structural mechanics have developed a number of strategies, using the geometry of components, to meet the requirements without greatly changing the materials. Stiffeners in the aeronautical industry are a well-known example, but the same strategies exist for the design of heat exchangers, or even microelectronics circuits. This strategy aims at distributing matter in a purposely heterogeneous and controlled manner at the level of the component. The strategy that this viewpoint set illustrates, "architectured materials", sits at the intermediate scale: developing either geometries or the association of materials, or microstructure gradients, at scales which are comparable to the scale of the component Figure 2. The result of this evolution is that the classical distinc-⇑ Corresponding author. E-mail: yves.brechet@simap.grenoble-inp.fr tion between "materials" and "structures" becomes blurred; the classical strategy of aiming at a homogeneous material is no longer operative, and the classical modelling tools always aiming at "homogenization" need to be adapted. The architectured materials are not totally new (all the surface-hardening techniques may be seen as examples of graded materials, and foams are clearly architectured materials) but the systematic use of a defined strategy to fulfil the conflicting requirements for niche applications is an emerging trend in materials science which needs to be brought to the attention of the general materials community. It is worth noticing (and this point is well illustrated in Ref. [3]) that the combination of microstructure gradients and wellcontrolled architecture is ubiquitous in natural materials and that the development of this strategy has a lot to learn from biomimetics approaches [3].

Architectured materials often require the provision of either multiple length scales in the structure at both the microscopic and mesoscopic level or a controlled spatial distribution of the constituent phases. Thus the develop-ment of architectured materials produces an important paradigm shift in materials engineering because it targets the development of tailored heterogeneous materials and changes as summarized in Figure 3 (a diagram comparing the usual structure-properties-processing triangle with one linking defined engineering objective to tailored structures to new processing routes).

Thus the production of tailored heterogeneous materials requires the development of new processing routes or important changes to conventional routes.

A number of the papers in this viewpoint set illustrate the important changes which can be made by modifying existing processing routes. The work of Lloyd [4] uses the development of the fusion process to illustrate how co-casting of two different alloys can produce valuable changes in the forming by bending operations. Similarly the paper by Ciccoria [5] shows the potential for expanding the conventional gas decarburization reactions to non-isothermal situations combined with masking techniques to develop spatially controlled mixtures of hard and soft phases in a variety of steels. The production of controlled architectures can often be accomplished by the combination of deformation and phase transitions in a given processing route. Some existing processing routes, such as the concept of ausforming in steels [START_REF] Zackey | High Strength Steels[END_REF], can be modified to produce materials with gradients of structure and properties.

However, new processes such as the surface deformation process discussed by Wang et al. [START_REF] Wang | [END_REF] use non-uniform plastic deformation of materials of low stacking fault energy to produce a range of new structures in which exciting combinations of strength and ductility can be achieved.

These types of localized deformation process can be combined with a variety of electrodeposition processes or pre-assembled structures to provide new processing routes.

The paper by Bouaziz [8] outlines a new approach to the development of architectured materials by utilizing not only the intrinsic structure of the material itself but also its geometric form to provide work-hardening capacity and toughness.

Thus in summary we can see that varying the architecture of materials provides an exciting new pathway to linking the overall structure of materials to design needs and that an essential part of this development will be the use of innovative new processing routes linked directly to the control of the spatial variations of structure and properties. A number of examples using combinations of matter and voids, and a variety of porous struc-tures (trusses [9], felts [10], hollow sphere stackings [11], foams [12]) provide structures for a variety of possible multifunctional applications such as electromagnetic damping [13]. It is not the aim of this viewpoint set to be exhaustive (see other examples in Ref. [START_REF] Brechet | Architectured Multifunctional Materials[END_REF]), but we hope to bring to the attention of the community an expanding field in which many new and innovative approaches are developing.

Figure 1 .

 1 Figure 1. Evolution of strategies for materials development.

Figure 2 .

 2 Figure 2. Architectured materials and length scales.


S Terzi

L Salvo

M Sue ´ry

N Limodin

J Adrien

E Maire

Y Pannier

M Bornert

D Bernard

M Felberbaum

M Rappaz

E Boller

In situ X-ray tomography observation of inhomogeneous deformation in semi-solid aluminium alloys

Keywords: In situ X-ray tomography, Aluminium alloys, Semi-solid state, Deformation, Pores

The aim of this paper is to report the very first in situ observations of the deformation behaviour of an Al-Cu alloy in the semisolid state by using ultrafast, high-resolution X-ray microtomography. It is shown that this deformation is non-homogeneous and involves an accumulation of liquid at an intergranular surface nearly perpendicular to the strain axis. Once the liquid is no longer able to feed such a region, micropores form and grow at this surface, finally leading to a crack.

During the solidification of alloys, various regimes of rheological behaviour have been described in the literature [1][2][3]; these involve several transitions, depending primarily on the type of mechanical interactions between the solid grains and on the ability of the liquid to flow in between the grains. In particular, at large solid fractions, continuous liquid films remain at grain boundaries and this situation is critical for the hot tearing phenomenon. Indeed, in this range, the local volume change accompanying solidification cannot be fed by intergranular liquid, and therefore pores nucleate and grow along grain boundaries, leading to the formation of hot tears.

The prediction of hot tearing requires knowledge of, on the one hand, the constitutive behaviour of the alloy while it solidifies [4], and, on the other hand, criteria for its formation [5,6]. Until now, these criteria have suffered from two main limitations: (i) they usually do not consider any localization (of strains or feeding) and (ii) they are based on observations carried out after complete solidification on ''cold" specimens. In addition to conventional methods, X-ray tomography [7,8] has been used recently to characterize hot tears in three dimensions. The only in situ observations of hot tearing formation were performed on transparent analogue alloys [9,10]. Although such alloys can mimic the solidification of metallic alloys (e.g. formation of dendrites), they do not have the same mechanical behaviour and the strain and strain rate were not carefully controlled in these experiments.

