Maciej Bednarczyk

Hassan Omran

Bernard Bayle

Passivity Filter for Variable Impedance Control

While impedance control is one of the most commonly used strategies for robot interaction control, variable impedance control is a more recent preoccupation. If designing impedance control with varying parameters allows increasing the system flexibility and dexterity, it is still a challenging issue, as it may result in a loss of passivity of the control system. This has an important impact on the stability and therefore on the safety of the interaction. In this paper, we propose methods to design passivity filters that guarantee passivity of the interaction. They aim at either checking whether a desired impedance profile is passive, or modifying it if required.

I. INTRODUCTION

Robots are increasingly present in the same space as humans. The development of collaborative robots designed to safely support operators during the execution of tasks has certainly played an important role in this evolution, with applications such as lifting of heavy tools and pieces, hand-guiding or collaborative assembly. In this context, the question of handling physical contact in the best possible way has become crucial. This can be obtained typically by controlling the compliance of the interaction between the robot and its environment, resulting in the classical impedance control (IC) scheme [START_REF] Hogan | Impedance control: An approach to manipulation[END_REF]. IC consists in imposing a mass-spring-damper behavior to the whole system, for attracting the robot to the desired trajectory and for managing the interactions, both with the environment and potential humans involved in a collaborative task.

One of the main challenges in interaction management using IC lies in the proper selection of the impedance parameters, not only according to the task but also to ensure stability for all possible parameters variations. Indeed, if using fixed impedance parameters makes the system passive and hence stable when interacting with a passive environment [START_REF] Colgate | Robust control of dynamically interacting systems[END_REF], [START_REF] Hogan | On the stability of manipulators performing contact tasks[END_REF], this property no longer holds for arbitrarily varying parameters [START_REF] Kronander | Stability considerations for variable impedance control[END_REF]. Not being able to modify the system impedance during the task is a notable drawback that may limit application ranges. For instance, human-robot interaction may benefit from the ability to adapt to human's force or fatigue, or to dynamically modify the interaction [START_REF] Ikeura | Variable impedance control of a robot for cooperation with a human[END_REF].

As a result, variable impedance control (VIC) has recently attracted more and more attention. VIC has been explored for challenging objectives, for example to deal with explosive movements [START_REF] Braun | Optimal variable stiffness control: formulation and application to explosive movement tasks[END_REF], to optimize the performance of hammering tasks [START_REF] Garabini | Optimality principles in variable stiffness control: The VSA hammer[END_REF], allow risk-sensitive interactions [START_REF] Medina | Risk-sensitive interaction control in uncertain manipulation tasks[END_REF] or maximize robot links velocities [START_REF] Haddadin | Optimal control for maximizing link velocity of robotic variable stiffness joints[END_REF]. VIC has been implemented using reinforcement learning [START_REF] Buchli | Learning variable impedance control[END_REF], as well as adaptive approaches for human-robot collaboration based on the estimation of the human arm stiffness, from the derivatives of force and position [START_REF] Tsumugiwa | Variable impedance control based on estimation of human arm stiffness for human-robot cooperative calligraphic task[END_REF], or from the measurement of muscle activity with electromyography [START_REF] Grafakos | Variable admittance control in pHRI using EMG-based arm muscles co-activation[END_REF]. However, in all these contributions, the resulting impedance of the robot is identified or learned during task execution, and therefore these methods cannot be used to check before execution whether a given impedance profile will lead to for stable execution.

Though in practice varying the stiffness does not necessarily lead to unstable behavior, several contributions addressed the issue of guaranteeing a stable execution of VIC, with a given impedance profile. In [START_REF] Ficuciello | Variable impedance control of redundant manipulators for intuitive human-robot physical interaction[END_REF], the authors showed that exploiting kinematic redundancies can ensure passivity for a larger panel of impedance profiles. Gain scheduling control has also been used to address VIC by interpolating feedback gains between operating points [START_REF] Lee | Force tracking impedance control with variable target stiffness[END_REF]. Virtual energy-storing tanks [START_REF] Ferraguti | A tank-based approach to impedance control with variable stiffness[END_REF] are another solution to modify the impedance model and guarantee passivity, by storing energy dissipated by the system. This stored energy can then be re-injected to simultaneously implement stiffness variations and in guarantee stable execution. This approach has been successfully implemented in several applications, as in [START_REF] Ferraguti | An energy tankbased interactive control architecture for autonomous and teleoperated robotic surgery[END_REF], [START_REF] Schindlbeck | Unified passivity-based cartesian force/impedance control for rigid and flexible joint robots via taskenergy tanks[END_REF], or [START_REF] Lutscher | Hierarchical force and positioning task specification for indirect force controlled robots[END_REF]. In [START_REF] Ferraguti | A tank-based approach to impedance control with variable stiffness[END_REF], it is used to deal with the interactions resulting from tissues and membranes penetration during a robotic needle insertion.The tank-based strategy has been shown very well suited for VIC, in spite of some difficulties to tune its parameters. Nevertheless, it is dependent on the states of the system, measured during task execution and so can only be applied online.

