Jeremy Begey

Loic Cuvillon

Maximilian Lesellier

Marc Gouttefarde

Jacques Gangloff

Loïc Cuvillon

Maximilien Lesellier

Dynamic Control of Parallel Robots Driven

Keywords: Parallel robots, Dynamics, Flexible cables, Singular perturbation, Motion control

published or not. The documents may come

I. INTRODUCTION

The main advantages of Cable-Driven Parallel Robots (CDPRs) include a large workspace, low inertia links, cost-effective scalability and potentially heavy payload capabilities. Their main drawbacks are high flexibility and relatively low positioning accuracy. Advanced dynamic control with exteroceptive feedback can compensate for these drawbacks by improving accuracy and rejecting the effects of flexibility. Some CDPRs use very flexible cables (very long cables or cables using a flexible material like polymer). This paper shows that flexibility can be an advantage from the control point of view since they yield lower frequency oscillations that can be more easily handled by position-controlled actuators.

Dynamic control of a Parallel Robot (PR) in the Cartesian space coordinates is more suitable than in the joint space coordinates as highlighted by Paccot et al. in [START_REF] Paccot | A review on the dynamic control of parallel kinematic machines: Theory and experiments[END_REF]. Computational efficiency, improved task robustness and better accuracy can be expected when using exteroceptive measurements of the end-effector pose. If the exteroceptive sensor is a camera and the task is defined in image space (2D) or Cartesian space (3D), the control law can be designed using the visual servoing framework [START_REF] Weiss | Dynamic sensor-based control of robots with visual feedback[END_REF], [START_REF] Chaumette | Visual servo control. I. basic approaches[END_REF], [START_REF] Espiau | A new approach to visual servoing in robotics[END_REF].

All control strategies reviewed in [START_REF] Paccot | A review on the dynamic control of parallel kinematic machines: Theory and experiments[END_REF] involve partial or complete inversion of the robot nonlinear dynamics. According to Cheng et al. [START_REF] Cheng | Dynamics and control of redundantly actuated parallel manipulators[END_REF], three control strategies may be implemented: Simple PD control in Cartesian space, augmented PD control and computed-torque control. Computed-torque control, also called inverse dynamics control or, in the most general context, feedback linearization, effectively achieves a decoupling and linearization of the manipulator dynamics.

Many of these control strategies suited for rigid-link PRs have been adapted to CDPRs. Compared to rigid-link PRs, a distinctive feature of CDPRs is the unidirectional nature of the forces applied by the cables on the end-effector [START_REF] Oh | Cable suspended planar robots with redundant cables: controllers with positive tensions[END_REF]. Consequently, a tension distribution algorithm must be implemented to ensure positiveness of cable tension in case of actuation redundancy [START_REF] Borgstrom | Rapid computation of optimally safe tension distributions for parallel cable-driven robots[END_REF], [START_REF] Taghirad | An analytic-iterative redundancy resolution scheme for cable-driven redundant parallel manipulators[END_REF], [START_REF] Gouttefarde | A versatile tension distribution algorithm for n -dof parallel robots driven by n + 2 cables[END_REF].

Several CDPR control studies are based on the assumption that cables are rigid. Some of them are controlling the CDPR in the joint space [START_REF] Kawamura | High-speed manipulation by using parallel wire-driven robots[END_REF], [START_REF] Fang | Motion control of a tendon-based parallel manipulator using optimal tension distribution[END_REF]. Others are controlling the robot in the Cartesian space [START_REF] Oh | Cable suspended planar robots with redundant cables: controllers with positive tensions[END_REF], [START_REF] Williams Ii | Translational planar cable-direct-driven robots[END_REF], [START_REF] Kino | Robust PD control using adaptive compensation for completely restrained parallel-wire driven robots: Translational systems using the minimum number of wires under zero-gravity condition[END_REF], [START_REF] Babaghasabha | Adaptive robust control of fully-constrained cable driven parallel robots[END_REF]. When controlling the CDPR in Cartesian space, pose feedback can be provided by forward kinematics [START_REF] Oh | Cable suspended planar robots with redundant cables: controllers with positive tensions[END_REF], [START_REF] Williams Ii | Translational planar cable-direct-driven robots[END_REF] or visual feedback [START_REF] Kino | Robust PD control using adaptive compensation for completely restrained parallel-wire driven robots: Translational systems using the minimum number of wires under zero-gravity condition[END_REF], [START_REF] Babaghasabha | Adaptive robust control of fully-constrained cable driven parallel robots[END_REF]. In all these works, the control input is a torque reference signal sent to current regulation of the winch actuator drive.

Other CDPR control studies are based on the assumption that is made in this work, i.e. that cables are elastic. None of them use visual feedback to measure the pose of the end-effector. Instead, forward kinematics is used to estimate this pose. An analysis of CDPR vibration modes is given by Diao et al. in [START_REF] Diao | Vibration analysis of cable-driven parallel manipulators[END_REF]. It shows that transversal cable flexibility can be neglected with respect to axial flexibility. Accordingly, in most of the papers cited below, the cables are modeled as straight line segments with an axial stiffness.

A cascade control architecture is proposed by Meunier et al. in [START_REF] Meunier | Control of an overactuated cable-driven parallel mechanism for a radio telescope application[END_REF] to control the end-effector pose of a CDPR with long elastic cables. The higher level controller (outer loop) regulates the distance between the winch output and the end-effector anchor point to obtain the desired pose. Its control signal is a vector of cable tension references. To ensure fast convergence of elastic cable tensions toward the references, an inner loop controls unstretched cable lengths by sending control signals to a winch angular position loop. In this work, the outer loop is a PID controller and the inverse dynamic model (IMD) of the end-effector is used to compute the desired cable tensions. This cascade controller is evaluated in simulation but no proof of stability is given for this architecture. Moreover, the dynamics of the winch angular position loop are neglected in the design of the inner loop.

A more complex lumped-mass cable model is used by Caverly et al. in [START_REF] Caverly | Dynamic modeling and noncollocated control of a flexible planar cable-driven manipulator[END_REF], [START_REF]Flexible cable-driven parallel manipulator control: Maintaining positive cable tensions[END_REF]. A model of the CDPR dynamics is derived taking into account the cables, the end-effector and the inertia of the torquecontrolled winches. A passivity-based control is proposed to ensure stable tracking of the desired CDPR tip position.

Khosravi and Jamshidifar explicitly addressed the rejection in Cartesian space of the vibrations induced by the axial flexibility of massless cables in [START_REF] Khosravi | Dynamic modeling and control of parallel robots with elastic cables: Singular perturbation approach[END_REF] and [START_REF] Jamshidifar | Vibration decoupled modeling and robust control of redundant cable-driven parallel robots[END_REF]. Similarly to the controller proposed by Ghorbel and Spong [START_REF] Ghorbel | Adaptive control of flexiblejoint manipulators[END_REF] for serial manipulators with flexible joints, a composite control involving a computed-torque controller and a constant feedback on the cable stretching velocity is studied in [START_REF] Khosravi | Dynamic modeling and control of parallel robots with elastic cables: Singular perturbation approach[END_REF]. No assessment is provided on a physical setup and no detail is given on how to measure or estimate the end-effector pose and cable stretching required for control. A detailed proof of stability based on singular perturbation theory [START_REF] Khalil | Nonlinear Systems[END_REF], [START_REF] Kokotovic | Singular Perturbation Methods in Control: Analysis and Design[END_REF] is provided, where high cable stiffness is assumed such that vibrations can be considered as fast dynamics. This proof extends the partial proof of stability given in [START_REF] Ghorbel | Adaptive control of flexiblejoint manipulators[END_REF] for serial robots, where it is assumed that the joint stiffness is large compared to other system parameters.

