Miassa Amira Taleb

Olivier Béthoux

Emmanuel Godoy

Amira Miassa

Taleb

Identification of a PEMFC fractional order model

Keywords: Polymer Electrolyte Membrane Fuel cell, PEMFC fractional order model, FC parameter identification, diagnosis 2010 MSC: 00-01, 99-00

The present paper addresses the important issue of monitoring the operating state of the Polymer Electrolyte Membrane Fuel Cell systems. The monitoring system takes a model based approach. Its originality lies in adopting a fuel cell fractional order impedance model which permits to provide a better insight into the fuel cell physical phenomena without increasing the number of parameters. This article first validates experimentally the accuracy of the suggested model, using a frequency identification method carried out by nonlinear optimization using single fuel cell experimental impedance spectroscopy data. In a second phase, time series identification is achieved using a least square method specifically designed for fractional order models. The latter method is first verified on registered data which represents a basic tool for off-line monitoring. Subsequently it is refined as a recursive tool permitting an on-line monitoring; it is validated on laboratory test bench.

Introduction

Electric vehicle (EV) is an emerging and growing market projects gradually emerge and fuel cell (FC) technology may boost the EV industry providing "zero emission" long-range vehicles and fast recharging ability. Indeed, in comparison with batteries, power and energy are independent:independent: the vehicle range is linked to the hydrogen reservoir which takes few minutes to fill while the onboard available power is dependent on the FC size. Consequently FC represents an attractive opportunity for the future development of EV since hydrogen allows a much larger cruising range than battery and convenient recharge. In this respect Polymer Electrolyte Membrane (PEM) FC systems operate at low temperature and high power density and are ideal candidates for this application, which results incaracterizedcharacterized by heavy environmental $ * Corresponding author Email address: Miassa.Taleb@supelec.fr (Miassa Amira Taleb) URL: Olivier.Bethoux@centralesupelec.fr (Olivier Béthoux), Emmanuel.Godoy@centralesupelec.fr (Emmanuel Godoy) constraints with intermittent and highly fluctuating operation stop/start conditions They represent an attractive opportunity for the future development of electrical vehicles since hydrogen allows a much larger cruising range 20 than battery and convenient recharge. Nevertheless, much remains to be done to overcome some of PEMFC technological obstacles which usually appear for automotive applications [START_REF] Jing | Local resolved investigation of pemfc performance degradation mechanism during dynamic driving 535 cycle[END_REF]. Water management (to avoid Fflooding and dehydrationrying) [START_REF] Steiner | Diagnosis of polymer electrolyte fuel cells failure modes (flooding & drying out) by neural networks modeling[END_REF], [START_REF] Bozorgnezhad | The experimental study of water management in the cathode channel of singleserpentine transparent proton exchange membrane fuel cell by direct visualization[END_REF], [START_REF] Fukuhara | A lumped fluidic model of an anode chamber for fault tolerant strategy design[END_REF], [START_REF] Kim | Numerical study on the effects of gas humidity 550 on proton-exchange membrane fuel cell performance[END_REF], [START_REF] Shimpalee | Numerical studies on rib & channel dimension of flow-field on pemfc performance[END_REF], [START_REF] Giurgea | An analysis of fluidic voltage statistical correlation for a diagnosis of pem fuel cell flooding[END_REF], [START_REF] Owejana | Effects of flowfield and diffusion layer properties on water accumulation in a pem fuel cell[END_REF] is one of the 25 most critical points to solve enhance. The principal means by which this will be achieved are both design optimization [START_REF] Bozorgnezhad | The experimental study of water management in the cathode channel of singleserpentine transparent proton exchange membrane fuel cell by direct visualization[END_REF], [START_REF] Shimpalee | Numerical studies on rib & channel dimension of flow-field on pemfc performance[END_REF], [START_REF] Yana | Ac impedance characteristics of a 2kwpem fuel cell stack under different operating conditions and load changes[END_REF] and health monitoring diagnosis and prognostics [START_REF] Jouin | Phm of protonexchange membrane fuel cells -a review[END_REF], [START_REF] Fouquet | Model based pem fuel cell state-of-health monitoring via ac impedance measurements[END_REF], [START_REF] Petrone | A review on model-based diagnosis methodologies for pemfcs[END_REF], [START_REF] Zheng | A review on non-575 model based diagnosis methodologies for pem fuel cell stacks and systems[END_REF], [START_REF] Giurgea | An analysis of fluidic voltage statistical correlation for a diagnosis of pem fuel cell flooding[END_REF] in order for the control system to maintain normal FC operation. This is where the fuel 30 cells PEMFC state operating real-time monitoring intervenes in order to establish a system diagnosis . Diagnosis methods [START_REF] Steiner | Diagnosis of polymer electrolyte fuel cells failure modes (flooding & drying out) by neural networks modeling[END_REF], [START_REF] Petrone | A review on model-based diagnosis methodologies for pemfcs[END_REF], [START_REF] Zheng | A review on non-575 model based diagnosis methodologies for pem fuel cell stacks and systems[END_REF], [START_REF] Wua | Diagnostic tools in pem fuel cell research: Part ii physical/chemical methods[END_REF], can be applied in real time, coupled with a FC control system (On Board diagnosis) or during regular planned maintenance.

35

Diagnosis proposed approaches may include signal processing techniques, experiential learning methods and model based procedures [xx on peut mme se citer!]. The latter has the advantage of providing a tool easier to adapt from one PEMFC design to another. But the main issues are to provide a reliable model [START_REF]Review of proton exchange membrane fuel cell models[END_REF] and to develop specific parameters identification tools. The aim of this paper is to propose accurate but compact model and robust identification tools which can be used for real time system diagnosis.