Other authors proposed an approach that can be used to check the passivity of a given time-varying impedance profile before the execution of the task [START_REF] Kronander | Stability considerations for variable impedance control[END_REF]. While this allows to check whether the interaction will have passive dynamics, it provides no solution to modify a profile that would not respect passivity conditions. The present paper aims to overcome this issue. Based on the combination of passivity conditions with an adaptation law on the impedance profile, the proposed solution allows checking whether a given profile is passive, and if it is not, it provides a method to modify it in a way to guarantee passivity. The proposed method can thus be used either offline, before the execution of the profile, or online.

The paper is organized as follows. Firstly, the control background and passivity analysis are introduced in Section II. Then, Section III describes the proposed design methods for passivity filters for VIC. In Section IV the features of the proposed passivity filters are explored, first in simulation and then using an experimental setup. Section V finally provides conclusions and perspectives of the work.

II. CONTROL BACKGROUND A. Rigid Body Dynamics and Linearized Model

Consider the model of a n-joint serial robotic manipulator H(q)q + c( q, q) + g(q) = τ c -J(q) T f ext [START_REF] Hogan | Impedance control: An approach to manipulation[END_REF] where q, q, q ∈ R n are joint accelerations, velocities and positions, respectively, τ c ∈ R n are the commanded joint torques, f ext ∈ R m is the end-effector wrench, J(q) ∈ R m×n is the robot Jacobian matrix and H(q) ∈ R n×n , c( q, q) ∈ R n and g(q) ∈ R n are the inertia matrix and the Coriolis and gravity terms, respectively. A classical solution to linearize equation ( 1) is to apply the computed torque control law, resulting in a double integrator model in joint space, or its equivalent in task space [START_REF] Siciliano | Springer Handbook of Robotics[END_REF].

B. Impedance Control

In order to dynamically link the system positions, velocities and accelerations with the external forces, the input of the linearized system is computed in order to obtain

M ëp + D ėp + Ke p = f ext (2)
with e p = p r -p ∈ R m , where p is the robot end effector pose and p r a reference motion. The interaction is then characterized by the impedance resulting from the apparent virtual mass M ∈ R m×m , the desired damping D ∈ R m×m and the desired stiffness K ∈ R m×m . Note that M , D and K are symmetric positive definite matrices and are chosen diagonal to simplify developments. For VIC, the case where D(t) and K(t) are time-varying is considered, while M is assumed to be constant. From now on, time dependency will generally not be mentioned in the notations for simplicity.

C. Passivity analysis

In order to investigate the stability of VIC given by (2) with varying stiffness and damping terms, the following storage function can be considered

V 1 = 1 2 ėT p M ėp + 1 2 e T p Ke p (3)
As K and M are symmetric, differentiating V 1 leads to

V1 = ėT p f ext + 1 2 e T p Ke p -ėT p D ėp (4)
If stiffness is constant or decreasing V1 ≤ ėT p f ext , which leads to the passivity condition

V 1 (t) -V 1 (0) ≤ t 0 ėT p f ext dτ (5)
Because of stiffness time-dependency in equation ( 4), the sign of the term in brackets may change, possibly leading to a violation of the passivity condition (5) in case of increasing stiffness. Passivity can only be guaranteed if the stiffness is either constant or decreasing. In this case, the storage function can be used as a Lyapunov-like function to show stability. Also, it is well known that the interaction with a passive environment yields a passive, thus stable, interaction.

The storage candidate function ( 3) is conservative and should not be used to conclude on system passivity for increasing stiffness. Alternative storage functions need to be explored. In order to facilitate the passivity analysis, the authors of [START_REF] Behal | Lyapunov-Based Control of Robotic Systems[END_REF] propose to use a filtered tracking error-like variable r ∈ R m r = ėp + αe p [START_REF] Braun | Optimal variable stiffness control: formulation and application to explosive movement tasks[END_REF] where α ∈ R + is a constant. As shown in [START_REF] Dawson | Nonlinear Control of Electric Machinery[END_REF], if r is bounded then e p , ėp are bounded and, in this case, if r → 0, then e p , ėp → 0. With this in mind, it is possible to reformulate the control objectives using r

t 0 r T f ext dτ = t 0 {r T (M ëp + D ėp + Ke p )}dτ = t 0 {r T (M (ë p + α ėp ) -αM ėp + D ėp + Ke p )}dτ
Using equation ( 6) and its derivative, it follows that

