(

  Re)conceptualizing the genesis of a "we is greater than me" psychological orientation: Sartre meets TomaselloLucia AngelinoTo cite this version:Lucia Angelino. (Re)conceptualizing the genesis of a "we is greater than me" psychological orientation: Sartre meets Tomasello. Journal of Social Ontology, 2022, 8 (1), pp.68-93. 10.25365/jso-2022-7311


Camilo Andres

Rey Arias

Wouter Weekers

Marco Morganti

Vincent Padois

Alessandro Saccon

Refined Post-Impact Velocity Prediction for Torque-Controlled Flexible-Joint Robots

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

I. INTRODUCTION

I MPACT-AWARE robotics refers to the exploitation of in- tentional collisions for achieving dynamic manipulation and locomotion [START_REF] Stouraitis | Multi-mode trajectory optimization for impact-aware manipulation[END_REF]- [START_REF] Gao | Impact-aware online motion planning for fully-actuated bipedal robot walking[END_REF]. This paradigm shift is envisioned to extend state-of-the-art physical interaction control approaches that resort to enforcing near-zero relative contact velocity between a robot and its environment, treating collisions as disturbances to be dealt with (cf. [START_REF] Salehian | A dynamical-systembased approach for controlling robotic manipulators during noncontact/contact transitions[END_REF], [START_REF] Englsberger | Threedimensional bipedal walking control based on divergent component of motion[END_REF]). This ambitious goal requires a new holistic framework including modeling [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF], learning [START_REF] Khurana | Learning to hit: A statistical dynamical system based approach[END_REF], planning [START_REF] Kong | Hybrid iLQR model predictive control for contact implicit stabilization on legged robots[END_REF], sensing [START_REF] Proper | Aim-aware collision monitoring: Discriminating between expected and unexpected post-impact behaviors[END_REF], and control aspects [START_REF] Van Steen | Quadratic programmingbased reference spreading control for dual-arm robotic manipulation with planned simultaneous impacts[END_REF], supported by collision-tolerant hardware [START_REF] Ostyn | Overload clutch with integrated torque sensing and decoupling detection for collision tolerant hybrid highspeed industrial cobots[END_REF]- [START_REF] Chiaradia | Compact series visco-elastic joint for smooth torque control[END_REF].

Impact-aware control strategies have to account for velocity jumps induced by intentional collisions that occur along a trajectory. These jumps, when they are not addressed directly [START_REF] Van Steen | Quadratic programmingbased reference spreading control for dual-arm robotic manipulation with planned simultaneous impacts[END_REF] or indirectly [START_REF] Yang | Impact invariant control with applications to bipedal locomotion[END_REF] by the control strategy, are seen as a sudden large disturbance in the system which results in poor tracking and even cause instability [START_REF] Rijnen | Reference spreading: Tracking performance for impact trajectories of a 1DoF setup[END_REF]. Accurate post-impact velocity prediction models for robotic manipulators [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF] can be used in trajectory planning to generate feedforward signals that lead to less aggressive post-impact control actions, since the postimpact system velocity matches the predicted feedforward, and limited feedback is thus required: reference spreading [START_REF] Van Steen | Quadratic programmingbased reference spreading control for dual-arm robotic manipulation with planned simultaneous impacts[END_REF], [START_REF] Rijnen | Control of humanoid robot motions with impacts: Numerical experiments with reference spreading control[END_REF], [START_REF] Van Steen | Robot control for simultaneous impact tasks via quadratic programming-based reference spreading[END_REF] is one of such impact-aware feedback control strategies that benefits from accurate post-impact feedfoward.

Manuscript received: October, 14, 2023; Revised January, 12, 2024; Accepted Jan, 31, 2024. This paper was recommended for publication by Editor Júlia Borràs Sol upon evaluation of the Associate Editor and Reviewers' comments. This work was supported by the Research Project I.AM. through the EU H2020 program under GA 871899. 1 Camilo Andres Rey Arias and Marco Morganti were at the time of performing this research with Franka Emika GmbH, 80797 Munich, Germany (camiloarac@gmail.com, marco.morganti@franka.de) 2 Wouter Weekers and Alessandro Saccon are with the Department of Mechanical Engineering, Eindhoven University of Technology (TU/e), 5600 MB Eindhoven, The Netherlands (w.weekers@tue.nl, a.saccon@tue.nl) 3 Vincent Padois is with Inria, 33405 Talence, France (vincent.padois@inria.fr) Digital Object Identifier (DOI): see top of this page. These models are also of prominent relevance for impact classification to distinguish between expected and unexpected impact scenarios [START_REF] Proper | Aim-aware collision monitoring: Discriminating between expected and unexpected post-impact behaviors[END_REF]. Other recent research in this direction, but for velocity-controlled robots, includes [START_REF] Wang | On inverse inertia matrix and contact-force model for robotic manipulators at normal impacts[END_REF], [START_REF] Wang | Predicting impactinduced joint velocity jumps on kinematic-controlled manipulator[END_REF].

In this work, we focus on post-impact prediction models with a single contact point, expanding on the seminal work [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF]. In [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF], impact maps based on readily available rigid-body robot models and nonsmooth impact laws are used to estimate the gross post-impact response of flexible joint robots impacting a rigid surface. The predictions are evaluated against real experiments by means of a novel quantitative comparison procedure [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF], derived by the idea that the rigid-robot response can be compared with the real robot response by suitably neglecting the post-impact higher-frequency behavior of the non-rigid system. The rigid impact map proposed in [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF] does not include any effect of either the low-level joint torque controller or the tangential friction between the end-effector and the impacted surface. We show here that incorporating these effects into an equivalent rigid-robot model with an adapted nonsmooth impact law significantly enhances prediction accuracy. In more detail, the contributions of this work are:

1) An improved post-impact velocity predictor, that incorporates the motor inertias, gear ratios, low-level joint torque-control gains, and tangential friction effects.

2) The experimental and numerical validation of the approach, showing the improvement with respect to two baseline approaches missing the tangential friction and low-level joint-torque control loop effects.

3) A post-impact velocity estimator to extract the rigidrobot post-impact response from recorded data. This estimator is based on a polynomial fitting procedure at position level and it is more robust in removing the postimpact oscillatory response than the one proposed in [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF]. Our work has also clear potential to contribute to robot safety literature (cf. [20, equation (4)]), as our results show how the low-level torque control loop affects the effective mass at impact times. The recorded data of the 50 impact experiments [START_REF] Rey Arias | archive containing impact experiments with a Franka Emika robot, This work is licensed under the Creative Commons Attribution Non Commercial Share Alike[END_REF], the code to compute the prediction and ground truth values of the post-impact velocity using the experimental data, and the illustrative 2D numerical simulation code [START_REF] Weekers | Refined post-impact velocity prediction for torque-controlled flexible joint robots[END_REF] are openly available for reproducibility and further investigation. The experimental impact motions are not only planar as in [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF], but they are fully 3D motions with changing motion directions and end-effector orientations. Figure 1 illustrates one such experiment.

This paper is structured as follows. Section II presents the system's dynamics including the low-level controller, along with the derivation of the post-impact velocity predictor. A simulation study for numerical validation of the effectiveness of the predictor is presented in Section III. Section IV describes the setup for the impact experiments conducted to validate the accuracy of the predictor for a 7 degrees-of-freedom (DoF) robot. Finally, conclusions are drawn in Section V.

II. CONTACT MODELING AND DERIVATION OF THE REFINED PREDICTOR

This section summarizes the standard flexible-joint robot model in Section II-A. This is then enriched with a continuous contact model in Section II-B, used in Section III for illustrating the novel post-impact velocity predictor by means of numerical simulations. In Section II-C, the novel equivalent rigid-robot model and the adapted impact map are detailed.