The aim of this paper is to propose a more compact and accurate model and robust identification tools which can be used for real time system diagnosis This method uses, as far as possible, the least number of sensors, by limiting the PEMFC state monitoring to the use of available data of the system such as the current and voltage. To this end, it is relevant to employ electric model built using the physical equations governing the FC process. Fontes [START_REF] Fontes | Interactions between fuel cell and 585 power converters influence of current harmonics on a fuel cell stack[END_REF], [START_REF] Fontes | Modelisation et caracterisation de la pile pem pour letude des interactions avec les convertisseurs statiques[END_REF] proposed a FC large signal nonlinear model represented by an electrical circuit using volt-55 age controlled current sources, reflecting the causality of phenomena. This model is commonly linearized to deduce an electrical impedance model. For improving Membrane Electrode Assembly (MEA) water management, N. Fouquet [START_REF] Fouquet | Model based pem fuel cell state-of-health monitoring via ac impedance measurements[END_REF] diagnoses the FC operating state based on the sensitivity of certain parameters to the MEA drying or flooding. He uses an equivalent electrical circuit model based on the Randles model. He also changes the double layer capacitor with Constant Phase Element (CPE). Alternatively I. Sadli [START_REF] Sadli | Behaviour of a pemfc supplying a low voltage static converter[END_REF], [START_REF] Sadli | Contributions of fractional differentiation to the modeling of electric double layer capacitance[END_REF], [START_REF] Sadli | Behaviour of a pemfc supplying a low voltage static converter[END_REF] models the FC behavior 65 by an impedance, substituting an equivalent transmission line with RC distributed cells for the classic Randles circuit. In this way he can take into account the critical AME convection-diffusion phenomenon in the electrochemical impedance. To interpret accurately it he needs to use a large number of individual RC cells. But partial differential equations correctly describe the convection-diffusion phenomenon. These latter leads to fractional order derives. That is the reason why Cao et al [START_REF] Cao | Dynamic modeling of electrical characteristics of solid oxide fuel cells using fractional derivatives[END_REF] proposes a fractional order model of solid oxide FC. The current work is also based on a model such as this. The key idea is to reformulate it so as to obtain a compact model characterized by a limited number of parameters. Experimental results conducted on a laboratory test bench will suggest that the proposed model reproduces reality very well. Furthermore a detailed parameter analysis will permit to reduce the number of parameters to follow.

The analysis of the FC impedance is an interesting solution for identifying failures through their signatures deformation of impedance spectrum [START_REF] Fouquet | Model based pem fuel cell state-of-health monitoring via ac impedance measurements[END_REF], [START_REF] Phlippoteau | Outils et methodes pour le diagnostic dun etat de sante dune pile a combustible[END_REF], [? ]. One of the most used methods for electrochemical system characterization and fuel cell diagnosis is the impedance spectroscopy [START_REF] Asghari | Study of pem fuel cell performance by electrochemical impedance spectroscopy[END_REF], [START_REF] Fouquet | Model based pem fuel cell state-of-health monitoring via ac impedance measurements[END_REF], [START_REF] Yana | Ac impedance characteristics of a 2kwpem fuel cell stack under different operating conditions and load changes[END_REF], [START_REF] Phlippoteau | Outils et methodes pour le diagnostic dun etat de sante dune pile a combustible[END_REF] For instance, Kurz et al. [START_REF] Kurz | An impedance-based predictive control strategy for the state-of-health of pem fuel cell stacks[END_REF] distinguish between flooding and drying modes using the analysis of two specific frequencies, one low and one high. But, since based on a harmonic perturbation, frequency domain analysis remains time consuming especially in the case of FC where the minimum frequency value is in the range of 100 mHz. That is the reason why it is crucial to develop alternative AC 95 impedance technique identification. This article intends to create a time domain rapid method able to monitor in real time the FC model parameters as a primary tool for diagnosing purpose and control modifications. For this purpose, the parameters are identified by least square method 100 adapted to the explicit fractional order FC model. In order to enable on-line investigations, the latter is adapted in a recursive form. Both time-series identification techniques represent basic tools for off-line and online diagnosis. They are experimented using a Pseudo Random 105 Binary Sequence (PRBS) small current perturbation and measuring the FC voltage response which can be technically possible on a fuel cell working in a car.

Identification results of the different methods are presented, analyzed and discussed. These methods estimate 110 the parameters describing the impedance spectrum evolution reflecting the fuel cell internal state. They represent basic tools for off-line and online diagnosis. Fractional order model identification results using these methods will judge the accuracy of the model to describe the fuel cell 115 behavior. The structure of the present In this paper, is as follows. After this brief introduction setting the background and the context of the work, Section II discusses a fuel cell FC impedance model and section III reformulates the non-linear model to obtain an explicit fractional 120 order transfer function. An analysis of the parameters sensitivity to flooding and drying will also be presented. Section IIIV presents the experimental setup used in the following section Part IV to identify the PEMFC model. In section IV an identification method based on frequency 125 measurements is presented to validate the model. The results obtained are compared with spectrum measurements obtained experimentally, concluding in the sufficient precision and accuracy about the accuracy of the presented model. Section VI presents a second identification method 130 of the fractional order model using time-series data. This method is derived from the classical least square and recursive least square methods and reformulated to be applied to the case of fractional order systems. Experimental results will be presented. Finally, the last section presents 135 some conclusions and perspectives.

introduction

PEMFC fractional order model

PEMFC impedance model

Over the years, many FC models have been developed 140 both to better understand the phenomena inside a FC system and to predict them [START_REF] Barbir | PEM Fuel Cells -Theory and Practice[END_REF], [START_REF] Gerard | Oxygen starvation analysis during air feeding faults in pemfc[END_REF]. Our goal is to determine a PEMFC model allowing to implement a real-time FC monitoring with to the aim of PEMFC reversible failures diagnosis, such as water flooding or drying. In this 145 context, a trade-off between accuracy and time execution has to be found. In the scientific literature, many studies use classical electrical impedance models to represent the FC system. For instance, Fontes [START_REF] Fontes | Interactions between fuel cell and 585 power converters influence of current harmonics on a fuel cell stack[END_REF], [START_REF] Fontes | Modelisation et caracterisation de la pile pem pour letude des interactions avec les convertisseurs statiques[END_REF]studies interactions between FC and power converters based on an Phlippoteau [START_REF] Phlippoteau | Outils et methodes pour le diagnostic dun etat de sante dune pile a combustible[END_REF] deal with the diagnosis of FC by using two different electrical impedance models, on dedicated to small magnitude signal and the other to large magnitude signal [START_REF] Turpin | Modelling and analysis of an original direct hybridization of fuel cells and ultracapacitors[END_REF]. In [START_REF] Hinaje | An equivalent electrical circuit model of proton exchange membrane fuel cells based on mathematical modelling[END_REF] and [START_REF] Noiying | Using electrical analogy to describe mass and charge transport in pem fuel cell[END_REF] Hinaje and al. use electrical analogy to describe mass and charge transport in FC and build a model well adapted to describe FC interacting with its electric load. Finally, Sadli [START_REF] Sadli | Contributions of fractional differentiation to the modeling of electric double layer capacitance[END_REF], [START_REF] Sadli | Contributions of fractional differentiation to the modeling of electric double layer capacitance[END_REF], [START_REF] Sadli | Behaviour of a pemfc supplying a low voltage static converter[END_REF] refines the PEMFC impedance model as a as a constants distributed transmission line so as to take into account gases diffusion inside the FC. The limitation of the methods described above is that they cannot reproduce some behaviors of the FC impedance spectrum, in particular with respect to the 45 • slope that appears around certain frequencies as noted in [START_REF] Sadli | Behaviour of a pemfc supplying a low voltage static converter[END_REF], [START_REF] Phlippoteau | Outils et methodes pour le diagnostic dun etat de sante dune pile a combustible[END_REF]and [START_REF] Iftikhar | Dynamic modeling of proton exchange membrane fuel cell using non-integer derivatives[END_REF]. As mentioned before [START_REF] Sadli | Contributions of fractional differentiation to the modeling of electric double layer capacitance[END_REF], to simulate reality much better, it is possible to add a large number of RC circuits, thus increasing the model order and therefore the number of parameters to identify as well as the model calculation time. Alternatively, recent works have suggested PEMFC modelling based on Fractional Order Models (FOM). This approach enables to reproduce more precisely the 45 • slope while being more compact than the previous models [START_REF] Iftikhar | Dynamic modeling of proton exchange membrane fuel cell using non-integer derivatives[END_REF], [START_REF] Sailler | Modelisation fractionnaire des piles a combustible de type pemfc[END_REF], [START_REF] Cao | Dynamic modeling of electrical characteristics of solid oxide fuel cells using fractional derivatives[END_REF].