t 0 r T f ext dτ = t 0 {r T M ṙ + ėT p (D -αM ) ėp +e T p (K + αD -α 2 M ) ėp + e T p (αK)e p )}dτ (7)
By introducing a symmetric, positive semidefinite and continuously differentiable matrix C, equation ( 7) can be written

t 0 r T f ext dτ = t 0 {r T M ṙ + e T p C ėp + 1 2 e T p Ċe p }dτ + t 0 { ėT p (D -αM ) ėp + e T p (αK - 1 2 Ċ)e p +e T p (K + αD -α 2 M -C) ėp }dτ (8)
which can be written as

t 0 r T f ext dτ = V 2 (t) -V 2 (0) + t 0 W 2 dτ (9)
with

V 2 = 1 2 r T M r + 1 2 e T p Ce p (10)
and

W 2 = ėT p (D -αM ) ėp + e T p (αK - 1 2 Ċ)e p +e T p (K + αD -α 2 M -C) ėp (11)
It is worth noticing that if we define C = K + αD -α 2 M , the candidate storage function [START_REF] Buchli | Learning variable impedance control[END_REF] is the same as in [START_REF] Kronander | Stability considerations for variable impedance control[END_REF], resulting in the stability the conditions

0 ≤D -αM (12) 0 ≤2αK -K -α Ḋ (13)
This particular choice leads to W 2 ≥ 0, allowing to conclude on system passivity, as in this case equation ( 9) leads to

V 2 (t) -V 2 (0) ≤ t 0 r T f ext dτ (14)
Passivity conditions ( 12) and ( 13) can be used in order to verify whether passivity is guaranteed for a given impedance profile. However, they do not give any information about how to modify a profile that is not passive. In the following, we will be interested in designing a method capable of guaranteeing passivity properties for any given impedance profile based on conditions ( 12) and [START_REF] Ficuciello | Variable impedance control of redundant manipulators for intuitive human-robot physical interaction[END_REF].

III. PASSIVITY FILTERS FOR VIC

This section introduces the design of a passivity filter, that makes it possible to modify a non passive impedance profile online, in order to guarantee passivity.

A. Guaranteeing passivity

Let us consider the change in system impedance, with the stiffness (respectively the damping) varying from K 0 to K 1 (respectively from D 0 to D 1 ). In the following only the case K 0 < K 1 is considered, as stiffness increase is the only case for which the passivity condition can be violated, as stated in Section II-C. Let us now define K and D such that

K = K(t) = K 0 + Γ(t)δK (15) D = D(t) = D 0 + Γ(t)δD (16)
with δK = K 1 -K 0 , δD = D 1 -D 0 and Γ a diagonal matrix where all diagonal terms γ i are differentiable gains such that 0 ≤ γ i ≤ 1.

As M , K, D were chosen to be diagonal, the impedance behavior can be decoupled. In order to ensure passivity using conditions on V 2 , K and D need to satisfy equations ( 12) and [START_REF] Ficuciello | Variable impedance control of redundant manipulators for intuitive human-robot physical interaction[END_REF]. Condition [START_REF] Grafakos | Variable admittance control in pHRI using EMG-based arm muscles co-activation[END_REF] allows choosing α easily. For instance, in [START_REF] Kronander | Stability considerations for variable impedance control[END_REF] it is chosen as the ratio between the smallest eigenvalue of D and the largest eigenvalue of M . Condition [START_REF] Ficuciello | Variable impedance control of redundant manipulators for intuitive human-robot physical interaction[END_REF] implies that ∀i ∈ {1, 2, . . . , n}

γi (αδd i + δk i ) -2α(k 0i + γδk i ) ≤ 0, (17)
with δk i and δd i the diagonal terms of δK and δD respectively, and k 0i the diagonal terms of K 0 .

Generally, an increase in the desired stiffness does not come with a decrease in the desired damping, which would be the most constraining case in terms of stability. Then, let us assume that δd i ≥ 0. As δk i > 0 and α ≥ 0, then αδd i + δk i > 0. Equation ( 17) can then be written as

γi ≤ a i γ i + b i (18)
with

a i = 2αδk i αδd i + δk i and b i = 2αk 0i αδd i + δk i
Combining equation ( 17) with a low-pass filter, in order to ensure smoothness of γ i , a passivity filter can be designed in such a way that it takes as input the desired switching profile γi and generates an output profile γ i that guarantees the system passivity. The filter can be then defined by

γi = min(β(γ i -γ i ), a i γ i + b i ) ( 19
)
where β is the filter parameter. The integration of (19) permits to obtain γ i that tracks γi in such a way that passivity conditions are respected. Note also that if the damping decreases while the stiffness increases, the previous reasoning can be applied, but should be adapted, based on the numerical values of δd i and δk i (the case αδd i +δk i = 0 is singular and should be avoided).

