W Qafsaoui
email: wqafsaoui@gmail.com

M W Kendig

H Perrot

H Takenouti
email: hisasi.takenouti@upmc.fr

Coupling of electrochemical techniques to study copper corrosion inhibition in 0.5 mol L -1 NaCl by 1-pyrrolidine dithiocarbamate

Keywords: EIS, EQCM, Oxygen Reduction Reaction, Surface film, Voltammetry 1 ISE member

In this work, as for the preliminary step to the protection of Al 2xxx alloys, the anticorrosion effect of environmentally friendly 1-pyrrolidine dithiocarbamate (PDTC) on Cu was investigated by combining various electrochemical methods in addition to SEM-EDS analyses and gravimetric measurements. The Cu / PDTC system was used as a model to simulate the inhibition of Cu-rich particles contained in Al 2xxx alloys. The corrosion test solution was 0.5 mol L -1 NaCl in presence of 10 -4 , 10 -3 , or 10 -2 mol L -1 PDTC. The immersion time of the copper electrode was extended generally up to 24 hours. By EQCM measurements, a marked protective effect of PDTC was observed for the highest PDTC used, and corroborated by voltammetry and EIS measurements. The cathodic reduction under constant current allowed determining the corrosion products remaining on the copper surface in terms of electrical charge. The protective effect of PDTC was explained by the formation of highly stable compounds on the copper surface, which decreases the rate of both anodic dissolution and oxygen reduction reaction.

Introduction

The 2xxx series of aluminium alloys obtained by addition of large quantities of copper to pure aluminium exhibit excellent mechanical properties. This copper addition leads to a heterogeneous microstructure constituted by copper rich precipitates. However, although the high strength acquired, these alloys suffer corrosion damages due to the formation of galvanic cells between the aluminium matrix and copper rich-particles, particularly in chloridecontaining medium [1,2]. These precipitates provide surfaces where oxygen can readily be reduced; therefore they constitute the cathodic area. Treatments involving the use of chromates and dichromates provide a highly efficient corrosion protection for aluminium alloys, particularly those containing copper [START_REF]Chemical Conversion Materials for Coating Aluminum and Aluminum Alloys[END_REF]. However, chromate solutions cause health and environment risks because hexavalent chromium compounds have an important toxic effect and are carcinogenic. Efforts are then made to replace Cr(VI) by non-toxic compounds able to specifically protect Cu-rich particles. Current attention of researchers is focused on the inhibition of electrochemical processes, especially the oxygen reduction reaction (ORR), on copper rich-particles to reach an action quite similar to that of Cr(VI). Corrosion inhibitors for copper, such as organic molecules, could then be considered as alternatives [START_REF] Qafsaoui | [END_REF]5].

Corrosion protection of metals by organic compounds depends mainly on the chemical structure of the inhibitor and its adsorption mode. Several groups of heterocyclic organic compounds have been reported to exert inhibitive effect on copper corrosion in different media [START_REF] Kuznetzov | Organic Inhibitors of Corrosion of Metals[END_REF][START_REF] Patel | [END_REF][8][9]. Benzotriazole (BTAH) is one of the most famous inhibitors for copper and copper alloys [10][11][12][13][14]. In previous works [START_REF] Qafsaoui | [END_REF]5], we showed that BTAH and 3-amino 1,2,4triazole (ATA) inhibit the dissolution associated with Cu-rich particles by the formation of protective complexes upon the particles. In the same way, Williams et al [15] showed that organic and inorganic anions inhibit pitting corrosion of Al 2024 T3 alloy in 5% NaCl, by film formation on Cu-rich particles and / or precipitation of copper cations. However, these compounds that act efficiently on Cu-rich particles are more or less toxic or noxious. In line with environmental protection requirements, toxic inhibitors that are widely used in industrial processes should be replaced with new environmentally friendly inhibitors.

Certain dithiols as 1-pyrrolidine dithiocarbamate (PDTC) inhibit ORR on Cu electrodes [16][17][18]. PDTC is a non-toxic product widely used in agricultural and medical applications [19] and may therefore provide an alternative to chromate-based inhibitors. PDTC was shown to inhibit corrosion of steel water pipes in different corrosion media [20]. Recent works on copper corrosion inhibition in 0.5 mol L -1 HCl by PDTC [21] showed that this compound is able to interact with copper to form a self-assembled monolayer allowing a high inhibition efficiency to be obtained (up to 95.3 %). Moreover, the authors showed that the sulphur atoms in the PDTC are the main active sites for the adsorption on the copper surface.