Following the analysis, the latter method is favored in this study. The PEMFC is modelled by a FOM. Then a finite fractional order transfer function is obtained approximating the diffusion impedance by Taylor series. This technique permits us to implement an innovative identification method easy to execute in real time and thus well-suited to the final on-line diagnosis goal. Figure 1 gives a basic schematic of a PEMFC. It is fabricated by stacking bipolar plates, Gas Diffusion Layer (GDL), Active Layer (AL) and a polymer proton exchange membrane. On both sides, gases flows and diffuses through GDL and AL. the two electrochemical reactions take place at the inter-phase between AL and membrane. From a user's perspective, a FC is modelled by the voltage at its terminals which is the sum of its theoretical voltage and the diffusion, activation and ohmic voltage drops. This voltage is given by:

V cell = U th -η act -|η dif f | -R m I (1)
First, the theoretical potential of the FC is a function of H 2 /O 2 Gibbs free energy which depends on the temperature and the partial pressures of oxygen in the cathode side and hydrogen in the anode side. It is given by the following relation:

U th = U 0 + RT nF log(P H2 (P O2 ) 0.5 ) (2)
Second, the kinetics of the redox reaction causes activation phenomena occurring in the active layers (AL). These phenomena result in losses represented by the Tafel law:

η act = - RT αnF log( I I 0 ) (3)
The linearization of equation ( 3) around a steady state operating current leads to describe activation phenomena losses using a resistance given by:

R t = ∂ |n act | ∂I act (4)
Third, ohmic phenomena are mainly due to the proton conduction in the membrane, depending on the rate of hydration and temperature. The FC ohmic resistance is represented by a resistor

R m = l Sσ(T, λ m ) ( 5
)
Where S is the membrane area and σ its protonic conductivity. This latter strongly depends on temperature T and membrane hydration rate λ m as illustrated by the following relations [START_REF] Springer | Polymer electrolyte fuel cell model[END_REF] σ = (0, 005139λ m -0.00326)e 1268( 1

303 -1 T ) (6)
The membrane hydration rate is given by [START_REF] Barbir | PEM Fuel Cells -Theory and Practice[END_REF] as the ratio between the number of water molecules and the number of sulphonic sites of the polymer:

λ m = n H2O n SO - 3 ( 7
)
Where n H2O is the number of water molecules in the membrane and n SO -

3
is the number of sulphonic sites of the polymer. The gases flow through the channels of the bipolar plates by convection as depicted in fig. 1. Once at the GDL, they diffuse to the AL and reach the reactive sites. This article assumes that the diffusion is uniform in the both layers, GDL and AL, and the consumption rate of gas is uniform over the entire active surface. The propagation direction is defined from the GDL to the membrane. Assuming that the species diffuses in a biphasic medium of liquid water and vapor and defining a diffusion coefficient to identify experimentally, most studies describe the diffusion using Fick's second law:

∂C i (x, t) ∂t = D i ∂ 2 C i (x, t) ∂x 2 (8)
Solving this diffusion equation in the Laplace domain defines an impedance of concentration-diffusion known as "Warburg impedance" [START_REF] Fontes | Modelisation et caracterisation de la pile pem pour letude des interactions avec les convertisseurs statiques[END_REF] given by:

Z W = R d tanh( √ τ s) ( √ τ s) (9)
Based on equations 2, 3, 4, 8, eaquation 1 can be represented by Fig. 2. Indeed, in this model, the activation phenomena losses and the membrane resistivity are described by two resistances Rt and Rm respectively, while the diffusion phenomena losses 8 are modeled by Warburg impedance Z W . Finally the accumulation of protons and electrons at both cathode and anode membrane/AL interfaces is modelled by the so-called double layer capacitor. In order to obtain an explicit I-V transfer function, the latter model will be refined.

PEMFC explicit fractional order impedance model

With the aim of on-line monitoring, a compact and simple mathematical model has to be exhibited. In particular, the Warburg impedance requires a specific attention. As it is a function of a hyperbolic tangent of a non-integer order, it can be approximated using Taylor series to deduce an explicit transfer function model. Indeed, Iftikhar [START_REF] Iftikhar | Dynamic modeling of proton exchange membrane fuel cell using non-integer derivatives[END_REF] and Sailler [START_REF] Sailler | Modelisation fractionnaire des piles a combustible de type pemfc[END_REF] proposed to approximate [START_REF] Szymon Racewicz | Half-order modeling of saturated synchronous machine[END_REF] the Warburg impedance by:

Z W = R tanh ( √ sτ ) √ sτ = R sinh ( √ sτ ) cosh ( √ sτ ) • 1 √ sτ ∼ = R √ 1 + sτ (10)
The square root will be approximated in this new relation by Taylor series to obtain distinct and explicit orders and deduce an explicit fractional order transfer function model. Considering the first three terms of the expansion, the root square in 10 can be written as :

√ 1 + x ∼ = √ x + 1 x 2 - 1 8 1 x 3 2 + O(x 3 ) (11)
Figure 3 shows the error between the precise expression of Warburg impedance 9 and its final approximation 10. It shows that it is very accurate except at very low frequency. From figure 2, the transfer function of the PEMFC impedance can be given by:

H(s) = 1 1 Rt+Zw + sC dc + R mem (12)
Consequently, based on 10 and 11, the suggested FC transfer function exhibits explicit orders:

H(s) = b 0 + b 1 s + b 2 s
a i = f i (R, R t , R mem , C DC , τ ); i = 0, ..., 4. b j = g j (R, R t , R mem , C DC , τ ); j = 0, ..., 5. (14)
In sum, this first part permitted to develop a PEMFC model closely based on the system structure and its physical phenomena. Hence, it can assist in understanding the FC evolutions. More specifically, as shown in tab.1, the 230 models parameters change regarding water flooded or dry operating conditions. For these very important PEMFC defaults, the most relevant parameters are a3, b2 and b4. Indeed, these three parameters increase significantly in the water flooded case, and decrease in the dry circumstances.