Strong X-ray beams now available at several synchrotron facilities open new perspectives for the study of hot tear formation. Recently, in situ observations of the evolution of the microstructure of aluminium alloys were carried out during solidification and partial remelting by using ultrafast X-ray microtomography at ESRF Grenoble [11][12][13].

In the present study, we report the first in situ X-ray tomography observations of tensile deformation in semi-solid specimens. For that purpose, a tensile testing machine was specifically designed to be fitted on the Xray beam line. This high-temperature tensile rig is rather similar to that described in Ref. [14]. The specimen of Al-8 wt.% Cu alloy was a cylinder of 2 mm in diameter and 8 mm in length, the extremities of which were in-serted into alumina tubes and fixed with ceramic glue. In order to localize the deformation of the specimen in the region of interest where X-ray images are recorded, the cross-section of the specimen was locally reduced to about 1.5 mm in diameter (i.e. a notched zone). The specimen was heated by an induction coil and the temperature was controlled and monitored by using a thermocouple 0.5 mm in diameter inserted into the lower alumina tube and in close contact with the specimen. The specimen was heated at 0.5 °Cs À1 up to 555 °C, i.e. above the eutectic temperature (548 °C), and then held isothermally for 3 min at this temperature to reach thermal homogenization. The tensile test was then performed at a displacement speed of 0.1 lms À1 which leads to an average strain rate equal to 2 Â 10 À4 s À1 , assuming that the gauge length of the specimen is equal to 0.5 mm.

During the tensile test, the tensile rig and the specimen were allowed to rotate through 180°in 16 s, while the heating coil itself was fixed. During this rotation, 400 transmission images of the zone of interest of the specimen were recorded. The pixel size was set to 2.8 lm. Such scans of the specimen were recorded every 27 s using the latest version of the Camera FReLoN F2kHD-Atmel with a fast readout of the CCD camera set at 20 Mpixels s À1 [15]. After data acquisition, three-dimensional (3-D) images of the specimen were reconstructed using standard reconstruction tools. In addition to direct post-processing of these images to quantify the evolution of the microstructure, specific volumetric digital image correlation (V-DIC) has been applied to measure 3-D displacement fields in the bulk of the specimen, from which local strain heterogeneities have been derived. A specific procedure has been developed to compute local strains from sparsely distributed displacement measurement points located at solid-liquid interfaces [START_REF] Pannier | Volumetric Digital Image Correlation Adapted to Images with a Spatially Non Uniform Contrast, Euromat[END_REF][START_REF] Pannier | Adapted Volumetric-DIC for Images with Sparse Local Contrast[END_REF].

Figure 1 shows a sequence of four 3-D images of the notched area taken at various times during deformation.

The liquid regions have been outlined in blue. At the onset of deformation, time t 0 , the liquid of near-eutectic composition is fairly homogeneously distributed within the whole specimen. The measured volume fraction of liquid, typically 7-8%, can be compared to the theoretical value predicted by two microsegregation models. At the temperature of 555 °C, the liquid fraction is equal to 9% under equilibrium conditions, whereas the Scheil-Gulliver model gives 17%. The measured value is closer to the equilibrium value, which seems to indicate that the heating plus holding times were sufficient to homogenize the initial copper concentration profiles of the solidified specimen. However, perfect homogenization should not be achieved if one calculates the corresponding Fourier number Fo (Fo < 0.1). Furthermore, the measured liquid fraction is even lower than the equilibrium value. This can be explained by the fact that some of the liquid remains in the form of intergranular films which are too thin to be detected by X-ray tomography.

The next three images of Figure 1 show the same notched region at various times (or levels) of deformation. As deformation proceeds, the fraction of liquid in the deformed zone increases. In other words, since the solid skeleton experiences plastic deformation with $Áv s > 0, liquid flows into the notched region (i.e. $Áv ' < 0) in order to ensure global mass conservation. At the longest time (i.e. about 900 s or 18% average deformation of the specimen) a few pores or cracks have already appeared in the notched area. This is shown in more detail in Figure 2, where a sequence of 2-D images has been extracted from the 3-D reconstructed views of the notched zone for a section parallel to the tensile direction. Before deformation is applied to the specimen, the liquid (white areas) is homogeneously distributed and a few isolated pores are initially present (black areas) (Fig. 2a). After 9 min of deformation (Fig. 2c), the liquid tends to accumulate in the centre of the notched area and a ''liquid path" has formed across the reduced section, nearly perpendicularly to the applied stress. Fluid motion and grain rearrangement/deformation are also revealed by the shrinking of the small isolated pores that were present in Figure 2a andb.

In order to assess more quantitatively the gradual inhomogeneous distribution of liquid in the specimen during deformation, six horizontal slices have been defined (labelled 1-6 and identified in Fig. 2d). These slices are about 125 lm thick and are selected such that slices 4 and 5 are symmetric on each side of the narrowest transverse section of the notch. Similarly, Sections 3 and 6 are also symmetric. In the central part of the notch (slices 4 and 5), the volume fraction of liquid was estimated to increase up to 25% when the local average equivalent strain reached about 10% (Fig. 2c). Since deformation is carried out isothermally, the total liquid fraction in the overall sample is constant under equilibrium conditions. Therefore, the increase in the volume fraction of liquid near the centre of the notch, which is much larger than that given by lever rule or even by the Scheil-Gulliver equation, is made at the expense of adjacent regions which become depleted in liquid. However, this is not clearly revealed by the direct observation of Figure 2. Again, this can be explained by the low resolution of the tomography, which did not allow the detection of the thin liquid films before deformation. This hypothesis is confirmed by the evolution of the average volume change (as determined by V-DIC) of each of the six transverse slices defined in Figure 2d. As shown in Figure 3, the central zone of the specimen (slices 4 and 5), where deformation is the highest, experiences a volumetric expansion. Sections 3 and 6, which are also symmetric about the notch, show differing behaviour at the onset of deformation but exhibit nearly the same expansion after 15 min. Finally, Section 2 shrinks throughout the deformation period, whereas Section 1 initially shrinks but subsequently experiences slight volumetric expansion. This complex behaviour of the semi-solid state is very different from that of a fully solid material strained at high temperature. If the elastic contribution is small or the Poisson coefficient is equal to 0.5, deformation should occur at constant volume (i.e. $Áv s = 0). Because of the local heterogeneities of the microstructure (solid bonds between grains, thickness and connectivity of the liquid films, etc.), this is no longer the case at a solid volume fraction g s % 0.9. In the weakest regions where dendrites (or grains) are pulled apart, liquid can flow in, leading to a volumetric expansion. This is done at the expense of other regions which are drained of their liquid.