Our previous works [START_REF] Qafsaoui | 213th Meeting of the Electrochemical Society, Symposium "Corrosion General Session[END_REF] with PDTC showed, besides inhibition of ORR, the formation of a highly barrier film on copper in 0.2 g L -1 NaCl. This film constitutes a partially blocking surface to diffusion-limited reactions. Preliminary tests on 2024-T3 aluminium alloy [START_REF] Kendig | Nace Research in Progress Symposium, session: Coatings and Inhibitors, Corrosion/08[END_REF] showed that PDTC decreases considerably anodic and cathodic current densities of the alloy in 0.2 g L -1 NaCl and shifts the open circuit potential towards cathodic values. However, PDTC has no inhibitive effect on pure Al in the same medium suggesting that PDTC interacts solely with the Cu-rich particles. To understand this interaction in more detail, it seemed important to estimate its action on copper in chloride media.

It is clear that the main objective of this work is to study Cu / PDTC system to understand the inhibitive behaviour of PDTC on Al 2xxx alloys in chloride media. However, this work is in itself relevant to inhibition of pure copper since corrosion inhibition of Cu by PDTC is scarcely reported in the literature.

By selecting PDTC, the major aim of the present work focuses on its influence on general corrosion of copper in 0.5 mol L -1 NaCl solution. Electrochemical experiments were carried out to study the behaviour of the copper / solution interface in the presence of different concentrations of PDTC.

Experimental

Electrodes

The disk electrodes were made of copper cylinder rod (Goodfellow) of 5 mm in diameter.

First, the lateral part of the cylinder was covered with cataphoretic paint (PGG W975 + G323) to avoid the electrolyte infiltration. The copper rod was then embedded into allylic resin or thermal shrinking sheath. The electrode surface was abraded, just before experiments by rotating emery paper up to 1200 grade under water flow, and then rinsed abundantly with deionised water.

For EQCM (Electrochemical Quartz Crystal Microbalance) experiments, two 0.2 cm 2 gold electrodes were vapour deposited on both the free faces of the quartz crystal blade to impose electrical field for oscillation. One of the faces was used as working electrode and, for this purpose copper was electro-deposited on the gold electrode from 0.5 mol L -1 CuSO 4 + 0.5 mol L -1 H 2 SO 4 solution by a cathodic current of 30 mA cm -2 for 6 min. In these conditions, the mass of deposited copper was about 3.5 mg cm -2 or 4 µm thick. The deposit thus obtained was polycrystalline and compact. The nominal resonant oscillation frequency of quartz blade was 6 or 9 MHz. For 6 MHz quartz crystals, experimental calibration measurements gave 1 µg cm -2 of the mass loss corresponding to the increase of oscillation frequency of 164 Hz [START_REF] Otmacic Curkovic | [END_REF][START_REF] Sauerbrey | Verwendung von Schwingquarzen zur Wägung dünner Schichten zur Mikrowägung[END_REF][26]. Measurements were carried out with a lab-made device and monitoring program.

The working electrode was set close to the centre of the electrolyzing cell, and under stationary conditions, i.e. without any electrolyte stirring.

The reference electrode was a calomel electrode in saturated KCl (SCE). All potentials are referred as measured, without any correction of liquid junction potential. No correction for the ohmic drop was made since its effect is negligible for analyses of polarization curves. The counter electrode was a platinum grid of large surface area set close to the cell wall.

Electrolytes

The blank corrosion test solution was 0.5 mol L -1 NaCl to which 10 -4 , 10 -3 , or 10 -2 mol L -1 of PDTC was added as inhibitor. 100 mL of electrolyte was used for each experiment. The corrosion test was carried out without purging of dissolved oxygen and maintained at 20 °C.

The molecular structure of PDTC is presented in Fig. 1. This molecule has three heteroatoms; one nitrogen and two sulphur atoms, thus possesses high possibilities of adsorption on metal surface conferring a good candidate for the corrosion inhibitor. This substance is considered an irritant but is otherwise inoxious [START_REF] Roth | Safety data card : 5486f[END_REF].

Surface analysis

The surface morphology of copper electrode was investigated with a LEICA STEREOSCAN 440 scanning electron microscope (SEM) and element analyses were performed with an energy dispersive X-ray analyzer (EDX; Princeton Gamma-Tech  ).

Electrochemical measurements

For coulometric experiments to evaluate the amount of copper corrosion products accumulated at the electrode surface, the specimen was first immersed in the corrosion test solution without or with PDTC during 1 or 24 hours at the open circuit conditions. Then, the copper electrode was immediately rinsed with deionised water and transferred to a borate buffer solution; 10 -2 mol L -1 H 3 BO 3 + 10 -2 mol L -1 Na 2 B 4 O 7 (pH 9). In this solution, corrosion products were reduced at the current density equal to -50 µA cm -2 . The solution was purged of oxygen by Ar bubbling before the experiments and then Ar was flowed over the electrolyte surface [START_REF] Qafsaoui | [END_REF].

