E Chabert

A Ershad Langroudi

C Gauthier

J Perez

The thermomechanical activation of deformation in amorphous and (38% crystalline) semicrystalline poly(ethylene terephthalate) (PET) has been investigated using dynamic mechanical tests and large strain experiments. Activation parameters for both materials are determined in the neighbourhood of the glass transition temperature using stress relaxation experiments. The complete mechanical behaviour of amorphous PET is then analysed in the frame of a molecular model or 'quasi point defect' theory. With this aim, a new method is proposed, based on only three isochronal measurements, and leading to the determination of the set parameters. This method makes it possible to reproduce the stress-strain curve over a range of temperatures, as well as the relevant activation parameters. Finally some qualitative explanations are given for the mechanical behaviour of semicrystalline PET.

Mechanical response of amorphous and semicrystalline poly(ethylene terephthalate) and modelling in frame of quasi point defect theory

INTRODUCTION

heated for 1 h at 359 K (10 K above the glass transition temperature T g ) and then quenched to room Poly(ethylene terephthalate) (PET) is extensively used temperature. The samples were characterised using today owing to its transparency and physical propwide angle X-ray scattering (WAXS) and density erties. The plastic deformation of PET, and especially gradient column measurements, to ensure that no its biaxial plastic deformation behaviour, is of great crystallisation occurred after this thermal treatment. industrial interest in the fields of films and bottles.

The 38% semicrystalline PET was obtained by iso-Moreover, PET can be considered as a model system thermal annealing at 430 K for 24 h and then quenchfor a fundamental study. It can be obtained either in ing to room temperature. The crystallinity was the completely amorphous state or, after suitable determined at 296 K using a density gradient column thermal processing, in the semicrystalline state, with containing ethanol and carbon tetrachloride, with variable crystallinity ratios and morphologies. Thus, densities d A =1•335 g cm-3 and d C =1•455 g cm-3 for like isotactic polystyrene and polycarbonate, PET the amorphous and crystalline phases, respectively.3 belongs to the 'low crystallinity' category of polymers

In the present paper, the amorphous and the 38% proposed by Boyd.1 Such polymers can crystallise to crystalline PET are designated A and C38. a limited extent (<50%).

Wide-angle X-ray diffractograms were recorded The aim of this work was to study the linear and at room temperature using a 'Rigaku-Geiger-flex' non-linear mechanical behaviour of amorphous and diffractometer (Cu K a1 , filtered radiation). Figure 1 semicrystalline PET in the domain of the glass transshows the WAXS profiles of A and C38, obtained at ition temperature, as well as the thermomechanical a scanning rate of 0•5°min-1. The amorphous PET activation of deformation. For that purpose, experisample shows two halos while the semicrystalline mental activation volume V exp and activation enthalpy sample exhibits three peaks characteristic of the tri-DH exp were determined with and without the presence clinic phase. The A and C38 samples were then of a crystalline phase. The whole mechanical response analysed using a DSC-7 Perkin Elmer differential of amorphous PET was then analysed using a molecscanning calorimeter at a heating rate of 10 K min-1 ular approach developed in the authors' laboratory, (thermograms are shown in Fig. 2). It can be seen and based on quasi point defects.2 Finally, the influthat the jump in heat capacity C p occurring at the ence of the crystalline phase on mechanical behaviour glass transition is smaller for semicrystalline PET and and deformation mechanisms is discussed.

is shifted towards higher temperatures. Determined from the midpoint of the endotherm, the T g s of A EXPERIMENTS and C38 are 349 and 358 K, respectively.

Materials

Mechanical experiments

The amorphous PET was supplied by Goodfellow Co., France in the form of 1 mm thick sheets. To Dynamic mechanical analysis erase physical aging and to obtain the same isoconfi-Dynamic mechanical spectrometry was performed on a home made inverted pendulum described in Ref. 4. gurational state for all the samples, these sheets were where s appl is the applied stress, c f is the Tresca the ductile behaviour of these materials was investifriction coefficient, and h is the sample thickness gated. Figures 5 and6 show tensile tests performed under compressive stress (h 0 at s appl =0). on A and C38 in the temperature range 293-413 K. Three pairs of dies (b=2, 3, and 4 mm) were used At temperature T <T g , the general features of the to determine c f at ambient temperature (0•013). stress-strain curve for glassy polymers are observed Molybdenum disulphide was used as a lubricant, so for both amorphous and semicrystalline PET (a sharp that friction between die and sample was considered yield point followed by a steady state plastic flow independent of temperature, to a first approximation.

regime and a gradual strain hardening before break).

In fact these curves are valid until the yield point,

RESULTS

beyond which the deformation becomes heterogeneous due to the appearance of macro shear bands Dynamic mechanical analysis and necking. When the temperature of the test is Figures 3 and4 show the variation of G∞ and tan w increased, the initial slope (relaxed modulus) and the with temperature for A and C38. It can be seen that yield stress s y decrease. At T >T g , the yield peak the characteristics of the b relaxation (width and vanishes and the stress increases monotonically with location of the peak temperature T b

) are similar strain. for the amorphous and the semicrystalline PET.

Results for plane strain experiments are presented Additionally, the activation energy U b (determined in Figs. 7 and8, for A and C38, respectively, near from curves obtained at three frequencies) is the same their T g s. As in tensile experiments, four domains of for both materials. The presence of the crystalline mechanical response can be distinguished: elasticity; phase induces only a decrease of the area of the b yield elongation; plastic deformation; and strain hardrelaxation peak. By contrast, the a relaxation is ening. It can be noted that the yield peak vanishes in strongly affected by the crystalline phase: the relaxthe presence of the crystalline phase. These large ation widens and becomes asymmetric, the peak T a is strain experiments (tensile and plane stress comshifted to a higher temperature, and its height pression) provide insights into the mechanical behavdecreases.

iour of amorphous and semicrystalline PET. , respectively. However, for higher temperatures that the decrease of s y with increasing temperature (generally considered to be above 0•7T g ),2 diffusion occurs more rapidly for the amorphous PET than for processes occur and the microstructure evolves under the semicrystalline PET. Moreover, C38 has higher stress; consequently, V exp and DH exp become physically yield stress values than A and also exhibits much meaningless. higher yield stresses in compression than in tension These activation parameters were measured using ( by nearly a factor of 2), while A has similar yield stress relaxation experiments, following a method stress values in tension and in the plane strain comproposed by Guiu and Pratt.7 Thus, the experimental pression mode.

activation volume V exp was determined from7

Thermodynamic parameters

s(0)-s(t)= kT V exp ln A 1+ t t p B . . . . . . (7)
To understand better the mechanisms of deformation in amorphous and crystalline phases, activation parameters were measured using plane strain tests (since where in this mode the deformation appeared to be homogeneous). The thermodynamic analysis of plasticity is (3) at time t (s(0) at t=0) and M is the modulus of the whole assembly machine-sample.

A preliminary study was carried out to optimise with the experimental conditions for determining

V exp . This DG a =DH a -T DS a =DH 0 -sV a -T DS a . . (4)
began with an investigation into the variation of V exp with the values of strain at which the crosshead was where e ˙is the average strain and e ˙0 is the strain at stopped. It was found that V exp decreases from an t=0; DG a is the free energy; DH a , DS a , and V a are initially infinite value, goes through a minimum in the enthalpy, the entropy, and the volume activation the yield region, and reaches a stable value in the parameters, respectively; DH 0 is the height of the plastic flow region. It was therefore decided to deterenthalpy barrier to be crossed by thermal activation; mine the stationary value in this latter region (corresand k is the Boltzmann constant. In the frame of this ponding to e#45%). The dependence of V exp on analysis, the stress sensitivity parameter relaxation time was then explored. Figure 10 shows the variations of s(t) and V exp (t) for the semicrystalline V exp Although this molecular theory has already been other work,8-10 suggests that plastic deformation is a applied successfully to various polymers,13-15 there is cooperative process involving many atoms. Above T g , a significant improvement in the present work since the recovered stress ceases to be monotonic after a the whole behaviour of the PET is modelled using certain period of time. This might be due either to parameters derived from dynamic mechanical analysis the very low stresses reached (the machine might not (DMA) and these parameters are determined from be sensitive enough), or to evolution of the microonly three isochronal measurements (instead of from structure in the case of semicrystalline PET. In these a master curve, which is much more time consuming). cases, the activation volume above T g was determined for periods of less than 1 h, using only the linear part Modelling of linear response of the log s v. log t curve.