After this first stage during which strain of the semisolid specimen is accommodated by liquid flow, solid grains may come into contact in the zones where $Áv s < 0, precluding further volume compaction in such zones where g s % 1. When these drained regions can no longer feed the liquid channel at the centre of the notch, further straining leads to the formation of pores. This can be observed in Figure 2d, 13.5 min after the beginning of straining. A pore located near the surface on the right side of the notch was already present in the specimen (see Fig. 2a-c) and has grown during deformation, but two ''new" larger pores have appeared in the liquid channel in Figure 2d, at the level of slices 3 and 4. Considering that the liquid readily wets its own solid (the solid-liquid interfacial energy c 's of aluminium is about one order of magnitude smaller than the gas-liquid interfacial energy c 'g ), it is easiest to nucleate and grow a pore in the widest liquid channel. As the specimen continues to be deformed, ''new" pores form in the liquid channel at the centre of the notch, while existing ones grow (Fig. 2e andf). This array of pores following the path of what was the main liquid channel now appears as a partially cracked surface, with a few solid bridges remaining across the gap. It therefore exhibits all the features typically observed in hot tears in metallic or organic alloys [9,10]. However, as we are observing a longitudinal section, it is not certain whether ''new" pores really nucleate in the liquid channel or are interconnected to existing ones through a tortuous 3-D path.

Figure 4 sheds some light on the interconnectivity of the pores formed at the level of the notch. This figure shows a 3-D view of the specimen cross-section after 22.5 min of deformation. Clearly, the pores seen in Figure 2 (and which are light grey in Figure 4) are in fact interconnected in three dimensions. They form along a surface (appearing as a channel in two dimensions) nearly perpendicular to the tensile axis and where liquid accumulates during the early stage of deformation.

In situ X-ray tomography observations, performed under isothermal conditions, have clearly demonstrated that deformation of a semi-solid material is highly inhomogeneous and the solid and liquid phases reorganize themselves throughout. In particular, the remaining liquid initially accumulates in a region where strains are highest and the solid skeleton is undergoing expansion, i.e. at an intergranular surface nearly perpendicular to the tensile axis. There is a global mass conservation so that regions undergoing expansion are fed by other regions where the solid is becoming compacted. Once these last regions are drained of their liquid, pores form in the region where nucleation and propagation is the easiest, i.e. in the intergranular region where the liquid layer is thickest. These pores, which appear disconnected in a 2-D metallographic section, are in fact interconnected and propagate as a void across the liquid surface. These observations clearly demonstrate the power of high-resolution X-ray tomography for the in situ study of hot tear formation in semi-solid alloys, and open new routes of investigations.

Figure 1 .

 1 Figure 1. Sequence of 3-D images of the notched zone of the specimen after various deformation times. The figure shows the variation of the liquid volume fraction in this zone (blue regions). (For interpretation of colour mentioned in this figure, the reader is referred to the web version of this article.)

Figure 2 .

 2 Figure 2. Sequence of 2-D images obtained from the 3-D reconstructed views of the notched area of the specimen. The tensile axis is vertical and the time after deformation is applied at time t 0 is indicated in each figure. In order to evaluate local strains at various positions along the sample, six horizontal slices (average vertical height of about 125 lm) have been defined and are indicated in (d).

Figure 3 .

 3 Figure 3. Evolution of the measured dilatation of the specimen in various transverse sections labelled 1-6 and identified in Figure 2d.

Figure 4 .

 4 Figure 4. 3-D image seen from the top of the pores (in light grey) formed after 22.5 min of deformation.

This work was carried out in the framework of the project ANR-05-BLAN-0286-01 ''TOMOSOLI-DAL" supported by the Agence Nationale de la Recherche, which is gratefully acknowledged. The authors thank all staff members of the ID19 beam line of ESRF Grenoble for their technical support.


Sylvie Crasquin

Marie-Béatrice Forel

Yuan Aihua

Galina Nestell

Merlynd Nestell

Species of Hollinella (Palaeocopida, Ostracoda, Crustacea) as stratigraphic indices of the Late Permian -Early Triassic post extinction interval

Keywords: Permian -Triassic extinction event, ostracods, Palaeocopida, biostratigraphic index

Members of the Ostracoda Order Palaeocopida, except for two very rare genera (Puncia and Manawa), disappeared in the stratigraphic record close to the Permian -Triassic boundary (PTB) event. Species of the genus Hollinella are often present just after the end Palaeozoic mass extinction event, in the latest Permian and earliest Triassic beds. They are one of the latest representatives of the Palaeocopida, the typical Palaeozoic straight dorsal border ostracods. The Early Triassic species are assigned by most authors to Hollinella tingi (Patte, 1935), but this assignment is incorrect. This species is present in the latest Permian and the earliest Triassic, and therefore is considered as a biostratigraphic index fossil of the postmass extinction interval. A revision of the Hollinella species from this interval is presented herein. Three Hollinella species occur in the earliest Triassic and are assigned to Hollinella panxiensis Wang 1978, Hollinella magninoda Wang 1978 and Hollinella (Hollinella) lungcamensis sp. nov. This new species of the post-PTB event and present in strata exposed in the northern part of Vietnam is described. These three species cross the PTB and H. (H.) lungcamensis is characteristic for the post extinction period (latest Changhsingian -earliest Induan).