To the best of the authors' knowledge, [START_REF] Jamshidifar | Vibration decoupled modeling and robust control of redundant cable-driven parallel robots[END_REF] is the most recent work on CDPRs dealing with vibration rejection control. Based on a first order linearization of the CDPR dynamics along a desired trajectory, a decoupled model of the vibration dynamics is extracted. This model is time variant due to the non-constant stiffness of the CDPR end-effector within its workspace. Thus, a linear parametervarying controller (LPV) is designed to guarantee the stability of the vibration rejection across the CDPR workspace. This approach was tested experimentally. Acceleration and velocity measurements of the end-effector needed for the LPV controller are provided by an embedded 6-axis IMU. A similar approach for vibration rejection has been proposed by Weber in [START_REF] Weber | Active vibration canceling of a cable-driven parallel robot in modal space[END_REF]. Instead of an LPV controller designed in Cartesian space, the controller is designed after mapping the motion into the modal space in order to decouple the vibration modes.

When controlling the actuator torque directly, as in the control of serial robots with flexible joints through a state feedback linearization [START_REF] Siciliano | Springer Handbook of Robotics[END_REF] and as in most of the previously cited works on CDPR, the bandwidths of the torque and of the motor current loop are the same, i.e. usually more than 1 kHz. In this case, it is completely justified to consider the actuator as a perfect torque generator. However, if the motor is associated with a gearbox as it is often the case except for rare direct-drive setups, frictions and especially dry frictions may add significant nonlinearities to the model. Cascade control may be a solution to reject these nonlinearities with a fast inner loop like in [START_REF] Meunier | Control of an overactuated cable-driven parallel mechanism for a radio telescope application[END_REF]. The present work shares the same root idea as Meunier et al. [START_REF] Meunier | Control of an overactuated cable-driven parallel mechanism for a radio telescope application[END_REF] that high cable flexibility may be an opportunity from a control point of view. In this case, the vibration modes of the CDPR have frequencies much lower than the bandwidth of low-level winch angular position control loops. These control loops have a linearizing effect on nonlinearities like dry friction.

The same idea is behind the inner velocity or position loops proposed among other cascade controllers in [START_REF] Vallery | Compliant actuation of rehabilitation robots[END_REF], [START_REF] Agarwal | Design, control, and testing of a thumb exoskeleton with series elastic actuation[END_REF] for the control of series elastic actuators (SEA) [START_REF] Calanca | A review of algorithms for compliant control of stiff and fixed-compliance robots[END_REF]. The system composed of the winch motor, its gearbox and the elastic cable can indeed be seen as a SEA. One benefit of SEA, that also applies to CDPR with a low cable stiffness, is the transformation of a force sensing problem into a position sensing problem. Indeed, the force can be measured accurately by the elongation of the elastic component, i.e. the cable in the case of CDPRs.

This paper is based on the novel idea that, for CDPRs with flexible cables, the winch angular position can be considered as the fast variable whereas the end-effector Cartesian pose is the slow variable in a singular perturbation framework. The main contribution of the paper is to prove the stability of the whole system controlled at the outer level with a classical feedback linearization scheme. This approach is validated experimentally on a planar 3-cable suspended CDPR. Feedback linearization needs pose and velocity feedback which are provided by a 500 Hz eye-to-hand camera and an embedded IMU.

This paper is organized as follows. Section II presents the modeling of CDPR dynamics with elastic cables. The control scheme is explained in section III. Proof of stability is derived in section IV. Experimental setup and results are presented in section V.

II. MODELING

A. Cable Modeling

The CDPR might be suspended or fully constrained. As shown in Fig. 1, the vector ui is a unit vector giving the direction of the i th cable. Points Ai and Bi are its attachment points to the base frame and to the end-effector, respectively.

𝑃 𝑅

# 𝑅 $ Inertial frame 𝐴 & 𝐵 & 𝑏 & 𝑢 &

End-effector frame

The approach proposed in this paper is suited for CDPRs whose cable sagging can be neglected. When sagging is negligible, the cable can be effectively modeled by a linear spring. Studies on cable characteristics show that cables usually have a nonlinear relationship between their tension and elongation [START_REF] Riehl | On the determination of cable characteristics for large dimension cable-driven parallel mechanisms[END_REF]. In the sequel, it is assumed that this relationship can be linearized, and thus that, within nominal tension bounds [START_REF] Riehl | Modélisation et design de robots parallèles à câbles de grande dimension[END_REF], the linear spring model is accurate.

Accordingly, the cable is modeled as a linear spring with a specific stiffness ks = es (N) related to its Young modulus e (N m -2 ) and its cross-section s (m 2 ). The axial stiffness of the cable is given by:

k = ks l2 = es l2 (1)
In ( 1), l2 denotes the free length of the cable, i.e. its length when its tension is equal to zero. Winding the cable around its winch reduces l2, and therefore increases the cable axial stiffness. Vibrations occurring at the CDPR end-effector are mainly due to the axial stiffness of the cables [START_REF] Diao | Vibration analysis of cable-driven parallel manipulators[END_REF].

B. Dynamics

1) End-effector: Newton-Euler equations ( 2) describe the combined translational and rotational dynamics of a CDPR end-effector written in an inertial frame Rg:

mI3×3 03×3 03×3 Ig p ω ω ω + 03×1 ω ω ω × Igω ω ω = F N + mg 03×1 ( 2
)
where m is the end-effector mass, g is the gravity acceleration vector, p is the acceleration of the end-effector center of gravity P , vector ω ω ω is the time derivative of the end-effector angular velocity, F and N are the force and moment vectors applied by the cables to the end-effector at point P . Matrix Ig is the end-effector inertia tensor.

The vectors and the inertia tensor are all expressed in the robot base frame Rg (see Fig. 1).

Let Rgr be the rotation matrix from the inertial frame (Rg) to the robot end-effector frame (Rr). Thus, the end-effector inertia tensor expressed in the inertial frame Rg is given by: Ig = RgrIrR T gr with Ir the constant inertia tensor of the end-effector expressed in Rr.

Let J be the inverse Jacobian of the robot mapping the velocity twist of the end-effector to the joint velocities l1

= [ • • • l1 i • • • ] T , l1 i being the time derivative of the distance AiBi between the cable end points: l1 = J ṗ ω ω ω (3)
Furthermore, J T maps the cable tensions τ τ τ to the wrench f = F T N T T applied by the cables to the end-effector:

f = -J T τ τ τ (4)
The inverse Jacobian transpose of a CDPR is given by:

J T = • • • g u i • • • • • • g b i × g u i • • • (5)
with u i the unit vector giving the direction of the i th cable from its end point on the base frame to its end point on the end-effector and b i = --→ P Bi the vector from the end-effector center of gravity P to the end point Bi (see Fig. 1).