The quasi point defect approach is based on three The activation energy DH exp was estimated by main assumptions. noting from equations ( 5) and ( 6) that 1. The amorphous material is described by a close packing of structural units in which some sites exhibit DH exp #-T V exp A ∂s ∂T B . . . . . . . . (9) a local excess of binding enthalpy and entropy. In these sites, called quasi point defects (qpd), the molecular mobility is higher than the average molecular where processes characterised by t b are assumed to be the origin of the b relaxation; t 0 is a timescale parameter; and x is the correlation parameter. The ANALYSIS OF BEHAVIOUR OF AMORPHOUS PET larger x is, the less important is the correlation effect;

x increases with the degree of disorder between 0 To analyse the mechanical behaviour of the amorphous PET in small and large strain experiments, a (perfect crystal, highest correlated movement) and 1 (perfect gas without any correlation). molecular model proposed by Perez et al.2,11,12 was These basic ideas lead to a macroscopic compliance to give the best fit. Finally, the decrease of the elastic modulus with temperature (resulting from a decrease in the mean curvature of the effective interaction potential17), is taken into account via a Relaxation

d3 J an2 J u A J an2 J r B -x A J vp2 J u B -x
U bk -U b )-exp[B Ub (U bk -U b )]} S k exp{B Ub (U bk -U b )-exp[B Ub (U bk -U b )]} . . . .
The analysis of the a relaxation is made using the

J el (T )=J el [1+1•4×10-3(T -T bmin )]
biparabolic expression (equation ( 14)). Thus, the Cole-Cole diagram provides most of the parameters: G r is the value of the rubber plateau (2•5 MPa); d is adjusted to the level of the G◊ peak (1•49); and x(T g ) and x∞ are determined by the angles h and h∞ made by the plot at the high and low temperature intercepts (x=2h/p and x∞=2h∞/p). In fact x∞, which controls the width of the distribution of viscoplastic relaxation times, can be very precisely adjusted to the height of the tan w peak at one frequency. Thus, it was found that x(T g )=0•34 and x∞=0•88. Parameter t 0 determines the timescale over which the global molecular motions occur: the smaller t 0 is, the closer together are the b and a relaxations; t 0 is fitted so that experimental and calculated temperatures of the maximum of one frequency relaxation peak coincide (t 0 = 2•6×10-15 s). Finally, a (equation ( 30) below) which is the origin of the usual William-Landel-Ferry like behaviour above T g , is adjusted to account for the B [START_REF]  | [END_REF] G◊ peak (since d and J an2 are connected) and t rm is evaluated to ensure a good temperature location of and the relaxation peaks. Thus, for amorphous PET

t rm (s)=x1/xt 0 A t b (s) t 0 B 1/x . . . . . . . . (26) J an2 =1•42 C dJ u (J r )x A J vp2 J u B x/x∞ D 1/(1-x) . . ( 22
)
where s 0 is the limiting yield stress necessary to and surmount the energy barrier for conformational change when the temperature is non-linear domain (s 0 , A an , and A vp ) that cannot be obtained from DMA measurements. These para-Finally, the calculated variations of G∞ and tan w with temperature are compared with the experimental meters are adjusted to match the s-e data at ambient temperature: s 0 is taken as being close to 260 MPa curves in Figs. 14 and16. Rather good agreement is observed between experimental data and simulated to fit the flow stress; A an is determined from the height of the yield peak as 0•65; and A vp is related to curves. Moreover, the biparabolic model (full lines) and the expression with a discrete distribution of the strain hardening effect and is found to be 0•1. At this point, it must be emphasised that there is relaxation times (dashed lines) gives very similar results.

K, i.e. s 0 #G 0 /2p t rm =2t a A J an2 J r BA J vp2 J u B 1/x∞ . . . . . . ( 23
no requirement for additional fitting parameters since On the other hand, the changes in the a relaxation with crystallinity cannot be described using the same activation volumes. When the temperature increases, mechanical model, since the crystalline phase causes the movements become translational and therefore the a relaxation peak not only to decrease in ampliinvolve more and more structural units, leading to tude, but also, at the same time, to broaden and shift higher activation volumes and activation enthalpies.

to a higher temperature. In the glassy plateau region, Above T g , the molecular mobility becomes so high the increase of modulus can once again be accounted that plastic deformation occurs throughout the for by a mechanical coupling effect. In addition, owing sample (V exp tends towards infinity) and the activation to physical crosslinking between the two phases, enthalpy decreases (the height of the barrier decreases crystallites may decrease the mobility of the amorphdue to increasing thermal activation).

ous phase, leading to the shift of T a to a higher As a conclusion, it has been shown that the qpd temperature and an increase in the shear modulus of theory describes the main aspects of the plastic behavthe rubbery phase. Thus, the behaviour of the iour of amorphous PET. The simulation of the whole amorphous phase in semicrystalline PET seems to set of data has been developed in a unique frame, differ strongly from the behaviour of fully amorphwith only one set of parameters, deduced principally ous PET. from three isochronal DMA curves. Currently the analysis seems to be one of the most successful Plastic behaviour approaches for combining all aspects of deformation Tensile and plane strain compression experiments behaviour in a unique selfconsistent theoretical frameperformed on amorphous and semicrystalline PET work. A study of the effects of crystalline phase on can be compared by considering first the modulus mechanical behaviour is considered below.

values, then the yield stress and strain (s y , e y ) values reached at the flow threshold, and finally, the experi-ANALYSIS OF BEHAVIOUR OF mental activation parameters (V exp and DH exp ) for SEMICRYSTALLINE PET plastic deformation. First, it can be observed that the moduli of A and Linear domain As shown in Figs. 3 and4, the crystalline phase does C38, determined from plane strain compression experiments, show the same dependence upon temper-not affect both relaxations in the same way. On one hand, its principal effect on the b relaxation is to ature as do their shear moduli G∞(T ) measured in DMA tests (Fig. 3). Furthermore, the decreases reduce its amplitude. This dependence of the b relaxation intensity upon crystallinity was therefore investi-in s y (T /T g ) for amorphous and semicrystalline PET (Fig. 9) are also similar to those observed in G∞(T ). gated for crystallinity X c ranging from 0 to 57%, and was then simulated using a mechanical reinforce-Hence, the crystalline phase appears to affect the modulus and the yield stress values principally by ment model: the auto-coherent model with three phases.29,30 Thus, taking a consistent value of crystal mechanical reinforcement at T <T g and by physical crosslinking of the amorphous phase at T >T g . modulus (for instance, 4 GPa, which allows fitting of tan w data for X c =38%), simulations in the b domain Moreover, it can be stated that the yield stress is related to the deformation of the amorphous phase. appeared to be in good agreement with experimental moduli G∞ and G◊ for all values of crystallinity. These However, it has been noted that the difference between tensile and plane strain yield stress data is greater for calculated results provide evidence that the amplitude of the b relaxation decreases through mechanical semicrystalline PET. For example, at 293 K, A reaches 52 MPa in the tensile mode and 56 MPa in the coupling effects. Consequently, it can be concluded that the b relaxation occurs only in the amorphous compression mode, while C38 reaches yield stresses of 77 and 120 MPa, respectively. This shift can be phase.

explained by hydrostatic pressure and/or morphologi-elastic behaviour (observed in small and large strain experiments), is governed by both reinforcement cal effects. In terms of coefficient of sensitivity to pressure, such values would lead to m=0•04 for A, effects and amorphous phase mobility. The shift between yield stress values in tensile and compression which is in agreement with other work,26 and m=0•2 for C38, which seems very high. Therefore, it is mode is larger for the semicrystalline PET than for the amorphous PET. Again, this can be attributed to suggested that the interaction between spherulites, occurring during compressive deformation, may interactions between the spherulites and the amorphous matrix. hinder the deformation of the amorphous phase, thus leading to higher yield stress values in plane strain Activation parameters were measured using stress relaxation experiments. Below T g , amorphous and tests than in tensile tests.

Finally, as highlighted by the changes in V exp (T /T g ) semicrystalline PET exhibited the same activation volume and enthalpy. Above T g , the activation volume and DH exp (T /T g ), the influence of the crystalline phase on deformation mechanisms can be discussed. Above of amorphous PET tends towards infinity, while the activation volume of the semicrystalline PET levels T g , is has been observed that the activation volume of the amorphous PET increases rapidly with temper-off. Then at high temperatures, plastic deformation occurs in both amorphous and crystalline phases, ature and tends towards infinity (the sample flows), while the activation volume of the semicrystalline whereas at low temperatures, plastic deformation is localised in the amorphous phase. polymer has a finite value. This suggests that at high temperatures, the semicrystalline phase might play a

The whole mechanical behaviour of amorphous PET was modelled using a molecular approach pro-part in plastic deformation, via dislocation propagation. Indeed, it has been shown that the thermal posed by Perez et al. Linear parameters were originally determined from three dynamic mechanical generation of dislocations is possible in crystalline polymers.31

analysis curves obtained at various frequencies. The same set of parameters has been used successfully to At low temperatures however, considering that amorphous and semicrystalline PET have very similar describe the plane strain and stress relaxation experiments. Moreover, predicted values of activation para-activation volumes and activation enthalpies and that the morphology is not strongly affected by plastic meters are in agreement with experiments. deformation (DSC experiments have shown that only the smallest lamellae are modified), it can be assumed REFERENCES that the plastic deformation occurs principally in the

  shear modulus of A (%) and C38 (#) PET This device works in a helium atmosphere in the at 1 Hz temperature range 100-700 K and frequency range 50 mHz to 5 Hz. The storage G∞ and loss G◊ parts of the dynamic shear modulus and hence the internal where L is the sample length during the test (L 0 at friction tan w (=G◊/G∞) were measured as a function time t=0), S 0 is the initial cross-sectional area of the of temperature with a heating rate of 15 K h-1, at sample, and F is the applied force. three frequencies (1, 0•1, and 0•01 Hz). Values of Young's modulus E, measured in tensile tests, were Plane strain compression tests used to normalise the values of G∞ in the glassy Plane strain compression tests were carried out on plateau, based on a correspondence between the an Adhamel Lhomargy DY25 machine, between dynamic data obtained at 1 Hz and tensile data for a 293 and 413 K. Parallelepiped shaped samples strain rate of 10-3 s-1. (33×17×1 mm) were placed between two parallel, Tensile tests flat polished dies (width b) and then transferred to Uniaxial tensile tests were performed on an Instron the thermally regulated cabinet of the compression 8561B machine over a temperature range of machine. The compressive force was measured using 293-373 K. Tests were conducted at a constant crossa piezoelectric gauge with a capacity of 20 kN and head speed, which corresponded to an initial strain an accuracy of ±5 N. The displacement DL of the top rate of 10-3 s-1. Samples were dumbbell shaped with piston was measured with an accuracy of ±0•005 mm. typical dimensions in the gauge portion of 4 mm Tests were carried out at a constant crosshead speed width and 25 mm length. The applied force was corresponding to an initial strain rate of measured using a 5 kN cell. Nominal strain e and 8•3×10-4 s-1. nominal stress s are given by the relationships The relationship between nominal stress s and nominal strain e is obtained using5 e= L -L 0 L 0 and s= 31/2/2s appl 1+(c f b/4h) . . . . . . . . . . (1)