235

They can be considered appropriate indicators of the critical track defects. That is why the study objective is to develop identification methods able to estimate the model's parameters and diagnose the operating state of the FC, referring to the parameters evolution. However for this pur-240 pose and greater certainty, experimental data are needed. That is why a test bench was developed and used.

Fractional order model validation

Before developing specific methods, our first attempt is to assess if the suggested compact model is representing the FC accurately. To do so it is necessary to obtain its ten parameters from data. In this section, the PEMFC FOM is 280 identified using a frequential identification method based on genetic algorithms: This method identify the model from the PEMFC impedance spectrum obtained using EIS measurements. A first identification by genetic algorithm is initiated to determine a suitable starting point for the 285 optimization algorithm. Which avoid the nonlinear optimization from starting the optimization randomly. The frequential methods aims here to validate the proposed fractional order model.

Frequential identification method 290

This section discusses the parametric identification using a multidimensional unconstrained nonlinear minimization. This approach is based on the minimization of a quadratic criterion which represents the difference between each real and imaginary part of the experimental 295 impedance and the identified impedance.

J = 0.5( ( Zexp -Z ident ) 2 + ( Zexp -Z ident ) 2 )
1 2

(15) This method is sensitive to its initial conditions. For this purpose, Phlippoteau [START_REF] Phlippoteau | Outils et methodes pour le diagnostic dun etat de sante dune pile a combustible[END_REF] proposes to initialize the identification algorithm using known parameters values given in the literature or deduced from the impedance spectrum geometry in the Nyquist plane. To go even farther in this approach, the suggested identification method relies on a two-step process: using a genetic based algorithm first and second operating the nonlinear optimization initialized with the first result. In this way nonlinear optimization algorithm, which will refine the genetics algorithm results while the genetic algorithm is used to prevent the optimization algorithm to stop in a local minimum point. Finally, the identification algorithm procedure is presented by figure 8.

Identification results

This method has been tested using impedance spectroscopy data made on a single FC using the test bench described previously. Figure 9 presents the identification results using only the genetic algorithms. Figure 10 presents the identification result using only the nonlinear optimization initialized with zero. Finally, figures 11 and 12 present the identification results using the optimization method described in this paper (genetic algorithm followed by nonlinear optimization technique) All the fig- 320 ures compare the identified impedance model's spectrum (red) to the experimental one (blue). The identification results using the suggested method are much more accurate than those based either on genetic algorithm or nonlinear optimization routine. Last but not least, the fractional 325 order model presents the advantage to be compact: it describes accurately the FC behavior using a small number of parameters. Moreover it represents exactly the 45 • slop which appears at high frequencies on the impedance diagram plotted in Nyquist plan and succeeds in following In a nutshell, section V shows that the FOM seems to be well adapted to describe the FC behavior. Furthermore, it is characterized by a low number of parameters. This proposal offers a good compromise between accuracy and 340 compactness. To address the FC diagnosis purpose, the issue is to find a viable solution able to extract in short

Time series parameters identification

345

A diagnosis method can be built by monitoring the evolution of the model parameters. Indeed, the parameters values change with the operating conditions. Using experimental data, the model parameters can be identified and compared to their reference values, in nominal, dry 350 and flooding cases. For online diagnosis in embedded applications, electrochemical impedance spectroscopy (EIS) is not well adapted to characterize the fuel cell. For online FC state monitoring, EIS is not well adapted. Indeed, it results in a slow process scanning harmonic sig-355 nals on a wide frequency range and moreover relates on a steady state assumption which is clearly unrealistic. Nevertheless,Alternatively temporal data series can be used to identify the model parameters. The vehicle load and. As a matter of fact, in many driving profiles, the de-360 manded power may present a sufficiently high spectral content which can be enough allows continuously updating extract FC fractional model parameters convenient using current/voltage data for this identification. Conversely, in steady state mode, a small amplitude PRBS current of 365 small amplitude can be superimposed to excite the system and extract dynamic voltage response. This method will be tested in this section.

In this section, the FOM will be reformulated to be adapted to classical identification methods as the least 370 square method. The present section is broken down in two parts. For offline diagnosis purpose, the model parameters will be first identified using a least square method. Later,Then, for online diagnosis application, a recursive least square method adapted to FOM will is derived to 375 identify the parameters.

Fractional order least square method for FOM offline diagnosis purposes

For offline diagnosis, least square method can be used to find the parameters values. Offline diagnosis can be 380 done post-use, in maintenance or retarded, after saving enough data needed to identify the parameters. Here, a least square method adapted to FOM is used to identify the model parameters [START_REF] Oustaloup | Systemes asservis lineaires dordre fractionnaire : theorie et pratique[END_REF], [START_REF] Malti | Developpements mathematiques autour des modeles fractionnaires application l'identification de systemes[END_REF], [START_REF] Victor | Identification par modle non entier pour la poursuite robuste de trajectoire par platitude[END_REF], [START_REF] Cois | Systemes lineaires non entiers et identification par modele non entier : application en thermique[END_REF], [START_REF] Victor | Parameter and differentiation order estimation in fractional models[END_REF]: This method uses time-series current and voltage. A PRBS 385 is applied at a polarization PEMFC current point. The PEMFC voltage response is then measured and used to identify the model's parameters.

A fractional order model can be described by the equation [START_REF] Cois | Systemes lineaires non entiers et identification par modele non entier : application en thermique[END_REF] :

390 L l=0 a l D na 1 y(t) = M m=0 b m D n bm u(t) (16)
With a 0 = 1 n a l , n bm : are real positive numbers, integers or nointegers (fractions) and supposed known. a l , b m : are the coefficients of the derivative operators supposed unknown.