Polarization measurements were carried out in a conventional three-electrode cell. The disk working electrode was stationary and faced towards the cell bottom. Electrochemical measurements were performed after 1 or 24 hours immersion (if not specified differently) in the test solution with or without the inhibitor.

The voltammetric and impedance measurements were carried out using a Gamry potentiostat / The impedance measurements were performed with 10 mV rms from 100 kHz to 10 mHz by 10 points per decade during 48 hours. For parameter regression calculation procedure, sometimes high frequency data (for instance beyond 40 kHz) were deleted since some instrumental and systematic error was observed, especially when the impedance spectrum exhibited a high capacitance (typically greater than 10 µF) at high frequency domain.

Results and discussion

To characterise the inhibitive effect of PDTC on the copper electrode, first the gravimetric measurements were carried out. Then, the amount of corrosion products accumulated on the copper electrode was evaluated by coulometry at constant cathodic current. These experiments were followed by volatmmetric measurements to determine the effect of PDTC on cathodic and anodic reactions. Finally the impedance spectroscopy was used to estimate the time change of inhibitive effect of PDTC at different concentrations.

EQCM

The results of gravimetric measurement obtained by Electrochemical Quartz Crystal Microbalance (EQCM) are presented in Fig. 2. Two behaviours can be distinguished:

-For the blank test solution and 10 -4 mol L -1 PDTC, the mass decreased continuously with time, indicating that the dissolution of Cu from the electrode surface is the predominant effect. However, an attentive examination of this curve with enlarged scale at the very beginning of the electrode immersion, revealed a slight mass increase observed in the blank test solution as illustrated in the insert. It can be also noticed that beyond 12 hours of immersion, the slope of m with respect to time t became steeper for the blank; whereas, the rate of m change decreased in the presence of 10 -4 mol L -1 of PDTC. Hence, it can be assumed that at this concentration of PDTC, its protective effect is insufficient, and only slows down the dissolution rate.

-For higher PDTC concentration (10 -3 and 10 -2 mol L -1 ), a global mass increase was observed,and m/t approached zero more or less quickly depending on the concentration.

When 10 -2 mol L -1 PDTC was present, the mass increase was faster in an early immersion period but becomes slower after half an hour. In presence of 10 -3 mol L -1 PDTC, the mass increase was slower than the most concentrated solution but this increase lasted for longer period of time, during ca. 6 hours. Consequently, the mass increase after 24 hours of immersion was greater in 10 -3 mol L -1 solution likely due to the formation of CuCl layer with some PDTC molecules at the electrode surface.

From these curves, the rate of mass change (that may directly be associated with the corrosion rate) was estimated from linear regression calculation for early (1 to 4 hours) and the late immersion period (21 to 24 hours). The time intervals at which these calculations were performed were indicated by dotted lines.

-For low PDTC concentrations, it was postulated that the copper dissolves as cuprous ion according to the following overall reaction [29][30]:

Cu + 2 Cl - CuCl 2 -+ e - (1)
It is important to remark that both Cu 2 O and CuCl [31] layers may form upon the electrode surface as can be estimated from an early mass change in the blank test solution. It is important to emphasize that the estimation of the corrosion current density (j corr ) by m(t)

curve postulates therefore that the amount of the surface species reached its steady state value. If the accumulation of the corrosion products is taking place together with the Cu dissolution, the corrosion current density evaluated from the rate of mass change should be underestimated. This may particularly be true for a short immersion period as seen in the insert. Indeed, during the first minutes of immersion, Cu 2 O and CuCl formation seems to overcome copper dissolution.

-In presence of higher inhibitor concentration, it is assumed that the mass increase is due only to the formation of CuCl layer [29], stabilized by PDTC.

Cu + Cl - CuCl + e - (2)
The mass change is then expressed in terms of j corr by applying the Faraday law. It is worth to notice however that the presence of PDTC was neglected though some SEM pictures showed, as shown below, few crystals.

The electrode mass remains constant, in the limit of the measurement resolution, above 12 hours in presence of 10 -3 mol L -1 PDTC and one hour for 10 -2 mol L -1 PDTC. That is, neither the dissolution of Cu through CuCl 2 -nor the formation of surface film become negligible.

For the first case (Eq. 1), only Cu atoms leaving the electrode were taken into account as the origin of the mass loss, whereas for the latter (Eq. 2), only Cl forming CuCl was involved.