1 10

 1 The relationship between yield stress and normalised tem-Tensile tests and plane strain experiments Continuing a previous study on the ductile-brittle transition of amorphous and semicrystalline PET,6 8 Plane stress experiments on C38 PET; e ˙= 8•3×10-4 s-1 D l/lo 6 Tensile tests on C38 PET; e˙ =10-3 s-Stress relaxation experiment on C38 PET previously deformed up to 45% at ambient temperature, evolutions of s (+) and V exp ($) v. time 9 Evolution of yield stress v. normalised temperature; tensile tests: A (%) and C38 (#) PET; plane strain tests: A (&) and C38 ($) PET the microstructure is constant and plastic deformation results from thermomechanically induced jumps of structural units over a well defined energy barrier, perature (T /T g ) is shown in Fig. 9 (for biaxial comthese experimental values V exp and DH exp are close to pression experiments, values are expressed in terms the activation volume V a and the activation enthalpy of equivalent compression stress). This figure shows DH a

  . . . . . . . . . . (8) based on the classical relationship6 where s(t) is the equivalent stress in shearing mode e ˙=e ˙0 exp A -DH a -T DS a kT B . . . . . .

  =kT A ∂ ln e ∂s B T,struct . . . . . . . . (5) PET compressed up to 45% at ambient temperature. It can be observed that V exp increases with time and and the experimental temperature sensitivity tends to a constant value ~1 h after the beginning of parameter stress relaxation. A time of 1 h of stress relaxation was then adopted in most of the experiments. In that DH exp =-kT 2 A ∂ ln e ∂T B s,struct . . . . . . (6) way, V exp at ambient temperature was found to be close to 2•4 nm3, which corresponds to approximately 10 monomer units. This value, in agreement with are generally measured. It is noteworthy that where 12 Evolution of DH exp v. normalised temperature for A (&) and C38 ($) PET 11 Evolution of V exp v. normalised temperature used, in which the macroscopic behaviour of a disfor A (&) and C38 ($) PET ordered material is related to local molecular motions.

H

  . . . . (20) and d depends on J an2 via 13 Distribution of b relaxation time in amorin b relaxation domain for where B Ub is the width parameter of the Gumbell amorphous PET; full lines: theoretical curve distribution and U b is the most probable value of U b . As an illustration, the calculated distribution of t bk is plotted in Fig. 13. where T bmin =95 K is the beginning temperature of The unrelaxed modulus G u and the elastic modulus the b relaxation. G el (respectively J-1 u and J-1 el ) are determined from Finally, using these parameters, curves of G∞ and the Cole-Cole diagram plotted in Fig. 14: G u is evalutan w in the b relaxation domain are calculated for ated from the inflection point between the a and the the three frequencies and compared with the experib relaxation and G el is estimated as the highest mental curves in Fig. 15. Figures 14-15 show a modulus value of the b relaxation. Thus, G u and G el reasonably good agreement between simulated and are close to 1800 and 865 MPa, respectively. The experimental isochronal curves in the b relaxation distribution width B Ub is chosen to account for domain. the width of the G◊ peak and found to be 9•8. The The contribution of the b relaxation is then taken parameter t b0 is precisely adjusted so that experiinto account in the expression of the global commental and calculated data for the G◊ peak obtained pliance (equation (12) or (13)) by replacing the unreat one frequency coincide, and the most probable laxed compliance of the a relaxation with value U b is fitted to account for the shift in temperature between the three frequency curves. Thus, a value of 0•66 eV for U b ( leading to an average value U b =0•6 eV) and 4×10-17 s for t . . . . . . . . (21)

14

  Cole-Cole plot of amorphous PET: (×) 1, shift in temperature between curves obtained at (%) 0•1, and (#) 0.01 Hz in b and a relaxation the three test frequencies. The fit of the component domains and theoretical curves (full lines: of the a relaxation lying above T g (T g =340 K for A biparabolic expression; dashed lines: discrete distribution of relaxation times) heated at 1 K min-1), requires a=0•0075. 17 Plane strain experiments, e ˙=8•4×10-4 s-1; experimental data (points) and theoretical 16 Isochronal curves of amorphous PET: (×) 1, (full lines) (%) 0•1, and (#) 0•01 Hz in a relaxation domain and theoretical curves (full lines: biparabolic Modelling of plastic behaviour expression; dashed lines: discrete distribution On the basis of equation (13), the qpd theory has of relaxation times) been further extended to the non-linear domain through an incremental technique that makes it possible to determine the anelastic and viscoplastic contri-As mentioned above, the extension to the nonbutions to a macroscopic deformation. The theoretical linear domain is based on a discrete distribution of developments have been developed in detail22-25 and times (which are much more convenient for the will only be outlined in the present paper. The evolincremental technique than the extended exponenution of the disorder (i.e. x) during the test is taken tials). Therefore, the DMA curves are fitted with into account as follows equation (13), with respect to the estimations of B tan , x(T , e)=x(T )+A an e an -A vp e vp . . . . . (24) B tvp , J an2 , and t rm given by equations (15) and (16). As explained, the distribution width of anelastic times where A an accounts for the increase of molecular B tan =x(T g ), while the distribution width of viscoplasmobility and consequently the increase of disorder tic relaxation times B tvp is close to x∞ (in fact, since as the number of smd increases. By contrast, A vp the exponent x∞ of the extended exponential varies accounts for the decrease in disorder due to chain from 0 to 1 and the distribution width B tvp varies orientation during elongation. from 0 to 2, then as x∞ becomes smaller, so B tvp Moreover, under high stress and under hydrostatic becomes closer to x∞). Therefore, the precise evalupressure, the thermomechanical activation of the b ation of B tvp is made on the level of the tan w peak. relaxation yields26,27 This gives B tvp =1•17x∞. Finally, J an2 , the amplitude of the anelastic process, is adjusted to the level of the t b

  ) (Frenkel argument) and V b represents the sensitivity of b movements to the hydrostatic pressure P. Measurements of the first pressure coefficient m on It should be noted that these values are very close to the estimates given by equations (15) and (16), amorphous PET ( m=0•05)28 show that V b is ~50 A ˚3. Thus, there are only three other parameters in the differing only by factors of 0•5 and 0•7, which confirms the consistency of this method of analysis.

  . . . . . . . . . (28) 18 Plane strain experiments showing evolution of flowing stress v. temperature for amorphous PET (&); theoretical with constant e ˙(+) and Finally, the anelastic deformation associated with with hypothesis of gradient of e ˙({6{); tensile each k process decreases to its stationary value e ank eq experiments (+) according to stress-strain curves have been simulated successfully de ank dt = e ank eq -e ank t an (s an ) . . . . . . . . . (29) at various temperatures using this unique set of parameters (see the experimental and predicted curves where the effective stress s an depends on s1 int and in Fig. 17

  ). The results are summarised in Fig.18, s2 int as follows: when e ank >e ank eq and e an >e an eq , then which shows the dependence of the simulated and experimental plastic flow stress s p on temperature. s an =(s1 int -s)A rec g ank +(s2 int -s) . . . . (30) Below T g , the model accounts rather well for the or, when only e ank >e ank eq , then decrease of s p with increasing temperature. Above T g however, while simulated s p values are in good agrees an =(s1 int -s)A rec g ank . . . . . . . . (31) ment with experimental tensile data, these values are where A rec accounts for the intensity of the internal lower than the values resulting from plane strain stress in the relaxation process. compression tests. Therefore, it is assumed that fric-In the same way, the characteristic time t b (s b) is tion between die and sample is so high in the high assisted by an effective stress s b given by temperature range, that the material is less mobile at the surface than it is in the centre of the sample. . . . . . .(32) There might be a gradient of strain rate in the rubbery region, which leads to greater deformation of the material than would be expected. Consequently, the The parameter A rec was adjusted to fit the active effective deformation rates encountered in plane strain unloading curve plotted in Fig.19to account for the experiments were adjusted in the sub T g temperature residual strain observed after the release of the stress. range, to account for the experimental s p values. Stress relaxation curves were then simulated and Finally, the same set of parameters was introduced activation volume and activation enthalpy were deterto describe the stress relaxation experiments. In these mined following the protocol mentioned above. By tests, the global deformation of the specimen (sum of comparing the experimental and simulated data on anelastic and viscoplastic deformations) is maintained V exp (T /T g ) in Fig.20, good agreement was observed constant. Since the viscoplastic component continues below and above T g . The simulation confirms that to increase (i.e. some smd coalesce), elastic energy is the activation volume of amorphous PET increases released and the stress decreases as a function of time.slightly with temperature below T g and then rapidly As a consequence, each k smd shortens under reaches high values above T g . retractive stresses related to the elastic energy stored It is also interesting to compare the evolution of by the dislocation loop bordering the smd s1 int and the activation enthalpy DH exp (T /T g ) with the predicalso to interactions with other shear microdomains tions given by the theoretical approach (see Fig.21). s2 int . The first contribution of the internal stress s1 int Below T g , both experimental and predicted values can be evaluated by noting that when s1 int =s, increase with temperature. In the vicinity of T g , simue ank =e ank eq =J an2 sg ank , and hence lated values reach a maximum and then drop when the material becomes rubbery. Thus the qpd theory makes it possible to predict the dependence of the s1 int = e ank J an2 g ank . . . . . . . . . . . (27) activation enthalpy on temperature. At low temperatures, plasticity results from local The second internal stress s2 int , which accounts, this time, for the global smd interaction, is given by molecular movements with small energy barriers and 21 Evolution of activation enthalpy DH exp