Introduction

Specimens of the genus Hollinella Coryell, 1928 (Palaeocopida, Ostracoda, Crustacea) are present in many sections of the Permian -Triassic boundary interval. Although the preservation is quite always very poor, these occurrences are remarkable because this genus is one of the latest representatives of the Palaeocopida, an emblematic Palaeozoic straight dorsal border ostracod group which disappears after the Permian -Triassic boundary event (PTBE). Three extremely rare genera, Promanawa, Puncia and Manawa, assigned to the Palaeocopida (Superfamily Puncioidea; [START_REF] Horne | Taxonomy, Morphology and Biology of Quaternary and living Ostracoda[END_REF] are still extant in the Southern Pacific area (e.g., [START_REF] Hornibrook | A new family of living ostracods with striking resemblances to some Palaeozoic Beyrichiidae[END_REF][START_REF] Hornibrook | The New Zealand ostracode family Punciidae[END_REF]. Some authors, however, consider that there are no living members of the Palaeocopida (see Martin & Davis 2001, p. 29-30). Except these three genera, the last well-dated Palaeocopida are known from the Early Anisian (genus Triassicindivisia; [START_REF] Forel | Lower Triassic ostracods (Crustacea) from Meishan section, Permian-Triassic GSSP (Zhejiang Province, South China)[END_REF][START_REF] Crasquin | Ostracods (Crustacea) through Permian-Triassic events[END_REF]).

Among the residual Early Triassic Palaeocopida [START_REF] Crasquin-Soleau | A new ostracod fauna from Permian -Triassic boundary in Turkey (Taurus, Antalya Nappes)[END_REF], only one species is supposed to cross the PTBE and it is assigned to Hollinella tingi [START_REF] Patte | Fossiles paléozoïques et mésozoïques du Sud Ouest de la Chine[END_REF].

H. tingi was described in 1935 by Patte from the Early Permian of South China, but it is not the species which occurs in the earliest Triassic. All specimens discovered in the earliest Triassic, which are quite often preserved as moulds, were assigned to H. tingi by nonspecialists, and this error was repeated many times. The aim of this paper is to clarify a very confused situation.

A new species, Hollinella (H.) lungcamensis sp. nov., is described from the post-PTBE of Northern Vietnam. [START_REF] Patte | Fossiles paléozoïques et mésozoïques du Sud Ouest de la Chine[END_REF] It is important to note that the genus Hollinella frequently shows significant intraspecific variability. Sexual dimorphism is apparent from different adventral structures (for example, a large and striated velum in females, which is absent or replaced by a small nodule row in males) and the stronger anteroventral convexity of the female carapacecorresponding to the brood pouch. The carapace shape of Hollinella shows also strong ontogenetic variations.

Original description of Hollinella tingi

Such variability is also recorded from many other palaeocopid taxa (for example, [START_REF] Sohn | Growth series of Ostracodes from the Permian of Texas[END_REF] for Kirkbyidae and Miltonellidae; [START_REF] Bless | Ostracodes of the Family Hollinellidae[END_REF] for Hollinellidae; [START_REF] Watabe | Ontogeny of a new species of Paraparchites (Ostracoda) from the Lower Permian Speiser Shale in Kansas[END_REF] for Paraparchitidae).

According to the revision of the Hollinellidae by [START_REF] Bless | Ostracodes of the Family Hollinellidae[END_REF], Hollinella tingi is very poorly known. The species was originally described as Beyrichia tingi in 1935 by Patte from the Early Permian T'ungstzu and Takuhesinch'ang Districts of South China. No holotype was designated by [START_REF] Patte | Fossiles paléozoïques et mésozoïques du Sud Ouest de la Chine[END_REF]. The preservation of this material is poor and its description was based on external moulds only. The original description in French is reproduced in the annexe.

The species was published once more by [START_REF] Hou | Some Lower Permian ostracods from Western Hupeh[END_REF]. Its description was written in Chinese and in English. The English version is not the exact translation of the Chinese one.

Note that in Hou's description and in the translation below, the orientation of the carapace is reversed, and males and females are confused. The English translation made by the third author of the present paper is given in the annex. [START_REF] Hou | Some Lower Permian ostracods from Western Hupeh[END_REF] figured the "holotype" (Hou 1954: pl. 1, fig. 2a). It is not clear if her "holotype" is the original and the true one, or if it is a lectotype. Hou found this species in Early Permian black shale of the Chihsia Formation (Western Hubei, South China).

In 1964, Ishizaki assigned specimens discovered in the Early Pennsylvanian (Late Carboniferous) of Japan to Hollinella tingi. The preservation of the figured specimen (Ishizaki 1964: pl. 1, fig. 1) is so poor that it is impossible to confirm its assignment. The stratigraphic range in the Early Pennsylvanian is therefore questionable.

"Hollinella tingi" and the PTBE

In studies of Permian-Triassic ostracods, specimens of Hollinella are frequently reported from the latest Permian and the earliest Triassic. In most of these papers, the specimens of Hollinella are assigned to H. tingi. As explained in [START_REF] Crasquin-Soleau | A new ostracod fauna from Permian -Triassic boundary in Turkey (Taurus, Antalya Nappes)[END_REF], all these specific assignments are incorrect. The first erroneous determination was made by [START_REF] Wang | Late Permian and Early Triassic ostracods of Western Guizhou and Northeastern Yunnan[END_REF], and thereafter repeated again and again by other authors.

In 1978, Wang figured and described four Hollinella species from the Late Changhsingian and Early Induan of Western Guizhou and North-Eastern Yunnan: H. tingi (Wang 1978: pl. 1, figs 5-7), and three new species including H. magninoda (Wang 1978: pl. 1, fig. 11), H. panxiensis (Wang 1978: pl. 1, figs 1-4, andH. capacilacuna (Wang 1978: pl. 1, figs 8-10). English translations are absent from Wang's paper. We made these translations and give them in the annexe (some passages are unclear in the original descriptions. Our translation is literal and without interpretation). With regard to these "descriptions" and the illustrations [START_REF] Wang | Late Permian and Early Triassic ostracods of Western Guizhou and Northeastern Yunnan[END_REF], we suggest that: Hollinella capacilacuna and H. magninodosa are valid species and are Late Permian in age.