The Cu atoms were just transferred from the metal substrate to the surface layer, thus the mass balance is naught. Table 1 summarizes the rate of mass change and the corrosion current density tentatively estimated with the hypotheses above. It is important to notice that in this consideration, the formation of Cu 2 O attested by coulometric measurements is neglected (see

paragraph 3.3).
From this table, it can be remarked that the corrosion rate decreased with immersion period and the corrosion current density in presence of 10 -2 mol L -1 PDTC was about 60 nA cm -2 ; the inhibitive efficiency was as high as 99 %.

From EQCM measurements it can be then concluded that the action of PDTC at low contents consists only in slowing down the copper dissolution rate. However, at high concentration, PDTC has a good inhibiting efficiency due to the stabilization of the surface layers. This effect is obtained easily and quickly for 10 -2 mol L -1 but lasts 6h for 10 -3 mol L -1 . Besides, it should be noted that these results were obtained assuming that corrosion products reached their steady-states. To correct these oversimplifications, the corrosion products accumulated on the electrode surface will be determined with cathodic reduction (section 3.3).

SEM observation and EDX analysis

Surface observations and analysis were performed on copper electrode (disk made of copper rod) after 24 hours immersion in 0.5 mol L -1 NaCl containing 10 -4 or 10 -2 mol L -1 of PDTC.

SEM images show a markedly attacked surface in 0.5 mol L -1 NaCl + 10 -4 mol L -1 PDTC (Fig. 3a). Two areas can be distinguished on the surface, area covered with a dark and coarse film and bare surface with ridge of abrading. EDX analysis (Fig. 3b) indicated the presence of S and Cl atoms at the dark film suggesting that this film is likely composed mainly of CuCl and Cu-PDTC complexes. However, this film is poorly adherent and as concluded from EQCM results, it has poor barrier property to protect the copper substrate from corrosion (Fig. 2).

As for the copper immersed in 0.5 mol L -1 NaCl + 10 -2 mol L -1 PDTC, Fig. 4a shows a ridge surface by abrading and some crystals above. In contrast to the case presented above, the ridge surface is also covered by a film likely formed of PDTC with CuCl but extremely thin since EDX analysis showed the presence of S with Cu. This film is too thin to perform a quantitative EDX elemental analysis. As for the crystals, the same elements are detected but S/Cu ratio is higher indicating a higher PDTC content. (Figs 4b and4c). PDTC will interact with copper atoms to form a complex on the electrode. Then one might conclude that a barrier film is formed on copper in presence of 10 -2 mol L -1 of PDTC in agreement with the results obtained by EQCM.

Coulometry under cathodic current

In order to evaluate the species accumulated on the Cu electrode during a short period (1 h) or 24 hours of immersion, coulometric measurements were performed. Fig. 5 presents the results obtained by galvanostatic reduction of surface layers. Whatever the length of the immersion period, two potential plateaux can be seen clearly for the electrode immersed in the blank test solution:

-The first one is located at about -0.58 V SCE , and this potential is close to that reported to the reduction of cuprous oxides in absence of chloride [START_REF] Qafsaoui | [END_REF]31]. This plateau is thus attributed to the following reaction:

Cu 2 O + H 2 O + 2 e - 2 Cu + 2 OH - (3)
The open circuit potential is actually located in the domain where cuprous species are

stable.

-The second potential plateau appeared when the copper electrode was left in chloride medium, thus this plateau can be allocated to the reduction of cuprous chloride.

CuCl + e - Cu + Cl -

When films are formed in the presence of PDTC, the current plateaux corresponding to Cu 2 O

and CuCl reduction are in some cases badly separated. Besides, except for the electrode immersed in 10 -4 mol L -1 PDTC, the plateau potentials were shifted towards more negative values for both immersion times.

Table 2 recapitulates the values of the plateaux potentials and their corresponding charges. In this table, the plateau potential was determined when one half of the corresponding species was reduced. It can be seen that this potential shifts towards more negative value for both plateaux when a higher PDTC content was added, and also for the longer immersion period.

The PDTC stabilized therefore the surface species, both CuCl and Cu 2 O, making the reduction reaction more difficult hence increases the overpotential for the reduction.

In terms of electrical charges corresponding to the species reduction, it can be seen from this table that the total charge involved after 1 hour immersion showed no clear tendency. In contrast, after 24 hours immersion, the total charge decreased significantly, essentially due to the first potential arrest plateau corresponding to Cu 2 O reduction. Indeed, without or with 10 -4 mol L -1 of PDTC, Cu 2 O amount increases significantly compared to that of CuCl, particularly for 24 hours of immersion. Moreover, the amount of Cu 2 O is lower in the layers formed on copper in 10 -3 mol L -1 or 10 -2 mol L -1 of PDTC. For these high concentrations, PDTC is involved in Cu 2 O formation making this film more stable and slowing down copper oxidation.