A. Torque-controlled flexible-joint robot dynamics

Consider an n-degrees-of-freedom flexible-joint robot, whose dynamics can be described as (cf., e.g., [START_REF] Albu-Schäffer | A unified passivity-based control framework for position, torque and impedance control of flexible joint robots[END_REF], [START_REF] De Luca | Robots with flexible elements[END_REF])

τ = K(θ -q), (1)
M(q)q + h(q, q) = τ + DK -1 τ + τ ext , (2)
B ρ θ + τ + DK -1 τ = τ act . (3)
Here, q ∈ R n denotes the link-side joint positions, θ ∈ R n denotes the motor positions including the gear reduction, M ∈ R n×n the joint space inertia matrix, h ∈ R n the Coriolis, centrifugal, and gravity effects, K = diag(k i ) ∈ R n×n the joint stiffness matrix, D = diag(d i ) ∈ R n×n the joint damping matrix, τ ext ∈ R n the externally applied joint torques, B ρ = diag(b ρ,i ) ∈ R n×n the motor inertia matrix including the gear reduction, and

τ act = B ρ B -1 θ u + I n -B ρ B -1 θ τ + DK -1 τ (4)
the actuation torques resulting from a low-level control loop, with u ∈ R n the desired link-side torques resulting from a high-level controller, and B θ = diag(b θ,i ) ∈ R n×n a positive definite matrix corresponding to the targeted closedloop behaviour of the joint torque, such that b θ,i < b ρ,i , i = {1, . . . , n}. Typical values for the ratios b ρ,i /b θ,i are between 4 and 6 [START_REF] Albu-Schäffer | A unified passivity-based control framework for position, torque and impedance control of flexible joint robots[END_REF]. The low-level torque control loop (4) can be shown to be equivalent to the PD + feedforward controller

τ act = u + K T (u -τ ) -K S τ (5)
with proportional torque gain matrix K T = diag(k T,i ) and derivative torque gain matrix K S = diag(k S,i ), by setting

K T = B ρ B -1 θ -I n and K S = K T DK -1 .
Note that, by substitution of (4) in (3), the motor-side dynamics is equivalently written as

B θ θ + τ + DK -1 τ = u, (6)
showing that the control parameter B θ can effectively be considered as a motor inertia matrix whose diagonal entries are reduced compared to those of B ρ . A flexible robotic joint and its associated parameters used in (1)-( 6) are shown in Figure 2, with τ J the reading of the link-side joint torque sensor.

B. Continuous contact model

Given the flexible-joint robot dynamics (1)-( 3), the contact dynamics -including impacts-can be incorporated for numerical simulation purposes by adding a continuous contact model. This is a direct extension of what is called Model A in [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF] to the case of a robot manipulator.

For reasons that will shortly be apparent, the exponentiallyextended Hunt-Crossley model [START_REF] Carvalho | Exact restitution and generalizations for the Hunt-Crossley contact model[END_REF] is chosen as the specific continuous contact model in simulation of Model A. Similar to the conventional Hunt-Crossley model [START_REF] Hunt | Coefficient of restitution interpreted as damping in vibroimpact[END_REF] used in [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF], the exponentially-extended Hunt-Crossley model prevents nonphysical force jumps when establishing contact with nonzero velocity. However, in addition to that, the exponentiallyextended Hunt-Crossley model prevents non-physical sticky contact forces to occur upon forced separation at high velocity and is thus preferred here. Explicitly, the contact model for inelastic impacts is given by

f N =      0, if δ < 0, k c δ η + d c δ η δ, if δ ≥ 0 and δ ≥ 0, k c exp dc kc δ δ η if δ ≥ 0 and δ < 0. (7)
In [START_REF] Khurana | Learning to hit: A statistical dynamical system based approach[END_REF], δ is the penetration depth, i.e., the relative displacement between the contacting surfaces in the contact normal direction, k c is the nonlinear stiffness, d c is the damping factor, and η is Hertz' geometry-dependent contact parameter. Given the expression for f N in (7), the contact and impact effects can be added to (2) by setting τ ext = J N (q) ⊤ f N , where J N (q) is the row of the contact Jacobian associated with the linear velocity in the contact normal direction.

C. Equivalent rigid-robot model and rigid impact map

A description of the rigid-joint robot dynamics based on the flexible-joint robot dynamics is used to predict the postimpact response via a rigid impact map (denoted as Model C Ideal gearbox b ρ

Reflected motor inertia

Link inertia

Input from the motor (τ act , θ)

k d τ + dk -1 τ (τ J , q) Fig. 2. Schematic diagram of a flexible robotic joint.
in [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF]). This model is obtained by substituting τ in (2) by its expression as a function of u, τ , and θ in (6), yielding

M(q)q + h(q, q) = u -B θ θ + τ ext . (8)
The above relationship is valid for any value of joint stiffness K. Therefore, assuming that the limit K → ∞ would exist, implying also that q ≡ θ, then one would obtain

(M(q) + B θ ) q + h(q, q) = u + τ ext . (9)
Equation ( 9), that we name the rigid-joint robot model associated to the torque-controlled flexible-joint robot model (1)-( 4), has a mass matrix which includes the apparent motor inertia B θ , that in turn depends on the actual motor inertia B ρ and the low-level torque control gain K T . Note how the mass matrix M+B θ appearing in ( 9), which we propose here in the context of impact dynamics for the first time, is different from the standard choices of M (ignoring the motor inertia, assuming full link-motor decoupling) or M + B ρ (adding the full scaled motor inertia to the links). In Sections III and IV, we will present numerical and experimental evidence that M + B θ provides indeed the best agreement with post-impact dynamics. Equation ( 9) has already appeared in [START_REF] Ott | Cartesian impedance control of redundant and flexible-joint robots[END_REF]Chapter 5] in the context of singular perturbation analysis of the torquecontrolled flexible-joint robot dynamics (1)-( 4). However, no contact dynamics was included in [27, Chapter 5] and a formal proof of the derivation of the reduced order model ( 9) from singular perturbation theory with intermittent contacts and impacts is left for future investigation (a possible direction would be to explore the results presented in [START_REF] Akhmet | Chattering as a singular problem[END_REF]). The observation that the mass matrix is corrected by the apparent motor inertia is important for the results discussed in Sections III and IV. However, it is important to recall that the rigid-joint robot model ( 9) is derived under the assumptions that (i) the lowlevel control loop (4) reduces the motor dynamics (3) to [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF], and (ii) the motor-side joint accelerations θ are equivalent to the link-side joint accelerations q. The validity of these assumptions especially under hard impacts is unclear, due to the short time scale typically associated with impacts (usually a few milliseconds) and the finite bandwidth of the low-level control loop. As previously mentioned, in Sections III and IV extensive numerical and experimental validation is performed to assess the ability of the rigid-robot model ( 9) to capture the gross velocity jump of the model ( 1)-( 4) after an impact, without explicitly modeling the flexible joint dynamics.

The rest of this subsection is devoted to the derivation of the impact map corresponding to the rigid-joint robot model expressed by [START_REF] Proper | Aim-aware collision monitoring: Discriminating between expected and unexpected post-impact behaviors[END_REF], following the standard approach in nonsmooth mechanics [START_REF] Glocker | An introduction to impacts[END_REF], [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]. To this end we start with the following impact equation (cf. [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF] for more details)

M θ ( q+ -q-) = J ⊤ N Λ N + J ⊤ T Λ T , (10)
where M θ = M(q) + B θ , Λ N and Λ T represent the normal and tangential impulsive forces, q+ ∈ R n the post-impact joint velocity, and q-∈ R n the ante-impact joint velocity. Equation ( 10) can be simplified using the definition of the frictional force

Λ T = -µu T Λ N , (11)
with µ the friction coefficient and u T the 2-dimensional direction of the frictional force determined using the anteimpact Cartesian velocity along the xy-plane, v - xy , as

u T = v - xy ||v - xy || . (12)
Substituting ( 11) into [START_REF] Van Steen | Quadratic programmingbased reference spreading control for dual-arm robotic manipulation with planned simultaneous impacts[END_REF], we obtain

M θ ( q+ -q-) = (J ⊤ N -µJ ⊤ T u T )Λ N , (13)
or equivalently

q+ = q-+ M -1 θ J⊤ µ Λ N , ( 14
)
where Jµ is defined as J N -µu ⊤ T J T . Considering a perfectly inelastic impact law in the Cartesian space

J N q+ = 0, (15)
and substituting ( 14) into ( 15), we get

Λ N = -(J N M -1 θ J⊤ µ ) -1 J N q-. (16)
Finally, substituting ( 16) into ( 14) leads to the post-impact velocity predictor that we will use in this work

q+ = I 7 -M -1 θ J⊤ µ (J N M -1 θ J⊤ µ ) -1 J N q-. (17)
Compared to the original expression for the post-impact velocity predictor in [6, Equation ( 5)], [START_REF] Van Steen | Robot control for simultaneous impact tasks via quadratic programming-based reference spreading[END_REF] contains two additional effects: first, the explicit dependence on K T and B ρ appearing via the mass matrix M θ ; second, the effect of tangential friction via the coefficient µ.

III. RIGID-ROBOT MODEL VALIDATION This section is devoted to presenting numerical evidence that the rigid-robot model ( 9) associated with the flexible-joint robot with joint-torque control (1)-( 4) is able to accurately predict the gross velocity jump after an impact. More precisely, the post-impact velocity prediction is obtained by employing the rigid-robot impact map [START_REF] Van Steen | Robot control for simultaneous impact tasks via quadratic programming-based reference spreading[END_REF] with µ = 0 (for simplicity).