DSC thermograms of A (%) and C38 (#) PET 4 DMA: tan w of A (%) and C38 (#) PET at 1 Hz (curves shifted vertically for clarity)


Stephane Bazeille
email: stephane.bazeille@gmail.com

Dr Vincent Lebastard

Frédéric Boyer

Stéphane Bazeille

Dr Stéphane Bazeille

Stéphane Bazeille

Underwater robots equipped with artificial electric sense for the exploration of unconventional aquatic niches

Keywords: Underwater robots, electric sense, autonomous navigation, environment monitoring

Underwater robots equipped with artificial electric sense

for the exploration of unconventional aquatic niches

Swarm exploration and monitoring

The novelty of subCULTron concentrates on the data collection that will be performed by a large robot swarm constituted of more than 130 entities (a swarm is a group or aggregation of free-swimming organisms). This artificial robot organization will be composed of 3 classes of cooperating robots (see figure 1): 5 floating platforms called artificial lily pads (aPad), 100 artificial mussels (aMussel) and 30 artificial fish (aFish) that will collaborate together to perform their mission.

aPad: The aPads are floating robots (see figure 2, 3.b). This class of autonomous robots globally localized by GPS and equipped with solar panels will be used as geolocalization satellites and recharge/gathering station for the underwater robots. They will communicate with the underwater robots though acoustic sensing. aPads will also maneuver actively and they have the longest runtime. aMussel: The aMussels is a class of underwater robots that sits on the ground (see figure 3.a and 3.b). These robots have an actuation limited to a buoyancy device. They will dive, and autonomously surface for recharge. They will be used under a low-energy consumption regime and they will harvest energy from the bacteria. These robots will self-distribute in the habitat, and then be used as landmark, and data storage station for aFish. aFish: The aFish are the most active robots (see figure 4). aFish will browses the habitat and organize to explore an area and collect data. As the most active robot, they have to autonomously manage their energy and to recharge their battery on aPads when they need. They will communicate with aMussels and aPads to localize and will exploit information collected by other aFish in order to actively explore the environment.

All experiments will take place in Italy in the Venice Lagoon (a shallow water area of 500 km 2 ). This place is a particularly challenging environment for common underwater robots as it is a large heterogeneous area constituted of: a network of turbulent canals, open waters, mud flats, tidal shallows or salt marshes. It presents many difficulties that have not been addressed: a ubiquitous turbidity, a fluctuating salinity, potential strong water currents and a large variability of biological activities. As common sensing cannot be used under these conditions, its exploration pushed us towards the use of a new perception sensor: the electric sense.

Robots sensors equipment

In the following we will concentrate our discussion on the underwater robots: the aFish and the aMussel. The aPad as a floating platform will not be equipped with electric sense. The underwater robots will dispose of 2 different set of sensor. A set of sensors to collect data about the environment (turbidity, pH, temperature . . . ) and navigation sensors. Among these navigation sensors, robots will dispose of common underwater sensors: an imu, an acoustic pinger and receiver, a camera, and a modulated light perception system. Besides this sensors underwater robots will include an artificial electric sensor that will be used as their main perception and communication sensor. Few reasons can explain this choice of electric sensing. First the omnipresence of suspended matters makes the water turbid or muddy, and the shallow waters rich in vegetation makes the environment cluttered and confined. These two specificities prevent robots from using common long and medium range underwater sensors such as sonar and vision. Vision, because the visibility is dependent on the available light energy which dramatically decreases in highly turbid turbulent and polluted waters. Sonar, because the reverberation of multiple echoes from the obstacles and the diffraction by suspended particles jam the sonar signals and makes their interpretation difficult. The electric sense giving an omnidirectional perception of the environment robust to the lack of light and the turbidity makes it perfectly suitable for our applications.

The electric sense in nature

Electric sense has been observed almost exclusively in aquatic or amphibious animals, the known exceptions being echidnas, cockroaches and bees [START_REF] Clarke | Detection and learning of floral electric fields by bumblebees[END_REF]. Underwater, several species of fish have the capacity to sense changes in electric fields in their vicinity [START_REF] Bullock | Electroreception[END_REF]. Among fishes, we can distinguish two typical modes of electroreception: some fish passively sense changes in the nearby electric fields (passive electric sense) [START_REF] Kalmijn | Electro-perception in sharks and rays[END_REF], some generate their own weak electric fields and sense the distortions of these fields with their skins (active electric sense) [START_REF] Caputi | The electric image in weakly electric fish: physical images of resistive objects in gnathonemus petersii[END_REF] [START_REF] Rasnow | The effects of simple objects on the electric field of apteronotus[END_REF].

Passive mode

Passive electroreception is the most common modality: the fish senses the weak electric field generated by other animals (prey or conspecifics) and uses it to locate them. These electric fields are typically generated by nearly any species due to the activity of their nerves and muscles. Passive electric sense is used by several species of sharks and rays that have evolved specific electro-receptors named Lorenzini ampullae [START_REF] Kalmijn | Electro-perception in sharks and rays[END_REF]. It has to be noted that these receptors are so sensitive that these fish also use them to orientate along the telluric fields or to sense the weak gradients of salinity. When hunting, the range of perception of passive electroreception is generally around few meters.

Active mode

In active electric sense, the fish called weakly electric fish, can sense their nearby environment by detecting the distortions of a self generated electric field. This electric field is generated thanks to an electric organ that is located at the base of its tail. It generates a dipolar shaped electric field around the fish which is the distorted by the nearby objects as it shown on figure 6. Then the distortions is measured using a dense array of electro-receptors distributed over the fish skin. This phenomena called "electrolocation" can be compared to the echolocation used by dolphins, where the carrier is no longer an acoustic wave but an electric field. Because electric emitters are dipoles instead of sources, electric sense has a much shorter range (around 1 body length) but it has the advantage to be omnidirectional. They can also use electric sense for passive sensing to hunt their preys or to escape from their predators. Most of electric fish are principally nocturnal and live in confined turbid waters of the equatorial forests [START_REF] Lissmann | The mechanism of object location in Gymnarchus Niloticus and similar fish[END_REF], that is to say waters that are rich in suspended particles with many obstacles such as the roots of the trees. Beyond perception, active electric fish can also communicate by modulating their electrical activity and they use this further ability for courtship behaviors or to mark out their territory. As an example, figure 5 shows the elephant-nose fish, the most clever electric fish. Recent behavioral experiments have shown that through active electric sense, this fish can localize preys, predators, conspecifics and also inert objects when they are electrically contrasted with respect to the water. Moreover, researchers showed that the fish can also identify the electric nature (conductive, insulating), and discriminate the sizes and shapes of objects [START_REF] Der Emde | Electric fish measure distance in the dark[END_REF]. Its polarization is performed in short pulses. Other fish of the family of Gymnotide use the same principle of reception but with a sinusoidal (alternative) electric field. These fish are named wave fish and their emission mode has inspired the artificial electric sensors [START_REF] Rasnow | The electric organ discharges of the gymnotiform fishes[END_REF]. 3 Artificial electric sense

Use of artificial electric sense for AUV

Electric sense is a short range omnidirectional perceptual ability well suited for confined environments and turbid or muddy waters. It is cheap, easy to integrate on a robot and can be used in different context such as reactive navigation, docking, obstacle avoidance and object recognition. Both active and passive modalities (active or passive) can be used depending on the application as the weakly electric fish do. When speaking about reactive navigation, a first example that clearly illustrates the advantage of the electric sense is the underwater docking of a passive robot on an active station for battery recharging or data exchange. Another possible use of the reactive navigation in the context of an active robot is to react to the electric field lines reflected by the nearby polarized objects to perform basic behaviors as obstacle avoidance or object seeking. Beyond these reflex behaviors, electric sense can be fused with other sensing modalities, as for instance inertial measurements to perform more complex cognitive tasks as shape recognition [START_REF] Lanneau | Object shape recognition using electric sense and ellipsoid's polarization tensor[END_REF] or mapping the environment while self-localizing, a topic never addressed in the artificial electric sense community. Disadvantages of electric sense is the short range and the electric measures potentially difficult to interpret when the environment is complex (more than one emitting agent, many objects for example). For these reasons, it has to be complemented with other sensors.

Electric sense in subCULTron

As discussed previously, electric sense being a short range sense, it will be supported with hydro-acoustic for long range localization, and complemented by modulated light at short range. Redundancy at short distance sensing has been chosen make the system more robust. Modulated light is an active sense as well. It is based on the emission of light that is reflected back by the obstacles and detected by some photoreceptors. Though being not much affected by the turbidity of the water, it can be severely disturbed by external light sources such as the sun when it is used close to the surface. In clear water, it has a little longer range than electric sense, typically about 1 meter but each photoreceptor has a small cone of perception then many devices are needed to obtain an omnidirectional perception without blind spots. Finally, 2 additional sensors will equip our underwater robots: an Inertial Measurement Unit (IMU) that gives the robot linear velocities and angular rates, and a pressure sensor that gives the depth. These proprioceptive information will be used for robot balancing (pitch, roll), control of the navigation (heading, velocity, depth). About electric sense, the two modalities will be used depending on the behaviours we need. Active mode will be extensively used to perceive the environment and navigate autonomously while avoiding obstacles. Passive mode will be used to navigate towards active robots (reach a base or follow a conspecific). In the following, we will consider passive electric sense as a particular case of active electric sense in which the electric field is no longer generated by the same sensor but by an external active dipole that can be another robot or a docking station for example. Passive electric sense will also be used as a way of communication between robots. However, the word communication will not refer to an exchange of an explicit information through coded messages, but will rather inform about the presence and position of others robots.