Though the usage of the proposed filter may guarantee passivity, the tracking performance highly depends on the conservatism of the passivity conditions. The constant value α plays an important role in the conditions resulting from function V 2 . In [START_REF] Kronander | Stability considerations for variable impedance control[END_REF] it is chosen in the worst case, thus reducing the variation range of the parameters. In the following, we will be interested in changing the candidate storage function in order to find less conservative passivity conditions resulting in better tracking performance of the switching function γ.

B. Modifying the storage function

In this section, we are interested in finding a novel candidate storage function that may allow us to conclude on system passivity in a less conservative way. To do so, let us fist modify the assumptions on (6) to replace α by a time-dependent matrix :

r = ėp + M -1 Be p ( 20
)
where B ∈ R m×m is a time-dependent diagonal matrix with bounded eigenvalues.

Lemma 1: Considering the filtered tracking error as defined in [START_REF] Behal | Lyapunov-Based Control of Robotic Systems[END_REF] with M a constant diagonal matrix and B a time-dependent diagonal matrix with bounded eigenvalues. If r is bounded then e p , ėp are bounded and that if r → 0 then e p , ėp → 0 (see Appendix A for the proof). In this case, using the same strategy as previously in equation ( 9) yields

t 0 r T f ext dτ = t 0 {r T M ṙ + e T p C ėp + 1 2 e T p Ċe p }dτ + t 0 { ėT p (D -B) ėp + e T p (BM -1 (K -Ḃ) - 1 2 Ċ)e p +e T p (K + BM -1 (D -B) -Ḃ -C) ėp }dτ (21)
that is

t 0 r T f ext dτ = V 3 (t) -V 3 (0) + t 0 W 3 dτ (22)
with

V 3 = 1 2 ėp e p T M B B C + BM -1 B ėp e p (23)
and

W 3 = ėT p (D -B) ėp + e T p (BM -1 (K -Ḃ) - 1 2 Ċ)e p +e T p (K + BM -1 (D -B) -Ḃ -C) ėp (24)
The choice C = K + BM -1 (D -B) -Ḃ allows canceling out the cross-terms. According to equation (24), the positivity of V 3 is obtained as a consequence of Schur lemma [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF] and the condition

0 ≤ C (25)
The positivity of W 3 is obtained if the following novel passivity conditions hold

0 ≤ D -B (26) 0 ≤ BM -1 (K -Ḃ) - 1 2 Ċ ( 27
)
It is worth noticing that the storage function V 2 of equation ( 10), initially defined by [START_REF] Kronander | Stability considerations for variable impedance control[END_REF], is a special case of V 3 defined in equation ( 23) with B = αM , C = K + αD and α constant. The analogy between the scalar α in equation ( 6) and the matrix DM -1 in equation [START_REF] Behal | Lyapunov-Based Control of Robotic Systems[END_REF], and then between αM and D suggests that the candidate storage function could be modified to limit the conservatism of the worst-case choice for α. This leads to set B = D and C = K -Ḋ. A fourth storage function is then defined, such that

V 4 = 1 2 ėp e p T M D D K + DM -1 D -Ḋ ėp e p ( 28
)
and the resulting passivity conditions are deduced from equations ( 25) to ( 27)

0 ≤ K -Ḋ (29) 0 ≤ 2DM -1 (K -Ḋ) -K + D (30)
As previously, all matrices are chosen diagonal and the problem can be decoupled, thus (30) is equivalent to

ki + 2 m i d i ḋi -di - 2 m i k i d i ≤ 0 resulting in -a 1i γi + a 2i γi + a 3i γ i γi -a 4i γ i -a 5i γ 2 i -a 6i ≤ 0 (31) with a 1i = δd i a 2i = δk i + 2 m i d 0i δd i a 3i = 2 m i δd 2 i a 4i = 2 m i (d 0i δk i + k 0i δd i ) a 5i = 2 m i δk i δd i a 6i = 2 m i k 0i d 0i
which are all positive since δK and δD are positive. Then, as a 2i > 0 and γ i is such that 0 ≤ γ i ≤ 1, ∀γ i ,a 2i + a 3i γ i > 0. This allows to write the passivity condition

γi ≤ a 1i γi + a 4i γ i + a 5i γ 2 i + a 6i a 2i + a 3i γ i h 1i (γ i , γ i ) (32)
Additionally, in the case where δd i = 0, equation ( 29) gives a second passivity condition for γi such that