Using equation ( 4), equation ( 2) becomes:

mI3×3 03×3 03×3 Ig p ω ω ω + 03×1 ω ω ω × Igω ω ω = -J T τ τ τ + mg 03×1 (6)
Let X = [p T φ φ φ T ] T be the vector describing the pose of the CDPR end-effector with respect to the inertial frame Rg. Vector φ φ φ contains angular coordinates (e.g. Euler angles) describing the rotation Rgr(φ φ φ). It is well known that the time derivative of a pose vector is related to the corresponding twist by a Jacobian matrix. Let us call S this 6 × 6 matrix such that [ ṗT ω ω ω T ] T = S Ẋ where:

S = I3×3 0 0 S(φ φ φ) (7)
and S(φ φ φ) denotes the matrix mapping the time derivative of φ φ φ to the angular velocity vector ω ω ω:

ω ω ω = S(φ φ φ) φ φ φ (8)
S is a 3×3 identity matrix when φ φ φ is zero, so that it is sometimes neglected in the literature since CDPRs are usually not performing large angular displacements. Here, for the sake of generality, this term is not neglected. Using (3), let us define J which maps the time derivative Ẋ of the pose to joint velocities l1:

l1 = J Ẋ (9)
with:

J = JS (10)
Using ( 8) in ( 6) yields:

mI3×3 03×3 03×3 IgS p φ φ φ + 03×1 Ig Ṡ φ φ φ + S φ φ φ × IgS φ φ φ = -J T τ τ τ + mg 03×1 (11)
Noting that JT = S T J T , equation ( 11) can be rewritten using J which gives:

mI3×3 03×3 03×3 S T IgS p φ φ φ + 03×1 S T (Ig Ṡ φ φ φ + S φ φ φ × IgS φ φ φ) = -JT τ τ τ + mg 03×1 (12)
Introducing M, C and G = -mg 03×1 , equation ( 12) can be written in the classical form:

M(X) Ẍ + C( Ẋ, X) Ẋ + G = -JT τ τ τ (13)
with

M(X) = mI3×3 03×3 03×3 S T IgS and C( Ẋ, X) = 03×3 03×3 03×3 S T Ig Ṡ + S T (S φ φ φ) [X] (IgS)
where

(S φ φ φ) [X] = (ω ω ω) [X]
is the skew-symmetric matrix associated to ω ω ω = [ωx ωy ωz] T defined by:

(ω ω ω) [X] =   0 -wz wy wz 0 -wx -wy wx 0  
With the assumption of elastic massless cables, the forces applied by the cables to the end-effector can be computed using the distances l1 between cable end points and the cable free lengths l2 (the length of the cable when its tension is zero). The free lengths can be calculated using the winch angular positions θ θ θ: l2 = rθ θ θ + l0, where l0 is the cable free lengths for the pose X = 0 and r is a scalar representing the winch radius.

The diagonal matrix of cable axial stiffness K = diag(ki) links cable tensions τ τ τ to cable lengths:

τ τ τ = K(l2)[l1 -l2] ( 14
)
where ki is the i th cable stiffness which depends on its free length l2 i (see Eq. ( 1)). All cables are assumed to be tensed. Dedicated algorithms have been proposed in the literature to guarantee nonnegative cable tensions in the CDPR workspace, e.g. [START_REF] Oh | Cable suspended planar robots with redundant cables: controllers with positive tensions[END_REF], [START_REF] Gouttefarde | A versatile tension distribution algorithm for n -dof parallel robots driven by n + 2 cables[END_REF]. Finally, from ( 13), the dynamic model of a CDPR with massless elastic cables becomes:

M(X) Ẍ + C( Ẋ, X) Ẋ + G = JT (X)K(l2)[l2 -l1(X)] (15)
Note that l1 depends only on X since it is obtained by the inverse kinematics (computing the distances AiBi being given the endeffector pose X). Furthermore, the dependence of JT on X has been made explicit in this equation.

2) Actuators:

We consider a simplified model of the winch actuators. Indeed, in the usual case of a high reduction gear ratio (e.g. 210 on the setup presented in Section V), inertial couplings between the end-effector and the actuator rotors can be neglected (see Eq. 13.9 page 293 in [START_REF] Siciliano | Springer Handbook of Robotics[END_REF] for more details). Furthermore, all actuators are considered to be identical, which is usually the case for CDPRs.

Let us define the following parameters:

• B the rotor inertia,

• Fv the viscous friction coefficient on the rotor axis,

• Fs the dry friction vector on the rotor axes,

• γ γ γ the vector of torques generated by electromagnetic interactions between the stator and rotor, • r the radius of all winches, • α α α the vector of rotor angular positions. The tension of the part of the cable wound on the pulley is assumed to be null. Hence, l2 can be expressed as:

l2 = rα α α (16)
Then, the resulting simplified actuator dynamic equations are given by:

B l2 + Fv l2 + rFs = rγ γ γ + r 2 K(l2) [l1(X) -l2] (17)

III. CONTROL

This section introduces the two levels of control loops that are proposed to drive the CDPR end-effector along a prescribed smooth Cartesian trajectory.

Classical computed torque does not require a low-level velocity or position loop at the actuator level. Instead, the torques computed by the feedback linearization are directly sent to high bandwidth current loops. Since the bandwidth of a current loop (equal to the bandwidth of torque for an electrical motor) is many orders of magnitude larger than the mechanical bandwidth of most robots, its dynamics can be neglected. Nevertheless, when using high reduction ratios, significant dry friction occurs. In this case, direct torque control would be highly inaccurate since dry friction can represent up to 30% of the actuator nominal torque yielding very poor performances of the computed torque approach. This explains why computed torque is best suited to direct drive actuation.

In this work, we consider standard high reduction ratio actuators controlled by an inner position loop at the drive level. This servo loop is usually sampled at a high rate and may be considered as almost continuous time. Besides, the usual assumption of a high stiffness spring modeling of the cables is not made in this paper. Especially when using long cables or cables made of polymer material, this assumption does not hold.

Here, we assume that the actuator position loop has high gains and so a high bandwidth. The gains are tuned in such a way that the equivalent damping coefficient of the second order closed loop transfer function is equal to 1/ √ 2 (optimal tuning to minimize the 5 % rise time). The bandwidth of this inner position loop is supposed to be higher than the highest oscillation frequency of the CDPR end-effector. This is a mandatory condition for the design of a control law that is supposed to cope with these frequencies (for direct velocity feedback see [START_REF] Goh | On the stability problem caused by finite actuator dynamics in the collocated control of large space structures[END_REF], for lead controller see [START_REF] Preumont | Vibration Control of Active Structures[END_REF]). This results in a system with a fast variable, the actuator position loop, and a slow variable, the end-effector Cartesian position, coupled with cables treated as linear springs with moderate stiffness. Such a system typically fits into the singular perturbation framework. The latter will be used to demonstrate the stability of the proposed control law. One main contribution of this paper, which could be extended to types of flexible link robots other than CDPRs, is to propose a cascade control scheme for feedback linearization. The inner loop is a high gain actuator position loop designed to reject nonlinearities like dry friction. The outer loop achieves the feedback linearization based on the inverse dynamic model of the CDPR. The stability of the whole control scheme (Fig. 3) is proved using the singular perturbation theory in Section V.