Basic principle

Two artificial electric sensing technologies inspired by electric fish exist today U -U [START_REF] Bai | Sensing capacitance of underwater objects in bio-inspired electrosense[END_REF] or U -I [3] [START_REF] Servagent | Electrolocation sensors in conducting water bio-inspired by electric fish[END_REF]. The first letter designates the first electric input controlling the electric emission (here a voltage U), the second, the measurement variable (I denotes a current). In both cases, the sensor is an insulating axisymmetric (plastic) shell on which a set of conductive electrodes are arrayed. The electric field is generated by setting a voltage on at least two electrodes in contact with the water. Though, both techniques (U -U and U -I) share this common emission principle, in the U -U mode, the other electrodes are paired floating potential electrodes between which the voltage is measured. While in the U -I mode all of the electrodes except the emitter are grounded, and the currents that flow across each of them are measured. It is worth noting here, that till today artificial electric sense still remains restricted to tap waters and electric sensors designs were built for planar navigation. The sensor [START_REF] Servagent | Electrolocation sensors in conducting water bio-inspired by electric fish[END_REF] will be used for some experiments presented in Section 5 is presented on figure 7. It is composed of an insulating shell with a set of 4 macro-electrodes denoted ε i , i = 0, 1, 2, 3. The shell and the array of electrodes obey to a bilateral (left-right) symmetry. The back macro-electrode, ε 0 stands for the emitter, while the others ε 1 , ε 2 , ε 3 are the receivers. The emitter ε 0 is set under a controlled voltage U with respect to all the receivers which are grounded. This voltage U is imposed through a sine wave generator. Note here that a continuous voltage would generate an undesirable electrolysis. The surrounding water being a conductive medium, such a polarization of the sensor generates an electric field in the sensor surrounding (see figure 6). When there is no object within the robot's range, this electric field is named "the basal field".

All macro-electrodes are divided into 2 electrodes as shown on figure 7. The basal currents measurements are gathered in the vector I 0 = [I 1 , I 2 , I 3 , I 4 , I 5 , I 6 , I 7 , I 8 ]. These currents are entirely modeled by the following the vector-equation:

I (0) = C (0) U, (1)
where C (0) is a 8 × 8 vector modeling the basal conductivity (indexed (0)) of the current paths between the emitter and the receivers, and U a vector of size 8 × 1 in which values are set to zero for the receiving electrodes and to U for the emitting ones. The currents measurement vector include 8 measurements, but it has to be noted that only 6 are useful because on the real robot due to an hardware limitation the current cannot be measured on the emitting electrodes. The currents vector only depends on the sensor's geometry and the medium's conductivity γ 0 through the relation:

C (0) = γ 0 S (0) , (2)
where S (0) is a matrix modeling the influence of the shape of the sensor on the conductance between the emitter and the receivers. For example, designing a sensor shape more and more complex, makes S (0) and so C (0) decrease since it obliges the electric current lines to be more and more curved. Another degree of freedom encoded into S (0) consists of the geometry of electrodes whose decreasing the size, makes C (0) decrease. Note also that C (0) can be obtained once for all either by numerical computation or in situ, through a preliminary calibration phase. Then, when an object (obstacle, other robot) appears in the sensor's surrounding, the vector of measured currents I becomes:

I = I (0) + δ I = (C (0) + δC)U = CU, (3)
where δ I represents the perturbative component of the measured currents which images the presence of the object, and δC is its contribution to the external conductivity between the emitter and the receivers noted C. This perturbative conductivity δC depends on the geometry of the object, its position, as well as the dimension-less number λ = γ/γ o , named contrast coefficient. In this respect note that when λ = 1, δC = 0, and the object is electrically transparent. 4 Electric sense for the subCULTron

Electric sensing in salt water

One of the main difficulty of the project is that we have to work in sea waters i.e. salt water, that is too say a water with a conductivity 100 higher than tap water.

The conductivity defined as a measure of the capability of a medium to pass electric flow, depends on many parameters in particular the chemical compounds of water. Salinity has the strongest influence, the temperature has also a non negligible influence. As an example, tap water has a conductivity that ranges in [0.005 -0.05] S/m and salt water ranges in [1 -10] S/m. The conductivity in the lagoon of Venice is highly varying temporally and spatially, it has been estimated in [START_REF] Bai | Sensing capacitance of underwater objects in bio-inspired electrosense[END_REF][START_REF] Boyer | Model for a sensor bio-inspired from electric fish[END_REF][START_REF] Boyer | Underwater navigation based on passive electric sense: New perspectives for underwater docking[END_REF][START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF][START_REF] Bullock | Electroreception[END_REF][START_REF] Caputi | The electric image in weakly electric fish: physical images of resistive objects in gnathonemus petersii[END_REF] based on data collected in the lagoon in 2015. For our robots we preferred to use the U-I mode as it has a greater range equivalent to the fish range and it is more adapted to a use in sea water. Through this implementation we consider that the electromagnetic waves in the water are in the range of electric field frequencies ( w 2π < 50kHz), and we restrict electric sense to a measure of the amplitude of electric current, the phase is neglected. As the conductivity is large and varying, the hardware presented in [START_REF] Servagent | Electrolocation sensors in conducting water bio-inspired by electric fish[END_REF] has been augmented with a micro-controller to maintain a suitable emission with respect to the conductivity changes. When the conductivity increases the sensor saturates so the voltage U of the emitter has to be decreased, on the contrary when the conductivity decreases the amplitude has to be increased in order to keep the signal noise ratio and the perception range.

The importance of the sensor morphology

For our 2 underwater robot, the morphology of the sensor has been also optimized in order to maximize the sensor range and maximize the information obtained from the current measures and ease the robot control. To maximize the sensor range we increased at the maximum the distance between the receivers and the emitters. On both robot the emitter and receivers are on opposite sides of the robots. Regarding the receiving electrodes we respected also a symmetric positioning constraint: aMussels are axisymmetric and aFish will have a left-right and up/down symmetry. Finally again to enhance the perception, the number of electrodes was chosen depending on the behaviours of each robot. For the aMussel which are vertically put on the seabed, we chosen to place the ring emitter at the bottom and 4 quarter of circle electrodes on the top as receivers to obtain directionality component from the current measures (see figure 8). For the aFish, we planned to have on a ring shape emitter on the tail and four electrodes on the nose (see figure 4). These receivers will be positioned in order to allow the detection of obstacles around the robot in 3D.

Experiments results based on electric sense

For all the experiments described in this section we consider the aMussel is a cylinder with a length l M = 50cm and a diameter r M = 12cm (see figure 8). Regarding the aFish, that has still not be produced we used the slender probe [START_REF] Boyer | Model for a sensor bio-inspired from electric fish[END_REF] presented on figure 7. It has a length l F = 20cm, a diameter of r F = 2cm. We will show 4 different use of electric sense illustrated using the subCULTron underwater robots. Among the experiments shown, the aMussels experiment have been done in salt water, all other experiments have been only performed in tap water. In this first experiment, we consider a passive robot and an active robot in its range of perception in salt water. Our goal is to estimate the direction and the distance of the active robot (emitting an electric field) from the passive one. We only dispose of the 4 measured currents on the passive robot that linearly depends on both the conductivity γ 0 , and the imposed voltage U. These two quantities are supposed unknown. To infer the distance and directionality, we computed the 2 following currents ratio from the 4 raw currents I i (i=1,2,3,4) measured on the 4 electrodes e 1 , e 2 , e 3 , e 4 (see figure 8):

Localizing an active robot from a passive robot

R 1,3 = I 1 I 3 -1, R 2,4 = I 2 I 4 -1 (4)
It has to be noted that computing the currents ratio makes our measurement independent from the conductivity. From these ratio we can estimate the directionality ϕ of the active robot defined by:

ϕ = ATAN2(R 2,4 , R 1,3 ) (5)
where ϕ is an angle that is defined a fixed reference frame attached to the passive robot. The direction estimated, we obtain the distance d using the following function:

d = 0.2 (ln R 2,4 ) 2 + (ln R 1,3 ) 2 (6)
This function is a parametrical model of the currents that has been computed in a preliminary phase from different measurements of the 2 currents ratio at different distances. As the currents ratio does not depend on the conductivity but only on the morphology of the aMussels and the distance between the 2 robots the calibration can be done once for all in any water medium. To illustrate this localization of an active robot from a passive one in salt water we present on figure 9 the raw currents measured on the passive robot while rotating it in place (changing w on figure 8). It can be see that as w changes the maximum current value changes from electrodes 1 to 4. It is maximum when an electrode of the passive robot is facing the active robot. On figure 10, we show the currents measured on the passive robot as we pull it away from the active one but keeping the orientation fixed. The currents are in this case decreasing as the distance increases. It can be seen the perception range of the sensor is about 0.8m. To summarize, by using such a method we can estimate the robot distance and direction of an active robot from a passive one just by measuring 4 currents and without the knowledge on the conductivity or the voltage U.