For the present authors, the specimens assigned to H. tingi and specimens assigned to H. panxiensis belong to the same species, H. panxiensis, which has no long posterodorsal spine, but just a small lateral flattening (pinch) of the posterior cardinal angle (Wang 1978: pl. 1, figs 1-4). L3 is elongated antero-posteriorly. When the velum is completely developed, it extends from the anterior cardinal angle to the posterior cardinal angle. Some small nodules could be present on the upper part of L3. The specimen illustrated by [START_REF] Wang | Late Permian and Early Triassic ostracods of Western Guizhou and Northeastern Yunnan[END_REF] on pl. 1, fig. 6 differs slightly by its higher height/length ratio possibly due to ontogeny and could be determined as H. panxiensis?

In 1981, Wei illustrated three specimens from the Induan -Olenekian interval of Sichuan (Wei 1981: pl. 1, figs 1-3), and assigned them to Hollinella tingi. The first two specimens (Wei 1981: pl. 1, figs 1 and 2) are very poorly preserved and the assignment appears dubious. The specimen figured on pl. 1, fig. 3 (ibid.) is not a Hollinella, but a Kloedenellid species.

In 1985, Kozur illustrated a specimen of Hollinella tingi (pl. 13, fig. 3) from the Werfenian (Induan) of the Bükk Mountains (Hungary). It is impossible to determine this corroded and abraded specimen at the specific level. In any case, it is not H. tingi. Furthermore, it is not reasonable to use this specific determination as a stratigraphic index for the Permian -Triassic boundary as suggested by [START_REF] Kozur | Neue Ostracoden -Arten aus dem Oberen Mittelkarbon (Höheres Moskovian), Mittel-und Oberperm des Bükk-gebirges (N-Ungarn)[END_REF].

In 1987, Shi & Chen illustrated specimens assigned to H. tingi (pl. 16, figs 2-5) from the early and middle Changhsingian of the Meishan section but gave no description. The specimen figured on pl. 16, fig. 2, is close to H. panxiensis; it apparently shows cardinal pinches, but the carapace is quite longer. The three other specimens (Shi & Chen 1987: pl. 16, figs 3-5) are poorly preserved and/or broken, but the three lobes are distinct and ornamentation could be observed on L3 (fig. 3). [START_REF] Shi | The Changhsingian ostracodes from Meishan Changxing, Zhejiang[END_REF] studied all Changhsingian ostracods of the Meishan section and did not report the species from the upper part of the section. We consider that these specimens belong to H. panxiensis. Hao (1992aHao ( , b, 1993[START_REF] Hao | The development of the Late Permian -Early Triassic ostracod fauna in Guizhou Province[END_REF][START_REF] Hao | Ostracods from the Upper Permian and Lower Triassic of the Zhenfeng section, South China[END_REF] illustrated specimens of Hollinella, including H. tingi, from the Late Permian and/or the Induan of Guizhou.

In 1992a, Hao presented eleven ostracod species from the Early Triassic, including three species of Hollinella: Hollinella cf. H. plana Jiang 1983 (p. 39, pl. 1, fig. 1) is not a species of Hollinella, but a Kloedenelid ostracod; Hollinella unispinata (a new species characterized by long spines at cardinal angles; p.39, pl. 1, fig. 23) is a synonym with Hollinella magninoda [START_REF] Wang | Late Permian and Early Triassic ostracods of Western Guizhou and Northeastern Yunnan[END_REF]. The specimens assigned to H. tingi (p. 39, pl. 1, figs 5-6) do not belong to that species. The first one has a very different lateral outline and is very badly preserved (pl. 1, fig. 5) and the second one (pl. 1, fig. 6) shows a strong reticulation in front of L1 and seems to have a ventral ridge. We assign it to H. panxiensis based on the presence of the pinch at the cardinal angles. The specimen of "H. tingi" illustrated in Hao (1992b;pl. 1, fig.1) from the Changhsingian of Guizhou could be conspecific with H. panxiensis due to its complete velum extending along the entire free margins. In a paper on the Changhsingian from Guizhou, [START_REF] Hao | Latest Permian Ostracoda from Zunyi, Guizhou[END_REF] assigned two specimens to H. tingi (Hao 1993: pl. 1, figs 1-2). The illustration is of poor quality and there is no description. As can be seen on plate 1 of [START_REF] Hao | Latest Permian Ostracoda from Zunyi, Guizhou[END_REF], the cardinal pinches are indistinct, but the specimens show an elongated antero-posteriorly L3 like the specimens of [START_REF] Wang | Late Permian and Early Triassic ostracods of Western Guizhou and Northeastern Yunnan[END_REF]. Furthermore, the velum begins at the anterior cardinal angle and ends at the posterior cardinal angle. We consider that these specimens could belong to H. panxiensis. In the same paper, Hao described a new species: H. echinata. The three illustrated specimens are quite small (L: 0.81-0.83mm). The description is a summary and very general.

The author just mentioned that the "border" of H. echinata possesses many short spines. In his publications from 1994 and 1996, Hao illustrated ostracods from the Permian -Triassic interval in Guizhou Province, but pictures of Hollinella are identical to those in the publications from 1992a and 1992b. [START_REF] Forel | Lower Triassic ostracods (Crustacea) from Meishan section, Permian-Triassic GSSP (Zhejiang Province, South China)[END_REF] indet. All the material is poorly preserved and does not allow a more precise determination.

Two tables are proposed in the annexe to summarize the distributions of all the listed species and the systematic attributions considered valid in this paper.