α α α 1 s 1 Bs+F v Fs Winch actuator + + - - - + + γ γ γ rK(l2)(l1(X) -l2) α α α * K p K v α α α

A. Actuator position loop

All actuators are assumed to be controlled by a low-level position control loop, as shown in Fig. 2. Furthermore, an even lower level motor current loop allows for controlling γ γ γ with a high bandwidth.

The control loop in Fig. 2 is usually carried out by a drive where the gains can be tuned through a dedicated software. The inner loop controls the rotor velocity and the outer loop controls the position.

Injecting the control law defined in Fig. 2 into the rotor dynamic model [START_REF] Caverly | Dynamic modeling and noncollocated control of a flexible planar cable-driven manipulator[END_REF] gives:

B l2 +Fv l2 = Kv Kp(l * 2 -l2) -l2 +r 2 K(l2) [l1(X) -l2]-rFs (18)
where l * 2 = rα * . By tuning the gains Kp and Kv, it is possible to shape the position loop response to a step on l * 2 .

B. Feedback Linearization

From [START_REF] Diao | Vibration analysis of cable-driven parallel manipulators[END_REF], let us define the following feedback linearization control law for l * 2 :

JT (X)K(l * 2 )[l * 2 -l1(X)] = M(X) Ẍ * (t) +Kvx( Ẋ * (t) -Ẋ) + Kpx(X * (t) -X) + C( Ẋ, X) Ẋ + G (19)
where X * (t) is the desired Cartesian trajectory, Kvx and Kpx are tunable diagonal gain matrices. In the following equations, the dependency on time of the reference trajectory X * (t) will be dropped for simplicity. Using ( 1) and ( 14), the control law can be written:

ks JT (X)       1 . . . 1    -diag(l * 2 ) -1 l1(X)    = f (20)
where

f = M(X) Ẍ * + Kvx( Ẋ * -Ẋ) + Kpx(X * -X) +C( Ẋ, X) Ẋ + G (21)
is the wrench to be applied to the end-effector to achieve the feedback linearization, and diag(l * 2 ) is the diagonal matrix constructed using the components of vector l * 2 . In the most general case of a redundant CDPR (with more cables than degrees of freedom), isolating the terms in l * 2 yields to the non-unique solution:

diag(l * 2 ) -1 l1(X) =    1 . . . 1    + 1 ks -JT (X) + f + τ τ τ Null τ τ τ (22)
where ( JT ) + is the Moore-Penrose pseudoinverse of the wrench matrix and τ τ τ Null ∈ Null( JT ) is a tension vector. The vector (-JT (X)) + f is the minimum-norm tension solution for (20), while the vector τ τ τ Null verifies JT τ τ τ Null = 0. Through various dedicated algorithms [START_REF] Oh | Cable suspended planar robots with redundant cables: controllers with positive tensions[END_REF], [START_REF] Gouttefarde | A versatile tension distribution algorithm for n -dof parallel robots driven by n + 2 cables[END_REF], the vector τ τ τ Null can be selected in the null space of the wrench matrix, such as to guarantee non-negative cable tensions τ τ τ ≥ 0. Note that this non-negative cable tension requirement enforces the requirement l * 2 ≤ l1(X). Finally, the analytic solution l * 2 (X, X * ) of the feedback linearization control signal is given by:

l * 2 = I + 1 ks diag -JT (X) + f + τ τ τ Null -1 l1(X) (23)
Note that this equation relies on the measurements of X and Ẋ. In this work, these measurements are obtained from an external camera looking at the end-effector (eye-to-hand configuration [START_REF] Chaumette | Visual servo control, part II: Advanced approaches[END_REF]). A camera is used to estimate the Cartesian position X by tracking a marker attached to the CDPR end-effector. The 6-dimensional Cartesian velocity Ẋ is estimated from the measurements of an Inertial Measurement Unit (IMU) embedded in the end-effector (Fig. 3).

IV. STABILITY ANALYSIS

A. Standard singular perturbation model

The position controlled actuator dynamics and the feedback linearized end-effector dynamics can be written using the singular perturbation formalism [START_REF] Khalil | Nonlinear Systems[END_REF]. The fast variable z is defined as follows:

z = z1 z2 =   (l * 2 -l2)/ l2   (24)
with the small variable defined as = 1/Kp where Kp is the high gain of the actuator position loop.

The slow variable x is the vector of the Cartesian position X of the end-effector and its time derivative Ẋ:

x = x1 x2 = X Ẋ (25)
Let us link gains Kp and Kv with the following equation:

Kv = 2BKp (26)
Thereby, it can be easily shown that the damping factor of the second order transfer function of the actuator position loop tends toward 1/ √ 2 when Kp goes to infinity. This pretty common tuning minimizes the 5% settling time of the step response.

Let the standard singular perturbation form be:

ẋ = f (t, x, z, ) (27) ż = g(t, x, z, ) (28)
Thus, using ( 15), ( 18) and ( 26), the model can be put into the following standard singular perturbation form:

ẋ =       x2 M -1 (x1) JT (x1)K(l * 2 -z1) [l * 2 -z1 -l1(x1)] -C(x1, x2)x2 -G       (29) ż =       l * 2 -z2 2(z1 -z2) -B -1 Fvz2 -r 2 K (l * 2 -z1) [l1(x1) -l * 2 + z1] + rFs       (30)

Desired Cartesian Trajectory

Inverse Dynamic Model T with z1 = z2 = l * 2 (x1, X * ) the quasi-steady state of the fast variable. Injecting these roots in [START_REF] Riehl | On the determination of cable characteristics for large dimension cable-driven parallel mechanisms[END_REF], and setting = 0 as well, yields the slow subsystem or reduced model:

τ τ τ Null -( JT ) + - + - + + + + + + + + M(X) 1 r f τ τ τ X Ẋ α α α α α α * l * 2 Ẍ * Ẋ * C(X, Ẋ) Ẋ +G(X) K vx K px X * Model (Eq.
ẋ =       x2 M -1 (x1) JT (x1)K(l * 2 ) [l * 2 -l1(x1)] -C(x2, x1)x2 -G       (31)
with l * 2 (X, X * ) verifying [START_REF] Khosravi | Dynamic modeling and control of parallel robots with elastic cables: Singular perturbation approach[END_REF]. When compared to [START_REF] Diao | Vibration analysis of cable-driven parallel manipulators[END_REF], the free length l2 has been replaced by the free length reference l * 2 (X, X * ), which means that the winch actuator position control loops have an infinite bandwidth compared to the second order dynamics of the end-effector Cartesian position. This model simplifies further when the feedback linearization law ( 19) is applied:

ẋ = x2 Ẍ * + Kvx( Ẋ * -x2) + Kpx(X * -x1) (32)
With the change of variable e = X * -X, the reduced model becomes:

ė ë = 0 1 -Kpx -Kvx e ė (33)
whose origin {e = 0 ; ė = 0} is an exponentially stable equilibrium point if Kvx > 0 and Kpx > 0.