Reactive navigation using active electric sense (memoryless)

In this simulated experiment we suppose that we use the robot presented on figure 7. The macro-electrode ε 0 is set to a potential U supposed known. On each of these receiving electrodes the current is measured and we obtained the following measures: I 1 , I 2 , I 3 , I 4 , I 5 , I 6 . These values can be re-parameterized by the following 6 scalars:

I lat,1 = (I 1 -I 2 ) 2 I lat,2 = (I 3 -I 4 ) 2 I lat,3 = (I 5 -I 6 ) 2 I ax,1 = (I 1 +I 2 ) 2 I ax,2 = (I 3 +I 4 ) 2 I ax,3 = (I 5 +I 6 ) 2 (7)
And, we define I lat = ∑ I lat,i , I ax = ∑ I ax,i I ax,i (axial current) represents the common part of the left and right currents flowing across the 2 electrodes of the macroelectrode i. The axial perturbative currents I ax,i is due to the variations of the total resistance of the scene. On the other hand, the lateral perturbative current I lat,i is proportional to the incident field. I lat,i (lateral current) represents the differential part of this left and right currents. This previous properties on I ax,i and I lat,i allows easily to determine some information of the environment. The axial current I ax,i is used to determine if the object is conductive or insulating. While the I lat,i currents is used to determine, with the knowledge on I ax,i , if the object is on the left or on the right hand side of the sensor. When I lat = 0, the sensor axis is necessarily aligned along the incident field (see table 1) Based on this 6 scalars defined in Eq. 7, by exploiting the morphology of the sensors (slender shape, bi-lateral symmetry), we can address the problem of navigation. A set of reactive control laws has been proposed [START_REF] Lebastard | Underwater electro-navigation in the dark[END_REF], the principle is based on the alignment of the body of the robot-sensor on the electric lines emitted by the polarized object. Remarkably, these strategies are used by fish that hunt by following the electric lines emitted by their prey [START_REF] Hopkins | Electrical perception and communication[END_REF]. If we assume that the robot is controlled using 2 parameters: its linear velocity V and its angular velocity Ω . We can apply the following control law on V , Ω (see figure 7):

V = C, Ω = K I lat , (8)
with C a constant positive value such that the sensor goes forward headlong and K a constant gain. By exploiting this control law, the robot just go straight at constant speed when there is not object in its surroundings but it will converge to an object or avoid an object depending on its electric properties (conducting and insulating) when such an object appears in its range of perception [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF]. In more details the robot follows the electric line reflected by the polarized object as shown on figure 6.down. Depending on the object electric properties the measured values δ I ax = I ax -I(0) and I lat varies following table 1.

Table 1 Sign of the perturbative axial and lateral currents δ I ax and I lat with respect to the electric properties of the object and its side in the sensor frame. Currents were measured by the head electrodes when the object is in the frontal part of the sensor [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF].

δ I ax > 0 for a conducting object δ I ax < 0 for an insulating object for a conducting object on the left of the sensor I lat > 0 or for an insulating object on the right of the sensor for a conducting object on the right of the sensor I lat < 0 or for an insulating object on the left of the sensor I lat = 0 for any contrasted object facing the sensor It is worth noting that δ I ax and I lat have a complementary role since the former can be used to determine if the object is conductive or insulating, while the latter allows us to determine if it is on the left or on the right hand side of the sensor. With our convention on the sign of the currents and angular velocities, taking K > 0 in (Eq. 8) ensures that when a conductive object is on the right (respectively on the left), the sensor turns to the right (respectively to the left). On the other hand, if the object is insulating and on the right (respectively the left), the control law makes it react as if there was a symmetric conductive object on the left (respectively the right). Thus, the controller repulsed the sensor from an insulating object. To invert the behavior, i.e. makes the sensor attracted by insulating objects and repulsed by conductive ones, the sign of K has to be changed in (Eq. 8). To summarize, depending on the parameter K, 4 reflex behaviours can be extracted: use to navigate autonomously reaching or avoiding obstacles. It has be mentioned that all these behaviours are fully reactive and do not need any prior knowledge.

Reactive navigation using active electric sense (memory based)

In the previous section we presented control strategies allowing the sensor to reach or avoid objects, here we will describe how to turn around an object based on the fact that we "recognized" it. This behaviour is based on a combination of reflex behaviours presented in the previous part. By sequentially ordering 3 reflexes we seek an object and then turns around it (see figure 11 from [START_REF] Lebastard | Reactive underwater object inspection based on artificial electric sense[END_REF]).

Fig. 11 Object exploration based on electric sense. It has to be mentioned on this illustration that the object is a conductive object [START_REF] Lebastard | Reactive underwater object inspection based on artificial electric sense[END_REF].

3. Follow the boundaries: From B to C on figure 11. The orbiting phase is obtained by applying the law Ω = K(δ I ax,1 -M). M is the value of δ I ax,1 measured at the last time of the previous phase.

It has to noted that the commutation between phases is ruled by events which only depend on time variation of the measurements (and not of their magnitude). Such a behaviour completely developed in [START_REF] Lebastard | Reactive underwater object inspection based on artificial electric sense[END_REF] shows that the robot can navigate avoiding, reaching and turning around objects only using active electric sense.

Reach an active robot by controlling a passive robot

We consider here a passive robot that tracks the electric lines of an electric field generated by another active robot. On figure 12.a, 2 external electrodes: emitter (circle 1) and receiver (circle 2) are representing the active robot that can be considered as a docking station [START_REF] Boyer | Underwater navigation based on passive electric sense: New perspectives for underwater docking[END_REF]. These electrodes are close to each other and located in one of the corners of the tank. By using the same reflex behaviours presented in the previous section we can navigate with a passive robot. This results is shown on figure 12 where the passive robot reach an emitter with 3 different starting positions [START_REF] Boyer | Underwater navigation based on passive electric sense: New perspectives for underwater docking[END_REF]. The sensor's path is represented by a dotted lines and letters correspond to intermediate poses. For all position on figure 12.a, the current I ax is positive along the path and monotonically increases (figure 12.b). On the other hand, I lat converges toward its desired zero-value. The changing sign is due to the presence of the perturbative repulsive walls (from 0 to C and from 0 to B) the probe to far from the emitter first avoid the insulating walls and converge towards the emitter as it is getting closer. As shown the robot is docking on the emitter in all 3 cases with different trajectories. This experiment first presented in [START_REF] Boyer | Underwater navigation based on passive electric sense: New perspectives for underwater docking[END_REF] show that navigation of a passive robot in the electric field generated by an active is also possible.

Object localization and recognition from an active robot

In this section we consider an active robot and an object in its range of perception than is electrically contrasted with the water (see figure 6). Our robot is moving straight alongside of the object (see figure 13) and we collect every millimeters 6 currents gathered in I lat , and δ I ax (see Section 5.2). Our goal here is to estimate the localization, the pose and the size the object O x that perturbates the electric field generated by the robot. To simplify the problem we will solve the problem only in 2D and we supposed that: the conductivity of the medium γ is known as well as the displacement of the robot along its trajectory in a global reference frame. We performed experiments with a conductive prolate ellipsoid (i.e. axisymmetric about its major axis), i.e., that can be described by 6 parameters: its localization x 0 , y 0 , its orientation θ 0 , its size a, b and its electric properties (conductive or insulating) σ . Based on Section 5.2 we can already estimate the electric properties of the object σ and its side with respect to the robot as soon as the object enter in the perception range of the robot. This can be obtained looking at the sign of δ I ax and I lat . Then, based on these information, we will estimate the missing parameters by using a greedy problem solving heuristic testing a set of potential solution in the parameter space and select the optimal solution. This method is based on the analytical model presented in [START_REF] Boyer | Model for a sensor bio-inspired from electric fish[END_REF] that models the dipolar electric response of an object immersed in a electric field produced by the robot of figure 7. Within this analytical we modeled the ellipsoidal object by its first order polarization tensor [1][11]. Giving this analytical model, we implemented a function f that receives as parameters: a robot position (x k , y k , θ k ) along the trajectory T (k) and an object O x in a global reference frame. The function returns the 6 currents described in Eq. 7. This function that estimates 6 values we named I(k) can be expressed as: ∀k∈ [1, n] we have:

f (T (k), O x ) = I(k) = I lat (T (k), O x ) δ I ax (T (k), O x ) (9)
where k is a position along the trajectory T (k) that is discretized into n positions. Finally, to estimate the ellipsoid that perturbates the electric field we compare for a set of possible object parameters the sum of the error between the real measurements and the estimated measures for each robot positions. The object with the smallest error is then chosen as the solution of our problem. This algorithm can be summarized by: the Eq. 10.

argmin

0 x n ∑ k=1 6 ∑ i=1 |I i (k) -f i (T (k), O x )| |I i (k)| (10)
Figure 14 shows 24 experiments performed with an conductive ellipsoid which size was 33 × 16 mm. The real ellipses are displayed in red and the estimated ellipses in blue. Each results has been obtained offline using data collected by the robot (following a straight line trajectory) at different 6 distances 50, 60, 70, 80, 90, 100mm and with 4 different object orientations. As it can be seen the localization and the shape can be estimated at a short distance. As we increase the distance with the object the recognition is unfortunately getting worst as the signal noise ratio decreases. It has to be noted that for the presented results the resolution for the localization and the shape estimate was 2mm and the angle resolution was 0.26 radians. Such a greedy method can be used to estimate our object in an acceptable time (few minutes) because all parameters are quite well constrained by the model assumption and the small range of the sensor. The position x 0 , y 0 are constrained by the range of the sensor which is about the sensor length l, so x 0 ∈ [-l, +l], and y 0 ∈ [0, ±l]. As well, θ 0 ∈ [0, π]. Then, the maximum size of the object is also known as our model consider small object (smaller than the robot length l). Then, the only parameter of the algorithm is the discretization step of the each parameters. In this last section we shown that just using electric sense we can also localize and estimate the shape of an object.