Hollinella species from the Permian -Triassic boundary in Vietnam

New material has been discovered in the latest Permian -Early Triassic boundary interval of the Hong Ngai Formation at the Lung Cam section, Northern Vietnam. In the Late Permian-Early Triassic this area belonged to the southern margin of the South China Craton and was a shallow marine carbonate platform [START_REF] Son | The Permian-Triassic boundary sections in northern Vietnam (Nhi Tao and Lung Cam sections): Carbon-isotope excursion and elemental variations indicate major anoxic event[END_REF]). The Lung Cam section was recently resampled and analysed in detail [START_REF] Nestell | High influx of carbon in walls of agglutinated foraminifers during the Permian-Triassic transition in global oceans[END_REF][START_REF] Wardlaw | Conodont biostratigraphy of the Permian -Triassic boundary sequence at Lung Cam, Vietnam[END_REF]. More than 200 specimens of Hollinella, associated with numerous other ostracod species (work in progress) were found in residues of 12 conodont samples (from VN 15 up to VN28) located above the extinction event and around the PTB (fig. 1).

This rich material is quite well preserved and permits a detailed study of a new species of Hollinella which is characteristic of the post-extinction event. The assemblage shows a high morphological diversity including all the transitional forms between the extremes (Figs 2 and5). Herein, 201 specimens were measured (H and L include here the adventral structure; fig.

3). The length (L) ranges between 426 and 1130µm and the height (H) between 282 and 807µm. Specimens of all this size range evenly occur in all the 12 samples.

Sexual dimorphism and ontogenetic variability in the Palaeocopida

Sexual dimorphism and ontogenetic variability in palaeocopids [START_REF] Van Veen | The identity of the genera Poloniella and Kloedenella[END_REF][START_REF] Jaanusson | Middle Ordovician ostracodes of Central and Southern Sweden[END_REF]Henningsmoen 1965;[START_REF] Kesling | Copulatory adaptations in ostracods. Part III. Adaptations in some extinct ostracods[END_REF]) and particularly in the Hollinellidae (see example on fig. 6) [START_REF] Jaanusson | Middle Ordovician ostracodes of Central and Southern Sweden[END_REF]Henningsmoen 1965;Bless & Jordan 1970[START_REF] Bless | Ostracodes of the Family Hollinellidae[END_REF] has been known for a long time.

The shape of the carapace changes during each moult. Some features develop gradually during the ontogeny whereas others appear suddenly at the transition to the adult stage. Five to six stages of growth, including the adult, are recorded for Hollinella. In the genus Hollinella the ventral lobe is further developed than L2 and L3 during early ontogeny, whereas L2 and L3 become more prominent in the latest juvenile stages. Early instars are often completely smooth, whereas latest ones become granulose. In the Carboniferous species, which show a tubulous layer during maturity, the last immature stage may possess a completely developed velum. Antero-and/or posterodorsal spines occur frequently in juvenile stages of Hollinella. They may or may not persist until maturity [START_REF] Bless | Ostracodes of the Family Hollinellidae[END_REF].

Sexual dimorphism usually appears in the adult stage, but some pre-adult dimorphism has been described from the last or two last juvenile stages [START_REF] Bless | On a case of dimorphism in the last juvenile stage of a hollinid ostracode from the Namurian B-C of the La Camocha Mine (Gijon, N. Spain)[END_REF][START_REF] Bless | Ostracodes from the Aegir Marine Band (basis Westphalian C) of South Limburg (The Netherlands)[END_REF]Forel et al. 2015 and references included). Such pre-adult dimorphism applies to the lobation, and the size and shape of the adventral structures. In the "true dimorphic" genera, for example of the Hollinidae family [START_REF] Bless | Classification of Palaeocopid ostracodes belonging to the families Ctenoloculinidae, Hollinidae and Hollinellidae[END_REF][START_REF] Bless | Ostracodes of the Family Hollinellidae[END_REF], the tecnomorph specimens keep the same morphology during ontogeny and the heteromorphs develop different adventral structures.

In the "trimorphic" genera [START_REF] Bless | Ostracodes of the Family Hollinellidae[END_REF], juveniles are clearly morphologically different due to the partial absence of secretion of calcite prims in juveniles, so that the velum is not formed. According to [START_REF] Bless | Ostracodes of the Family Hollinellidae[END_REF], presumed heteromorphs differ from presumed adult tecnomorphs by the following characters: in heteromorphs L3 is larger than the domicilium (part of the carapace exclusive of projecting velate structure) and the velum width is smaller than the domicilium. The space between the right and left velums, the antrum, can be ornamented in heteromorphs and smooth in tecnomorphs.

Herein, the ontogenetic variability is expressed by the progressive calcification of the spines forming a continuous velum with all the transitional forms in between during the ontogeny. Sexual dimorphism is sometimes observable at some of the largest specimens.

Some males show a greater height and some females are more elongated with the maximum of height in the posterior part. But it is difficult to assign gender for a great majority of the adult specimens. Sexual dimorphism appears during ontogeny before the adult stage (and sexual maturity), as observed in material from North Iran [START_REF] Forel | Precocious sexual dimorphism and Lilliput effect of ostracods (Crustacea) through the Permian -Triassic events in Neo-Tethys[END_REF]. This dimorphism is characterized by the larger PB radius of curvature and the greater posterior carapace thickness at the females.

Dimensions. L = 426 -1130µm; H = 282 -807µm; W = 419-488µm (fig. 3).

Remarks. Hollinella (Hollinella) lungcamensis sp. nov. is very similar to H. ulrichi [START_REF] Knight | Some Pennsylvanian Ostracodes from the Henrietta Formation of Eastern Missouri (part 1)[END_REF] from the Late Carboniferous -Early Permian of the South-Central United States, Wales, the Netherlands, Belgium and Germany (see complete synthesis on H. ulrichi [START_REF] Knight | Some Pennsylvanian Ostracodes from the Henrietta Formation of Eastern Missouri (part 1)[END_REF] The new species is attributed to the subgenus Hollinella (Hollinella) in regard of the clearly identified L2, the velum which ended at the lower part of BP and the presence of a row of ventral spines in larval stages (see fig. 6).