In this simulation study, we consider a planar RRRmanipulator, based on joints 2, 4, and 6 of the Franka Emika robot. We use a task-based QP controller with a sample rate of 1 kHz to generate the desired joint torques u needed to follow a linear trajectory from an initial point to a point below the contact surface, to impact the surface with a desired impact velocity and approach angle. The low-level controlloop (4), which is implemented with a sample rate of 4 kHz (mimicking the robot's internal low-level torque controller), uses the desired joint torques u to compute the real actuation torques based on the motor accelerations and the measured joint torques. The torque measurements are filtered with a firstorder low-pass filter with a cut-off frequency of 300 Hz to include the lag effect that the sensors add on the real robot.

The dynamic parameters, i.e., link lengths, masses, rotational inertias, and center of mass positions, are determined by lumping the corresponding parameters identified in [START_REF] Gaz | Dynamic identification of the Franka Emika Panda robot with retrieval of feasible parameters using penalty-based optimization[END_REF] of the Franka Emika robot for the 1) second and third, 2) fourth and fifth, and 3) sixth, seventh and end-effector links. The values are used in the equations of motion derived with the Euler-Lagrange method using the MATLAB symbolic toolbox. The values of the flexible joint parameters such as joint stiffness (k i ), joint damping (d i ), and motor inertia (b ρ,i and b θ,i ), are defined as order of magnitude estimates provided by Franka Emika. Finally, the parameters of the continuous contact model are computed as follows. The contact nonlinear stiffness (k c ) is computed based on Hertz contact theory for sphere-half space steel-wood interaction, and the contact damping factor (d c ) is chosen empirically by observing for which value the simulations resulted in fully inelastic impact. The values of these parameters are available on [START_REF] Weekers | Refined post-impact velocity prediction for torque-controlled flexible joint robots[END_REF] for the interested reader.

The simulations clearly support the claim that the rigidjoint robot model ( 9) with the impact map ( 17) is capable of capturing the main post-impact response of a flexible-joint robot, as long as the right inertia matrix M θ = M+B θ is used. In particular, the following two key observations can be made. First, the magnitude of the velocity jump observed for the associated rigid-robot model closely resembles the magnitude of the velocity jump of the flexible-joint model with continuous contact. Second, the flexible-joint model response oscillates around the response of the rigid-robot model. To illustrate these facts on a specific simulation, the joint velocities for the representative impact scenario with an approach angle of 30 degrees with respect to the horizontal plane and an impact velocity of 0.1 m/s in the direction of motion visualized in Figure 3 are shown in Figure 4. Besides the aforementioned observations, simulations with different inertia matrices in the impact map (using M, M + B θ , or M + B ρ as values for M θ in ( 17)) show that using M θ = M + B θ indeed leads to the most accurate rigid-robot model. Comparing the different post-impact responses one observes that the magnitude of the velocity jump does not match the velocity jump of the flexiblejoint model in case M or M + B ρ are used, as shown in Figure 4, nor does the resulting post-impact response in these cases describe the response around which the flexible-joint model oscillates. IV. EXPERIMENTAL VALIDATION In the following, given the numerical evidence provided in Section III, we detail the validation of the post-impact velocity predictor [START_REF] Van Steen | Robot control for simultaneous impact tasks via quadratic programming-based reference spreading[END_REF] on experimental data, including the new method to extract the equivalent rigid body post-impact motion from post-impact oscillatory responses. These validation results provide an indication of the expectable accuracy, and two ablation studies showing the importance of choosing the correct mass matrix (recall the M + B ρ vs. M + B θ discussion in the previous section) as well as the relevance of accounting for the tangential friction force, via friction coefficient µ in [START_REF] Van Steen | Robot control for simultaneous impact tasks via quadratic programming-based reference spreading[END_REF]. A set of 50 experiments was recorded where a 7 DoF Franka Emika robot arm with a spherical aluminum end-effector impacted a horizontal plate made of steel, as illustrated by the sequence shown in Figure 1. During the recording process 10 different initial robot configurations along with the respective target force sent to the Franka Control Interface (FCI) 4 were set. The target force applied at the end-effector is transformed via the end-effector Jacobian to joint-torque commands, which are then sent to the robot via the FCI. The target force is set to zero before impact, causing the robot to move freely towards the steel plate under pure gravity compensation and thus being influenced during impact only by the low-level torque control loop at 4 kHz, but not by any external higher-level controller. The impact causes a jump in the velocity signal, with a postimpact response with short-lived oscillations, as illustrated by the joint velocity of a representative impact experiment in Figure 5. These oscillations damp out after about 100 ms, after which the arm continues to move until being stopped by friction in both its own joints and the contact with the surface.

For each of the 10 sets of initial conditions, where each set defines an experiment type, 5 repetitions were run in order to prove the repeatability of the experiment type and little sensitivity of the prediction for very similar responses. These 10 groups of experiments can be subdivided in three main subtypes depending on motion direction, each in turn subdivided according to different end-effector orientations. An overview of all types of experiments and additional details are given in Table I. These experiments are similar to the simulation shown in Figure 3, especially for experiments of type 1 to 4 (Table I), because the motion direction and the

Post-impact fit to q6 Fig. 5. Joint velocity of joint 6, q6 , for a representative impact experiment (type 1 in Table I), together with the joint position q 6 (black line), postimpact polynomial fit and its derivative (dashed lines), and gross velocity jump estimation q+ 6 .

motion plane are the same but with different approach angles, i.e., 55°for type 1, 30°for type 2, and 26°for types 3-4. Moreover, the ante-impact velocities for these experiments are within a span of 0.1 to 0.2 m/s.

A. Impact detection and post-impact velocity estimate

A practical offline impact detector was implemented for the automatic identification of the impact time, to avoid manual data annotation. The detector identifies sharp changes in the joint velocity, q, using a joint acceleration estimate, computed using central differences to eliminate phase lag. The detector was applied in parallel to joints 1 to 6 because it often retrieved false positives when applied to joint 7. From the six estimates the one that happens the earliest was selected.

In regard to the impact time detection, this study also revealed that different joints exhibit a gradual lag in the time of impact, with joints farther from the impact surface presenting a larger lag. The simplifying modeling assumption that velocity is jumping at each joint at the same time is therefore not fully satisfied at millisecond scale and, therefore, having picked for simplicity just one impact moment, one can see small inaccuracies as visible in Figure 5. Therein, the selected impact time used for polynomial fitting in all joints is slightly delayed with respect to when the velocity really starts to change for the specific joint 6. This leads to small inaccuracies in extracting the ante-and post-impact velocities qand q+ for some joints, that are however at par with average velocity prediction errors z-axis aligned with gravity 9

Lateral (negative direction along y-axis) -30°about x-axis 10 z-axis aligned with gravity (≈ 1 • /s according to the error definition in the following subsection) and therefore have been deemed not worth of further refinement. Following the approach that was proposed in [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF], we are mainly interested in predicting the gross velocity jump after an impact. Therefore, we need a method to filter out the impact-induced oscillations shown, e.g., in Figure 5, and keeping lower frequency content. Differently from [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF], where a nonlinear least-squares fitting was performed on the velocity signal, here a simpler and more robust approach based on a polynomial least-squares fit on the post-impact joint position time signal is used. The gross velocity jump is estimated by simply evaluating the derivative of the fitted polynomial at the estimated impact time. The improved robustness of this method is provided by the fact that it results in a linear least-squares problem, which is convex. Hence, its performance does not depend on the parameter initialization and it is not prone to getting stuck in local minima, as opposed to the nonlinear least-square fitting used in [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF].

Given the time-scale of the post-impact oscillation at hand, the fitting is applied over a window that spans 100 ms, starting at the estimated impact time. This window was selected as a result of the qualitative analysis of the velocity signals throughout the 50 recorded experiments, that showed that 100 ms does correspond in all cases to the approximate duration of the oscillations present in the post-impact response of the robot (cf. oscillations in Figure 5). We have found that a third order polynomial is sufficient in capturing the evolution of joint position trajectories.

As in [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF], the gross velocity jump estimation provided by the polynomial least-squares fit on the position data is used as ground truth in the assessment of the prediction accuracy as explained later in detail.

B. Selection of motor inertia constant

The numerical validation presented in Section III shows that using M + B θ as the inertia matrix within the post-impact velocity predictor delivers the most accurate results. In the following, we provide experimental evidence that this is indeed the case, showing that the naïve choice of using M + B ρ leads to worse results (as well as using M, a fact that was not understood in [START_REF] Aouaj | Predicting the postimpact velocity of a robotic arm via rigid multibody models: An experimental study[END_REF]).

The predictions over the whole set of 50 experiments were assessed based on the quantitative comparison procedure proposed in [6, Section IV] that enables a quantitative evaluation of the prediction by comparing the experimental post-impact data, containing oscillations, and the post-impact velocity estimated by the predictor derived using nonsmooth mechanics theory. In this comparison we used the absolute error metric

e i = | q+ i - △ q + i |, (18)
where, for each joint i, q+ i is the virtual rigid-body postimpact velocity obtained via the polynomial fit procedure of Section IV-A on experimental data and △ q + i is the predicted rigid-robot post impact velocity for joint i obtained via [START_REF] Van Steen | Robot control for simultaneous impact tasks via quadratic programming-based reference spreading[END_REF].

