A Lachhab

E Robin

J.-B Le Cam

F Mortier

Y Tirel

F Canevet

Energy stored during deformation of crystallizing TPU foams

Keywords: energy stored, infrared thermography, intrinsic dissipation, strain-induced crystallization, TPU foam ORCID

The hysteresis loop observed in the mechanical response of elastomers is classically assumed to be due to viscosity. In this study, a complete energy balance is carried out during cyclic deformation of compact and foamed crystallizing thermoplastic polyurethanes. Results show that viscosity is not the only phenomenon involved in the hysteresis loop formation: A significant part of the mechanical energy brought is not dissipated into heat and is stored by the material when the material changes its microstructure, typically when it is crystallizing. Some of this energy is released during unloading, when melting occurs, but with a different rate, which contributes to the hysteresis loop. The part of the mechanical energy stored by the material has been quantified through the 𝛾 se ratio (Loukil et al. (2018) European Polymer Journal, 98, 448-455) to investigate the effects of the loading rate and the void volume fraction on the energetic response of thermoplastic polyurethane.

INTRODUCTION

The thermoplastic polyurethanes (TPUs) exhibit specific properties such as chemical resistance, biocompatibility, and high elasticity and are therefore good candidates for many applications: medicine, transport, agri-food, and military industries, to name a few. From a mechanical point of view, they are subjected to many anelastic phenomena such as mechanical hysteresis, residual stretch, and softening. [1][2][3][4][5][6] These phenomena strongly depend on the material formulation.

When foamed, TPU becomes more deformable and are still dissipative. [START_REF] Alzoubi | Compression and Hysteresis Curves of Nonlinear Polyurethane Foams Under Different Densities[END_REF][START_REF] Blatz | [END_REF][9][10] Even though the energy dissipation capacity of TPU is classically measured as the mechanical hysteresis area, no study investigates the physical origin of this dissipation.

Different phenomena can be responsible for the mechanical hysteresis.

(a) The intrinsic dissipation (d 1 ), due to internal friction and/or damage, which leads to self-heating; (b) the thermal dissipation (d 2 ; in nonadiabatic test conditions). If heat is exchanged with the exterior of the specimen, a hysteresis loop in the stretch-stress relationship forms, even for purely elastic materials. In most of the tests performed, the loading rate is sufficiently high and the thermal dissipation does not really contribute to the mechanical hysteresis; (c) the part of the mechanical energy used by the material to change its microstructure W structure . In this case, all the work done to the system is not measured as a temperature change (see, for instance, the recent study by Mott et al. [11] on polyurea). The material stores part of the mechanical energy brought and releases it (or a part of it) with a different kinetics. [12,13] Such analysis is classically carried out in metallic materials, since the pioneering work by Farren and Taylor [14] and Taylor and Quinney [15] who measured the latent energy remaining after cold working in a metal under quasistatic monotonous loadings. Today, the fraction of the anelastic deformation energy rate irreversibly converted into heat is studied through the Taylor-Quinney ratio. [16][17][18][19][20] Polymers have then benefited from this approach to improve the understanding of the deformation processes. [21][22][23][24] Concerning elastomers, only two recent studies by Le Cam and co-workers investigate the energetic behaviour and the energy storage during deformation. [12,13] In unfilled natural rubber, the stored energy is mainly due to strain-induced crystallization (SIC)/melting process. No irreversibilities take place, and no energy is dissipated into heat. In the case of filled uncrystallizable nitrile butadiene rubber (NBR), only a part of the mechanical energy is dissipated into heat, due to viscous effect induced by carbon black fillers. [13] Indeed, adding fillers also induces energy storage. To further discuss on the relative contribution of the energy stored in the hysteresis loop of rubbers, authors proposed a ratio 𝛾 se , written in terms of energies over one mechanical cycle as follows [13] :

𝛾 se = W c𝑦cle stored W c𝑦cle h𝑦st .
(

) 1
W c𝑦cle stored is the energy stored over one cycle, and W c𝑦cle h𝑦st is the energy of the hysteresis loop. • If 𝛾 se tends to be 0, no energy is stored during the deformation process. The whole hysteresis loop is due to the intrinsic dissipation, and the corresponding energy is dissipated into heat, • If 𝛾 se tends to be 1, the whole hysteresis loop corresponds to energy stored, and no intrinsic dissipation is detected. This is typically the case in unfilled natural rubber. [25] The physical meaning of this ratio is different from the Taylor-Quinney coefficient, because it is not measured from a latent energy remaining after cold working, because the energy stored in the elastomer network can released. Estimating such a ratio is of importance to better compare material formulations, especially in relation to their fatigue resistance. In case of TPU materials, no study in this field has been reported in the literature while they are submitted to mechanical cycles in many applications. In the present work, a complete energy balance is carried out in order to determine the effects of the loading rate, the number of cycles (accommodation) and the void volume fraction on the energetic behaviour of compact and foamed TPUs. For this purpose, infrared thermography was used to determine heat sources using the heat diffusion equation. The paper is organized as follows. Section 2 recalls the thermodynamic framework. Section 3 presents the technique used to elaborate the samples, the loading conditions, and the thermal measurement during the mechanical tests. Section 4 gives the results and compares the TPUs with different densities according to their mechanical and calorimetric responses. Calorific effects of SIC are more particularly addressed. Concluding remarks close the paper.

THERMODYNAMIC FRAMEWORK

Identifying the phenomena involved in the mechanical energy dissipation requires to define the different quantities needed for energy balances, especially the mechanical energy, the heat sources, and the intrinsic dissipation. They are introduced in the following.