Conclusion

In this paper, we presented an overview of electric sense based perception and navigation algorithms for the underwater robots that will be used in the context of the EU project subCULTron. We presented different algorithms based on electric sense that will be used by our robots to perform reactive navigation, obstacles avoidance, objects exploration and finally object shape estimation. These methods have been illustrated by some simulation and experimental results. From these results we can conclude that in many cases electric sense can supplement common underwater sensing devices in harsh conditions that is to say confined, low light and turbid underwater environment. In the new future, this work on electric sense will be extended towards collective based behaviours and localization and mapping (SLAM).

Fig. 1

 1 Fig. 1 Illustration of an experiment using the 3 different robots developed within the subCULTron project.

Fig.

  Fig. An aPad in the Arsenale in Venice.

Fig.

  Fig. Left: An aMussel, Right: An aMussel recharging on an aPad.

Fig.

  Fig. First design of the aFish (artificial fish).

Fig. 5 A

 5 Fig. 5 A weakly electric fish: Gnathonemus petersii.

Fig. 6

 6 Fig. 6 Up. Basal electric field produced by the weakly electric fish. Down) Electric field re-emitted by the conductive polarized cylinder situation on the right side of the fish.

Fig. 7

 7 Fig. 7 Left: Artificial electric sensor [17], Right: Design of the sensor with our notations.

Fig. 8

 8 Fig. 8 aMussel experiment set up.

Fig. 9

 9 Fig. 9 Raw currents measured on the 4 electrodes of the passive aMussel with 4 different orientations (0 • , 90 • , 180 • , 270 • ) at the same position.

4 Fig. 10

 410 Fig. 10 Raw currents measured on the 4 electrodes of the passive aMussel while increasing the distance between the 2 aMussels.

1 Fig. 12

 112 Fig. 12 Starting from 3 different initial poses (dashed lines), the sensor seeks the emitter until it touches it following 3 paths (a) Scene and paths P1, P2 and P3. (b) Axial currents (I ax ) and (c) lateral currents (δ I lat ) for the 3 paths. A, B, C, D indicate poses where the sign of lateral currents change [4].

Fig. 13

 13 Fig. 13 Experiment set up with the aFish.

Fig. 14

 14 Fig. 14 Localization and estimation of a 33 × 16 conductive ellipsoid at 5 different distance. Real ellipses (red), estimates (blue).

subCULTron (2015-2019), EU-H2020, FET PROACT-2-2014 under grant agreement no 640967. http://www.subcultron.eu

Seek an object: From A to B on figure11the robot is seeking an electrically non-transparent object by applying the attractive behavior with K = k/δ I ax,1 and k > 0. This behavior is maintained until δ I ax,2 changes its sign. The reason of this is that when getting closer to an object, more and more electric field lines are captured by the object if it is conductive. Thus, the electric field lines are concentrated on the front electrodes. On the contrary, if the object is insulating the electric field lines are pushed backward along the sensor axis. Thus, in both cases at some point I ax,2 changes of sign.

Flee from the electric influence: From B to B on figure 11. It corresponds to the initialization of the orbiting motion of the sensor around the object. It is obtained by applying the repulsive behavior, i.e. K = k/ | δ I ax,1 | and k > 0 until | δ I ax,1 | reach its minimum.

Acknowledgements SubCULTron (SUBmarine Cultures perform Long-Term Robotic Exploration of unconventional environmental Niches) is supported by European Unions Horizon 2020 research and innovation program under the grant agreement no 640967. http://www.subcultron.eu.

The project is coordinated by the Artificial Life Lab of the University of Graz (Austria)


Fatima Zohra Belaimeche

Abderrahim Bentaallah

Sarra Massoum

Patrice Wira

A comparative study between a simplified fuzzy PI and classic PI input-output linearizing controller for the wind-turbine doubly fed induction generator

Keywords: Doubly-Fed Induction Generator (DFIG), Input-Output linearization, Fuzzy Logic Controller (FLC) Rezultati simulacije potrjujejo, da predlagano krmiljenje ponuja nadzor odklopa, beleženje

The paper presents a comparative study of a linearizing control with classic PI and fuzzy PI controllers of the active and reactive stator power of a doubly fed induction generator (DFIG) applied to a wind-energy conversion systems (WECS). The paper discusses the operating principles of the power-generation scheme. Simulation results show that the preented input-output linearizing control provides a decoupled control, perfect tracking of the generated active and reactive power and robustness the active-and reactive-power variations.

INTRODUCTION

Wind energy is one of the most important and promising sources of the renewable energy all over the world, mainly because it is considered to be nonpolluting and economically viable. At the same time, there has been a rapid development of the related windenergy conversion technology [START_REF] Doumi | Nonlinear Integral Backstepping Control of Wind Energy Conversion System Based on a double-fed Induction generator[END_REF].

In terms of the wind-power generation technology, because of the numerous technical benefits (higher energy yield, reduced power loses and improved supply), the modern MW-size wind turbines always use a variable-speed operation which is achieved by the converter system [START_REF] Bekka | Power control of a wind generator connected to the grid in front of strong winds[END_REF].

The used converters are typically associated with individual generators and they contribute significantly sto the costs of wind turbines. Among the variable-speed wind-turbine generators, doublyfed induction generators (DFIGs) and permanentmagnet synchronous generators (PMSGs) with primary converters are emerging as the preferred technologies [START_REF] Bekka | Power control of a wind generator connected to the grid in front of strong winds[END_REF].

DFIG is widely used for the variable-speed generation, and it is one of the most important generators for the wind-energy conversion systems.

Both the grid-connected and stand-alone operation are feasible through an AC/DC/AC frequency converter [START_REF] Doumi | Nonlinear Integral Backstepping Control of Wind Energy Conversion System Based on a double-fed Induction generator[END_REF][START_REF] Rekioua | Wind power electric systems: Modeling, simulation and control, Green energy and technology[END_REF]. The major DFIG advantage is that the power electronic equipment has to handle a fraction (20-30%) of the total system power in order to guarantee the stability in acceptable conditions [START_REF] Doumi | Nonlinear Integral Backstepping Control of Wind Energy Conversion System Based on a double-fed Induction generator[END_REF][START_REF] Md | Vector control of a DFIG based wind turbine[END_REF].

In order to improve control of the active and reactive power generated by DFIG [START_REF] Doumi | Nonlinear Integral Backstepping Control of Wind Energy Conversion System Based on a double-fed Induction generator[END_REF], the paper proposes a robust simplified input-output linearizing Fuzzy-PI controller. The controller exhibits excellent dynamics and steady-state performances.

The paper presents a comparative analysis of a simplified input-output linearizing control with a proportional integral (PI) controller and a fuzzy-PI controller for the doubly-fed induction wind-energy conversion system (WECS). Theoretical analysis, modeling and simulation results are provided. A control strategy is developed to control the active and reactive power in order to maximize the wind energy production.

Fig. 1 shows the DFIG wind-energy conversion system structure Figure 1. Wind-energy conversion-system-based DFIG.

TURBINE MODEL

Wind turbines convert the wind kinetic energy into mechanical energy by producing a torque. Since the wind-energy is in the form of the kinetic energy, its magnitude depends on the air density and the wind speed.The wind power developed by the turbine is given by equation (1) [START_REF] Krichen | A fuzzy logic supervisor for active and reactive power control of a fixed speed wind energy conversion system[END_REF][START_REF] Boudjema | Nonlinear control of a doubly fed induction generator supplied by a matrix converter for wind energy conversion systems[END_REF]:

  3 2 2 1 V R C P p t     (1)
where ρ is the air density, R is the radius of the wind turbine, V is the speed of the wind, Cp (λ,β) is the power coefficient, β is the blade pitch angle, and λ is the tip speed ratio of the rotor blade tip speed to the wind speed and is defined by [START_REF] Doumi | Nonlinear Integral Backstepping Control of Wind Energy Conversion System Based on a double-fed Induction generator[END_REF]:

V R t    (2)
The expression of the turbine torque:

2 1 3 ) , ( 2
1 V R C C m t      (3)
In the model, the Cp (  ,β) value of the turbine rotor is approximated using a non-linear function according to [START_REF] Doumi | Nonlinear Integral Backstepping Control of Wind Energy Conversion System Based on a double-fed Induction generator[END_REF].

      6 i 5 4 3 i 2 1 p C C exp C C C C ) , ( f C                       (4)

MATHEMATICAL MODEL OF THE TURBINE DFIG MODEL

The most significant feature of the wound-rotor machine, which is widely used for the wind-power generation, is that it has to be fed from both the stator and the rotor side. Normally, the stator is directly connected to the grid and the rotor is interfaced through a variable-frequency back-to-back AC-DC-AC power converter to provide a bidirectional rotor power flow [START_REF] Meny | Modélisation et réalisation d'une chaîne de conversion éolienne petite puissance[END_REF].

The DFIG operating principle can be analyzed using the classical theory of rotating fields and the well known d-q model, as well as both the three-to-two and the two-to-three axes transformation. In order to deal with the machine dynamic behavior in the most realistic possible way, both the stator and rotor variables are referred to their corresponding natural reference frames in the developed model. In other words, the stator-side current and voltage components are referred to a stationary reference frame, while the rotor-side current and voltage components are referred to a reference frame rotating at the rotor electrical speed [START_REF] Meny | Modélisation et réalisation d'une chaîne de conversion éolienne petite puissance[END_REF][START_REF] Multon | Etat de l'art des aérogénérateurs électriques[END_REF].