The discretization of a Fractional Order Derivative (FOD) is done using Grunwald's [START_REF] Cois | Systemes lineaires non entiers et identification par modele non entier : application en thermique[END_REF], [START_REF] Victor | Identification par modle non entier pour la poursuite robuste de trajectoire par platitude[END_REF] approximation

[Dx(t)] n = 1 h n K k=0 (-1) k n k x[K -k] (17)
The discretized model obtained with Grunwald formula can be written as in Eq 18, where the output sample y[K] is a function of the previous inputs and outputs.

y[K] = - L l=0 a l h na l K k=1 (-1) k C k na l y[K-k] L l=0 a l h na l + M m=0 bm h n bm K k=0 (-1) k C k n b m u[K-k] L l=0 a l h na l (18)
This relation presents a non-linearity with respect to parameters. To deduce a linear form, a variable change is used to define a new parameters set (a 0 , ..., a L , b 0 , ..., b M ) where [START_REF] Victor | Identification par modle non entier pour la poursuite robuste de trajectoire par platitude[END_REF]:

a l = a l h na l L i=0 ai h na i ; b m = bm h n bm L i=0 ai h na i (19)
The new model expression is then:

y[K] = - L l=0 a l Y l [K] + M m=0 b m U m [K] (20)
With,

Y l [K] = K k=1 (-1) k C k na l y[K -k], U m [K] = K k=0 (-1) k C k n b m u[K -k] (21) 0 ≤ l ≤ L, 0 ≤ m ≤ M ( 22
)
Equation 21 is developed as :

ŷ[K, θr ] = -â 0 Y 0 [K] - L l=1 â l Y l [K]+ M m=0 b m U m [K] (23)
Using the constraint

L l=0 a l = 1 permits to eliminate a 0 ŷ[K, θr ] = -1 - L l=1 â l Y 0 [K]- L l=1 â l Y l [K]+ M m=0 b m U m [K] (24)
The model output is hence linearly expressed with respect to the vector of new parameters:

410 ŷ[K, θr ] = -Y 0 [K] - L l=1 â l (Y l [K] -Y 0 [K]) + M m=0 b m U m [K] (25)
For N measurement points between Kh and (K + N )h linear matrix equation is given by:

Ŷ ( θr ) = -Y 0 + ϕ θr ( 26
)
Where φ is composed by

-Y i [K] + Y 0 [K] to -Y 1 [K + N ] + Y 0 [K + N ], i = 1...L and U j [K] to U j [K + N ]
The vector of parameters is given by:

415 θropt = (ϕ T ϕ) -1 ϕ T (Y + Y 0 ) (27)
Finally initial parameters are computed by reversing the previous variable change using the following relations [START_REF] Cois | Systemes lineaires non entiers et identification par modele non entier : application en thermique[END_REF], [START_REF] Victor | Identification par modle non entier pour la poursuite robuste de trajectoire par platitude[END_REF] :

A =      (a 1 -1)h -na 1 • • • a 1 h -n aL a 2 h -na1 • • • a 2 h -na L . . . . . . . . . a L h -na1 • • • (a L -1)h -na L      ⇒ A *      a 1 a 2 . . . a L      =      (28)
b m = b m L l=0 a l h n bm -na l (29)
This algorithm was implemented to identify the PEMFC FOM parameters. The identified model simulation results 420 are presented in the next section.

Offline identification results

The identification data had been obtained on a signal cell using using the PEMFC response to a PRBS current excitation of 1 A around a 7A polarization current. The 425 identification results using the previous method experimental results are presented in figure ??. The identified model reproduces correctly the FC voltage response.

To test the parameters values, identification data had been simulated using the model identified previously. The 430 PEMFC response to a PRBS current excitation of 1 A around a 7A polarization current had been simulated . From the parameter evolution given by 4 in the dried and flooded cas, three models had been simulated to generate identification data in these cases. The data had been used 435 later to identify the parameters in the three cases (nominal, dry, flooded).

Tab 1 present the comparison between the identified parameters and the model's parameters (used to simulate the identification signal), in the dry, flooded and nominal 440 cases. As an example, the parameters a 3 are identified correctly. With regards to the identification results, it can be concluded that the time-series identification results are a first step to off-line diagnosis and reinforce the idea that the recursive least square method adapted to fractional order models can be applied for on line diagnosis using the previous model.

With this method, the parameters can be identified off line (post-operating) using the experimental data saved during the operating period. The identified parameters can be used to establish an offline diagnosis by comparing their identified values to their references values (in a nominal, dried and drown operating cases)

on line identification for diagnosis purpose :purpose: MCR

An on line diagnosis method can be build based on recursive identification methods, which identifies the model parameters online. Recursive least square (RLS) method can be used [START_REF] Djouambi | Recursive prediction error identification of fractional order models[END_REF] to identify the model parameters using the model reformulation given previously by the equations 16 to 24.

The equation 24 has the following linear form:

y((k + 1)h) = θ t φ(k) (30)
θ is the vector of parameters to be identified and φ(k) the measurement matrix , such that: The prediction equation is

φ(k) = [-Y 1 (k), . . . , -Y 5 (k), U 0 (k), . . . , U 6 (k)] (31)
ŷ(k + 1) = θt k+1 φ(k) (32)
The parameter vector is estimated by minimizing a quadratic criterion:

θk = arg min θ 1 k K k=1 [y(k) -ŷ(k, θ)] 2 (33)
The solution to this problem [START_REF] Djouambi | Recursive prediction error identification of fractional order models[END_REF] is given by: θ

(K) = K k=1 φ (k) φ T (k) -1 K k=1 φ (k) [y(k) -Y 0 (k)] (34)
The algorithm of the recursive least square method is 470 then implanted using the following relations at each iteration:

θk = θk-1 + F (k-1)φ(k)(y(k)-[ θT k-1 φ(k)-Y0(k)]) 1+φ T (k)F (k-1)φ(k) F (k) = F (k -1) -F (k-1)φ(k)φ T (k)F (k-1) 1+φ T (k)F (k-1)φ(k) (35)
The data used to identify the FOM, with the previous method, are current/voltage temporal series measurements. The control signal is a Pseudo-Random Binary 475 Sequence (PRBS) with an amplitude δI = 1(A), applied around a current of polarization I 0 = 7 [A].