Total strain energy density and hysteresis loop

The strain energy density W strain (in J∕m 3 ) is the energy brought mechanically to deform the material. It corresponds to the area under the load (unload) strain-stress curve and is calculated as follows:

W load strain = ∫ load 𝜋 d𝜆 and W unload strain = ∫ unload 𝜋 d𝜆. ( 2
)
𝜆 is the stretch defined as the ratio between current and initial lengths. 𝜋 is the nominal stress, defined as the force per unit of initial (undeformed) surface. If the material's behaviour is purely elastic and if the test is carried out under adiabatic loading conditions, the mechanical response obtained during a load-unload cycle is such that no hysteresis loop forms (W load strain = W unload strain ). If a hysteresis loop forms, the mechanical energy dissipated over one cycle W c𝑦cle h𝑦st is defined as follows:

W c𝑦cle h𝑦st = W load strain -W unload strain .

(3)

From this energy, a quantity P c𝑦cle h𝑦st is calculated in W∕m 3 . It is obtained by dividing W c𝑦cle h𝑦st by the cycle duration. It is therefore an energy density per time unit or an energy rate.

Heat sources

During the mechanical cycle, the material produces and absorbs heat. Under nonadiabatic conditions, corresponding temperature variations are influenced by heat diffusion effects. To account for these effects, a simplified formulation of the heat diffusion equation is used (see Chrysochoos and Louche [26] for further details), considering the heat power density field as homogeneous during uniaxial tensile loading:

𝜌C ( θ + 𝜃 𝜏 ) = s, ( 4
)
where 𝜌 is the initial density, C is the specific heat, 𝜃 is the temperature variation, and 𝜏 is a characteristic time that is identified from a natural return to ambient temperature. The right-hand side of Equation 4 corresponds to the heat sources. It can be divided into two terms that differ in nature:

• the intrinsic dissipation d 1 : This positive quantity corresponds to the heat production due to mechanical irreversibilities during the deformation process, for instance viscosity or damage; • the thermomechanical couplings s tmc : They correspond to the couplings between the temperature and the other state variables and describe reversible deformation processes. Consequently, their integration with respect to time over one cycle is null.

Therefore, the temporal integration of Equation 4 over one mechanical cycle provides the energy density due to intrinsic dissipation W c𝑦cle intrinsic :

W c𝑦cle intrinsic = ∫ c𝑦cle (s tmc + d 1 ) dt = ∫ c𝑦cle d 1 dt. ( 5
)
W c𝑦cle intrinsic is divided by the cycle duration to obtain the energy rate P d 1 due to intrinsic dissipation or the average intrinsic dissipation over one cycle. It should be noted that d 1 is possibly nonconstant during the mechanical cycle.

Energy balance

The energy balance consists in comparing the energy contained in the mechanical hysteresis (W c𝑦cle h𝑦st ), more precisely P c𝑦cle h𝑦st , with P d 1 . Using the energy rate instead of the energy itself enables us to compare the energetic response from one cycle to another one. If both quantities are equal for a given cycle, all the mechanical energy of the hysteresis loop is converted into heat. Else, it is necessary to distinguish the part of the mechanical energy converted into heat from the one stored by the material to change its microstructure. For that purpose, the ratio 𝛾 se recently proposed by Le Cam and co-workers [13] is used:

𝛾 se = W c𝑦cle stored W c𝑦cle h𝑦st , ( 6
)
where W c𝑦cle stored = W c𝑦cle h𝑦st -W c𝑦cle intrinsic . The subsequent question is whether all (or a part of) the energy stored is released or not during unloading. For crystallizable unfilled natural rubber, it has been shown that all the energy stored is released, but with a different rate, which forms a hysteresis loop. At the end of the mechanical cycle, the material returns to the same thermal and deformation states (see more details in Chrysochoos et al. [27] ). This will be discussed for the case of the TPU studied in the following.

EXPERIMENTS

Materials and experimental setup

The material considered here is a TPU referred to as Irogran A87H4615 TPU marketed by the Huntsman corporation (The Woodlands, Texas). It is elaborated by reacting together a diisocyanate, a macrodiol (long chain diol), which is a polyester, and a small molecule chain-extender (butane diol). [28] This TPU has a melting temperature of 180 • C, a density equal to 1.20 kg/dm 3 , and a hardness of 85 Shore A at 23 • C. The TPU is mixed with carbon dioxide in its supercritical state during injection, in order to form cells and to significantly decrease its density. The forming process is based on two patents [START_REF] Colton | Microcellular foams of semi-crystaline polymeric materials[END_REF][START_REF] Martini-Vvedensky | [END_REF] and is more precisely described in Primel et al. [28] and Lachhab et al. [3] The samples were obtained by cutting a bar with a water jet cutting device. Three densities were compared: 1.2, 0.7, and 0.5 kg/dm 3 , the corresponding samples are denoted, respectively, d 1.2 , d 0.7 ,andd 0.5 in the following. The density equal to 1.2 corresponds to the compact form, obtained without mixing the TPU with carbon dioxide. The sample geometry is presented in Figure 1.

It can be noted that this geometry prevents the sample from any sliding in the grips. Figure 2 presents micrographs of the cellular microstructure of specimens d 0.7 and d 0.5 in the vicinity of the specimen surface. A skin effect is observed close to the surface. The skin contains a very few number of cells. It is therefore stiffer than the rest of the sample and could influence significantly the mechanical response of the foamed samples. Its thickness is about 1 mm for both densities. This is the reason why the skin formed during the moulding was removed from the bar by using water jet cutting.

Loading conditions

Uniaxial tensile loadings were carried out with a 5543 Instron testing machine. Tests were performed under prescribed displacement. The signal shape was triangular to ensure a constant strain rate during loading and unloading. As shown in Figure 1, mechanical tests were performed by applying five mechanical cycles at four increasing maximum stretch levels (𝜆 1 = 1.5, 𝜆 2 = 2, 𝜆 3 = 2.5, 𝜆 4 = 3). The number of cycles at a given stretch was chosen to ensure the mechanical response to stabilize, in order to investigate softening effects. Two loading rates were applied, ±100 and ±300 mm/min.