γi ≤ δk i δd i γ i + k 0i δd i h 2i (γ i ) ( 33
)
Using passivity conditions (32) and (33), a new passivity filter similar to [START_REF] Siciliano | Springer Handbook of Robotics[END_REF] could be defined such that

γi = min(h 1i (γ i , γ i ), h 2i (γ i ), β(γ i -γ i )) (34)
Condition (32) is however not well suited for practical implementation as it requires derivating γi , which is the output of a min-switch and therefore can be discontinuous. For this reason, we choose to use a more conservative but practically implementable constraint, such that

γi ≤ a 4i γ i + a 5i γ 2 i + a 6i a 2i + a 3i γ i h 3i (γ i ) (35)
In fact, when the constraint is active, we have γi ≥ 0 and (35) becomes an equality. In this case it can be shown that

γi = 2a 5 a 2 γ i + a 5 a 3 γ 2 i + (a 2 a 4 -a 3 a 6 ) (a 2 + a 3 γ i ) 2 γi
and

a 2 a 4 -a 3 a 6 = 4 m 2 i δk i ( m i 2 d 0i + δd i k 0i + d 0 2 i δd i ) ≥ 0
resulting in γi ≥ 0. This leads to the fact that if γi respects the constraint given by equation ( 35), then it also respects the passivity condition (32). Finally, the proposed passivity filter resulting from conditions (33) and ( 35) is defined by

γi = min(h 2i (γ i ), h 3i (γ i ), β(γ i -γ i )) (36)
IV. VALIDATION

A. Simulations

In order to illustrate the advantages of the proposed approach, simulations analyzing passivity properties on a onedimensional mass-spring-damper system were performed.

The simulated system has a constant mass m = 10kg, a stiffness varying between k 0 = 2N/m and k 1 = 22N/m and a damping ratio of 0.1. The considered reference trajectory was set such that p r = 10 sin 0.1t. The stiffness is expected to vary according to the switching function γ illustrated in Fig. 1 in dashed gray. The passivity filters [START_REF] Siciliano | Springer Handbook of Robotics[END_REF] and (36), associated to storage functions V 2 and V 4 , respectively, are tuned with the parameter β = 10. The resulting switching functions γ are represented in Fig. 1. One can see that even if the outputs of both filters first evolve similarly, the output of filter (36) converges much more rapidly to the reference switching variable. This illustrates the fact that the conditions given by V 4 are less conservative than those given by V 2 . Note that no external force is applied to the system, and in this case passivity is guaranteed if the derivative of the storage (Lyapunov-like) function is negative. As shown in Fig. 2, when comparing the evolution of V1 , V2 and V4 in the case where γ is calculated using passivity condition [START_REF] Siciliano | Springer Handbook of Robotics[END_REF] (based on V 2 ), one can see that in contrary to function V 1 which is too conservative, V 2 and V 4 guarantee passivity. On the other hand, using passivity condition (36) (based on V 4 ), one can see in Fig. 3 that functions V 1 and V 2 cannot guarantee passivity, whereas V 4 is capable of doing so.

B. Experiment

The passivity filter (36) was then tested experimentally in a needle insertion task, inspired by the scenario described in [START_REF] Ferraguti | A tank-based approach to impedance control with variable stiffness[END_REF]. A 16G medical needle was placed in a needle holder, mounted at the end-effector of a KUKA iiwa 14 robot, equipped with an ATI-Mini40 force-torque sensor (Fig. 4). The needle punctured a 4-layer silicon phantom, whose layers imitate skin, fat, muscle and liver, respectively. The muscle is the stiffest layer, followed by skin, fat and liver. Generally, during needle insertion procedures, no rotations of the needle are desired, thus we choose to change impedance profiles only in translation. The impedance parameters are chosen such that all inertial terms m i = 5kg and the damping ratio is set to be 1. The desired stiffness varies with the depth in the phantom such that ki = 3000, 3500, 4000N/m for fat/liver, skin and muscle, respectively. Outside of the phantom, the system starts insertion with ki = 1000N/m and extracts the needle with ki = 3000N/m. The filter parameter is β = 50. The VIC algorithm implemented in C++ generates torque commands fed to the system at 500Hz, using KUKA's Fast Robot Interface protocol.

V. CONCLUSIONS AND PERSPECTIVES

In this paper we propose a design method for filters that permit the tracking of impedance profiles for variable impedance control, while guaranteeing passivity. Two filters are derived based on two candidate storage functions. The proposed passivity filters were validated in simulation and in an experimental setup featuring a KUKA iiwa collaborative robot executing a needle insertion task.