The parameters identification using this method and time series experimental data permit to mimic correctly the experimental output voltage for slow and quick vari-480 ations, and enable reproducing correctly the shape of the impedance spectrum, as shown in fig. 14 and15, respectively. The parameters identified using this method are composed by the physical parameters of the PEMFC. The so-485 lution of equations to go back to the FC physical parameters (Rd, Cdc, Rt..etc) is not unique. It is therefore not

Conclusion

This paper presented the modeling and identification of a PEMFC fractional order model in the perspective of the 500 PEMFC state monitoring. It presented a new explicit fractional order transfer function model. To identify directly the impedance spectrum form and validate the presented model, a frequency identification method had been used to find the FOM parameters which allow reproducing the impedance form. The nonlinear optimization key point is the initialization of the identification algorithm. The paper discusses then an initialization strategy which allows finding better results. The frequency identification results had been compared with experimental impedance 510 spectrum which leads to the conclusion that the fractional order model is well adapted to describe a PEMFC impedance. For on-line diagnosis method, only time-series measurements can be technically used. The second part of this paper presented an identification method based on 515 time series data. A least square method, adapted to fractional order models, had been implemented. The obtained results shown a good consistency with PEMFC simulated voltage response. This method does not use directly the FC experimental impedance spectrum data, but is technically feasible on a FC placed in a car for an on-line diagnosis goal, unlike an EIS. Time-series identification by least square method adapted to fractional order gives good results which suggests that for future on line monitoring, recursive least square method is well suited for frac-525 tional order systems. As on-line monitoring methods can only rely on time-series measurements, these first pieces of evidence are key-points for the ongoing work. They indeed confirm that the FOM mimics well the FC behavior and with the objective of on-line diagnosis, recursive least 530 square adapted to FOM can be an efficient tool to identify the model's parameters.

Figure 1 :

 1 Figure 1: Fuel cell

Figure 2 :

 2 Figure 2: PEMFC equivalent electrical circuit model (Randles)

  220

Figure 3 :

 3 Figure 3: error of Warburg impedance approximation

3 2 + b 3 s 2 + b 4 s 5 2 + b 5 s 3 1 + 5 2 + a 4 s 3

 325153 a 1 s + a 2 s 2 + a 3 s

  ) is characterized by only ten coefficients a i and b j 225 which are obviously function of the physical parameters.

Figure 4 :

 4 Figure 4: Explicit fractional order parameters evolution in dry and water flooded operating conditions

  Figure 4 and 5 shows a picture and a block diagram of the test bench developed at the GeePs. Finally the electronic load can be monitored by Fuel Cell Impedance Analyzer FC350 designed by the Gamry Company. It allows to impose a static steady current and superimpose a sinusoidal current signal from 10 mHz to 300 kHz. It also enables electric (VFC, IFC) data acquisition and processes these signals to calculate the FC impedance. This latter technique is called the Electrochemical Impedance Spectroscopy (EIS). EIS is performed over a frequency range of 0.1 Hz to 1 kHz around several operating points: from 5 A to 40 A with 5 A increments. Indeed, the FC impedance spectrum changes with the considered operating point.

Figure 5 :

 5 Figure 5: Test bench picture at GeePs

Figure 6 :Figure 7 :

 67 Figure 6: Block diagram of the test bench

  310

Figure 8 :

 8 Figure 8: : Genetic algorithm and nonlinear optimization flowchart

330

  the real FC at low frequency.

Figure 9 :

 9 Figure 9: : Genetic algorithm results, for an impedance spectroscopy measured at 10A

Figure 11 :

 11 Figure 11: : Frequential identification results using impedance spectroscopy data measured at 10A

Figure 13 :

 13 Figure 13: : Identification results using least square method for FOM and experimental data measured at 7A

Figure 14 :

 14 Figure 14: : online identification validation of the FOM using RLS method for FOM

Figure 15 :

 15 Figure 15: : online identification frequential validation of the FOM using RLS method for FOM

Table 1 :

 1 Comparison between the identified parameters and the model's parameters

	a 3	N ominal f looded dry
	M odelP arameter	?	?	?
	Identif iedP arameter ?	?	?

electrical impedance model. Similarly Fouquet[START_REF] Fouquet | Model based pem fuel cell state-of-health monitoring via ac impedance measurements[END_REF] and


Mohsen Kaboli

Gordon Cheng

Novel Tactile Descriptors and a Tactile Transfer Learning Technique for Active In-Hand Object Recognition via Texture Properties

This paper proposes robust tactile descriptors and ,for the first time, a novel online tactile transfer learning strategy for discriminating objects through surface texture properties via a robotic hand and an artificial robotic skin. Using the proposed tactile descriptors the robotic hand can extract robust tactile information from generated vibro-tactile signals during in-hand object exploration. Tactile transfer learning algorithm enables the robotic system to autonomously select and then exploit the previously learned multiple texture models when classifying new objects with a few training samples or even one. The experimental outcomes demonstrate that employing the proposed methods and 10 prior texture models, the robotic hand could identify 12 objects through their surface textures properties with 97% and 100% recognition rate respectively with only one and ten training samples.

I. INTRODUCTION AND BACKGROUND

Tactile information is crucial for autonomous robots for detecting and learning the physical properties of objects. The performance of tactile systems depends not only on the technological aspect of the sensory device, but also on the design of the learning methods that interpret information contained in tactile data [START_REF] Kaboli | Humanoids learn touch modalities identification via multi-modal robotic skin and robust tactile descriptors[END_REF]. The object material can be characterized and differentiated based on surface texture, stiffness, and thermal information obtained through tactile sensing. However, to the best of our knowledge, so far there is no research paper addressing object discrimination via their physical properties while the objects are in the hand of a robot. Jamali et al. fabricated a biologically inspired artificial finger composed of silicon within which were two PVDF pressure sensors and two strain gauges. The finger was mounted on a robotic gripper and was scraped over eight materials. The Majority voting learning method was employed to find the optimal technique for the texture recognition problem [START_REF] Hu | A fingershaped tactile sensor for fabric surfaces evaluation by 2-dimensional active sliding touch[END_REF]. Hu et al. used Support Vector Machine (SVM) to classify five different fabrics by sliding a finger-shaped sensor over the surfaces [START_REF] Dallaire | Autonomous tactile perception: A combined improved sensing and bayesian nonparametric approach[END_REF]. Dallaire et al. [START_REF] Giguere | A simple tactile probe for surface identification by mobile robots[END_REF] managed to classify 28 different surfaces such as Aluminum, Plexiglas and Kitchen towel via Bayesian non-parametric learning approach. In this respect, a three axis accelerometer was placed on a stylus, which was then mounted above a rotating turn-table on which the surface was placed. Ten different surfaces were detected through an artificial neural network by sliding an accelerometer mounted prob over the surfaces such as wooden flooring, short hair carpet, and tile linoleum flooring [START_REF] Liu | Surface material recognition through haptic exploration using an intelligent contact sensing finger[END_REF]. Liu et al. employed an intelligent contact sensing finger to classify surface materials with Naive Bayes classifier [START_REF] Windau | An inertia-based surface identification system[END_REF]. Hu et al. used Support Vector Machine (SVM) to classify five different fabrics by sliding a finger-shaped sensor over the surfaces [START_REF] Dallaire | Autonomous tactile perception: A combined improved sensing and bayesian nonparametric approach[END_REF]. A robot actively knocks on the surface of the experimental objects with an accelerometer-equipped device to discriminate stone, mulch, moss, and grass from each other with a lookup table and k-nearest neighbors (K-NN) techniques [START_REF] Xu | Tactile identification of objects using bayesian exploration[END_REF]. A multimodal tactile sensor called BioTac was used to perceive tactile information. In this experiment one BioTac sensor was placed on a customized tool and a vibration-free linear staged was used to slide textures under the tactile sensor. In [START_REF] Kaboli | Humanoids learn object properties from robust tactile feature descriptors via multi-modal artificial skin[END_REF], the Shadow Hand with the BioTac sensor on the index fingertip and Bayesian exploration technique were employed to discriminate 10 different objects from each other by executing exploratory movements over the objects surface. However, the existing texture classification methods are not able to re-use the past experience or prior learned texture models (tactile transfer learning). The focus of this study is to propose a set of robust tactile feature descriptor for active inhand object recognition via surface texture properties. For the first time in the community of the tactile object recognition, we propose an online tactile transfer learning method to enable robotic hands to re-use their prior tactile knowledge to discriminate new in-hand objects (through their textural properties) with a few available training samples or even one (one-shot tactile learning).