Thermal measurement

Temperature field measurements were performed using an FLIR X6540sc InSb infrared camera (640 × 512 pixels, wavelength ranges between 1.5 and 5.1 𝜇m, detector pitch of 15 𝜇m), with an integration time equal to 1,000 𝜇s. The acquisition frequency was set at 50 Hz. The thermal resolution, namely, the noise-equivalent temperature difference, was equal to 20 mK at 25 • C. The calibration of the camera detectors was performed with a black body using a Non-Uniformity Correction procedure. Temperature measurement was performed at the specimen centre, by averaging the temperature in a small zone of 5 × 5 pixels at the centre of the sample. Temperature variation was obtained by subtracting the initial averaged temperature to the current one. As the zone moves during the test, a suitable movement compensation technique enabled us to track this zone during the test (see Pottier et al., [START_REF] Pottier | [END_REF] Toussaint et al., [32] or Samaca Martinez et al. [33] for further information on this technique). To process IR images, emissivity of the polyurethane surface was set at 0.9 for compact TPU and 0.62 for TPU foams.

FIGURE 1 Sample geometry and loading conditions

RESULTS AND DISCUSSION

Mechanical responses

The mechanical responses obtained are presented for samples d 1.2 , d 0.7 ,andd 0.5 at ±100 and ±300 mm/min in Figure 3(a) and 3(b), respectively.

These figures give the nominal stress, defined as the force per initial surface ratio, in relation to the stretch. For each loading condition and density, typical effects in the mechanical response of elastomers are observed: softening, residual stretch, and hysteresis. These effects have been widely investigated in the literature. [2,5,6,[START_REF] Grady | The science and technology of rubber[END_REF] Concerning more particularly the present TPU, an exhaustive analysis is provided in Lachhab et al. [3] and is therefore not precisely detailed here. The main results can be summed up as follows:

• decreasing the density leads to a decrease in the global stiffness and in the hysteresis area, slightly increases the residual stretch, and does not affect significantly the stress softening; • the loading rate has no significant influence on the stress softening, although it influences more the residual stretch and the hysteresis area. It should be noted that the loading rate does not affect significantly the stiffness, which is a less intuitive result.

These mechanical curves are used to determine the strain energy densities needed for energy balances.

Calorimetric responses

Temperature measurements have been performed during the mechanical tests. Parameter 𝜏 have been characterized for each test configuration and material tested, and the corresponding heat sources have been determined (see Equation 4).

It should be noted that, in practice, 𝜏 is usually identified from a natural return of the specimen to room temperature after a heating. As it was shown in Samaca Martinez et al., [25,[START_REF] Samaca Martinez | [END_REF] 𝜏 strongly depends on the stretch. The identification of 𝜏 was therefore performed for the different densities and stretch levels applied.

Results obtained are presented in Figure 4(a) and 4(b) for the three densities tested at ±100 and ±300 mm/min, respectively.

FIGURE 2 SEM micrographs of the skin

As shown in these figures, the heat power density increases when the TPU is stretched and decreases when the stress is released. The effect of the entropic coupling is therefore the most significant. The higher the density, the higher the maximum heat sources produced. As highlighted by the mechanical responses, the material accommodates each time the maximum stretch is increased, which explains that the highest value of the heat sources is reached for the first mechanical cycles of each set.

Let us now consider the calorimetric response versus the stretch for the three densities at the two loading rates (Figures 5 and6). In this analysis, the calorific response of the compact TPU d 1.2 at ±300 mm/min (Figure 5a) is chosen as the reference one. For the first two sets of stretches applied (𝜆 = 1.5and𝜆 = 2), heat power densities are symmetrical with respect to the abscissa axis, meaning that the elastic coupling is the preponderant contribution to the calorific response. Therefore, no significant intrinsic dissipation is produced as the curves are not shifted towards the positive heat power densities. For sets at a maximum stretch superior to 2, a strong change in the curve slope is observed during the loading (at about 𝜆 = 2.1 for the third set at 𝜆 = 2.5). As the mechanical response does not exhibit such a singularity, this cannot be due to the effect of the entropic coupling. This strong increase in the heat production is analogue to what is observed in NR when crystallizes [12,25] and highlights that the present TPU is crystallizing under strain. The higher the maximum stretch applied, the higher the stretch at which the strong increase in the heat production is observed, and the higher the residual stretch. This means that SIC in TPU, more precisely the beginning of SIC, should be mainly driven by the stretch. For the last set of maximum stretch applied (𝜆 4 = 3), the maximum value of the heat sources reached is not higher than the previous sets, meaning that a saturation effect as observed in natural rubber (or a crystallinity rate decrease) could be obtained in TPU. During unloading, a strong dissymmetry, comparable to what is obtained in NR, is observed. In the case of NR, X-ray diffraction (crystallinity versus stretch curves) shows that crystallinity is higher during unloading than during loading [36,37] and that the dissolution of the crystalline order in NR is faster than its establishment after a strain step. [38] This demonstrates that crystallization continues during unloading in NR and this is probably what occurs in the case of the present TPU. During unloading, a strong heat absorption is observed at a stretch inferior to that at which SIC starts. This is once again analogue to NR. The next paragraph details the effect of the density and the loading rate separately.

FIGURE 3

Mechanical responses (stress vs. stretch) for the three densities

Effect of the density

Let us consider the foamed TPUs at ±300 mm/min (Figure 5b and5c). For d 0.7 , the previous observations apply, except that the high heat production due to SIC is lower. This seems to indicate that the crystallinity is lower. For the first set of mechanical cycles, SIC starts at a stretch of about 1.4 for 𝜆 1 = 2, which increases for sets at higher maximum stretches, as the residual stretch also increases (SIC being driven by the strain). The fact that the stretch at which SIC starts is lower FIGURE 4 Heat sources versus time for the different densities at the two loading rates FIGURE 5 Heat sources versus stretch for the different densities at ±300 mm/min in the foamed is explained by the fact that the cavity wholes concentrate the strain. Nevertheless, the macromolecular mobility is decreased, which decreases the ability of the macromolecule network to form crystallites and explains why the heat production is lower than in compact TPU. These reasons explain why for d 0.5 , it was not possible to detect SIC with the technique used. Increasing the void volume fraction is in a strong analogy with the addition of fillers in crystallizable rubbers in terms of effects on crystallinity. Note that it induces the reverse effect on the stiffness.