Hollinella from the South China Block

Hollinella tingi has also been reported, but not illustrated, from marginal marine environments of the so-called Permian-Triassic Transition Beds and the overlying Lower Triassic Kayitou Formation of Yunnan (Mide section) and Guizhou (Tucheng section), South China (e.g., [START_REF] Yu | Annalepis, a pioneering lycopsid genus in the recovery of the Triassic land flora in South China[END_REF]. Recent fieldwork enabled the second author to collect abundant Hollinella specimens from the lowermost Kayitou Formation of the Mide section (preliminary analysis in [START_REF] Bercovici | Terrestrial paleoenvironment characterization across the Permian-Triassic boundary in South China[END_REF]. These specimens are mostly preserved as moulds on the surface of cracked siltstone/fine sandstone. Abundant specimens occur with the two valves still attached dorsally in a butterfly shape indicating very quiet hydrodynamic conditions at the time of deposition and rapid burial. Although extremely abundant, these Hollinella specimens are poorly preserved and corroded so that the original surface structure is usually not observable.

Despite poor preservation, all observed specimens could correspond to Hollinella panxiensis described above. The size range of all specimens of this assemblage suggests that several ontogenetic stages are present. The individual stages can, however, hardly be differentiated because specimens cannot be extracted from the matrix and are often only partially exposed.

Sexual dimorphism is clearly visible in the Hollinella specimens from Mide.

No ostracods have been found so far in the underlying Late Permian Xuanwei Formation in the Mide section. The monospecific assemblage of Hollinella panxiensis from this very marginal marine succession seems to result from a short proliferation event during unstable conditions.

Conclusion

Hollinella tingi [START_REF] Patte | Fossiles paléozoïques et mésozoïques du Sud Ouest de la Chine[END_REF] does not occur in the latest Permian -Early Triassic. Three

Hollinella species are recognized in the latest Permian -earliest Triassic and assigned to Hollinella panxiensis Wang, 1978, Hollinella magninoda Wang, 1978 andHollinella (H.) lungcamensis sp. nov. [START_REF] Nestell | High influx of carbon in walls of agglutinated foraminifers during the Permian-Triassic transition in global oceans[END_REF]. The detailed geochemical analysis of foraminiferal tests and pyrite clusters led [START_REF] Nestell | High influx of carbon in walls of agglutinated foraminifers during the Permian-Triassic transition in global oceans[END_REF] to evoke euxinic conditions at the seabed and a carbon-enriched sea-water column. The source of the carbon could have been global coal fly ash or forest fire-dispersed charcoal, or a combination of both, same samples, however, indicates that the bottom water could not be anoxic nor dysoxic. The analysis of the total ostracod assemblages will improve these ecological data.

Legend of figures

  Hollinella (H.) lungcamensis sp. nov. (see description below) shows different morphotypes. A first group has free margins ornamented by several rows of spines (see figs.2, 4, 5) from anterior cardinal angle (ACA) to the posterior cardinal angle (PCA). The second group has a well-developed velum which is completely calcified forming a continuous blade from the ACA to the posteroventral part of the carapace. The posterior part of the carapace is ornamented by spines. All transitional forms between these two extremes are represented. If we plot the size of specimens with spines and specimens with a complete velum on an H/L diagram (fig.4), it becomes clear that this morphological variability is due to ontogenesis.

Figure 1 :

 1 Figure 1: Location map and stratigraphic column of the Lung Cam section in Northern Vietnam with sample numbers (log modified after Nestell et al. 2015).

Figure 2 :

 2 Figure 2: Interpretative drawing of some specimens of Hollinella (H.) lungcamensis sp. nov. A, B, C: adults with well-developed velum as adventral structure; A: female in left lateral view; B: male in left lateral view; C: male in right lateral view; D, E, F: juveniles with spines as adventral structures; D and F in left lateral view and E in right lateral view. Scale bar is 100µm.

Figure 3 :

 3 Figure 3: Height/length diagram of Hollinella (H.) lungcamensis sp. nov. from Vietnam with indication of individual samples.

Figure 4 :

 4 Figure 4: Height/length diagram of Hollinella (H.) lungcamensis sp.nov. with indication of specimens with velum or spines on free margins and indication of identified males and females.

Figure 5 :

 5 Figure 5: Specimens of Hollinella (H.) lungcamensis sp. nov. from the latest Permianearliest Triassic of the Lung Cam section, Northern Vietnam. All the specimens are at the same scale and scale bar is 200µm. When possible, the gender of specimen is indicated. The numbers with the prefix P6M correspond to the collection numbers of Pierre et Marie Curie University (Paris) Micropalaeontology collection.

Figure 6 :

 6 Figure 6: Schematic representation of ontogenetic variations in some genera and subgenera of Hollinellidae (modified after Bless & Jordan 1970).

Figure 7 :

 7 Figure 7: Schematic representation of Hollinella (Hollinella) lungcamensis sp. nov. with descriptive terms and abbreviations used in text. The small arrows indicate the anterior part of the carapace. A. external view of right valve: S1, S2, S3: sulcus from anterior to posterior; L1, L2, L3, L4: lobes from anterior to posterior; L: maximum length of carapace; H: maximum height of carapace; small dots indicate the location of maximum of convexity of AB, PB and VB. B. external view of right valve: AB: anterior border, PB: posterior border; DB: dorsal border; VB: ventral border; ACA: anterior cardinal angle; PCA: posterior cardinal angle. C. Ventral view; W: maximum width of carapace. D. Dorsal view: L1 to L3 and S1 to S3, lobes and sulcus same as A.

  Type level. Sample VN20, Hong Ngai Formation, latest Changhsingian, latest Permian.