Future work aims at expanding the proposed filter design with other passivity conditions in search of even less conservative passivity guarantees. Also, it can be investigated how unusual impedance profile variations, such as increasing stiffness while decreasing damping, impact on the passivity guarantees and the design of the passivity filters. As the matrices M and B are chosen to be diagonal, the problem can be decoupled accordingly to the diagonal terms. Hence ∀i ∈ {1, 2, . . . , n} :

1) if r i is bounded then e p i , ėpi are bounded : the solution of the differential equation (t-τ ) r i (τ )dτ || 2 is bounded as ||r i || 2 is bounded. This shows that ||e p i || 2 is bounded. Using Corollary 1.1 from [START_REF] Dawson | Nonlinear Control of Electric Machinery[END_REF], as e p i , ėpi and ||e p i || 2 are bounded, then lim t→∞ e p i = 0. Finally, from (40) with lim t→∞ e p i = 0 and lim t→∞ r i = 0, one can conclude that lim t→∞ ėpi = 0.

r i = ėpi + b i m i e p i (

Fig. 1 .

 1 Fig. 1. Switching variable γ resulting from filter with conditions from V 2 and V 4 and its reference.

  (a) Storage functions. (b) Derivative of storage functions.

Fig. 2 .

 2 Fig. 2. Time evolution of storage functions for γ calculated to respect passivity constraints (19) (based on V 2 ).

  (a) Storage functions. (b) Derivative of storage functions.

Fig. 3 .

 3 Fig. 3. Time evolution of storage functions for γ calculated to respect passivity constraints (36) (based on V 4 ).

Fig. 4 .

 4 Fig. 4. Experimental setup.

  (a) Stiffness and damping profiles for needle insertion and extraction. (b) Passivity constraints and γi during stiffness increase. Condition h2 i increases to high values and was cut-out of the scope for readability.

Fig. 5 .

 5 Fig. 5. Impedance profile and evolution of γi . The effective stiffness and damping (k i and d i ), generated by the passivity filter are shown on Fig 5a. The influence of the passivity filter on the impedance profile can be observed when the needle punctures the skin, at t = 9.35s, and the stiffness is increased from 1000 to 3500N/m. In this case, the velocity of the switching variable γ i increases freely until it meets the constraint h 3i imposed by the passivity filter (36) (Fig 5b) that shapes its further increase rate. This allows an important stiffness increase while guaranteeing passivity.

  ACKNOWLEDGMENT This work was supported by the French government research program Investissements d'Avenir under the Robotex Equipment of Excellence and Labex CAMI (ANR-10-EQPX-44 and ANR-11-LABX-0004) and by the IDEX of the University of Strasbourg. APPENDIX A. Proof of Lemma 1.

  r i (τ )dτ(38) As b i ∈ R + is considered to be bounded such that b i ≤ b i ≤ bi , (38) can be upper bounded by|e p i | ≤ |e p i (0)| + τ ) |r i (τ )|dτ ≤ |e p i (0)| + sup t (|r i (τ )|) if r i is bounded then e p i is bounded.Additionally, using (37) one get that| ėpi | ≤ bi m i |e p i | + |r i (t)|(40)that can be written using (39)| ėpi | ≤ bi m i |e p i (0)| + sup t (|r i (τ )|) |r i (t)|what shows that if r i is bounded then ėpi is bounded.2) if r i → 0 then e p i , ėpi → 0: Considering the 2-norm for a scalar functionf ∈ R defined by ||f || 2 = ∞ 0 f 2 (τ )dτ , one can show that ||e p i || 2 is bounded. Hence ||e p i || 2 = ||e τ ) r i (τ )dτ || 2 as b imi is constant, following the same steps of the proof of Lemma 1.6 in[START_REF] Dawson | Nonlinear Control of Electric Machinery[END_REF], one can show that ||e

ICube, UMR 7357, University of Strasbourg, CNRS, 1 place de l'Hôpital, 67091, Strasbourg, France. m.bednarczyk@unistra.fr


Nicole Séon Clavaud
email: nicole.clavaud@inserm.fr

Quentin Guillon

Vergne Judith

Sandrine Sonié

Christina Schmitz

Judith Vergne

Impact of motor experience on perception/action coupling in children and teenagers with Autism Spectrum Disorder (ASD): an eye-tracking study

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

Considering Age as a fixed effect in our model did not improve model fit significantly (compared to a null model without Age as a fixed effect), indicating that Age is not a significant predictor (i.e. Age does not account for significant variation in the likelihood of looking at Backward vs. Forward actions). This was the case for both the TYP(X 2 = 3.06, p = 0.080 and ASD (X 2 = 1.12, p = 0.291) in both groups.