II. SYSTEM DESCRIPTION A. Robotic Hand

The Shadow Hand is a dexterous Robotic Hand System with five fingers and 20 active degrees of freedom in total, which enables the robot to have a range of movement equivalent to that of a human hand (see Fig. 1).

B. Multi-Modal Artificial Skin

The BioTac is a multi-modal tactile sensor. When the sensor moves over an object, caused vibration can be measured by a dynamic pressure signal (P AC ) with the sampling data rate of 2 KHz. Moreover, BioTac has 19 impedance-sensing electrodes (E 1 , ..., E 19 ) measuring the deformation that arises when normal forces are applied to the surface of the skin with a 50 Hz sampling rate (see Fig. 1).

C. Properties Of Experimental Objects

In this work 22 everyday objects were selected. 10 objects with an identical geometrical shape including a Red and a Yellow ball with almost similar smooth surface texture, a Rough textured ball, an Orange, an apple, a Colorful ball with smooth and non-uniform texture, a Rough spherical sponge, a Pine apple textured ball, a String ball, and Mirror ball (see Fig. 1-Prior Objects). Also, 12 objects with different shapes including a Soft Sponge, a Memory sponge (nonuniform texture), a Toothbrush (non-uniform texture), a Floor brush, a Rough textured star, a Soap, a Spray, a Coffee capsule, a Paper box, a Cream tube, a Plastic baby feeder, a Metal ruler (see Fig. 2-New Objects).

D. Data Collection With Prior Objects Set

In this scenario, the Shadow Hand held each of the spherical shaped prior objects (see Fig. 1-Prior Objects) in palm with three random fingers. Afterwards, the robotic hand explored the texture of each in-hand object by randomly moving the remaining free fingers (two fingers) over the surface of the object for 3 seconds. The texture exploration was repeated 50 times for each prior objects with random orientation. The collected tactile data for each object then randomly divided in tow sets, one with 30 samples for the training and the other one with 20 trials for the testing. Altogether, 300 training and 200 testing samples for 10 prior objects.

E. Data Collection With New Objects Set

In this case, the Shadow Hand used three fingers to hold each complex shaped object (Fig. 2-New Objects). The exploration carried out with the remaining two fingers by sliding over the surface texture of the object for 3 seconds. The rest of data collection procedure remained as same as described before.

III. Proposed Feature Descriptors

The generated tactile signals were measured by pressure sensor (P AC ) and the impedance sensing electrode array (E 1 , ..., E 19 ). To extract robust information from vibro-tactile signals we propose a set of parameters, called Activity, Mobility, Complexity, linear correlation coefficient Eq.( 4), and non-linear correlation coefficient Eq.( 5) [START_REF] Kaboli | In-hand object recognition via texture properties with robotic hands, artificial skin, and novel tactile descriptors[END_REF], [START_REF] Kaboli | Re-using prior tactile experience by robotic hands to discriminate in-hand objects via texture properties[END_REF].

Activity = 1 N N ∑ n=1 (S n -S) 2 (1)
Mobility = Var( dS n dn ) Var(S n ) (2)
Complexity = mobility( dS n dn ) mobility(S n ) (3)
Pcorr(P AC , E K ) = ∑ N i=1 (P AC i -PAC ).(E K i -ĒK ) σ (P AC ).σ (E K ) (4)
Scorr(P AC , E K ) = 1 - 6 ∑ N i=1 R 2 i N(N 2 -1) (5)
where S n is the input signal, N is the number of data points, K is the number of impedance electrodes, and R i is the difference between the rank of (P AC ) i and the rank of (E K ) i .

IV. PROPOSED TACTILE TRANSFER LEARNING TECHNIQUE

Consider a scenario in which the Shadow Hand has already constructed a set of learning models to discriminate k = 10 different surface textures using Least Squared Support Vector Machine learning algorithm with sufficient available training samples (in our case 300 training samples) as well as high enough measured tactile information (tactile data collected by five fingertips). Now the new task of the Shadow Hand is to classify N = 12 new surface textures with one or a very few available training samples while re-using the prior texture models in an online manner.

1) Constructing Prior Tactile Models: LS-SVM was trained with k prior object textures (in our 10 objects Fig. 1-Prior Objects) to construct surface texture models as a prior knowledge . By slightly modifying the regularization term in LS-SVM, it is possible to construct new discriminating texture models for the new objects (see Fig. 2-New Objects) close to the already constructed prior models.

2) Prior Tactile Knowledge Selection: Suppose there are already k = 10 constructed models, and N new object textures with T available training samples for each new object textures (x t , y t ) t = 1, . . . , T . The optimization problem has the same cost function as LS-SVM in which the regularize term has been modified to impose closeness between the new object texture models and a linear combination of prior models.

3) Online Tactile Transfer Learning: The proposed online tactile transfer learning in [12] is a hybrid algorithm which integrates the prior and new models via the adapted version of LS-SVM in order to properly initialize the PA algorithm (see Algorithm 1). w 1 is composed of two parts. The first part is the linear combination of the weighted prior models where w s is the prior model, λ s is the scaling factor, and k is the number of prior models. The second part represents the received new training samples in which T is the number of the samples. Now, the PA algorithm uses the new initial models w 1 instead of the w 1 = (0, . . . , 0) to learn from the (t + 1) -th new incoming samples. So far, we initialized the PA leaning algorithm by integrating the prior and new models. But, still, the prior models are not directly re-weighted during the on-line learning process. We describe here how during the on-line learning progressively update the prior and new models weights in time. In this case, the prediction can be made on each new incoming samples by means of the current constructed models (see Algorithm 1) as w 1 • x t . The results of the prediction σ k,t will be cropped between -1 and 1 and will be used as the (d + k)-th element in the feature vector descriptor of x t .