Effect of the loading rate

Heat sources increase with the loading rate, which is due to the couplings between temperature and stretch and to intrinsic dissipation if any. When the loading rate is increased by a factor 3, the heat sources also. As the mechanical response is quite the same between the two loading rates, this means that the heat sources are mainly due to the coupling between temperature and stretch, in other terms the intrinsic dissipation is very low. Nevertheless, a hysteresis loop is observed in the mechanical response, which means that the material uses a part of the mechanical energy to change its microstructure. This will be discussed in the next paragraph. Moreover, conclusions drawn on the effect of the density on the calorific response are not changed by decreasing the loading rate.

Energy balance

To further discuss on the energetic behaviour and especially on physical origin of the mechanical hysteresis, P c𝑦cle h𝑦st and P d 1 over each cycle are plotted in the same diagram in relation to the density and for the two loading rates (Figures 7 and8). In F7 F8

the case where the mechanical hysteresis is due to viscosity only, P c𝑦cle h𝑦st and P d 1 are equal, that is, all the energy contained in the hysteresis loop is converted into heat. If P d 1 is inferior to P c𝑦cle h𝑦st , the material uses a part of the mechanical energy to change its microstructure. In other words, the material stores energy. In Figures 7 and8, the green zones stand for the energy density per time unit due to microstructure changes (P c𝑦cle stored ), the red zones for P d 1 . Figure 7, giving results for the lowest loading rate, shows that P d 1 is not the main contribution to P c𝑦cle h𝑦st , whatever the density and the stretch considered. Thus, the material uses part of the energy brought mechanically to change its microstructure. The main physical origin involved in the hysteresis loop is therefore not viscosity in the range of the loading rates applied. This is in the same line as results recently obtained by Mott et al. [11] who have shown that during extension, the polyurea they tested uses a significant part of the mechanical energy to change its microstructure, whatever the stretch level considered. This is analogue with what was observed in NR by Le Cam. [12] P c𝑦cle h𝑦st increases with the stretch level and the density. Furthermore, the higher value of P c𝑦cle h𝑦st is reached at each first cycle, and we consider that it is stabilized from the second cycle of each set of maximum stretch. This has to be put into perspective with the strong residual stretch observed at the first cycles. This is not observed the next cycles (no increase in the residual stretch, no decrease in the value of P c𝑦cle h𝑦st ). As P d 1 is very low whatever the cycles considered, this might mean that some latent heat remains in the material after the first cycles, in other words, all the crystallites do not melt. This assumption has to be verified by further X-ray diffraction experiments. * These results are then compared with those obtained at the highest loading rate (see Figure 8). For samples d 1.2 and d 0.7 , *It should be noted that if all the crystallites formed do not melt, all the heat produced during crystallization is not reabsorbed during melting, meaning that an additional apparent P d 1 can be detected FIGURE 9 𝛾 se for the three densities at ±100 mm/min (primary axis on the left) and ±300 mm/min (secondary axis on the right) the viscous effects are found to increase with the loading rate, the density, and the maximum stretch applied. The increase of the viscous effect, that is, the decrease in the energy stored, has already been highlighted in other polymer materials such as PA66. [39] This is not observed for sample d 0.5 for which P d 1 remains low. This is a remarkable result that was previously observed in Lachhab et al [3] by considering the thermal response. Indeed, for this density level, no self-heating was observed, even when the loading rate was increased by a factor 3. Such a result is of importance for designing TPU parts submitted to cyclic loadings. In order to summarize these results, the ratio 𝛾 se has been calculated for every second cycles, for which the mechanical stabilization is obtained. This leads to the diagram in Figure 9, given 𝛾 se in relation to the density, the loading rate, and the stretch. It should be noted that the calorific response is not stabilized from the second cycle, especially for results obtained samples d 0.5 at ±300 mm/min. In this case, P d 1 tends to zero, which means that 𝛾 se would be close to 1 whatever the stretch applied, similarly to what is observed at ±100 mm/min. Such a representation opens a new way to investigate the thermomechanical and energetic behaviour of elastomers. Further investigations are currently carried out in our laboratory to investigate the physical meaning of the values of 𝛾 se in relation to the material microstructure and the experimental parameters.

CONCLUSION

The present study investigates the thermomechanical and calorific behaviour of TPU foams. Complete energy balances have been carried out in order to highlight the physical origin of the hysteresis loop. Different densities were tested, including the compact state, which ensures a relevant characterization of the effect of the cell volume fraction on the TPU's formulation used. A series of cyclic uniaxial tensile tests were carried out at different loading rates and different sample densities. The softening was found to be only dependent on the maximum stretch applied. Contrarily, the residual stretch and the hysteresis are clearly affected by changes in the loading rate and the density (the amount of viscous matter available). The calorimetric response has shown that the material's elasticity is mainly entropic, but it uses mechanical energy brought to it to change its microstructure. This is in good agreement with measurements recently performed in a compact polyurea in Mott et al. [11] For samples of the highest density, viscous effects are found to increase with the loading rate, the density, and the maximum stretch applied. This is not observed for sample of the lowest density, for which the intrinsic dissipation remains low. Therefore, for the lowest density, no self-heating was observed, even when the loading rate was increased by a factor 3. To fully address viscoelastic effects, complementary experiments at loading rates of one order of magnitude higher are currently carried out. Furthermore, the fact that the material is able to store part of the mechanical energy is in a strong analogy with what is observed in natural rubber. This probably explains why both materials are among the most resistant to the crack growth. Such a result is of importance when designing foamed TPU parts submitted to cyclic loadings.