To study the fast transient of the variable z and its effective convergence toward its quasi-steady state z, a "stretched" time variable t = (t -t0)/ is introduced [START_REF] Kokotovic | Singular Perturbation Methods in Control: Analysis and Design[END_REF]. In this new time scale and at an initial time t0, the dynamics of z is called the fast subsystem or boundary layer model of [START_REF] Riehl | Modélisation et design de robots parallèles à câbles de grande dimension[END_REF] and is given by:

dy dt = -y2 2(y1 -y2) ( 34
)
with y = [y T 1 y T 2 ] T , y1 = z1 -z1 and y2 = z2 -z2 a change of variable which purpose is to shift the equilibrium point toward zero. The boundary layer model ( 34) is a linear system. The eigenvalues of the matrix A of its state-space representation are equal to -1 ± j. Thus, the origin of the boundary layer model is exponentially stable with a damping factor equal to1 √ 2 .

B. Tikhonov's Theorem

Theorem 11.2 in [START_REF] Khalil | Nonlinear Systems[END_REF] states that the solution of ( 29) and [START_REF] Riehl | Modélisation et design de robots parallèles à câbles de grande dimension[END_REF] approaches the solution of the reduced model [START_REF] Preumont | Vibration Control of Active Structures[END_REF] if:

• the origin is an exponentially stable equilibrium point of the boundary-layer model uniformly in (t, x);

• the origin is an exponentially stable equilibrium point of the reduced model; • on any compact subset including the origin, the functions f , g, their first partial derivatives with respect to (x, z, ), and the first partial derivative of g with respect to t are continuous and bounded, h(t, x) and [∂g(t, x, z, 0)/∂z] have bounded first partial derivatives with respect to their arguments, and [∂f (t, x, h(t, x), 0)/∂x] is Lipschitz in x, uniformly in t; the initial data are smooth functions of .

The first two points were verified in the previous section. The conditions in the third point can be easily verified if the reference trajectory X * is within the wrench-feasible workspace of the CDPR.

If all conditions are verified, the theorem states that, for gains Kv and Kp high enough ( small enough), for t > t0 > 0, the singular perturbation problem ( 29)-( 30) has a unique solution x(t, ) and z(t, ) such that:

x(t, ) -x(t) = O( ) (35) z(t, ) -h(t, x(t)) -ŷ(t/ ) = O( ) (36)
with x the solution of the reduced model [START_REF] Preumont | Vibration Control of Active Structures[END_REF] and ŷ the solution of the boundary layer [START_REF] Jiang | Trajectory generation for three-degree-offreedom cable-suspended parallel robots based on analytical integration of the dynamic equations[END_REF] or, equivalently, using the variable e:

e(t, ) -ē(t) = O( ) (37) z(t, ) -h(t, ē(t)) -ŷ(t/ ) = O( ) (38)
where ē is the solution of [START_REF] Chaumette | Visual servo control, part II: Advanced approaches[END_REF]. Furthermore, a t b exists such that for t > t b > t0:

e(t, ) = ē(t) + O( ) (39) z(t, ) = h(t, ē(t)) + O( ) (40) = l * 2 (X, X * ) O + O( ) (41)
Consequently, the feedback linearized system controlled with lowlevel high gain position loops is stable at the origin since limt→∞ ē(t) = 0 and e(t, ), the trajectory tracking error, is at an order of magnitude O( ) equal to ē(t).

V. EXPERIMENTS 1

A. The PiSaRo2 robot

The PiSaRo2 is a 3-cable suspended planar CDPR inspired by [START_REF] Jiang | Trajectory generation for three-degree-offreedom cable-suspended parallel robots based on analytical integration of the dynamic equations[END_REF] (see Fig. 4). The PiSaRo2 structure is made of Matrix™ aluminum beams. This allows for rapid prototyping of its architecture while keeping a decent global rigidity. Its workspace has been optimized to cover a 1.2 m by 0.96 m white board.

Three winches are driven by three Dynamixel™ XM430-W210-R actuators mounted on the end-effector. The inner loop of the proposed control strategy sends control inputs to the built-in velocity control loops of the Dynamixels™, which have been activated (the gain of this loop is set to 180 at the Dynamixel™ register address 78).

The position loop is tuned to maximize the bandwidth while keeping the step response overshoot around 5% yielding Kp = 125. With this setting, the winch position loop exhibits a -3 dB bandwidth of 21 Hz. These actuators are connected to a Raspberry Pi model 3 (RPi) by a 1Mbps RS485 serial bus. The RPi is a target for a dedicated toolbox developed in our lab called RPIt 2 . With RPIt, it is possible to use Matlab's™ hardware-in-the-loop Simulink Coder™ rapid prototyping toolbox with a RPi target. Generic blocks are available to control Dynamixel actuators, to read measurements from standard IMU chips connected to the RPi i2c bus or to interact through TCPIP sockets with a dedicated vision PC.

The winches were printed on a Stratasys 4 Polyjet™ 3D printer using VeroClear or VeroWhite polymer. A helicoidal track at the bottom of the winch groove helps to guide the cable winding, avoiding sudden cable jump from one layer to another. Brakes have been added to prevent the robot from falling in case of power loss due to its own mass (see Fig. 5). The robot is suspended with three fishing wires (Caperlan™ Line Clear 0.55 mm, 13.8 kg). Cyclic tension tests show that after a running-in procedure, its specific stiffness ks tends toward 500 N.

Aluminum profile

A picture of the end-effector is given in Fig. 6. Four red LED markers arranged in a square pattern are used to simplify the pose estimation algorithm on the vision PC. An Invensense™ MPU 9150 IMU is positioned approximately at the end-effector center of gravity. A USB wireless Xbox 360™ controller receiver is used for manual remote control of the robot.

B. 500 Hz visual feedback

A vision PC is dedicated to the image processing at a high sampling rate (500 Hz). It is connected through USB3 to a Ximea xiQ™ MQ003MG-CM VGA monochrome 500 frames per second camera mounted on a tripod and pointed to the CDPR end-effector. It acts as a server for a special RPIt block that communicates using TCPIP sockets. To keep the system fully wireless while minimizing jitters and delays, a dedicated 5GHz MIMO WiFi hotspot is used to relay the packets from the RPi to the vision PC. Delays were assessed and long lasting high rate ping sessions show that they are less than 1 ms.

The visual control loop runs on the RPi with a sampling rate of 100 Hz. It periodically sends measurement requests through a socket to the vision PC. Since the visual measurement runs five times faster than the visual control loop and since it is not synchronized with the camera clock, there is a random delay in the visual feedback loop ranging from 0 to 2 ms with an average of 1 ms. We assume that this delay is negligible with respect to the control loop sampling period (10 ms).

The image processing is a simple OpenCV blob detector that robustly detects 4 LED spots in the image. With the pose estimation, it takes less than 400 µs on the vision PC (Intel Core™ i5-6500 CPU @ 3.20GHz). Video stream buffering delay of the Ximea Linux device driver has also been assessed and can be considered as negligible.