Algorithm 1 : Proposed Online Tactile Transfer Learning

For t = 1 Initialize PA: w k,1 = (w 1 , 1) ∈ R d+k where where w 1 = (∑ k s=1 λ s ŵs + ∑ T t=1 α i x i ) For t = 1, 2, ..., T T = 10 Input to PA:

x t New Comming Samples

x t = (x t , σ 1,t , . . . , σ k,t ) ∈ R d+k , η = 1

Augmented Samples

where:

σ k,t = max{-1, min{1, w k,1 • x t }} ⇒ σ k,t ∈ [-1, 1] • Prediction: ŷt = sign(w k,t • x t ) • Suffer loss: t = max{0, 1 -y t w k,t • x t } • update: 1.set θ t = min η, max{ t } x t 2
2.update: w k,t+1 = w k,t + θ t y t x t

V. EXPERIMENTAL RESULTS

A. Constructing Prior Tactile Model

In order to build up a 10 prior models with the collected training samples during the prior object textures exploration the LS-SVM classifier was employed. The entire training samples (30 training samples for each prior texture and all together 300 training samples) were split in to parts, 70% for training and 30% for the test. Five-fold cross validation was applied in order to find the optimal kernel parameter and regularizer value C. LS-SVM was re-trained with entire collected training data and the optimal found parameters to construct 10 prior leaning models. The learning models (w, b) ∈ R 10 were evaluated by predicting on unseen collected test data (20 test samples for each class of prior texture and altogether 200). The LS-SVM could manage to classify successfully 10 prior textures with 100% recognition accuracy.

B. Evaluation of proposed Online Transfer Learning

In this experiment, the Shadow Hand used the proposed tactile transfer learning technique to recognize 12 new surface textures while re-using ten k = 10 already learned prior models together with learning from a very few training samples. In this scenario, the collected training samples (see Fig. 2-New Objects) during new textures exploration entered to the proposed hybrid online transfer learning sequentially one after one to construct new hybrid learning models. At each time t = 1, ..., 10, the constructed leaning models were evaluated by predicting on unseen test data collected during the new object textures exploration (20 test samples per new textures). The prediction results were reported as a recognition rate in Fig. 3.

C. Baseline

In order to compare our obtained results the traditional PA algorithm was employed to construct surface texture models while receiving new training samples continuously over time (one new texture per time t (t = 1, • • • , 10)). The new constructed learning models at each time t were evaluated by predicting on unseen test data (20 test samples per new textures). The classification results were reported as a recognition rate in Fig. 3. The value for η was fixed to 1 in both hybrid online transfer learning and PA online leaning (base line). Fig. 3 shows that using our proposed hybrid online transfer learning method the Shadow Hand could discriminate 12 new texture with 97% recognition accuracy while using only one training sample plus 10 prior models. By increasing the number of training samples from one to ten, the Shadow Hand achieved 100% recognition accuracy. The results in Fig. 3 illustrates that our proposed method outperforms the traditional online learning. Although the PA achieved lower classification accuracy compare to the proposed methods, PA achieved 78% and 97% recognition accuracy while learning from one and ten training samples respectively. The obtained high classification performance by PA is due to the fact that our proposed feature descriptors provided the PA algorithm with information-rich tactile data.

1) Negative Knowledge Transfer Consistency:

In transfer learning scenario the constructed prior models are not always relevant for new object models. If the prior models are dissimilar to the new models, brute force transfer can degrade the recognition performance generating so called negative knowledge transfer. Ideally, a transfer learning method should be beneficial between prior and new models while avoiding negative transfer when the object surface textures are not a good match. We show that our proposed tactile transfer learning technique is robust against of the negative knowledge transfer. In this respect, Expectation Maximization algorithm was employed to find out which of the new object textures are similar or dissimilar to the prior textures. In this case, the EM was trained with entire training samples (10 samples per each texture) to cluster all available 20 objects (both prior and new object textures). The EM then was evaluated by unseen test data (20 samples per each texture). In this scenario, Spray, Metal ruler, Pine apple, and String ball were selected as a set of new textures while the prior textures were remain same. The hybrid online transfer learning was employed to discriminate the four new textures and traditional PA was used as a base line. The procedure was the same as describe above. Fig. 4 shows the classification results in terms of recognition accuracy. The illustrated results clearly shows that the obtained recognition performance while using the proposed hybrid online transfer learning is similar to the performance achieved while using the traditional PA. This means that our algorithm stopped transferring irrelevant prior knowledge to new task (see Fig. 4).

VI. CONCLUSION

We proposed robust tactile descriptors for active in-hand object recognition task. Furthermore, for the first time, we designed an online tactile transfer learning methods to provide the robotic systems with the ability of re-using previously learned tactile models (prior models) to discriminate new in-hand objects with a very few available training samples. In this study, the distributions of the tactile information in both prior knowledge and new tasks were similar.

Fig. 1 .

 1 Fig. 1. This figure shows the uniform in-hand objects considered as Prior Objects. The Shadow Hand with the BioTac sensors is exploring the texture properties of the in-hand objects with an identical shape. It moves any of the fingertips to slide over the objects surface and uses the proposed tactile descriptors to extract robust tactile information.

Fig. 2 .

 2 Fig.2. This figure demenstrates the complex shape in-hand objects considered as New Objects. Employing the proposed tactile transfer learning the Shadow Hand discriminates new objects from their texture with a very few trials whilst re-using prior knowledge.

Fig. 3 .

 3 Fig. 3. This figure shows the recognition results on a separate test data for the online tactile transfer learning and traditional PA online learning (No-Transfer) methods. In this experiments 10 prior models were re-used by the Shadow Hand as prior tactile knowledge.

Fig. 4 .

 4 Fig. 4. This figure shows the recognition results corresponding to hybrid tactile transfer learning and traditional transfer learning (No-Transfer) in which the new surface textures were dissimiar to the prior textures. The recognition results on the test set were plotted as a function of the number of the trainig samples

Mohsen Kaboli and Gordon Cheng are with the Institute for Cognitive Systems, Faculty of Electrical Engineering and Information Technology, Technical University of Munich (TUM)-Germany. Email: mohsen.kaboli@tum.de(http://www.ics.ei.tum.de)

ACKNOWLEDGMENT

This work is supported by the European Commission under grant agreements PITN-GA-2012-317488-CONTEST.