FIGURE 6

 6 FIGURE 6Heat sources versus stretch for the different densities at ±100 mm/min

FIGURE 7 FIGURE 8

 78 FIGURE 7 Maps of the strain power density and intrinsic dissipation over each cycle in relation to the density and the stretch (±100 mm/min)

ACKNOWLEDGEMENTS

The authors thank the Cooper Standard France company for supporting this work and for fruitful discussions. The authors thank also the National Center for Scientific Research (MRCT-CNRS and MI-CNRS) and Rennes Metropole for supporting this work financially.


Student Member, IEEE L Seddiki

Member, IEEE Kevin Guelton
email: kevin.guelton@univ-reims.fr.

S Leteneur

J Zaytoon

Concept and simulation of "Sys-Reeduc": Closed Muscular Chain Lower Limbs Rehabilitation Device

This paper deals with a concept of a lower limbs closed loop rehabilitation device. The human force applied to this device is considered as an external disturbance to the system dynamics. A control structure was proposed to track desired trajectories and to attenuate the external disturbances.

Stability conditions are given in terms of Linear Matrix Inequalities (LMI) using a quadratic Lyapunov function and an H-infinity criterion. Finally, the simulation of the designed device on an isokinetic movement illustrates the efficiency of the proposed control structure.

I. INTRODUCTION

T ECHNICAL assistance to functional rehabilitation was the subject of many studies during last years, see e.g. [START_REF] Dallaway | Rehabilitation robotics in europe[END_REF][2] and references therein. Thus, a lot of rehabilitation devices were designed within specific wishes expressed by the clinicians. For instance, Lokomat is a device for the gait rehabilitation of patients with neuromuscular traurnatisms [START_REF] Jezemik | Automatic gait-pattern adaptation algorithms for rehabilitation with a 4-dof robotics orthosis[END_REF]. Another example is an open muscular chain lower limbs rehabilitation device developed by the CReSTIC and named Multi-ISO [START_REF] Moughamir | Conception et developpement d'une machine de reeducation des membres inforieurs[END_REF] [START_REF] Moughamir | A system approach for control development of lower-limbs training machines[END_REF]. The design of a new device is made starting from specific needs, expressed by clinicians, in rehabilitation. Thus, the concerned muscular complex and the various rehabilitation techniques have to be taking into account with special cares. In this study, we are interested in knee rehabilitation and consequently, to the global lower limb muscular complex.

Two rehabilitation techniques can be used: the first one is named open muscular chain (OMC) which is characterized by strengthening isolated muscular group whereas the second one, named closed muscular chain (CMC), is characterized by recruiting both the agonist and antagonists muscular groups that contributes to the movement. These two techniques are complementary and bring in various stages of rehabilitation protocols established by the clinicians.

For technical reasons, it should be difficult to design a device that is able to reproduce simultaneously these two techniques. The only device that is able to reproduce both the techniques is named Motionl\1aker but remains to a very complex and expensive robotic structure that should only be used in the context of research studies [START_REF] Metrailler | Improvement of rehabilitation possibilities with the MotionMaker™[END_REF]. OMC devices are commonly used in clinical context for the lower limbs rehabilitation. This is the case of isokinetic devices such as Cybex, Biodex or Multi-Isa [START_REF] Moughamir | Conception et developpement d'une machine de reeducation des membres inforieurs[END_REF] [START_REF]Les appareils d'isocinetisme en evaluation et en reeducation musculaire: interet et utilisation[END_REF]. More recently, CMC rehabilitation have been the subject of a particular attention because of its aptitude to stabilize the targeted joint while ensuring similar exercises used in the Daily life (gait, sit to stand, etc ... ). For instance, Moflex, Contrex LP or Erigo devices are CMC rehabilitation devices [8][9][10]. In this study, we propose the design of a new CMC lower limbs rehabilitation device named "Sys-R66duc". This apparatus is designed to allow the rehabilitation of each lower limb independently. The goal of designing a new rehabilitation device is not to compete with the previously quoted devices. In fact, the objective is to mix their advantages and to bring some new specifications to allow a precise rehabilitation when soliciting the 6 degrees of freedom of the knee joint [11].

The paper is structured as follows. First, we will give some advantages and drawbacks of both rehabilitation techniques in order to justify the choice of the designed structure. Then, the dynamical model and the control structure based on disturbance attenuation will be presented. Finally, simulation results will illustrate the tracking performances.

II. "SYS-REEDUC" DEVICE CONCEPT

A. Definitions , advantages and draw backs of OMC and CMC rehabilitation techni ques

The exercises in open muscular chain are defined by the contraction of the agonist muscular group that allows the movement of the considered segment. In this rehabilitation mode, the foot is considered "free", figure 1. This approach is efficient and allows a fast recovery when strengthening isolated muscular groups. However, OMC has a major drawback, it causes constraints localized on the anterior cruciate ligament and on the patellar tendon which can generate significant pains during an intensive use [START_REF] Lutz | Comparaison of tibio-femoral joint forces during open kinetic chain and closed kinetic chain exercices[END_REF] [START_REF] Bynum | Open versus closed chain kinetic exercices after anterior cruplate ligament reconstruction», A prospective randomized study[END_REF]. Moreover, this technique tends towards to joint imbalance due to asymmetrical tension of the targeted joint complex.

Exercises in CMC are defined by a simultaneous contraction of both the agonist and antagonists muscles for a considered movement. In the most case, these exercises are possible when the foot is in contact with a support, figure 2. Thus, the whole muscles contribute to the joint's actuation. Consequently, the musculo-skeletal tension is done in a symmetrical way and contributes to increase the joint stability. Thus, patellar pains are reduced during CMC rehabilitation compared to the OMC rehabilitation [START_REF] Steinkamp | Biomechanical consideration in patellofemoral joint rehabilitation[END_REF] [START_REF] O'connor | Can muscle co-contraction protect knee ligaments after injury or repairn[END_REF]. Note that the benefit of CMC rehabilitation can be obtained for a longer period. Finally, OMC and CMC rehabilitation techniques are complementary and make it possible to reach specific rehabilitation goals regarding to different pathologies. For instance, during long-term rehabilitation protocols, one can imagine to propose, in a first step, CMC exercises for stabilizing the whole joint complex and retaining from constraints generated by the OMC. In a second step, when the patient's clinical state makes it possible, rehabilitation in OMC could be proposed for a specific muscular strengthening as the joint balance achieved in CMC is maintained.