C. Experiments

The feedback linearization of ( 23) is implemented. The reference trajectory X * describes a 21 cm × 21 cm square on the white board. Though twice differentiable with respect to time, X * has high accelerations around the square corner in order to excite the flexible modes of the robot. Dynamic Visual Servoing (DVS) implementing the control law in ( 23) is compared with two other control schemes: Inverse Kinematics Control (IKC) and Kinematic Visual Servoing (KVS).

IKC calculates offline the inverse kinetostatic model of the desired Cartesian trajectory X * . It provides an equivalent quasi-static trajectory at the actuator level which is replayed with proprioceptive (encoder) feedback but without exteroceptive (vision) feedback.

KVS assumes that the robot is a perfect Cartesian motion device, i.e., dynamic effects are neglected (see [START_REF] Chaumette | Visual servo control. I. basic approaches[END_REF]). A position-based visual servo loop is implemented that uses the Jacobian J as the interaction matrix (see [START_REF] Gouttefarde | A versatile tension distribution algorithm for n -dof parallel robots driven by n + 2 cables[END_REF]). The visual feedback provides the measurement of X with respect to 3 degrees of freedom: the two translations within the robot plane and the rotation around an axis normal to this plane.

First experiments of DVS with the control law (23) exhibit significant tracking errors (up to 2.5 cm). As the feedback linearized outer loop is equivalent to a double integrator system, adding an integral gain in the proportional derivative controller should be worthless. However, an integral effect improved tracking error by compensating for small unmodeled phenomena in this control strategy [START_REF] Paccot | A review on the dynamic control of parallel kinematic machines: Theory and experiments[END_REF]. Consequently, to cope with uncertainties on the cable stiffness amongst others, the control law [START_REF] Ghorbel | Adaptive control of flexiblejoint manipulators[END_REF] has been modified accordingly with:

f = M(X) Ẍ * + Kvx( Ẋ * -Ẋ) + Kpx(X * -X) +Ti (X * -X)dt + C( Ẋ, X) Ẋ + G ( 42
)
where Ti is the diagonal matrix of the integral term gains. In order to prove the stability of the whole system, the stability of the reduced model [START_REF] Goh | On the stability problem caused by finite actuator dynamics in the collocated control of large space structures[END_REF] with the modified PID feedback linearization (42) should be proved. This can be easily done since the resulting system is linear.

In our experiments the PID gains have been tuned as follows:

Kpx = diag(144, 144, 80) Kvx = diag(10, 10 , 5)
Ti = diag(300, 300, 300)

These gains where tuned experimentally to minimize the tracking error along the trajectory (a 21 cm × 21 cm square described at an average speed of 0.11 m s -1 ), while avoiding actuator saturation. These gains yield a -3 dB bandwidth of 0.41 Hz for the two translational control loops and 1.75 Hz for the rotational control loop. Note that the -3 dB bandwidth of the winch angular position loop (21 Hz) is much larger. This verifies the main assumption of a twotime-scale system used by the singular perturbation theory to achieve the stability proof.

Estimates of the Cartesian position X and velocity Ẋ are required to compute (42). The Cartesian position is measured by the vision system whereas the velocity is estimated from the measurements of an IMU. One axis of the IMU gyro gives the velocity directly around the axis normal to the robot plane. The translational velocities are obtained after integration over time of the accelerometer axes which are in the robot plane. In order to avoid a drift of this estimate over time, it is processed by a high-pass digital filter. Fig. 7, 8 and 9 show the 2D end-effector trajectory and the quadratic tracking error for a square-shaped X * (t) that is followed at an average speed of 0.11 m s -1 . These plots show clearly the superiority of the DVS control law. KVS has increased significant errors over time due to the absence of oscillation rejection (the robot dynamics are neglected). IKC has greater tracking error than DVS as well due to its lack of exteroceptive feedback. Indeed, modeling errors that impact the inverse kinetostatic model cannot be rejected with this open-loop approach. On the other hand, it yields less oscillations than KVS since its stability is not compromised by a feedback loop designed for a simplified model. The RMS errors of the end-effector position along the whole trajectory are respectively 32.6 mm, 11.2 mm and 1.7 mm with KVS, IKC and DVS. The mean anglular errors along the whole trajectory are respectively 36.3 8.6 mrad and 7.8 mrad with KVS, DVS and IKC.

VI. CONCLUSION

In this paper, a new approach to dynamic control of CDPRs using visual feedback has been proposed. CDPRs are usually controlled using a torque control input through the actuator drive current loop, thereby acting on the cable force. This approach is relevant for robots with high stiffness cables. In this case, the proof of stability using the singular perturbation theory is almost the same as for other kinds of flexible robots, parallel [START_REF] Sun | Singular-perturbation-based nonlinear hybrid control of redundant parallel robot[END_REF] or serial [START_REF] Ghorbel | Adaptive control of flexiblejoint manipulators[END_REF], [START_REF] Siciliano | A singular perturbation approach to control of lightweight flexible manipulators[END_REF], and the small variable is usually chosen inversely proportional to the stiffness coefficient. However, when using long cables or low stiffness cables, like cables made of polymers, this assumption does not hold anymore. Thus, as it is usually the case with industrial robots, we propose to control the CDPR in a cascaded way, with a linearizing inner loop controlling the winch angular position and an outer visual loop controlling the end-effector Cartesian position. A new singular perturbation small variable is introduced i.e. the inverse of the position loop gain. Using Tikhonov's theorem, the stability of this dynamic visual servoing control scheme implementing standard feedback linearization was demonstrated in this paper without requiring a high-stiffness cable assumption. Experiments validated the approach on a planar 3cable CDPR, showing the added value of a dynamic control law with respect to a kinematic one when using high frame rate visual feedback.

Fig. 1 .

 1 Fig. 1. CDPR general definitions.

Fig. 2 .

 2 Fig. 2. Actuator position control loop.

Fig. 3 . 1 zT 2 ]

 312 Fig. 3. Cartesian model-based control of elastic cable CDPRs, with an actuator position inner control loop.

Fig. 4 .

 4 Fig. 4. PiSaRo2 cable configuration.

Fig. 5 .

 5 Fig. 5. PiSaRo2 winches.

Fig. 6 .

 6 Fig. 6. PiSaRo2 end-effector. This robot weights 3.2 kg, can theoretically carry a load of 2 kg within a 1.2 m × 1 m workspace, has a maximum speed of approximately 0.25 m s -1 and a maximum position accuracy of 0.1 mm (related to encoder resolution).

Fig. 7 .

 7 Fig. 7. End-effector trajectory with IKC (green), KVS (blue) and DVS (red).

Fig. 8 .

 8 Fig. 8. Quadratic tracking error: IKC (green), KVS (blue) and DVS (red).

Fig. 9 .

 9 Fig. 9. Absolute value of the angular error: IKC (green), KVS (blue) and DVS (red).

J. Begey, L. Cuvillon and J. Gangloff are members of the ICube laboratory, University of Strasbourg, France. M. Lesellier and M. Gouttefarde are members of LIRMM, Université de Montpellier, CNRS, France.