The stator and rotor voltage components are:

                          
I R V dt d I R V dt d I R V dt d I R V               ) ( ) ( (6)
where:

sd V , sq V , rd V , rq V , sd I , sq V , rd V , rq V
represent the direct and quadrature voltage and current for the stator and rotor respectively.

The magnetic equations are:

               sq m rd r rq sd m rd r rd rq m sq s sq rd m sd s sd I L I L I L I L I L I L I L I L     (7)
The expression of the electromagnetic torque based on the dq stator fluxes and dq rotor currents is:

) I I ( L L p C sd rq sq rd s m e      (8)
The DFIG active and reactive power of the stator and rotor of the are:

             sq sd sd sq s sq sq sd sd s I V I V Q I V I V P (9)

GENERAL CONCEPT OF THE INPUT-OUTPUT LINEARIZING CONTOL

In order to linearize the system, the MIMO system is considered [START_REF] Boyette | Direct and indirect control of a Doubly Fed Induction Generator wind turbine including a storage unit[END_REF]:

      ) ( . ) ( x h y u g x f x  ( 10
)
where x is the state vector; u is the output; f and g are smooth vector fields; h is a smooth scalar function.

In order to obtain the input-output linearization of the multi-input multi-output system, output y of the system is differentiated until the inputs appear:

u x h L x h L y g f ) ( ) (    (11)
where Lf and Lg represent the Lie derivative of h(x) with respect to f(x) and g(x) respectively. If Lgihj(x)=0 for all i, then the inputs do not appear and we have to differentiate again [START_REF] Boyette | Direct and indirect control of a Doubly Fed Induction Generator wind turbine including a storage unit[END_REF]: where r is the relative rank of y. Performing the above procedure for each input yi, we get a total of m equations in the above form written as [START_REF] Boyette | Direct and indirect control of a Doubly Fed Induction Generator wind turbine including a storage unit[END_REF].

u x h L x h L y g f r ) ( ) ( 1    (12)
                            m m r r u u x E x A y y m ... ... ) ( ) ( ... ... 1 ) ( 1 ) ( (13)
where the m*m matrix E(x) is defined as [START_REF] Boyette | Direct and indirect control of a Doubly Fed Induction Generator wind turbine including a storage unit[END_REF][START_REF] Poitiers | Advanced control of a doubly-fed induction generator for wind energy conversion[END_REF]:

                   m m f gm m r f g r f gm r f g h L L h L L h L L h L L x E
  T m f r f r h L h L x A m ... ... ) ( 1  (15)
Matrix E(x) is the decoupling matrix for the system. If E(x) is nonsingular, then original input u is controlled by the coordinate transformation [START_REF] Carrasco | Power-electronic systems for the grid integration of renewable energy sources[END_REF]:

v x E x A x E u ) ( ) ( ) ( 1 1     (16)
where The procedure of the input-output linearizing control is shown in Fig. 1.

  T m v v v ... ...

A SIMPLIFIED INPUT-OUTPUT LINEARIZING CONTROL WITH A PI CLASSIC CONTROLLER

In the stator flux-field-oriented frame According to (21) the direct and quadrature components of the stator and rotor currents are linear and so the state vectors are [START_REF] Carrasco | Power-electronic systems for the grid integration of renewable energy sources[END_REF]:

s sd    and 0  sq  (19)      0 sq s sd V V V
    T rq rd T i i x x x   2 1 (22)
By substituting ( 7), (19), and (21) into (6), we get the following aquations hold:

                                                  s s m s rd s m r s rq s m r rq r rq rq s m r s rd s m r rd r rd L V L g I L L L g I dt d L L L I R V I L L L g I dt d L L L I R V 2 2 2 2   (23)
Arranging (23) as in [START_REF] Mj | Design and Implementation of Neuron-Fuzzy Vector control for Wind-Driven Doubly-Fed Induction generator[END_REF]: Since the rotor-side controller decouples the active and reactive power, the stator active and reactive powers are selected as the output [START_REF] Carrasco | Power-electronic systems for the grid integration of renewable energy sources[END_REF][START_REF] Boyette | Direct and indirect control of a Doubly Fed Induction Generator wind turbine including a storage unit[END_REF][START_REF] Poitiers | Advanced control of a doubly-fed induction generator for wind energy conversion[END_REF]:

v Nonlinear System u Linearization Controller u   E -1 (x)A(x)  E -1 (x)v x •  f(x)  g  u y  h( x)                                 
                      
L u L L u L f u f u L L i u L L i L L u y u L u L u L L u L f u f u L L i u L L i L L u y       ) ( ) ( ) ( ) ( 2 1 1 2 1 1       (28)
Rewriting (28) in the form of ( 13):

              2 1 2 1 ) ( ) ( u u x E x A y y   (29)
where:

                     ) ( ) ( ) ( ) ( ) ( 2 1 2 1 2 1 2 1 f u f u L L x u L L x L L u f u f u L L x u L L x L L u x A
L u L L L u L L u L L L u L x E     ) ( (31)
Since E(x) is nonsingular, the control scheme is given from ( 16) as:

                      2 1 1 ) ( ) ( v v x A x E u u rq rd (32)
To track the control and to obtain a robust control of the parameter variations, the input system is [START_REF] Boyette | Direct and indirect control of a Doubly Fed Induction Generator wind turbine including a storage unit[END_REF]:

                     dt e k e k y dt e k e k y v v i p i p 2 1 2 2 * 2 1 1 1 1 * 1 2 1   (33)
where e1 is the error between the demanded and the achieved active power, and e2 is related to the reactive power [START_REF] Multon | Etat de l'art des aérogénérateurs électriques[END_REF][START_REF] Carrasco | Power-electronic systems for the grid integration of renewable energy sources[END_REF].

SIMPLIFIED INPUT-OUTPUT LINEARIZING CONTROL WITH A FUZZY-PI CONTROLLER

This type of control system is based on the fuzzy logic that makes use of the tolerance, uncertainty, imprecision and fuzziness in the human decisionmaking process, offers a very satisfactory performance with no need of a detailed mathematical model of the system. As shown in Fig. 5, our focus is on the fuzzy logic control based on mamdani system. This system has three main parts. First, by using the input membership functions, the inputs are fuzzified, then based on the rule base and inference system, outputs are produced and finally the fuzzy outputs are defuzzified and applied to the main control system. At any time interval,the error and the error change rate are chosen as inputs. Fig. 4 shows a block diagram where the fuzzy controllers are integrated into the rotor side converter to control the DFIG. The main objective of this part is to control the active and the reactive power.

FLC DESIGN

The inputs of the fuzzy controller are the error (e) and the error change rate (Δe) and its output is (Δu). The universe of (e), (Δe), and (Δu) are partitioned into three fuzzy sets, i.e. N (negative), Z (zero) and P (positive). Each fuzzy set is represented by either a triangular or a trapezoidal membership function. The FLC rule base contains nine rules based on IF-THEN [START_REF] Md | Vector control of a DFIG based wind turbine[END_REF].

SIMULATION RESULTS AND DISCUSSION

Some illustrations will be introduced now in order to show the dynamic performances of the proposed control system. The controllers are tested at reference tracking and robustness to parameter variations. Our simulations are made on a 1.5 MW generator connected to a 398 V/50 Hz grid.

The DFIG parameters are: Rs = 0.012 Ω, Ls = 0.0137 H, Rr =0.021 Ω, Lr = 0.0136 H, Lm = 0.0135 H, F = 0.0024 Nm/s, J = 0.0031 kg. m 2 and R = 35.25 m.

Pursuit test

The aim of test is to the study the behaviour of the two controllers at reference tracking with the machine speed constant at its nominal value. As seen from the simulation results shown in Fig. 6, the active and reactive powers of the two controllers track almost perfectly their references, contrary to the FLC controller where the coupling effect between the two axes is very clear.

Sensitivity to the speed variations

The aim of this test is to analyze the impact of speed variations on the DFIG active and reactive powers.

Power curves at variation show important oscillations of the PI controller system with, while they are almost negligible for the fuzzy PI system. Here, there are any variations and the power variations are very small. This result is attractive for the wind-energy applications for ensuring stability and quality of the generated power at speed variations.

CONCLUSION

A simplified input-output linearizing fuzzy control applied to a turbine DFIG is propose. A simulation study is made to use it on DFIG of a wind-energy conversion system. The performance of classic PI controller and fuzzy controller used in a wind-power generation are compared. While the design parameters of the PI classic controller have to be tested and adjusted, the fuzzy controller shows strong robustness to parameter of the control system. Comparing the simulation results shows that the fuzzy controller outperforms the PI classic controller. The results are highly consistent with the theoretical calculations and validate correctness of the presented simulation system.

Figure 2 .

 2 Figure 2. Power coefficient as a function of the speed ratio (λ) and the angle (β).

Figure 3 .

 3 Figure 3. Schematic diagram of the input-output linearizing control.

1 

 1 ) into[START_REF] Kerboua | hybrid fuzzy sliding mode control of a doubly-fed induction generator in wind turbines[END_REF] provides a linear differential relation between the output y and new input v:

Figure 4 .

 4 Figure 4. DFIG control diagram using a simplified inputoutput linearizing control

Figure 5 .

 5 Figure 5. Structure of the proposed fuzzy logic controller.

Figure 6 .Figure 7 .Figure 8 .Figure 9 .Figure 10 .

 678910 Figure 6. Active and reactive powers for PI classic controller

Fatima Zohra Belaimeche received the license and the Master degrees in Automatic from the University of Sidi Bel-Abbes (Algeria). Currently, she is a PHD student at the same university. She is a member of the