	Material. Holotype: P6M3775, right valve (fig. 5C), sample VN20. Paratypes: P6M3776, left
	valve (fig. 5A), sample VN20; P6M3777, left valve (fig. 5E), sample VN25. Additional material:
	201 complete valves and carapaces, many fragments.
	Occurrences. samples VN15, VN18, VN19, VN20, VN21, VN22, VN23, VN24, VN25, VN26,
	VN27, VN28, Lung Cam section (fig. 1), Northern Vietnam, Hong Ngai Formation, latest
	Changhsingian (latest Permian) -Griesbachian (parvus zone), earliest Triassic.
	Description. Species of Hollinella (Hollinella) with long straight DB (68 -80% of total L) and
	clear morphological variations between dimorphs occurring early in ontogenesis.
	ACA very obtuse (130-140°); lobation well developed: L1 quite always visible, anterior part is
	smoothed out, top of L1 located at anterior fourth of L, not passing over hinge line, occasionally
	with spines; S1 quite deep, oriented backwards, base at 27-30% of H and 36-38% of L; L2 always
	distinct, vertical and narrow, axis at 42-45% of L; S2 narrow, vertical, base located high (26-
	Late ontogenetic stages and adults (L>750µm and H>550µm): AB with large radius of
	curvature with maximum of curvature at 51-58% of H; S3 and L4 visible in most specimens. The
	velate structure is a large smooth velum which begins a little below ACA. Some spines could
	be present just at the anterior end of the velum. VB is largely rounded and bordered by the
	velum which decreases and disappears at the junction with PB where it is replaced by spines;
	maximum of curvature of VB is located at 33-38% of L; velum relatively broad comprising 17-
	23% of H; PB with large radius of curvature with maximal convexity located at 25-29% of H, PB

30% of H); L3 bulbous, rounded, large (diameter 37-39% of L), highest part just at hinge line (quite not overpassing), covered with spines which can be long; spines are also distributed on the surface of the valves and on free margins of PB; ventral lobe poorly developed. Dorsal view: diamond-like carapace outline, hinge line straight.

Early ontogenetic stages: ACA very obtuse; maximum of convexity of AB located above mid H; S3 and L4 invisible; the free margins are covered with several rows of isolated spines. When the specimens reach 790-800µm, the space between velar spines is progressively calcified, calcification forming a continuous velum. Carapace outline in dorsal view biconvex with maximum width at mid L. covered by numerous spines; PCA 105-110°; carapace outline in ventral view diamond-like with large antrum space anteriorly, right and left velums distinct at AB, posterior part of AB covered by spines; maximum width located at the posterior third of L.

in

  [START_REF] Bless | Ostracodes of the Family Hollinellidae[END_REF]. The two species have the same ornamentation at the free margin and the same ontogenetic variability and sexual dimorphism. The main difference is the fusion of L1 and L2 and the spines which are more irregularly distributed at the ventral margin of H.

	ulrichi.

H. spinulosa

[START_REF] Demanet | Contribution à l'étude de la microfaune marine du Westphalien de la Campine[END_REF]

from the Late Carboniferous of Southern Limburg, the Netherlands and Campine Basin of Belgium (see synthesis in

[START_REF] Bless | Ostracodes of the Family Hollinellidae[END_REF]

shows less and wider arranged spines on the free margins and a reticulate carapace surface.

H. echinata

[START_REF] Hao | Latest Permian Ostracoda from Zunyi, Guizhou[END_REF]

from the Changhsingian (Late Permian) of Guizhou, South China has the same lobes, the same spines as H. lungcamensis sp. nov., but its velum runs from ACA to PCA and the H/L ratio is lower.

H. panxiensis

[START_REF] Wang | Late Permian and Early Triassic ostracods of Western Guizhou and Northeastern Yunnan[END_REF]

from the Wuchiapingian (Late Permian)-Induan (Early Triassic) of South China has a complete velum extending from ACA to PCA, a distinct ventral lobe and flattened cardinal angles.

H. magninoda

[START_REF] Wang | Late Permian and Early Triassic ostracods of Western Guizhou and Northeastern Yunnan[END_REF]

from the Wuchiapingian -Induan of South China has long spines at ACA and PCA and a broad velum (1/3 of H).

  The first two species have quite a long stratigraphic range extending from the Wuchiapingian to the Griesbachian. H. (H.) lungcamensis, for the time being, is only present in the Hong Ngai Formation in Vietnam, which is of topmost Changhsingian and earliest Induan (H. parvus zone) age. H. (H.) lungcamensis is characteristic for the post-

	extinction period. In the Lung Cam section, H. (H.) lungcamensis is associated with many other
	palaeocopid and podocopid ostracod taxa (work in progress) and with foraminifers of the
	genera Geinitzina, Nodosaria, Globivalvulina, the species Hyperammina deformis,
	Ammodiscus kalhori, and microconchids

Acknowledgments:

The authors thank Mark Williams (University of Leicester, UK), anonymous reviewer and the editors for their constructive comments and corrections on the first version of the manuscript. We thank Justine Jacquot and Charlène Letenneur (CR2P -MNHN) for drawing of figures 6 and 7 and Alexandre Lethiers (CR2P -UPMC) for the production of figure 5. Y.A. is supported by the Chinese projects number B08030 (111 Project), 41572001 and 41430101 (NFSC Project).

Repository

All the specimens are deposited in the collections of Pierre et Marie Curie University, Paris, France, under the prefix P6M. Suborder Beyrichicopina [START_REF] Scott | Suborder Beyrichicopina Scott, n.suborder[END_REF] Superfamily Hollinoidea [START_REF] Swartz | Revision of the Primitiidae and Beyrichidae with new ostracoda from the Lower Devonian of Pennsylvania[END_REF] Family Hollinellidae [START_REF] Bless | Classification of Palaeocopid ostracodes belonging to the families Ctenoloculinidae, Hollinidae and Hollinellidae[END_REF] Genus Hollinella [START_REF] Coryell | Some new Pennsylvanian Ostracoda[END_REF]emend. Kellett, 1929 Subgenus Hollinella (Hollinella) [START_REF] Coryell | Some new Pennsylvanian Ostracoda[END_REF] Hollinella (Hollinella) lungcamensis Crasquin, sp. Type locality. Lung Cam section, Northern Vietnam (23°14'30"N; 105°13'20"E)

Systematics