This paper has supplementary downloadable material available at http://ieeexplore.ieee.org. It includes a video presenting the setup used to carry out the experiments and showing the main experimental results, a summary file and a readme file. The size of this material is 33.2 MB.

https://github.com/jacqu/rpit

ACKNOWLEDGMENT

This work is part of project DexterWide (ANR-15-CE10-0006) funded by the French National Research Agency (ANR).


Andreas Seifert

Keywords:

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

Author's version accepted

Claire Clivaz, "Review. Der Markusschluss: Narratologie und Traditiongeschichte, written by Andreas Seifert", Novum Testamentum, Brill Academic Publishers, 2019, 62 (1), pp.107-108. This book is the published version of a PhD defended in 2018 at the University of Bochum, Germany, under the supervision of Prof. Dr. Reinhard von Bendemann. Andreas Seifert argues that "the short Mark ending is not a 'foreign body' in the Markan story, but can be considered as belonging to the entire story, from the point of view of the content as well as of the functionality'" (p. 243). If the Gospel according to Mark is, in its core, a narrative Christology, it focuses for Seifert also on the figures of the women and the disciples: the reader is invited to act differently from them; the Markan kerygma has "a pedagogic and paranetic effect" (p. 285). Chapters 2 and 3 present the narrative methodology used by the author to reach this conclusion: he combines the Aristotelean tripartite analysis and the work of Constanze Krings on the short English 19th century endings. Chapter 4 analyses the literary genre of Mark, and chapters 5 to 7 compare Mark's ending to the book of Jonas, to Philo Vita Mosis and to Plutarchus Demosthenes Life. Chapter 8 presents the state of art on Mk 16,1-8, before a complete narrative analysis of Mk 16,1-8 in chapter 9.

Book Notes for

As such, Seifert's thesis does not sound particularly innovative in Markan studies. Notably Norman Petersen, or the French-speaking scholars Corina Combet-Galland and Benoît Standaert -not quoted by the author -have supported similar narrative readings in the 80ies. But his comparison with the endings of the book of Jonas, the Vita Mosis and Demosthenes Life is a happy new contribution to the field. Moreover, this book is of interest since it illustrates a clear challenge: the necessity for exegetes to read also in detail at Mk 16,1-8 textual criticism, a steps too often delegated to the specialists. Seifert presents in about twenty pages a good synthesis of the Mark ending textual criticism, even if he misses crucial elements of the file, like the 2nd century attestations of the longer ending by Irenaeus, Justin and Tatien. Moreover, a discrepancy appears between a historical approach and a quite narrow narrative methodology, for example when Seifert concludes that some manuscripts do not make sense «from a narrative perspective » (p. 190). Other elements are minimized by him to support his conclusions, such as the differences of the two lists of women's names in Mk 15,47 and 16,1, or Karen King's comparison with the Gospel according to Judas, reported but not discussed (p. 207).

In summary, the reader will find in this book a fresh overview on the Mark ending, an exhaustive analysis of the narrative elements of Mk 16,1-8, and an a contrario award: the perception of the unavoidable role of history and documents beyond narrative patterns.

Claire Clivaz, Swiss Institute of Bioinformatics (Lausanne, CH), claire.clivaz@sib.swiss

  Novum Testamentum Andreas Seifert, Der Markusschluss. Narratologie und Traditiongeschichte (Stuttgart: W. Kohlhammer, 2019), 314 pp. ISBN 978-3-17-036434-9 (=Beiträge zur Wissenschaft vom Alten und Neuen Testament 220), 64 euros.


INTRODUCTION

The Agly river is located in the Southern part of France, in the Mediterranean region characterized by extreme flood events. The lower flood plain of Agly river is constituted of sediments brought from the upper catchment during centuries. Because of the intense use of the flood plain for agriculture, tourism and urban growth of Perpignan city, levees were built along the river and progressively reinforced, which did not prevent from occasional flooding during extreme floods. Many civil engineering structures were constructed along the river for various aims: flooding protection, dams, railway, roads,…The flood plain materials are characterized by fine sediments while the bed materials are dominated by coarse sediments.

In this paper, the authors focus on the 13 kilometre long downstream reach, which is fitted out to the levees constructed in 1970 with a design discharge of 1300 m 3 /s. Moreover, the simulated reach was extended by approximately 10 km upstream to eliminate any effect of the upstream boundary condition on the results.

The two more important floods after the levees building are studied. The first one was in 1992 with a return period of about 20 years, which means that the peak flow is close to the design flow discharge of the levees project; the second one was in 1999 corresponding to a 100-year return period and for which the levees are overtopped.

Inundations generated from river overflow are usually simulated by 1-D or 2-D numerical hydraulic models. The 1-D model is dedicated to the areas such as the river main channel in which the flow remains parallel to a hydraulic axis during the flood events; conversely, the 2-D model is limited to the areas such as flood plains in which the flow pattern evolves with time.

In this paper, throw numerical modelling of these two Agly floods, the main channel bed evolution is studied to find out the eroded and deposited zones. The authors compare also the calculated flooded zones with observations. In addition, a comparison between a coupled 1D/2D model and a full 2D model is carried out. The coupled 1D/2D model uses 1D modelling of the main channel between levees and 2D modelling for the flow and sediment transport in the floodplain; ABSTRACT: The authors discuss the use of a 1-D model or a 2-D model for the evolution of the bed between the levees while the spreading of water and sediment over the floodplains is simulated using a 2-D model. A 1-D model cannot represent the asymmetrical behaviour of the flow in the bends but makes easier the estimate of the variations of the sediment transport capacity along the river if the bed geometry is complex. For the case of the Agly river floods, results in terms of water elevation and bed evolution are compared for a simplified geometry of the bed. The two models (coupled 1-D/2-D or full 2-D) provide quite similar processes in the flood plain but different erosion rates of the main channel bed. Because differences of both results to observations are large, one cannot decide which model provides the more relevant results. with this coupled method, one can reduce the calculation time avoiding more refined cells across the main channel.

MATERIAL AND METHODS

2.1 Software 2.1.1 One dimensional model with sediment transport -RubarBE RubarBE software can simulate the flow and the bed evolution using 1D Saint Venant equations coupled with Exner equation [START_REF] Exner | Über die wechselwirkungzwischenwasser und geschiebe in flüssen (On the interaction between water and sediment in rivers)[END_REF], El kadi & Paquier 2009). The model solves these equations by a finite volume method with an explicit numerical scheme in which the time step is limited by the usual Courant-Friedrichs-Levy (CFL) condition (maximum Courant number selected: 0.9 here). The shear stress calculation is obtained throw the geometrical method called Merged Perpendicular Method [START_REF] Khodashenas | A geometrical method for computing the distribution of boundary shear stress across irregular straight open channels[END_REF]. This latter calculation depends on the water depth so it permits more detailed modelling; especially in bends and where there are geometry variations (El kadi. & Paquier 2009).