The influence of the selected dynamic model motor inertia on the prediction of the post-impact velocity can be assessed quantitatively joint-wise. Comparing the mean values obtained for the experiments confirmed that the prediction based on

q- i q+ i △ q + i B ρ △ q + i B θ
Fig. 6. Mean values of the virtual rigid-body post-impact velocity, q+ i (blue line), the measured ante-impact velocity, qi (black line), and the predictions, △ q + i , for joints 1 to 7 using either M θ = M+B θ (green) or Mρ = M+Bρ (red) (averaged over 5 experiments of type 1, see Table I).

B θ is more accurate overall than the prediction based on B ρ . This means that the predictions using B θ are in general closer to the post-impact velocity obtained via the polynomial fit, as illustrated by the representative results shown in Figure 6, where an experiment of type 1, see Table I, is depicted. In both predictions, surface friction at impact is accounted for (the importance and effect of surface friction on the post-impact velocity prediction will be discussed later).

For each of the 10 types of experiments reported in Table I, a comparison of the absolute prediction error using B θ and B ρ is reported in Table II. The errors are averaged over the 5 experiment repetitions. The table shows that for most joints there is not a significant difference in the absolute prediction error when using B θ and B ρ in [START_REF] Van Steen | Robot control for simultaneous impact tasks via quadratic programming-based reference spreading[END_REF]. We have highlighted the cells in the table where the difference between the magnitudes of the absolute error in the predictions for B θ and B ρ goes over a threshold of 1.0 • /s. These cells represent the essential differences. This threshold is selected based on a qualitative analysis of plots such as the one displayed in Figure 6, where differences below 1.0 • /s are hardly noticeable. For error prediction differences above this threshold, we highlight in green the case where the prediction based on B θ outperforms that of using B ρ and in red the opposite case.

Although there is not a significant difference in the prediction accuracy when considering all joints, for joint 6, which as expected is the most affected by the impact, the inclusion of B θ in the predictor improves the accuracy by at least a factor of ≈ 2 (1.88) for all experiments where the difference in the prediction is significant (7 out of 10). This provides evidence that a prediction based on B θ is to be preferred over a prediction based on B ρ . To provide the reader with the ability to also assess the relative error in the prediction, Table II provides, besides the absolute error, the ante-and post-impact velocities extracted from experimental data.

It has to be reported that the prediction based on B θ appears sometimes (5 out of 10 experiments) to worsen the prediction for joint 5, with an error difference of about 2 • /s (red cells in the respective column of Table II). Our claim is that (this joint's) motor static friction -currently not accounted for in our nonsmooth modeling-plays a role in this apparent inconsistency, but a thorough investigation of this fact is left for further research. This decrease in prediction accuracy for joint 5 is however a smaller effect when compared with the gain in prediction accuracy that one obtains in joint 6.

C. Robot-surface tangential Coulomb friction

A value of 0.3 was identified for the friction coefficient between the spherical aluminum end-effector and the impacted steel plate. This parameter was tuned through the minimization of the prediction error for one of the experiments (type 4 in Table I) as a function of the selected friction coefficient included in Jµ for the predictor expression [START_REF] Van Steen | Robot control for simultaneous impact tasks via quadratic programming-based reference spreading[END_REF]. The friction coefficient retrieved by the optimization was the same using either of the two model motor inertia constants, i.e., either B θ or B ρ . Moreover, the identified magnitude of the friction coefficient was verified using a force gauge, by fixating one end to the spherical aluminum end-effector and pulling from the other end of the gauge. This measurement provided a similar value (difference below 10%).

The tuning of the friction coefficient already shows that a minimum is obtained at a value that is different from zero, but in order to provide evidence that the addition of friction to the impact modeling is providing a substantial increase in prediction accuracy, we report the prediction errors corresponding to the case where no friction is considered (µ = 0) and the one where µ = 0.3. In both predictions we employed M θ = M + B θ as the inertia matrix.

The predictions including friction are much closer to the virtual rigid-body post-impact velocity obtained in experiments, as illustrated by the representative mean results of the experiments of type 4 in Figure 7. In this figure, the green dots representing the predictions with friction are closer to the virtual rigid-body post-impact velocity, shown in blue, than the predictions without friction, shown in red. Furthermore, a comparison of the average absolute prediction error with and without friction shows that the inclusion of the tangential Coulomb friction term significantly improves its accuracy, as illustrated by the values provided in Table III. The absolute prediction error is defined, as before, according to [START_REF] Wang | On inverse inertia matrix and contact-force model for robotic manipulators at normal impacts[END_REF].

Cells highlighted in green indicate a significantly better performance (> 1.0°/s) of the predictor that accounts for friction, and cells in red indicate that the opposite is true. In this q + i without friction △ q + i with friction Fig. 7. Mean values of the virtual rigid-body post-impact velocity, q+ i (blue line), the measured ante-impact velocity, qi (black line), and the predictions, △ q + i , for joints 1 to 7 with (green) and without (red) considering tangential friction (averaged over 5 experiments of type 4, see Table I).

Fig. 1 .

 1 Fig. 1. Impact experiment example, recorded at Franka Emika: (1) The robot stands still at initial configuration; (2) The end-effector moves downwards and away from the robot base, as indicated by the red velocity arrow; (3) It impacts the steel plate; (4) After impact, it slightly bounces and then slides.

Fig. 3 .

 3 Fig. 3. Simulated impact scenario with an approach angle of 30 degrees and an impact velocity of 0.1 m/s. Time evolves from left to right, and from top to bottom. The impact location is indicated by the vertical dashed line.

Fig. 4 .

 4 Fig.4. Comparison of the post-impact joint velocities for a flexible-joint robot with a continuous contact model and three instances of a rigid-joint robot with different inertia matrices for the impact map (impact as in Figure3). Results obtained with M + B θ yield the best asymptotic behaviour w.r.t. the flexible joints + continuous contact model which serves as a reference.

  THE MOTION DIRECTION IS DEFINED W.R.T. THE BASE FRAME DISPLAYED IN FIGURE 1, WHEREAS THE ORIENTATION IS DEFINED W.R.T. A FRAME LOCATED AT THE CENTER OF THE END-EFFECTOR SPHERE AND ALIGNED WITH THE BASE FRAME along x-axis) -30°about y-axis 2 -20°about y-axis 3 -10°about y-axis 4 z-axis aligned with gravity 5 Backward (negative direction along x-axis) 30°about y-axis 6 20°about y-axis 7 10°about y-axis 8

TABLE I TYPES

 I OF PERFORMED IMPACT EXPERIMENTS (EACH REPEATED 5 TIMES), WITH DIFFERENT MOTION DIRECTIONS AND ORIENTATIONS (ROTATIONS) PRIOR TO IMPACT.

Further details on https://frankaemika.github.io/docs/overview.html.

V. CONCLUSION AND FUTURE WORK A refined post-impact velocity predictor for torquecontrolled flexible-joint robots, that aimed to improve the prediction accuracy, has been presented. Through simulation and experimental validation, the proportional gain of the lowlevel torque control loop has been shown to have an important effect on the impact dynamics of this type of robots. The inclusion of this effect in conjunction with the surface friction brings the absolute error of the post-impact velocity prediction down from tens of degrees per second to values below 7.0 °/s as shown in Tables II andIII. This reduction is achieved with respect to the previously proposed method by the authors. Moreover, an equivalent, but simpler and more robust virtual rigid-robot post-impact velocity estimate has been introduced, based on polynomial fitting. Follow-up studies will explore methods to further improve the accuracy of the impact time detector, as well as validating post-impact velocity predictors for simultaneous contact locations, focusing on inelastic impacts enabled by damped and compliant robot hardware that is becoming available and is designed for dynamic interaction.


• Theory Darwinienne de l'évolution par sélection naturelle.

• Evolution graduelle de caractères quantitatifs sous l'effet de la sélection naturelle. • Mécanisme de sélection naturelle : « Survival of the fittest » : processus d'évolution conduits les organismes a être adaptés à leur environnement. • « On the Origin of Species », 1869, C Darwin.

Les travaux de Darwin

Définitions

• La génétique des populations correspond à l'étude de la variation génétique au sein des populations, et aussi à caractériser l'évolution de la structure génétique des populations au cours du temps.

• La génétique des populations a été élaborée par Fisher Haldane et Wright, basée sur des mathématiques appliquées à l'hérédité mendélienne.