B. Kinematical concept of "Sys-Re educ"

At the CReSTIC, an isokinetic rehabilitation device, named Multi-Iso, is available [START_REF] Moughamir | Conception et developpement d'une machine de reeducation des membres inforieurs[END_REF]. In order to extend to a large panel of rehabilitation protocols, our objective is now to design an isokinetic CMC rehabilitation apparatus. By analogy with robotic systems, the mechanical complex {lower limb, rehabilitation device} must constitute a closed kinematical chain. For each lower limb, a closed kinematical chains, presented at figure 3, is proposed and composed of the links E1, E2, E3, E 4 and E5 that are respectively the apparatus base, the thigh, the leg, the foot and the mobile foot's support. The joint between two links i and j is noted L,.;. �.3 L i .2 According to the kinematical chain depicted in figure 3, the design of a new device remains in defining the nature of the joints �.2, L2.3, L3. 4 , L 4 .5 and L5.1 . In that way, we assume that the human body consists in a set of rigid poly articulated segments. In the sagittal plane, the whole kinematical structure, presented in figure 4, is supposed to be composed of perfect joints such as:

-Lu is the joint between the base and the thigh. The patient sits on a chair fixed to the base of the system. In that case, the trunk and the pelvis are also supposed to be fixed to the base. Then, L i .2 is supposed to be a spherical joint corresponding to the coxo-femoral joint (hip) .

-L2.3 corresponds to the knee. This one is the main joint we wish to rehabilitate. Its complex kinematics was the subject of a previous study [START_REF] Petermann | Efficacy of linear isocinetic versus dynamic auxotonic muscle training for the shoulder-A prospective, randomised, controlled study in men[END_REF] and will not be detailed in this paper.

In order to simplify the specification of the «Sys-Reeduc» kinematical concept, the knee will be represented as a hinge joint around the main lower limb flexion-extension axis.

-The joints L3_ 4 and L 4 _5 can be reduced to an unique joint �.s. To do that, a design constraint is to be made: the ankle axes and mobile support hinge joint, in the sagittal plane, are perfectly aligned with those of the foot mobile support. This constraint allows removing the foot from the kinematical chain. Then, L3.5 consists of two hinge joints. The first one is the plantar flexion/dorsal flexion (L 1 d ankle axis that is collinear with the main lower limb flexion-extension axis).

The second one is the ankle internal-external rotation ( L , .

ankle rotation m the tibia axis). Note that the adduction/abduction of the ankle was voluntarily omitted, insofar as it doesn't influence a lot the knee rehabilitation.

-The joint L5.1, between the foot's mobile support and the base is realised by a prismatic joint along the axis defined by the lower limb in complete extension. Note that, according to the kinematics defined in figure 4, an external disturbance J P ( t) to be attenuated or to be rejected. In [START_REF] Seddiki | H-infinity Takagi Sugeno fuzzy control of a lower limbs rehabilitation device[END_REF], the synthesis of a robust nonlinear control law with an H00 criterion was proposed. Note that, this level does not allow the voluntary control of the system by the patient.

-A discrete level is devoted to provide the desired trajectory y d (t) to be tracked by the output y(t) of the continuous dynamical system. This one is based on clinical specifications (isokinetic, isometric, isotonic ... ).

The principle remains on a discrete states machine [START_REF] Moughamir | A system approach for control development of lower-limbs training machines[END_REF][17] and allows the voluntary control of the system by the patient. Thus, according to the effort applied by the patient to the device J P ( t), a desired and coherent trajectory y d (t) is generated. (along the x axis and located by the variable q1 ( t)) are sufficient to solicit the knee in its six degrees of freedom [11]. So these two axes will be motorized and the plantar flexion/dorsal flexion will remain passive (free or forced by a mechanical brake). In terms of rehabilitation, the motorization of the joint L5•1 brings about a movement of the lower limb along the medial-lateral axis (flexion extension movement). This allows the rehabilitation of the flexor muscular complex and limits the constraints applied to the cruciate ligaments. Moreover, the foot's mobile support rotations allow the knee internal/external rotation and the plantar flexion/dorsal-flexion. These make possible a selective and precise muscular constraint. Then, rehabilitation or muscular strengthening can be practiced in a specific way during a desired movement. For instance, the lower limb movement with the foot in external rotation and in extension helps the stretching of medial hamstrings.

III. MODELLING AND CONTROL OF «SYS-REEDUC»

A. Generic control structure for rehabilitation devices

A generic control structure for rehabilitation devices is presented in figure 5. This one was developed and experimentally validated in the case of OMC rehabilitation on Multi-Iso [16] [20]. Thus, the proposed control structure of such a rehabilitation device is based on two levels depicted figure 5.

-The continuous control level is devoted to stabilize the «Sys-Reeduc» device in closed-loop. To provide a safe behaviour, this level must ensure the stability of the mechanical part whatever the patient's effect to the machine is. In this case, we consider the human effect as In the following, we will only consider the continuous level.

Let us recall that various generic rehabilitation modes have been already translated in terms of discrete states machine [START_REF] Moughamir | A system approach for control development of lower-limbs training machines[END_REF][17] and will not be included in this article.

B. "Sys-Reeduc" dynamical model

The model is obtained through the Newton-Euler's equations. Figure 6 presents the mechanical scheme to be modeled. C M 1(t) and C M 2(t) are respectively the motors torques that allow the degrees of freedom movements associated with the variables q1 ( t) and q2 ( t) . In order to synthesize the control law, we consider that the device is to be controlled via the motors torques while the patient applies external efforts (the force J P (t) along the x axis and the torque C P ( t) around the z, axis).