For Agly calculations, the assumption is that the local bed variation dz is proportional to the local difference between effective and critical shear stresses with a 1.5 power. However, there are other possibilities proposed by the model. The section bed change is obtained from the difference between the upstream sediment discharge and the sediment transport capacity considering a lag distance. Several sediment layers are defined in each point of the geometry, each layer being characterized by three parameters that can evolve with time: a layer thickness, a mean diameter d50 and a standard deviation σ. The evolution of the mean diameter and the standard deviation of the sediment carried out by the flow (sorting effect) is governed by an exponential law with specific lag distances [START_REF] Camenen | 1D morphodynamic modelling using a simplified grain size description[END_REF]).

Two dimensional model with sediment transport -Rubar20TS

Rubar 20TS software (El Kadi et al. 2009) solves 2-D shallow water equations coupled to a sediment transport equation. It uses the same explicit Godunov scheme that the one in RubarBE and requires the same type of time step limitation (Courant number limited to 0.9 here). According to the complexity of the flow [START_REF] Paquier | A method for coupling 1D and 2D models used to simulate floods and its application to the Niger inner delta[END_REF], the Riemann problem that calculates the flux across an edge can be replaced by a weir type equation (particularly relevant for a hydraulic structure). There is possibility to use quadrilateral or/and triangular meshes according to the geometry in order to adapt to field complex geometries. Similarly to RubarBE, several sediment layers are defined at each point of the geometry and time evolution of the sediment features depends of sediment balance and sorting lag distances. The rate of erosion and deposition at each point is calculated from the difference between the sediment discharge and the sediment transport capacity taking into account a lag distance.

Coupled model

The studied area is shared between a 1-D area and a 2-D area. Along the boundary between the two areas, coupling is performed at each time step. The 1D model and the 2D model are compiled as one single model, which permits quicker data transfers between the two models. The exchanged terms between the two models are the fluxes of mass (water and sediment) and momentum. Usually, the fluxes are calculated as solutions of a 1-D Riemann problem in the direction perpendicular to the boundary between the two models similarly to what is used in the 1-D and 2-D numerical schemes; then, hydraulic jump can be integrated as an ordinary calculation point even at the boundary between the two models (Paquier 2019).

Building the model

The data used to build the models were from [START_REF] Serrat | Genesis and dynamics of a Mediterranean river system: the Agly basin (France)[END_REF] for sediment and the 1995 geometry of the main channel, from Water Authorities for the 1970 project geometry and 2000 geometries of the main channel, the flood plain geometry and the flow discharge hydrographs while the limits of the flooded areas were obtained from aerial photos.

For computation, the initial geometry for the main channel was the 1970 project geometry simplified to 8 points cross sections in order to have exactly the same geometry in 1-D and 2-D models.

Sediment layers are represented with mean diameter d50 = 2 mm, standard deviation σ=10 and not dimensional critical shear stress 0.047. To be able to represent the variations of sediments during the flood events, four erodible layers, the thickness of each one is 1 m, were set in the model. The space step is 100 m except where constructions obliged to reduce it. The lag distance that is the distance in which the transport reaches its capacity is fixed at 200 meters while the lag distance for sediment sorting is fixed at 1000 meters [START_REF] Camenen | 1D morphodynamic modelling using a simplified grain size description[END_REF]. The presented results come from sediment transport capacity calculated by [START_REF] Meyer-Peter | Formulas for bed load transport[END_REF] equation using the total shear stress (because using a reduced shear stress does not provide significant erosion of the main channel bed).

RESULTS

Main channel evolution

For the two floods, Figures 12show good agreement between the calculated longitudinal profile of maximum free water surface profile using one or the other model and the measured one. However, the 2-D model seems to underestimate the peak water elevation in the downstream part of the reach although the Strickler coefficients used for bed friction are the same ones and were selected on the basis of a 2-D calculation with fixed bed. For the 2-D model, the bed during the 1992 and 1999 floods (Fig. 3) tends to an erosion in the most part of the longitudinal profile of the bed between levees like what is deduced from the comparison of geometries measured in 1995 and 2000 and the 1970 project geometry. For the coupled model, erosion alternates with deposition but represents the actual streamwise variations of the erosion or deposition depths in a better way. On Figure 3, note that calculation starts from 1970 project geometry and considers one flood (either 1992 or 1999) while observations considers changes over two periods 1970-1995 (during which the main flood is the one of 1992) and 1995-2000 (during which the main flood is the one of 1999).

Flood extent and floodplain deposits

For the 1992 flood, the inundations are seldom as expected because the peak flow discharge corresponds to the design flow of the levees project. The calculated inundated area for the 1999 flood with 2-D or coupled 1-D/2-D models shows good agreement with the delimitation from the aerial photos in red in Figures 45. But the 2-D model (that was calibrated on observations) inundates more upstream than the coupled model, may be because the erosion of the main channel is lower at the locations in which overflow occurs. The area flooded through a breach in the levee is represented in a correct way if the breaching model (used in both models) provides enough erosion for a correct development of the breach. Figure 6 shows the erosion and sedimentation areas in the floodplain at the upstream part of the reach between levees calculated with the 2D model Rubar20TS and the 1D/2D coupled model. The same processes can be found in the two models, with larger bed evolutions presented by the 2D model that inundates more.

The main processes are quite similar in the two models and the slightly better representation of the 1999-flooded area using the 2-D model is likely to come from the calibration performed on the 2-D model (parameters being extended to the coupled model). Deposits and erosions in the floodplain are quite similar in the two models although the 2-D model that inundates more consequently changes the bed elevation more. The trend and the amplitude of bed elevation changes are quite different in the main channel for the two models; although the 1-D and the 2-D models use the same parameters, the location of the sediment balance (one cross section in 1-D, one cell in 2-D) results in different behaviours of the main channel while the differences in the flood plain morphological changes only come from the location and intensity of overflow. Because the available data are not enough accurate and because the model results depend strongly on sediment parameters, one cannot decide which model provides the more relevant results. However, the time calculation is reduced by about 50% using the coupled model, which thus represents the more practical tool.

Figure 1 .

 1 Figure 1. Comparison between measured and calculated longitudinal profile of peak water surface during the 1992 flood.

Figure 2 .

 2 Figure 2. Comparison between calculated longitudinal profile of peak water surface during the 1999 flood and measured ones at the right and the left banks.

Figure 3 .

 3 Figure 3. Calculated bed evolution during the 1992 and 1999 floods.

Figure 4 .

 4 Figure 4. Calculated flooded area during the 1999 flood with 2D Rubar20TS model (peak water depths in meters; red line shows observed limit of flood, x and y coordinates in meters).

Figure 5 .

 5 Figure 5. Calculated flooded area during the 1999 flood with the coupled 1D/2D model (peak water depths in meters; red line shows observed limit of flood, x and y coordinates in meters).

Figure 6 .

 6 Figure 6. Zoom at erosion and sedimentation areas in the upstream part of reach between levees for the 1999 flood with the 2D model Rubar20TS (left) and with the 1D/2D coupled model (right) (dz = deposit thickness in meters, x and y coordinates in meters).

ACKNOWLEDGEMENTS

The PROFAS B+ program for the scholarship (Algerian-French cooperation) funded the stay of first author at INRAE.