UE Genet. Pop. 24/1/2023 p.3

Les travaux précurseurs

• Suite aux travaux de Georges Mendel (1865), repris à partir de 1900, a donné lieu à la naissance de la génétique. • Travaux de Johannsen (1909) sur les graines de haricots (Phaseolis vulgaris) : ségrégation de nombreux genes pouvant expliquer la variation du phénotype "taille de haricot" pour expliquer l'hérédité. • Termes "Génotype" et "Phénotype".

W. Johannsen • Electrophorèse : migration de protéines (ou acides nucléiques) dans un champ électrique. • Principe de l'électrophorèse : mise en évidence de variation (dans la chaine d'acides aminés) qui vont affecter la mobilité de la protéine. • On observe ainsi des différences de migration entre individus si les variations dans la séquence protéique affecte la mobilité.

Pinsons

  UE Genet. Pop. 24/1/2023 p.2

•Ø

  Terme fréquence de gènes dans les populations avec observations que de 'nouveaux génotypes' pouvaient remplacer les 'anciens génotypes'. • La génétique des populations est alors née, dans les années 1920 : conciliation théorie darwinienne de l'évolution et la transmission des caractères héréditaires (RA Fisher, JBS Haldane, S Wright). • Construction de modèles mathématiques décrivant l'évolution des populations. • But étant de décrire l'évolution des changements, au sein et entre les populations. On observe un polymorphisme de taille des fragments de restriction entre les individus A et B (Restriction Fragment Length Polymorphism : RFLP).

•

  Protéine visible par stimulation d'une activité enzymatique et coloration. • Pour une migration d'ADN, des différences de taille de fragments d'ADN induiront des temps de migration différents, ADN visible par un intercallant de l'ADN tel que BET (Bromure d'ethidium) ou SYBR safe (à base de cyanine).

	Exemple 1 : RFLP Notion d'echantillonnage Estimateurs et probabilités Séquençage
	Migration de protéines enzymatiques révélées, espèce diploïde • Observation au sein de populations échantillonnées • Paramètres et estimateurs ADN Séquençage Sanger
		Echantillon monomorphe	'Fixation' = fréquence de 1
			Annotation du génotype de
			chaque individu (A, B, C, D, E, F). Séquençage Illumina
	vrais calculs	estimations et probabilités
	3 allèles Ø Somme des fréquences alléliques a 1 locus (diploïde, bi allélique) : p = 1 -& 3 allèles Echantillon polymorphe Echantillon 𝑞 Séquençage Pacbio
	enzyme monomérique Ø Fréquence de l'allele A1 : Ø Erreur standard : 𝑠 = Population	p p & 𝑞 / 𝑛 Séquençage minION enzyme dimérique
			UE Genet. Pop. 24/1/2023 p.11 UE Genet. Pop. 24/1/2023 p.12

UE Genet. Pop. 24/1/2023 p.10 Exemple 2 : Allozymes UE Genet. Pop. 24/1/2023 p.13 • Notion de phénotype et polymorphisme Polymorphe / Monomorphe UE Genet. Pop. 24/1/2023 p.14 UE Genet. Pop. 24/1/2023 p.15


Hamid Amir
email: hamid.amir@unc.nc

Thomas Crossay
email: thomasjc.crossay@gmail.com

Functional and practical importance of AMF mixed inoculants for plant development

Keywords: Arbuscular mycorrhiza, AMF mixture, multispecies inoculum, functional complementarity, commercial inoculant

Until a recent past, arbuscular mycorrhizal fungi (AMF) were generally used as single species inoculants carefully selected to enhance plant growth. However, in ecological perspective, it is expected that a mixture of different AMF species would be more efficient to occupy the ecological niches of the symbiosis and would adapt better to environmental fluctuations. A limited number of studies have reported complementary effects of different AMF taxa on plant development.

Experiments were performed in different conditions. In most cases, combinations of AMF showed better results than single species. Particularly clear results were obtained in ultramafic soils characterized by multiple stress factors: low level of main mineral elements, high concentrations of potentially toxic metals (Ni, Co, Cr, Mn), and critically low Ca/Mg ratio. These studies reported synergistic effects of mixed isolates belonging to different families. The most performant mix showed quantities of P and K absorbed per plant more than 10 times higher than non-inoculated plants and more than two times higher than the most performant AMF isolate. Inversely, the translocation factors for potentially toxic metals (particularly Ni and Co) were reduced three to ten times. Mathematical modelization of the effects of 12 AMF inoculants, including six mixes on plant growth, revealed that the predicted value of the performance based on the properties of the different AMF was very high (R2 = 0.90; p < 0.00001), indicating that the improvement of plant biomass was nearly totally explained as a resultant of the complementary effects of the 5 isolates. Field experiments in natural conditions and commercial trials confirmed the greenhouse results obtained in sterilized soils and revealed large positive effects on plant development of selected AMF mixes. For such success, it is suggested that the isolates used for the mixed inoculants must belong to different families, and must be selected for their synergistic effects on plant development.

Introduction

Arbuscular mycorrhizal fungi (AMF) are ubiquitous plant symbionts with great potential as biofertilizers for biotechnological improvement of plant development in environmental and agricultural sustainable practices. The mastery of AMF inoculants is an important goal for the coming decades [START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF][START_REF] Berruti | Arbuscular Mycorrhizal Fungi as Natural Biofertilizers: Let'sBenefitfromPast Successes[END_REF]. This mastery can help solve global problems such as increasing soil degradation, concerns related to chemical fertilizers such as water pollution and phosphorus depletion, and adaptation to climate change. However, due to the high complexity and diversity of the soils, different difficulties are still limiting commercial use of AMF to optimize agroecosystem services [START_REF] Sportes | A historical perspective on mycorrhizal mutualism emphasizing arbuscular mycorrhizas and their emerging challenges[END_REF].

The effects of AMF on plant development are highly variable in relation to plant taxa, fungal isolates, and substrate composition [START_REF] Wang | Phylogenetic distribution and evolution of mycorrhizas in land plants[END_REF][START_REF] Hoeksema | A meta-analysis of context-dependency in plant response to inoculation with mycorrhizal fungi[END_REF][START_REF] Smith | Plant performance in stressful environments: interpreting new and established knowledge of the roles of arbuscular mycorrhizas[END_REF][START_REF] Faye | Evaluation of commercial arbuscular mycorrhizal inoculants[END_REF][START_REF] Koch | Evolutionary asymmetry in the arbuscular mycorrhizal symbiosis: conservatism in fungal morphology does not predict host plant growth[END_REF]. The efficiency of the isolates in plant growth promotion can vary from negative to highly positive effect [START_REF] Smith | Fresh perspectives on the roles of arbuscular mycorrhizal fungi in plant nutrition and growth[END_REF]. The number of studies concerning the efficiency of AMF inoculants has increased in the last decade. A search on Google Scholar with the keywords "arbuscular mycorrhizal inoculants" and "arbuscular mycorrhizal inoculum" from 2000 to 2010 gives a result of 402 papers; from 2011 to 2021 the search leads to 579 papers. The number of companies producing AMF inoculants grew from 10 in 1990 to 75 in 2017 [START_REF] Sportes | A historical perspective on mycorrhizal mutualism emphasizing arbuscular mycorrhizas and their emerging challenges[END_REF]).

Thus, from the 2000s onwards, more and more companies developed end-use formulations with varied success (Duffy and Cassels 2000;[START_REF] Ijdo | Methods for large-scale production of AM fungi: past, present, and future[END_REF][START_REF] Faye | Evaluation of commercial arbuscular mycorrhizal inoculants[END_REF][START_REF] Berruti | Arbuscular Mycorrhizal Fungi as Natural Biofertilizers: Let'sBenefitfromPast Successes[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF][START_REF] Srivastava | Review of patents for agricultural use of arbuscular mycorrhizal fungi[END_REF]. [START_REF] Srivastava | Review of patents for agricultural use of arbuscular mycorrhizal fungi[END_REF] reported an analysis of 696 patents published in the two last decades showing that AMF have been used consistently as biofertilizers and bioremediators. However, in a meta-analysis of 631 papers relating to mycorrhiza inoculation, from 1970 to 2014, [START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF] showed that there is yet a lot to do before mastering correctly commercial inoculants at large scale. The inefficiency of different commercial AMF inoculants is sometimes simply due to the absence of viable propagules [START_REF] Salomon | Global evaluation of commercial arbuscular mycorrhizal inoculants under greenhouse and field conditions[END_REF]) related to bad technical practices. But there are also more complex difficulties that need new investigations [START_REF] Berruti | Arbuscular Mycorrhizal Fungi as Natural Biofertilizers: Let'sBenefitfromPast Successes[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF].