Note that the goal is now to validate in simulation the control structure proposed for the "Sys-Reeduc" concept and not to provide a fine modeling of a mechanical structure that is to be build (and not available at the moment). Then, in the following modeling, we neglect the frictions and backlash and only inertial masses will be tacking into account. Moreover, using the control scheme depicted in figure 5, only the mechanical part less the human lower limb of the CMC kinematical structure have to be controlled as the patient effect to the device is considered as an external disturbances. In that case, the modeling remains to a robot with open chain mechanism. After isolating the foot's mobile support, the movement equations of the robot effectors (mobile support) are given by:

C. Control law design

The objective is to guarantee the trajectory tracking for the system described by [START_REF] Riener | control of robots for rehabilitation[END_REF]. Then, the goal is to ensure the convergence of the tracking error defined by [START_REF] Dallaway | Rehabilitation robotics in europe[END_REF] e(t)=x(t)-xd(t) with xd(t) the state vector of desired trajectory.

with a the band wheel radius of the system realizing the translation along the x axis, m and J the foot's mobile support mass and inertia.

We suppose that the state is available, the model is subject to external disturbances cp( t) to be attenuated. In order to synthesize a control law that stabilizes the system subject to external disturbances, a robust control approach is needed.

Thus, an H00 criterion may be used since it allows Naturally, the model ( 1) can be rewritten as the following attenuating external disturbances and ensures the security of state space representation:

the patient. This one is given by [START_REF] Zhou | Essentials of robust control[END_REF]:

if if !x(t ) = Ax(t )+Bu( t) + H cp ( t) y(t) = Cx(t)
(2) f e7 ( t ) Q e(t )dt � r; 2 f [ r 7 ( t) r (t )+ <p 7 ( t ) cp ( t) ]dt m,J with Q > 0 the weighting matrix and r; the attenuation rate.

••-•••-•••-••-••-••-•• i Motor Belt Mobile support
We consider the control law given by:

u(t) = -Ke(t) (4)
where KE 91 2 x 4 is the feedback gain matrix.

The dynamic of the desired trajectory xd ( t) is given by the following state equation:

(

) 5
with Ad E 91 4 x 4 an Hurwitz matrix and r ( t) E 91 4 the input of the reference model.

Figure 7 shows the trajectory tracking control scheme with the reference model which will be used in the following.

q.i ( t )

-----------x( t )

y ( t )

Note that, for technical aspects related to the motors capability, the control signal is saturated.

The tracking error dynamic can be written as:

e ( t) = (A + BK) e ( t) + (A -Ad ) x d ( t) + H cp ( t) -r ( t) (6)
Let us consider x , ( t ) = [e ( t ) xd ( t ) T E'.R8 as an extended state vector. Its dynamic is given as:

x, (t) = A,x, (t)+ H , ¢ , (t)

[A+BK with A = e Q 4x4 [ rp ( t ) ]
and ¢ , (t)= O M . r (t)

A-Ad ] A , d (7)
H = [H e Q4x 2
In this case, the H00 performance described by the inequality ( 3) , can be rewritten according to the extended state x , (t) such as:

If tf f( X , T ( t) Q, X , (t) d /) c; 7/ f ¢; (t) ¢, ( t)dt ( 8
)
where

Q , = [ Q 0 4 ' 4 ] Q4x4 Q4x4
and ¢; (t) !/J. (t) = r7 (t)r(t)+ rp 7 (t) rp (t).

The stability of ( 7) will be verified using the following Lyapunov [ , , ,, . ,, ,

+z1K7 B7 +BKz1 Az 2 -Ad z 2 H -/4x4 +z1 QZ 1 c; o ( 14) z2A7 -z 2 AJ z2AJ + Adz2 Q4x4 /4x4 H T Q4x4 -r; 2 I 4x4 Q4x4 -/4x4 /4x4 Q4x4 -r; 2 J4x4
In ( 14) , z1, z2, K , Ad and the attenuation rate 77 are the variables to be find. The inequality [START_REF] Steinkamp | Biomechanical consideration in patellofemoral joint rehabilitation[END_REF] is obviously nonlinear within these variables. In this case, this cannot be v ( x , (t)) = x ; (t)Px , (t) solved using classical convex optimization tools such as the one describe in [START_REF] Gahinet | LMI control Toolbox[END_REF]. To obtain an Linear matrix inequality (LJ'v1I) formulation, some matrix transformations are needed.

Thus, one first apply the Schur complement on the first (9) diagonal bloc of [START_REF] Steinkamp | Biomechanical consideration in patellofemoral joint rehabilitation[END_REF] , it yields with P = P7 > 0 a ll18 x 8 matrix.

Indeed, the extended system described by ( 7) is stable along the trajectory x , ( t) if v( x , ( t)) decreases for t E '.R+ within the H00 performances (8), i.e. if there exist a matrix P = P7 > 0 satisfying the following constraint: Finally, the control law design consist in solving the LMI [START_REF] Seddiki | H-infinity Takagi Sugeno fuzzy control of a lower limbs rehabilitation device[END_REF]. Thus, the gain matrix K , the Hurwitz matrix Ad and the attenuation rate µ = r/ are automatically obtained by means of the matrices M1, M 2, z1 and z2 solving by LMI algorithms such as the Matlab LMl toolbox [START_REF] Gahinet | LMI control Toolbox[END_REF]. This ensures the trajectory tracking closed loop stability with attenuation of the external disturbances.

IV. SIMULATIONS RESULTS

The Trajectory tracking of the mechanical system (2) is simulated in closed loop via the control law ( 4). The model parameters used in this simulation are a= 0.025(m) , m = 5 ( kg ) and J = 1 ( kg I m2) . The design of the control law is obtained by solving the linear matrix inequality ( 16) with the Matlab LMl Toolbox [START_REF] Gahinet | LMI control Toolbox[END_REF]. Therefore, the performances was imposed by fixing the weighting matrix Q = diag[0.899 0.899 0.899 0.899). Thus, the matrices and the attenuation rate leading to a feasible solution of ( 16) are given by: One of the characteristics of the proposed LMl approach is that it allows obtaining automatically an Hurwitz reference matrix Ad whereas in the literature concerning trajectory tracking, its choice is difficult and often arbitrary realized.