Further exploitation of these results allows calculating the mass accumulated on the copper surface from the electrical charge. The quantities of O and Cl contributing to the formation of Cu 2 O and CuCl were then respectively determined from the charges of the first and the second plateaux. The results are shown in Table 3.

From this table, it can be seen that after 1 hour immersion, no clear effect of PDTC can be seen especially for 10 -3 and 10 -2 mol L -1 , which is in agreement with EQCM results. Indeed, as can be seen from the insert of figure 2, for the higher concentrations and after 1h immersion, films are still growing and have not yet reached their final state. On the contrary, after 24 hour immersion, the higher the PDTC concentration is, the smaller are both corrosion products (Cu 2 O and CuCl) accumulated on the copper surface with some exception.

Combination of EQCM and coulometric results

It is obvious that coulometric measurements allow evaluating only the quantities of products accumulated on copper surface, and therefore do not enable determining the corrosion rate of copper with or without PDTC. On the other hand, EQCM experiments showed a mass loss in a low PDTC concentration and the mass gain when its concentration is greater than 10 -3 mol L -1 . However, corrosion products form on copper even without or with low content of PDTC. By EQCM measurements, the masse loss (dissolution of copper) and the mass gain (formation of Cu 2 O and CuCl) are not separated. Thus, the combination of these two techniques may allow us to calculate the corrosion rate of copper at each concentration of PDTC. This approach, to our best knowledge, is the first case where EQCM and galvanostatic reduction are coupled to evaluate the corrosion current density.

The EQCM data can be split into three contributions:

m EQCM = m Cu + m O + m Cl ( 5
)
where m Cu , m O , and m Cl represent respectively the mass change induced by the dissolution of the copper (thus negative), the mass of oxygen by Cu 2 O and that by CuCl. The coulometic measurements allowed evaluating m coulom = m O + m Cl accumulated at the copper surface.

Consequently, the mass loss due to the copper dissolution alone can be calculated by the following equation:

m Cu = m EQCM -m coulom ( 6
)
m EQCM is determined on Fig. 2 and m coulom displayed in Table 3 as "m total". Table 4 displays the copper mass loss and the corrosion current density <j corr > calculated by applying the Faraday law with Cu. <j corr > represents the mean value for 1 or 24 hours immersion and may then decrease progressively with time. The values obtained are thus overestimated.

Besides, for low PDTC concentrations, the corrosion current density is positive and this approach supplies rather good results. However, negative <j corr > was obtained in high PDTC concentrations, in total contradiction to the corrosion process. This contradictory result can be explained by four manners:

-As mentioned above, the SEM pictures showed the presence of PDTC crystals, thus the EQCM experiments overestimate the amount of corrosion products at the copper surface.

-Two coppers used were slightly different in nature, thus the corrosion rate might be different between two measurements.

-The reproducibility of the experimental results is far from ideal, and thus some discrepancy is expected in these data.

-More likely, at high concentrations of PDTC, the whole surface products were not reduced entirely when the electrode potential reached the hydrogen evolution reaction domain because PDTC makes the surface species highly stable. Evaluation of the surface species accumulated on the copper surface seemed then to be too small leading to the mass gain of copper and therefore a negative value of <j corr >.

All these reasons may explain the fact that despite of the combination of EQCM and coulometric measurements, <j corr > values are not entirely reliable. However, globally, it is important to remark that the corrosion current density thus estimated decreased with time, that is, the inhibitive effect settled slowly with time.

Voltammetries

Two approaches were used by voltammetry to characterise the effect of PDTC on the corrosion of copper; a wide potential scan where its effect on the electrode kinetics can be visualised readily and a narrow potential scan to evaluate the corrosion current density j corr by Stern -Geary relationship [32].

Wide potential scan

Fig. 6 presents the polarisation curves obtained in presence of PDTC at different concentrations in 0.5 mol L -1 NaCl after immersion for 1 or 24 hours. The anodic and cathodic characteristics were collected by independent cathodic and anodic scan experiments as mentioned above.

Except for 10 -4 mol L -1 PDTC, for both anodic and cathodic branches, the higher the concentration of the inhibitor is, the smaller is the current density in the whole potential domain.

Cathodic reactions:

At high overpotentials, beyond -1.2 V, the current density decreased in absolute value, that is the reduction process is hindered in the presence of PDTC.

For less cathodic potentials, several observations can be made:

-The two current peaks can be guessed one around -0.5 V and another at -0.8 V. They may correspond to Cu 2 O and CuCl reduction respectively according to coulometric measurements. The difference is that the products are reduced, here, by cathodic potential sweep.