Until a recent past, AMF were generally commercialized as single species inoculants carefully selected to enhance plant growth and concerned only a few well-studied fungal species relatively easy to produce, such as Rhizophagus intraradices, R. irregularis, Funneliformis mosseae, and Claroideoglomus etunicatum [START_REF] Ijdo | Methods for large-scale production of AM fungi: past, present, and future[END_REF][START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF]Berruti et al. 2017;[START_REF] Basiru | Analysis of arbuscular mycorrizal fungal inoculant benchmarks[END_REF]. However, as stressed by [START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF], there is no evidence that a given selected AMF could be "superior" to the others in diverse soil and environmental conditions. This use of a single "performant" isolate is now undergoing change because in ecological perspective, it is expected that a mixture of different AMF species would be more efficient to occupy the ecological niches of the symbiosis and adapt better to environmental fluctuations. This appeared clearer when it was evidenced that each plant can host many AMF species [START_REF] Jansa | Soil Tillage Affects the Community Structure of Mycorrhizal Fungi in Maize Roots[END_REF][START_REF] Vandenkoornhuyse | Active root-inhabiting microbes identified by rapid incorporation of plant derived carbon into RNA[END_REF]. From 2001 to 2015, about 21% of AMF inoculation experiments concerned multispecies inoculants, and 18 % tested multispecies and monospecies inoculants [START_REF] Berruti | Arbuscular Mycorrhizal Fungi as Natural Biofertilizers: Let'sBenefitfromPast Successes[END_REF]. However, a limited number of studies reported complementary effects of different AMF taxa on plant development. [START_REF] Kiers | Reciprocal Rewards Stabilize Cooperation in the Mycorrhizal Symbiosis[END_REF] suggested that plant species select AMF partners by transferring more carbohydrates to the symbionts that are better for their development. Inversely, AMF reinforce this selected relation by transferring more nutrients to plants that feed them better. These selective interactions suggest how AMF diversity linked to one plant can reach a balanced state optimizing the functional traits of the symbiosis. In practice, only a part of the comparative studies showed that AMF mixed inoculants were more efficient than single inoculants [START_REF] Jin | Impact of arbuscular mycorrhizal fungal inoculants on subsequent arbuscular mycorrhizal fungi colonization in pot-cultured field pea (Pisum sativum L.)[END_REF][START_REF] Chen | Combined Inoculation with Multiple Arbuscular Mycorrhizal Fungi Improves Growth, Nutrient Uptake and Photosynthesis in Cucumber Seedlings[END_REF][START_REF] Yang | Taxonomic resolution is a determinant of biodiversity effects in arbuscular mycorrhizal fungal communities[END_REF][START_REF] Crossay | Is a mixture of arbuscular mycorrhizal fungi better for plant growth than single-species inoculants?[END_REF][START_REF] Fathalla | Inoculation of drought-stressed wheat plant (Triticum aestivum L.) with single and combined inoculants of Arbuscular mycorrhizal fungi[END_REF][START_REF] Martignoni | Coinoculation with arbuscular mycorrhizal fungi differing in carbon sink strength induces a synergistic effect in plant growth[END_REF]. Different factors can explain the variations in conclusions concerning the AMF-mixed inoculants in relation to soil or substrate variability, specific conditions of the experiments, AMF origin, and selection criteria of the isolates.

It is then of major importance to synthetize studies that can allow to understand more clearly these variations and their consequences on the efficiency of AMF inoculants. This chapter aims to highlight the interest of AMF mixed inoculants and how they should be selected. A particular focus is made on using mixed inoculants in extreme soils for plants facing multi-stress environments. These environments are considered preferential targets for commercial use of AMF inoculants because of their clear effects in these conditions [START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF]. This is particularly the case of ultramafic soils characterized by nutrient deficiencies and high concentrations of potentially toxic metals [START_REF] Brooks | Serpentine and its vegetation: a multidisciplinary approach[END_REF][START_REF] Jaffré | Conditions de milieu des terrains miniers[END_REF].

Interest of mixed inoculants

Main problems posed by inoculation

In natural conditions, AMF inoculant faces niche competition with local AMF communities for root colonization [START_REF] Jefwa | Benefits and potential use of arbuscular mycorrhizal fungi (AMF) in banana and plantain (Musa spp.) systems in Africa. Proc. IC on Banana & Plantain in Africa[END_REF][START_REF] Faye | Evaluation of commercial arbuscular mycorrhizal inoculants[END_REF]) and can also be affected by other microbial taxa in the rhizosphere, a very dense microenvironment with a high level of antagonism and stress [START_REF] Hinsinger | Rhizosphere: biophysics, biogeochemistry and ecological relevance[END_REF][START_REF] Miransari | Arbuscular Mycorrhiza and soil microbes[END_REF][START_REF] Velásquez | Mycobization as a biotechnological tool: a challenge[END_REF]. Thus, the success in the field of a selected AMF inoculant is not guaranteed, particularly for mono-species ones and non-native isolates generally less adapted to local conditions (Henkel e al. 1989;[START_REF] Faye | Evaluation of commercial arbuscular mycorrhizal inoculants[END_REF][START_REF] Pellegrino | Enhancing ecosystem services in sustainable agriculture: biofertilization and biofortification of chickpea (Cicer arietinum L.) by arbuscular mycorrhizal fungi[END_REF]. To reduce this difficulty, AMF inoculation is generally performed with a sterile or poorly colonized substrate in a nursery (Velàszquez and Cabello, 2010;[START_REF] Amir | Arbuscular mycorrhizal fungi and sewage sludge enhance growth and adaptation of Metrosideros laurifolia on ultramafic soil in New Caledonia: A field experiment[END_REF]) before transferring the plants to the field. Even in these conditions, AMF inoculant can decline in roots of inoculated plants, few months to few years after transfer [START_REF] Pellegrino | Establishment, persistence and effectiveness of arbuscular mycorrhizal fungal inoculants in the field revealed using molecular genetic tracing and measurement of yield components[END_REF][START_REF] Amir | Arbuscular mycorrhizal fungi and sewage sludge enhance growth and adaptation of Metrosideros laurifolia on ultramafic soil in New Caledonia: A field experiment[END_REF]. A progressive decline of the AMF inoculated population was also reported in greenhouse conditions in the presence of a pre-established arbuscular mycorrhizal fungal community [START_REF] Janoušková | Effects of inoculum additions in the presence of a preestablished arbuscular mycorrhizal fungal community[END_REF]. According to this latter study, the decline varied with AMF species and was observed 12 to 24 weeks after inoculation. A combination of different AMF taxa can allow a better adaptation and then better stability in the ecosystem with the occupation of more ecological niches [START_REF] Maherali | Influence of Phylogeny on Fungal Community Assembly and Ecosystem Functioning[END_REF][START_REF] Yang | Taxonomic resolution is a determinant of biodiversity effects in arbuscular mycorrhizal fungal communities[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF][START_REF] Crossay | Is a mixture of arbuscular mycorrhizal fungi better for plant growth than single-species inoculants?[END_REF]. Indeed, as suggested by [START_REF] Kiers | Reciprocal Rewards Stabilize Cooperation in the Mycorrhizal Symbiosis[END_REF], the interactions between AMF communities and plant host result in a selection of a number of AMF taxa that can optimize the symbiosis. Furthermore, [START_REF] Bennett | Trade-offs between arbuscular mycorrhizal fungal competitive ability and host growth promotion in Plantago lanceolata[END_REF] found that AMF isolates that have a high competitive ability can be bad mutualists, and good mutualists can have low competitive ability. A trade-off between these two abilities may then occur between AMF communities during the competition for root colonization. These selective interactions vary depending on soil, plant, and environment [START_REF] Bever | Host-specificity of AM fungal population growth rates can generate feedback on plan growth[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF]. For all these reasons, it is important to test the performance of AMF inoculants, not only in simplified conditions but also in varied natural conditions before their use on a large scale, as also suggested by other authors [START_REF] Faye | Evaluation of commercial arbuscular mycorrhizal inoculants[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF].

Several types of difficulties can explain the variations in conclusions of experiments that compare the efficiency of AMF inoculants. Differences in symbiosis performance can be due to variations in selection criteria used to perform the AMF mixtures. Thus, [START_REF] Maherali | Influence of Phylogeny on Fungal Community Assembly and Ecosystem Functioning[END_REF] found that the more the phylogenetic distance between AMF of the inoculant consortium was, the more the efficiency of the symbiosis was, due to functional complementarity and reduced competition between the isolates. [START_REF] Chen | Combined Inoculation with Multiple Arbuscular Mycorrhizal Fungi Improves Growth, Nutrient Uptake and Photosynthesis in Cucumber Seedlings[END_REF] and [START_REF] Crossay | Is a mixture of arbuscular mycorrhizal fungi better for plant growth than single-species inoculants?[END_REF] also confirmed that AMF consortium consisting of phylogenetically distant AMF species have a better effect than a single isolate or closely related ones. Fathallah (2020) found that an inoculant with two AMF isolates belonging to different families showed better root colonization and was more efficient in increasing wheat grain production than single species or close species. In a meta-analysis, [START_REF] Yang | Taxonomic resolution is a determinant of biodiversity effects in arbuscular mycorrhizal fungal communities[END_REF] reported that the performance of the symbiosis was positively correlated to AMF family richness and not to their species richness. According to [START_REF] Gosling | Evidence for functional redundancy in arbuscular mycorrhizal fungi and implications for agroecosystem management[END_REF], the failure of mixed inoculants compared with single ones is generally linked to functional redundancy of phylogenetically close species.