Centre de Recherche Neurosciences Lyon

The strength of the perception/action coupling was modulated by the motor experience specific to each participant in typically developed children, with a strong individual variability. This effect was not found in children with ASD. The timing of events within each trial (picture, video) is depicted with solid horizontal lines. The red dots along the x-axis mark the time points at which the two pupil size differ significantly as indicated by a bootstrap t-test and based on a cluster threshold method to control for multiple comparisons. The likelihood of looking at Backward vs. Forward actions at any given trial during the preference phase was modelled using Generalized Linear Mixed Models. Action familiarity, determined on an individual basis (Figure 3 and 4), was entered as a fixed effect. As random effects, we included a by-participants and by-items random intercept and a by-participants random slope.

Results indicated that the likelihood of looking at Backward vs. Forward videos was greater than chance for the TYP Group (OR = 1.84, z = 3.74, p < .001) but not for the ASD group (OR = 1.31, z = 1.86, p = 0.063).

The by-participants random slope allowed us to model individual variation for the effect of familiarity (Figure 5 and 6). Removing this effect from the model significantly reduced the model fit in both groups, indicating that there is a reliable by-participant variation for the effect of familiarity.

Interestingly, accounting for this individual variation, the effect of familiarity remained significant in the TYP groups but not in the ASD group.

  1 girls and 18 boys (Non verbal IQ ≥ 50). Control group: 20 girls and 16 boys Stimuli: • 16 videos of daily actions (e.g. lace up his/her shoe) presenting with different degrees of motor familiarity to the participant. • Displayed : o Forward = strong perception/action coupling o backward = weak perception/action coupling Protocol: Participants were comfortably seated in a chair (Fig.1) and were simply asked to watch 16 videos of daily actions, for 10 minutes. a) familiarization phase  fixation times and pupil diameter within AOI (Figure 2) b) visual preference test  fixation times within AOI (Figure 2) Analysis and Statistics: Data were recorded and extracted with EyeLink® Data Viewer, and processed with Matlab®. Statistics were performed with R®.

2

  Age did not influence the measure of the perception/action coupling (even if a tendency was found in typically developped children) 3 The time course if the pupil diameter variation reflected the strengh of the perception/action coupling in typically developed children. Such a difference was not found in children with ASD. 4 Individual variation for the effect of familiarity across TYP participants (each couple of point = one participant) Individual variation for the effect of familiarity across ASD participants (each couple of point = one participant) Significant time points Average time course of the pupil diameter across participants for backward (blue line) and forward (red line) displays during the Familiarization phase. Shaded areas denote the 95% confidence intervals around the trimmed mean established at each time point (20Hz).

  Pupil Diameter Time Course during backward and forward video displays (robust mean and 95% Confidence Interval by condition) Time in ms Baseline-corrected pupil size Time in ms Pupil Diameter Time Course during Backward and Forward video displays (robust mean and 95% Confidence Interval)

Fig. 3 Fig. 4

 34 Fig. 3

  Fig. 5 Fig. 6


Didier Louis

Cindy Lombart

Fabien Durif

Fabien

teaching and research institutions in France or abroad, or from public or private research centers.

Le vrac, une bonne stratégie pour développer une offre à impact positif ?

Didier LOUIS (IUT Saint-Nazaire, LEMNA), Cindy LOMBART (Audencia, In Situ Lab) et Fabien DURIF (ESG-UQAM, OCR) L'article « L'offre de produits en vrac : les implications pour le consommateur et pour l'enseigne » a permis de comprendre les tenants et aboutissants de l'offre en vrac pour le consommateur mais également pour l'enseigne. Si les contraintes inhérentes au vrac ne doivent pas être ignorées, cette stratégie de présentation des produits en magasin confère un avantage certains pour des activités entrepreneuriales à impact positif.

Interrogeons-nous

L'humain a naturellement tendance à choisir l'option la plus simple (1). Compte tenu de ce constat, qu'est-ce qui motive le consommateur à adopter l'achat en vrac alors que cela lui demande un effort supplémentaire ? Quelles sont les motivations sous-jacentes à l'achat en vrac ? La motivation à acheter en vrac est-elle homogène parmi tous les profils de consommateurs ?

Revenons aux fondamentaux

Une étude quantitative menée, au Québec, auprès de 1407 acheteurs et consommateurs de produits en vrac montre qu'il existe trois profils de consommateurs (2):

-les convaincus achètent du vrac souvent ou régulièrement. Ils sont les plus sensibles au gaspillage alimentaire, les plus résistants à la société de consommation et les plus socialement responsables ; -les méfiants achètent rarement ou parfois des produits en vrac. Ils sont les moins sensibles au gaspillage alimentaire, les moins résistants à la société de consommation, les moins socialement responsables et les moins sensibles aux prix ; -les pragmatiques se situent entre les deux groupes précédents. Ils sont les plus sensibles aux prix. Le vrac est, pour eux, un bon moyen de découvrir de nouveaux produits et souhaitent d'ailleurs se servir eux-mêmes.