-These current peaks shift towards more negative direction when PDTC concentration increased because surface products are stabilized by PDTC and thus a higher overvoltage is necessary for their reduction.

-These peaks move also towards more negative direction when the immersion period increased. It is also important to remark that the charge involved in cathodic peaks is the greatest at the PDTC concentration equal to 10 -4 mol L -1 . Then it decreased significantly in the presence of this inhibitor.

-The current peaks were higher when the immersion period increased.

All of these changes are in agreement with the coulometric curves observed by the cathodic reduction of corrosion species. These observations show then that PDTC inhibit ORR on Cu in 0.5 mol L -1 NaCl and may explain our previous preliminary inhibition tests with PDTC on Al 2024 in 0.2 g L -1 NaCl [START_REF] Kendig | Nace Research in Progress Symposium, session: Coatings and Inhibitors, Corrosion/08[END_REF]. It is important to recall that no inhibitive effect of PDTC was observed on pure Al, therefore it can be concluded that PDTC interacts with Cu-rich particles to hinder ORR, and consequently the corrosion rate of the aluminium alloy.

Anodic reactions:

-In the blank solution or by addition of 10 -4 mol L -1 of PDTC, the current densities tend to a plateau at high overpotential corresponding to the reaction rate determined by a diffusion process. The anodic current peak at about 0 V could be noted for these two mediums. For the highest concentration examined, however, this peak disappears completely, and a current plateau similar to passive behaviour was observed.

-The concentration at 10 -3 mol L -1 for one hour immersion constituted a transition state.

From one experiment to another, two behaviours were recorded: either the appearance of current peak at about 0 V or its disappearance similar to the polarization curves obtained at 10 -2 mol L -1 . For the former, the corrosion potential E corr (open circuit one) is located more positive value than for the latter. Besides, it was remarked that for the former, the cathodic current density near E corr was higher and depicted a marked cathodic current peak, indicating the formation of greater corrosion products. Fig. 7 illustrates this situation. From the data shown in this figure, one can estimate that E corr differed by about 0.13 V and j corr changed more than ten times from experiment to experiment. The polarisation curve drawn in Fig. 6a corresponds to the case where E corr is more positive thus exhibiting the anodic current peak.

The interface behaviour observed with 10 -3 mol L -1 PDTC suggests that the film formed after 1 hour immersion is not always protective. Thus, one hour is a transition time for the formation of a barrier film which corroborates the results obtained by EQCM. A higher concentration, or a longer immersion time are then required to form a protective layer. When these conditions are met a large passivity-like plateau approaching, in some cases, 1 V width is obtained with low current densities. Since the open circuit potential did not change substantially, this behaviour can be explained by a marked barrier property of the surface film formed by corrosion products. This formation of barrier film can be exploited when copper is used in oxidizing conditions.

The results obtained in this section show that PDTC decreases, when added sufficiently, the current densities of both anodic and cathodic reactions. Therefore, this inhibitor will be considered as a mixed inhibitor.

Narrow potential scan

Polarization curves of narrow potential range were performed on copper for different concentrations of PDTC (Fig. 8). The sample was left in the solution during 1h or 24h and then the polarization curve was plotted ± 30 mV with respect to the open circuit potential, from cathodic to anodic potentials.

The corrosion current density can be evaluated by applying the Stern -Geary relationship [32]. The regression calculation using non-linear least-square method was applied to evaluate the corrosion parameters: j, j corr , E corr , b a and b c that are respectively the experimentally observable current density, corrosion current density, corrosion potential where the overall current is zero, the Tafel coefficient of anodic and cathodic process. The results are summarized in Table 5. Calculated polarization curves were overlaid by solid lines on experimental data presented by symbols. It can be seen that the Stern -Geary relationship suitably represents the experimental polarization curves.

The higher the inhibitor concentration is, the smaller is the corrosion current density as well for the short as for the long immersion period. For a given inhibitor concentration, the corrosion rate decreased with the immersion period, thus, as stated above, the corrosion

Electrochemical Impedance Spectroscopy

EIS measurements were carried out during 48 hours, and the impedance data were collected every hour. As for an example, Fig. 9 presents the impedance spectrum obtained after 3 hours immersion in the presence of 10 -3 mol L -1 PDTC. Though it is difficult to see the presence of a capacitive loop in high frequency domain in the Nyquist plot (Fig. 9a), the Bode plot (Fig. 9b) shows clearly the presence of a high frequency capacitive loop above several kHz. The impedance spectra obtained involves therefore three time constants.