In order to illustrate the disturbances attenuation, the simulation is realized by imposing external efforts that mimic the physiological constraint that is assumed to be maximal regarding to the effort that can be apply by a patient to "Sys-Reeduc". These are given by: f, (t) = 37 • 5 sin(6t) and C (t) =20sin(5t) .

r a r (17)
Note that the disturbance amplitudes correspond to the maximal motor loads and the frequency is quite slow ( � lHz).

In figure 8, the simulated movement corresponds to an isokinetic extension of the lower limb with the velocity q1 = 0. 8 m Is realized between t = 1 s and t = 2 s from the initial position q1 (0) = 0.2 m . This translational movement is combined to a sinusoidal internal-external rotation q 2 = -( 7r I 4) cos(2t) , presented in figure 9.

Desired isokinectic trajectory Now, the following simulation, presented figure 10 for the lower limb isokinetic extension, will illustrate the performances of the proposed control law on the same movement but with "non physiological" external disturbances. That is to say a signal f (t) that outperforms the effort that can be applied by a patient:

f ( t) = 500sin ( 6t) + 250sin(30t) +125 sin( 60t) + rand (t) (18)
In this signal, the function rand ( t) is used to simulate a noise measurement of the force applied by the patient. This one is a Gaussian disturbed signal with a variance of 1000 at a sample time of O.Ols.

l : :r r . These simulation results show that the H00 control law successfully attenuates the external disturbances by means of compensations in the input signal.

V. CONCLUSION

In this paper, after having expressed the various advantages and drawbacks of the OMC and the CMC rehabilitation techniques, a new device concept for the lower limbs rehabilitation was proposed. This one allows, by means of a CMC rehabilitation, the six degrees of freedom solicitation of the knee joint. The control structure is based on two levels. The first one is a continuous level that ensures the stability of the mechanical system and the second level is based on a discrete state machine to generate the desired trajectories.

In this paper, the continuous level was studied. The efforts apply by the patient to the rehabilitation device were considered as external disturbances. Then, an H 00 control law has been designed in order to stabilize the system as well as attenuate the external disturbances. Stability conditions have been derived in terms of LMl. These one were verified with the considered system and simulation results have shown the efficiency of the proposed approach. Note that attenuating the external disturbances endows a safe behavior with the patient use of the rehabilitation device.

In a next paper, we will specify the discrete part according to many rehabilitation modes (isokinetic, isotonic, etc.). These trajectories will be parameterized in order to help the clinicians to specify an adapted rehabilitation protocol m reply to the considered pathology.

Fig. 1 .

 1 Fig. 1. Exercise in OMC.

Hamstring

Fig. 2 .

 2 Fig. 2. Exercise in CMC.

Fig. 3 .

 3 Fig. 3. Kinematic chain of a rehabilitation device of the lower limbs in CMC.

Fig. 4 .

 4 Fig. 4. Kinematic of Sys-Reeduc device in the sagittal plane.

  External p (t) the internal-external ankle rotation L;. (along the z, axis f Y d (t) 1 disturbance Closed loop y(t) and depicted by the variable q2 (t)) and the translation L5.1 � - Trajectory generator rehabilitation \___; device plant \___;

Fig. 5 .

 5 Fig. 5. Generic control structure for rehabilitation devices.

( 3 )

 3 where the vector cp (t)=[fp(t) Cp(t) Y is considered asan external disturbance Vector, U ( t) = [ C M l ( t) C M 2 ( t) ris the input vector, x(t)=[q1(t) q2(t) q1(t) q2(t)T is the state vector, y ( t) = [ ql ( t) q2 ( t) r is the output vector, state matrix, the input matrix and the transfer matrix of the external disturbance, and C = [ 0 2 x 2 I 2] the output matrix.

Fig. 6 .

 6 Fig. 6. Mechanical principle from the rehabilitation device.

Fig. 7 .

 7 Fig. 7. Trajectory tracking control scheme with reference model.

1 > 0 ,

 10 after left and right multiplying[START_REF] Lutz | Comparaison of tibio-femoral joint forces during open kinetic chain and closed kinetic chain exercices[END_REF] Consider and H = [ H 0 4 ,2 ] with z 1 = z� > 0 and z2 = z � , ll1 4x4 matrixes.[START_REF] Bynum | Open versus closed chain kinetic exercices after anterior cruplate ligament reconstruction», A prospective randomized study[END_REF] can be developed with the matrixes defined below (7) as follows:

  ) + x ; (t) Q,x; (t)-r/¢; (t)¢ , (t) c; 0 ( 10)c an be developed in an extended form such as: [x;(t )][ A;P+PA, + Q, PH, ][ x, ( t) ] c; o ¢; (t) H;P -1/I ¢, (t) ( 11)i s obviously verified if: [A;P+PA, + Q, PH , ]< O H;P -1/I -use the bijective change of variables M1 = Kz 1, M2 = Adz2 and µ = 77 2 . Then ( 1 5 ) becomes:

  Note that the choice of the weighting matrix Q is based on a compromise between the performances and the technical capabilities of the "Sys-Reeduc" device. Thus, the maximal motor load capacities are C M 1 = 37.5Nm for L5 _ 1 and C M 2 = 20Nm for L".

-Fig. 8 .Fig. 9 .

 89 Fig. 8. Tracking trajectory oftranslation of the mobile support.

Fig. 9 .

 9 Fig. 9. Tracking trajectory for rotation of the mobile support.

VI. ACKNOWLEDGEMENT

The authors would like to thank :tvlrs. Maggy Bowl for their valuable advices relating to lower limb rehabilitation.

VII. REFERENCES

This work was supported by the French Ministry of research and the "Region Champagne-Ardennes" within the framework of the CPER "Sys-Reeduc".