Conversely, [START_REF] Martignoni | Coinoculation with arbuscular mycorrhizal fungi differing in carbon sink strength induces a synergistic effect in plant growth[END_REF] studied the effects on plant productivity of two AMF isolates differing for their carbon demand (carbon sink strength) and showed their functional complementarity when mixed. The isolate with high carbon sink strength was able to establish a fungal community more quickly and improved the nutrition and the growth of the plant, whereas the isolate with low carbon sink strength inflicted lower carbon costs to the host plant and supported maximal plant productivity once the plant biomass was large.

However, the only consideration of taxonomic and functional resolution may not be sufficient to explain the relative success or failure of a mixed inoculant. For example, [START_REF] Malicka | Differences in the effects of single and mixed species of AMF on the growth and oxidative stress defense in Lolium perenne exposed to hydrocarbons[END_REF] compared the effects of three native AMF isolates belonging to three different families and their mix on the growth of Lolium perenne in pots with a mixture of sand and bentonite contaminated with phenol and polynuclear aromatic hydrocarbons. They showed that mixed inoculant was less performant than the single ones. Here, we can suspect that using a simplified substrate instead of the soil from where AMF were isolated could be a cause of these results. Indeed, the abiotic and biotic conditions, and consequently the substrate's ecological niches, are very different from the original soil. Thus, differences in experimental conditions, particularly AMF origin and soils or substrates used to test them, can influence the relative performance of the inoculants. The relevance of comparisons on a unique soil of AMF isolates of different origins is also questionable for the same reasons [START_REF] Faye | Evaluation of commercial arbuscular mycorrhizal inoculants[END_REF]. [START_REF] Gosling | Evidence for functional redundancy in arbuscular mycorrhizal fungi and implications for agroecosystem management[END_REF] suggested that soils characterized by multiple stress factors would reveal the benefits of AMF diversity more clearly. All these reasons can explain why the effectiveness of commercial inoculants is often not apparent and vary depending on conditions [START_REF] Faye | Evaluation of commercial arbuscular mycorrhizal inoculants[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF]).

Case studies on ultramafic soils

The use of AMF inoculants in stressful environments, particularly for restoring degraded ecosystems and soil remediation, is considered as important [START_REF] Valliere | Preparing for the worst: Utilizing stress-tolerant soil microbial communities to aid ecological restoration in the Anthropocene[END_REF]) and commercially less difficult [START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF]. It could have different advantages, not only for plant growth promotion and adaptation but also to reduce soil erosion, due to soil binding capacity of extraradical mycelium [START_REF] Mozafar | Effect of Heavy Metal Contaminated Shooting Range Soils on Mycorrhizal Colonization of Roots and Metal Uptake by Leek[END_REF][START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF]) and glomalin production [START_REF] Rilling | Arbuscular mycorrhizae, glomalin, and soil aggregation[END_REF][START_REF] Vodnik | The contribution of glomalin-related soil protein to Pb and Zn sequestration in polluted soil[END_REF]. As stressed by [START_REF] Gosling | Evidence for functional redundancy in arbuscular mycorrhizal fungi and implications for agroecosystem management[END_REF], the benefits of AMF mixed inoculants, compared with single species ones, could become clearly apparent when the plant host faces multiple stress factors. Ultramafic soils can then be considered as good models to highlight the interest of mixed inoculants, with at least three types of abiotic stresses [START_REF] Brooks | Serpentine and its vegetation: a multidisciplinary approach[END_REF][START_REF] Proctor | Vegetation and soil and plant chemistry on ultramafic rocks in the tropical Far East[END_REF]; Jaffré and L'Huillier 2010): i) main element deficiencies (particularly P and K), ii) potentially toxic metals (Ni, Co, Cr, Mn), and iii) low Ca/Mg value (< 1) which is a limiting factor because Ca absorption is restricted by the competition with Mg cations. In these ecosystems, AMF symbiosis has been shown to be abundant and essential [START_REF] Perrier | Occurrence of mycorrhizal symbiosesin the metal-rich lateritic soils of the Koniambo massif[END_REF][START_REF] Doubkova | The symbiosis with arbuscular mycorrhizal fungi contributesto plant tolerance to serpentine edaphic stress[END_REF][START_REF] Doubkova | Arbuscular mycorrhizal symbiosis alleviatesdrought stress imposed on Knautia arvensis plants in serpentine soil[END_REF][START_REF] Amir | Arbuscular mycorrhizal fungi from New Caledonian ultramafic soils improve toleranceto nickel of endemic plant species[END_REF][START_REF] Amir | Les bactéries et les champignons du sol surroches ultramafiques[END_REF][START_REF] Lagrange | Mycorrhizal status ofCyperaceae from New Caledonian ultramafic soils: effects of phosphorusavailability on arbuscular mycorrhizal colonization ofCostulariacomosa under field conditions[END_REF].

Five new species of AMF from New Caledonian ultramafic soils, recently described, Rhizophagus neocaledonicus, Acaulospora saccata, A. fragilissima, Scutelospora ovalis, and Pervetustus simplex (Blaszkowski et al. 2017;[START_REF] Crossay | Four new species of arbuscular mycorrhizal fungi (Glomeromycota) associated with endemic plants from ultramafic soils of New Caledonia[END_REF], and a native isolate of Claroideoglomus etunicatum, have been tested on host plants [START_REF] Crossay | Is a mixture of arbuscular mycorrhizal fungi better for plant growth than single-species inoculants?[END_REF][START_REF] Crossay | Combinations of different arbuscular mycorrhizal fungi improve fitness and metal tolerance of sorghum in ultramafic soil[END_REF]. Crossay et al. ( 2019) have compared the effects of these 6 isolates inoculated separately, or in combinations, on the development of Metrosideros laurifolia (an endemic shrub), in greenhouse conditions. The tested isolates differed for their performance features: root colonization, sporulation, effects on N, P, K, and Ca plant uptake, influence on Ca/Mg value of plant organs, and effect on potentially toxic metal translocation. Two AMF mixes showed synergistic effects on the growth and adaptation of the plant (Fig. 1a). The most performant treatment (mix 5) was a consortium of 5 isolates belonging to 5 different families (Glomeraceae, Claroideoglomeraceae, Acaulosporaceae, Gigasporaceae and Pervertustaceae), confirming that combinations with phylogenetically distant isolates are more efficient, as suggested by other authors [START_REF] Chen | Combined Inoculation with Multiple Arbuscular Mycorrhizal Fungi Improves Growth, Nutrient Uptake and Photosynthesis in Cucumber Seedlings[END_REF]Maherali and Kliromonos, 2007;[START_REF] Yang | Taxonomic resolution is a determinant of biodiversity effects in arbuscular mycorrhizal fungal communities[END_REF]. Mix 5 was more efficient than the other mixes for most of the tested properties (Fig. 1b,1c). For this mixed inoculum quantities of P and K absorbed per plant were more than 10 times higher than non-inoculated plants and more than two times higher than the most performant AMF isolate.

Ca/Mg values of plant organs were also significantly higher than plants inoculated with a single AMF.

The translocation factors for potentially toxic metals (particularly Ni and Co) were reduced three to ten times. Moreover, [START_REF] Crossay | Combinations of different arbuscular mycorrhizal fungi improve fitness and metal tolerance of sorghum in ultramafic soil[END_REF] showed that this last effect was clearer for sorghum (a low metal-tolerant plant), than for M. laurifolia. This result reinforces the conclusion that the symbiosis involving AMF efficient consortium can optimize plant adaptation to the soil in which it develops. [START_REF] Crossay | Is a mixture of arbuscular mycorrhizal fungi better for plant growth than single-species inoculants?[END_REF] also reported a mathematical model demonstrating the synergistic effects of the native AMF isolates (Fig. 1 d). The correlation between plant biomass (variable to be explained) and the predicted value of the performance based on the properties of the different AMF was very high (R2 = 0.90; p < 0.00001), indicating that the improvement of plant biomass was nearly totally explained as a resultant of the complementary effects of the 5 isolates.