Though the time constants are not always clearly separated as illustrated here, especially in an early immersion period in the blank test solution, all experimental spectra contained three time constants, typical for the copper or copper alloy electrode in a mildly aggressive medium in the presence of a CuCl or a patina layer [START_REF] Otmacic Curkovic | [END_REF][33][34]. Therefore, the electrical equivalent circuit illustrated in Fig. 10 was used for the parameter regression calculation using a Simplex method. This remark suggests that the addition of PDTC in the corrosion test solution does not alter the structure of the electrode surface but modifies its reactivity or the barrier property of the surface film in agreement with the polarization curves obtained with wide potential scan.

The physical nature of these elements is as follows.

 The high frequency loop R f -C f :

-R f : Film resistance representing the ionic leakage through the surface film.

-C f : Film capacitance due to the dielectric nature of the surface film.

 The medium frequency loop R t -C d :

-R t : Charge transfer resistance describing the sum of all charge transfer processes taking place at the electrode interface.

-C d : Double layer capacitance at the electronic conducting and ionic conducting interface.

 The low frequency loop R F -C F :

-R F : Faradaic resistance associated specifically with the rate of oxidation-reduction process which constitutes the difference with R t .

-C F : Faradaic capacitance due to the oxidation-reduction process. Corrosion products accumulated at the electrode surface play as reservoir of the electrical charge.

 n f , n d , n F : Cole-Cole coefficients representing the depressed feature of the capacitive loops in Nyquist diagrams [START_REF] Barsoukov | Impedance spectroscopy, Theory, experiment, and applications[END_REF][START_REF] Cole | [END_REF].

This depressed shape was expressed by the equation identical to the Cole-Cole frequency dispersion of dielectrics:

  n j 1 C R R Z       (8)
Figs 11, 12, and 13 present respectively for the highest, the medium and the lowest frequency capacitive loops, the results of a regression calculation with an in-house Simplex program based on the down-hill algorithm [START_REF] Press | Numerical Recipes[END_REF].

R f -C f couple

In previous work on the patinated and bare Cu-Sn bronze, it was concluded that the film resistance is closely related to the patina layer and not to the cuprite (Cu 2 O) layer [34,[START_REF] Muresan | [END_REF]. Thus, it is very likely that this resistance is to be associated with CuCl layer, and its barrier effect was reinforced by PDTC.

Note that R f increased with increasing PDTC concentration except for 10 -4 mol L -1 PDTC for a short immersion period. After 48 hours immersion in the solution containing 10 -2 mol L -1 PDTC, R f reached 386 k cm 2 whereas R f for the blank solution reached only 9.81  cm 2 . On the contrary, the addition of 10 -4 mol L -1 PDTC increased this resistance only slightly. This observation is in agreement with a poor protective property of surface film when PDTC concentration is low as stated above (c.f. §3.2.).

The film capacitance C f , decreased when the PDTC concentration increased, indicating thickening of the surface film (cf. Table 6). It is important to note that changes in R f and C f were slower in the presence of 10 -3 mol L -1 PDTC than those observed with 10 -2 mol L -1 in agreement with EQCM and coulometric measurements though for the former experiments the mass change was induced both by Cu 2 O and CuCl layer formation (cf. Table 3) whereas the R f -C f is associated to CuCl layer alone. Hence, by combining the results obtained by EQCM, coulometry, and EIS, one can assume that the stabilization of corrosion products by PDTC is more significant on CuCl film and seems to take more time at 10 -3 mol L -1 . By using the planar condenser model, the film thickness d was evaluated by the following equation:

f 0 C d     (9)
Here,  and  0 stand respectively for the relative dielectric constant and the dielectric constant of vacuum. The value of , was considered to be equal to 7.5 [39].

As for the comparison, the film thickness calculated from the coulometric measurements of the second plateau by galvanostatic reduction, i.e. the charge necessary to reduce the CuCl layer (cf. Table 3 indicated as the mass of Cl at the surface) is also presented in Table 6.

Cl CuCl 2 ) ( M M m Q d    ( 10
)
Where  represents the specific density of CuCl (4.15 g mL -1 ), m being the masse of Cl calculated from coulometric measurements (µg cm -2 , table 3), and M CuCl and M Cl denote respectively the molecular mass of CuCl and Cl respectively.