L'offre de produits en vrac permet de fidéliser tous les consommateurs L'offre de produits en vrac permet de rapprocher les consommateurs de l'enseigne : les trois groupes de consommateurs identifiés dans le cadre de l'étude menée en 2021 partagent avec l'enseigne les mêmes valeurs et estiment disposer d'informations sur le fonctionnement du magasin qui leur garantit la qualité des produits et du service offert ; les convaincus et les pragmatiques se sentent plus proches du personnel de vente. Ce rapprochement permet ensuite d'accroître la satisfaction, la confiance et les intentions d'achat de ces consommateurs à l'égard de l'enseigne. Par conséquent, l'offre de produits en vrac est un outil stratégique pour les enseignes qui permet de créer ou de renforcer les relations nouées avec les consommateurs en vue de les fidéliser, quel que soit le profil des consommateurs (de la découverte des produits en vrac à une habitude d'achat bien établie).

(1) Cheval, B., Tipura, E., Burra, N., Frossard, J., Chanal, J., Orsholits, D., ... & Boisgontier, M. P. (2018). Avoiding sedentary behaviors requires more cortical resources than avoiding physical activity: An EEG study. Neuropsychologia, 119, 68-80.


Marcela V C Machado

Artemio Plana-Fattori

Kevin Lachin

Stéphane Vitu

Cristian Trelea

Mikel Leturia

Elias Daouk

Khashayar Saleh

Marie Debacq

Stephane Vitu

Numerical and experimental analysis of spherical and non-spherical particles in a rotary drum: Metamodels for calibrating the friction coefficients in a rotary drum from DEM simuations

Problem in this approach:

Possibility of non-unicity in the set of parameters.

❖ To study the influence of static friction (SF) and rolling friction (RF) coefficients in both interactions particleparticle (PP) and particle-wall (PW) on the dynamic repose angle of quinoa seeds in a rotary drum.

❖ To obtain metamodels, which allow analyzing the interaction between the parameters and their significance.

❖ To validate the four parameters set found in other speed rotations.

Rolling regime

Quinoa seeds

Variable

Coded levels -α -1 0 +1 + α SF PP 0.300 0.350 0.400 0.450 0.500 SF PW 0.350 0.400 0.450 0.500 0.550 RF PP 0.005 0.070 0.135 0.200 0.265 RF PW 0.300 0.350 0.400 0.450 0.500

DIRECT COMPUTATION Lagrangian Method or Discrete Element Method (DEM):

• All the particles trajectories are tracked.

• Temporal integration is applied to each of them using Newton's laws:

SF PP SF PW a A x RF PP RF PW -             -       = = =       -             SF-PP 0.

PARAMETERS IDENTIFICATION

(3) Optimization: resolution of the metamodels and determining the 4 parameters optimized.

(1) 4 Central Composite Designs: parameters: SFPP, SFPW, RFPP, RFPW 26 simulations for each speed rotation.

(2) 4 metamodels: verification of the interactions between the parameters.

(4) Validation: Comparison: angles calculated by metamodels, angles simulated with parameters optimized, and angles measured experimentally. • The rolling friction influence more on the dynamic angle of repose than the static friction, mainly in PP interactions, because it allows particles roll over each other, enabling better rearrangement and redistribution of stress, leading to a more stable packing.

MODEL PARAMETERS

• Drum: diameter 0.084m and length 0.020m.

• Quinoa seeds: d = 0.0017m, ρ = 1058 kg/m 3 .

• Speed rotation: 6, 8, 10, 12 rpm (rolling regime).

• Time-step = 8.6x10 -6 s -1 (5% of Rayleigh time).

• Software: LIGGGHTS®.

• Linear regression in the mean interface of 200 images of the bed drum, implemented by a MATLAB code.

••

  Particulate systems + fluids • Isothermal or non-isothermal • A large number of industrial processes: rotary kilns, dryers, mills, etc • But lack of simple tools for the Understanding the flow material behavior with more details • Overcoming industrial safety limitations CALIBRATION STEP: adjusting the parameters until the model response corresponds to the equivalent experimental responses.

  Figure 1:

  Figure 2:

  Figure 4:

  Figure 5:

4.5. Validation: comparison between angles provided by metamodels, angles simulated with parameters optimized, and angles measured experimentally

	4.1. Experimental dynamic angles of repose
	Speed rotation	6 rpm	8 rpm	10 rpm	12 rpm
		Angle (°)	31.96	32.46	33.14	33.74
				at 12 rpm:
		3226
	SF-PW	0.3938
	RF-PP	0.0665
	RF-PW	0.4227