As noted before, the inoculated AMF are generally progressively replaced by native taxa in the field. However, contrary to the assertion of [START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF], this could not be a problem. Indeed, the inoculant primary effect can continue indirectly to stimulate plant development because the better health of the inoculated plants allows a higher metabolism and photosynthesis and then a better ability to feed AMF communities [START_REF] Amir | Arbuscular mycorrhizal fungi and sewage sludge enhance growth and adaptation of Metrosideros laurifolia on ultramafic soil in New Caledonia: A field experiment[END_REF]. These latter authors performed a field experiment to restore an ultramafic area. They used degraded topsoil containing a low initial level of AMF viable spores. Metrosideros laurifolia plants, inoculated with a mix of three selected AMF isolates showed a clearly higher level of mycorrhizal colonization than non-inoculated ones after 528 days of field plantation, while other taxa have mainly replaced AMF inoculant. Indeed, after only 335 days, the inoculated taxa represented not more than 9% of AMF communities inside plant roots. However, AMF diversity was significantly higher in the roots of inoculated plants, which weighed four times more than in control plots.

The precedent studies allowed the selection of the better AMF consortium improving plant adaptation and growth on ultramafic soils. After this work, a company developed a dixenic production protocol for the selected AMF species to market a suitable product (Fig. 2). Today, different companies use the AMF commercial inoculant for the restoration of mine-degraded areas.

These case studies clearly show the interest of AMF native mixed inoculants with isolates phylogenetically distant. The main practical conclusion is that the selection of AMF isolates aiming to perform as an efficient mixed inoculant must be based on their complementary features and effects on different aspects of plant development. However, more field trials must be assessed to evaluate the effect of AMF selected consortia in diverse situations. More generally, rigorous field experiments are still needed to demonstrate the benefit of AMF inoculation, at a large-scale, for plantations and ecosystems [START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF][START_REF] Berruti | Arbuscular Mycorrhizal Fungi as Natural Biofertilizers: Let'sBenefitfromPast Successes[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF], and check the absence of ecosystemic perturbations linked to interactions between introduced and native AMF over several years (Antunes andKoyama 2017, Hart et al. 2017).

Towards more complexity in practices

As just illustrated, there are different advantages of using multispecies inoculants; however, there are also some difficulties. The most important is that only a reduced number of AMF species can be produced aseptically [START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF][START_REF] Berruti | Arbuscular Mycorrhizal Fungi as Natural Biofertilizers: Let'sBenefitfromPast Successes[END_REF], particularly with root-organ monoxenic culture. Therefore, it could be challenging to guarantee the total safety of the inoculants in terms of deleterious microorganisms [START_REF] Ijdo | Methods for large-scale production of AM fungi: past, present, and future[END_REF][START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF][START_REF] Berruti | Arbuscular Mycorrhizal Fungi as Natural Biofertilizers: Let'sBenefitfromPast Successes[END_REF] when at least a part of AMF consortium was not produced in that safety conditions. However, this problem will likely be solved with the improvement of AMF culture technologies. The other difficulty is the more complex and longer selection strategy of complementary AMF isolates compared to single inoculants. Nevertheless, this constraint could be easily accepted if the superiority of the mixed inoculants, when correctly selected, is demonstrated in diverse conditions.

If the ecological point of view supports the use of inoculants containing functionally and phylogenetically distant mixed AMF, it would also suggest, for the future, the use of more complex inoculants affecting more ecological niches. Indeed, different studies (Vasquez et al. 2000;[START_REF] Miransari | Arbuscular Mycorrhiza and soil microbes[END_REF][START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF][START_REF] Bizos | The Role of Microbial Inoculants on Plant Protection,Growth Stimulation, and Crop Productivity of theOlive Tree (Olea europea L.)[END_REF][START_REF] Bourles | Co-inoculation with a bacterium and arbuscular mycorrhizal fungi improves root colonization, plant mineral nutrition, and plant growth of a Cyperaceae plant in an ultramafic soil[END_REF] suggest the use of AMF isolates mixed with other microorganisms, such as mycorrhiza helper bacteria (MHB), plant growth promoting rhizobacteria (PGPR), and/or saprophytic or endophytic performant fungi such as Trichoderma and Fusarium. [START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF] conclude their synthesis on mycorrhizal biotechnology and industry by indicating that mycorrhizal inoculation is not a panacea and that "protection and proper management of native AMF populations in soils is a primary tool to exploit positive effects of mycorrhizal symbiosis phenomena." It is indeed possible to optimize mycorrhizal symbiosis without inoculation.

For example, [START_REF] Brito | Managing the functional diversity of arbuscular mycorrhizal fungi for the sustainable intensification of crop production[END_REF] developed an agricultural strategy that stimulates AMF extension as a soil health condition [START_REF] Abbott | Mycorrhizal fungi as a potential indicator of soil health[END_REF] for a sustainable development perspective. The authors [START_REF] Brito | Managing the functional diversity of arbuscular mycorrhizal fungi for the sustainable intensification of crop production[END_REF]) used plants that favor AMF development with non-tilled soil or appropriate tillage methods that avoid perturbation of extra-radical mycelium. They showed that these conditions induce a higher AMF diversity, with better plant growth and higher tolerance of plants to abiotic and biotic stresses, including tolerance to manganese toxicity and fungal diseases. This is an excellent example of integrated use of natural AMF diversity to improve plant production.

Conclusion and perspectives

Biotechnologies for commercial production of AMF inoculants are undergoing processes of improvement. In the context of a fast-changing world in reaction to the global ecological crisis, their use for ecosystem restoration, soil remediation, and sustainable agriculture may be of high interest.

However, different challenges are still running before large-scale use of these new practices. The success of AMF inoculants as biofertilizers will thus depend on a few points that need more focalized research. These points are mostly listed in different reviews (Idjo et al. 2011;[START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF][START_REF] Berruti | Arbuscular Mycorrhizal Fungi as Natural Biofertilizers: Let'sBenefitfromPast Successes[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF], and it is not necessary to report them here. Overall, the two major difficulties of any microbial inoculation aiming plant development consist of i) an inoculant adequation with functional and ecological niches that can receive the entering microorganisms; ii) a rigorous selection of the isolates to stick with the set ends. These are highly challenging goals that need quite complex studies for each inoculant before reaching the commercial stage. Here we want to stress the aspects mainly discussed in this text. Technologies generally use very tight approaches, but, in this case, it is necessary to follow a systemic approach to answer correctly the difficulties we just outlined. In this perspective, the use of AMF consortia correctly selected for their synergistic effects may significantly improve the inoculant efficiency. More molecular research is also needed to track correctly each isolate of the consortium among soil AMF communities and monitor their persistence over several years [START_REF] Thonar | Real-time PCR to quantify composition of arbuscular mycorrhizal fungal communities marker design, verification, calibration and field validation[END_REF][START_REF] Berruti | Applicationoflasermicrodissectiontoidentifythemycorrhizalfungithatestablisharbusculesinsidero otcells[END_REF][START_REF] Hart | Fungal inoculants in the field: Is the reward greater than the risk?[END_REF].

As suggested by some reviews [START_REF] Vosátka | Development of arbuscular mycorrhizal biotechnology and industry: current achievements and bottlenecks[END_REF][START_REF] Berruti | Arbuscular Mycorrhizal Fungi as Natural Biofertilizers: Let'sBenefitfromPast Successes[END_REF], commercial inoculants' production probably needs to be more localized, with selection of AMF isolates for a type of soil or ecosystem in a given region. This will allow a better adaptation of the inoculant and prevent a possible ecosystem transformation in relation to changes in mycorrhizal communities [START_REF] Antunes | Chapter 9 -Mycorrhizas as Nutrient and Energy Pumps of Soil Food Webs: Multitrophic Interactions and Feedbacks[END_REF][START_REF] Islam | Survival of a commercial AM fungal inoculant and its impact on indigenous AM fungal communities in field soils[END_REF]. At least in some cases, producers may adapt their inoculants so that the inoculant efficiency could be totally mastered and ecologically indisputable. In all cases, this will need more research to develop practical but complex methods to perform such selection, particularly for mixed inoculants. The AMF features could then be determined in relation to the type of expected conditions and effects: type of environment and soil (mineral deficiencies, toxicity of particular elements, drought limits, salinity, etc.), agricultural production to be optimized (whole plant, grains, fruits or other organs), etc. Therefore, AMF isolates to be included in the mixed inoculant must be selected for these features, and must present complementary performances so that their resultant effect on plant health and production could be optimized. A mathematical modelization of the effects of the different variables influencing the symbiosis efficiency could be produced as approached by [START_REF] Crossay | Is a mixture of arbuscular mycorrhizal fungi better for plant growth than single-species inoculants?[END_REF]. The selection tests could then be standardized, and a software application able to give a performance value for each AMF consortium could be created. This will allow adapting more easily the inoculant production to the different conditions as noted above while keeping the commercial inoculants in an acceptable bracket of prices.