Note that the film thickness evaluated from C f is thinner than that estimated from the coulometric measurements for the blank and 0.1 mM PDTC whereas the opposite trend is observed for 1 and 10 mM. According to the thickness calculated from EIS measurement, it increased with PDTC concentration and immersion time. On the contrary, the coulometric measurements showed its thinning of the CuCl layer. This divergence can be explained, as it was the case for sulphate patina formed on Cu-Sn bronze [34], only the inner part of the salt layer exhibits the insulating dielectric property. This inner dielectric layer, only a few molecular layers at the initial stage without inhibitor becomes thicker with the increase of PDTC concentration and also with the immersion time. Over this layer, a thick and coarse

CuCl layer was likely formed by the corrosion process. In other words, by EIS measurements, C f takes into account the thickening of CuCl by PDTC exhibiting the barrier property. On the contrary, the thicknesses evaluated by coulometric measurements correspond to CuCl layer that can be reduced cathodically which becomes thiner with the increase of PDTC concentration. On the other hand, the thinning of CuCl observed by coulometric tests could be explained, as postulated above, by the fact that in presence of high concentration of PDTC, the reduction of CuCl layer is not complete when the hydrogen reduction reaction starts to take place.

Note also that between the blank and 10 -2 mol L -1 PDTC solution, the thickness of the dielectric layer estimated from C f increases by about 50 and 10 times respectively for 1 and 24 hours immersion, whereas the R f increased by three or four orders of magnitude. That is, the resistivity of the inner compact layer increased substantially in presence of PDTC.

R t -C d couple

As can be seen in Fig. 12a, the charge transfer resistance R t increased with immersion time. In addition, R t increased with the PDTC concentration except for the 10 -4 mol L -1 PDTC solution where no significant difference was observed as compared to the blank test. At the 48 th hour immersion time, R t was 20 times greater at the concentration of 10 -3 and 100 times as great at 10 -2 mol L -1 PDTC compared with the blank test. Therefore, the adsorption of PDTC at the Cu surface hinders strongly its electro-reactivity in NaCl medium.

C d in 10 -2 M PDTC decreased very quickly from the initial time indicating that the adsorption of PDTC at the copper surface was rather fast, whereas R t increased slowly. The same behaviour, less pronounced, is observed at 10 -3 M PDTC. Thus, the inhibitive effect of PDTC to the reaction rate evolved more slowly with immersion period. At the concentration of 10 -2 mol L -1 PDTC and at the 48 th hour of immersion, C d decreased by more than 2000 times, thus the surface area where the electrolyte was in contact with electronic conducting species becomes very small.

R F -C F couple

This couple was attributed mainly to the kinetics of the reversible redox process between Cu and Cu 2 O (Eq. 3) as can be deduced from Table 2, and Fig. 5 showing the amount of Cu The EIS data allowed us to summarize the effect of PDTC as follows: The formation of CuCl -PDTC complex improves its ionic barrier property according to R f change. In parallel, as attested by R t change, PDTC will be adsorbed directly on Cu surface making its corrosion resistance higher. Besides, PDTC make lower the electrochemical reactivity of Cu 2 O layer.

Polarization resistance R p

It was shown that in the presence of redox processes, the polarization resistance, and not the charge transfer resistance, is the most closely correlated with the corrosion rate [34,40]. Since R f is non-faradaic entity, the polarization resistance is defined as:

F t p R R R   (11)
Fig. 14 presents the variation of R p with respect to the immersion time in the presence of different concentrations of PDTC. It can be seen that R p increased with immersion time and also with PDTC concentrations as expected from the results presented above. Up to 24 hours immersion, 10 -4 mol L -1 PDTC has little inhibitive effect, but beyond this period, its protective effect appears. More precisely, this improvement takes place beyond 12 hours as we have seen through EQCM results (Fig 2 ) and the inhibiting efficiency reached 91 % at 48 hours immersion. At a PDTC concentration equal to 10 -2 mol L -1 , the inhibitive effect reached 99.8 % after 2 days immersion.

Comparison of the inhibitive effect evaluated with different methods

Various methods were applied as described above to evaluate the inhibitive effect of PDTC to the corrosion of copper in 0.5 mol L -1 NaCl solution in the presence of various concentration of PDTC. Table 7 summarizes the inhibitive efficiency calculated from different experimental methods employed in this study. For the blank test solution, the corrosion current density, instead of the inhibiting efficiency, as determined by these techniques is displayed.

By taking into account of the corrosion current density in presence of PDTC also, not presented here, it was remarked that the EQCM technique did not allowed the correct estimation of the corrosion rate j corr . The corrosion current densities were generally too small due to the over simplification of the hypotheses used. By combining EQCM and coulometric results, the corrosion current density was calculated rather correctly in absence of the inhibitor. However, the copper mass change (m(Cu), table 4) was sometimes evaluated as positive, since as assumed in section coulometry, corrosion products were not completely reduced probably because of their stabilization by PDTC. The positive mass loss leads to the calculated inhibitive efficiency greater than 100. The voltammetry and EIS supply consistent data. The protective efficiency is estimated to be greater than 99 % when 10 -2 mol L -1 PDTC was added in 0.5 mol L -1 NaCl solution.

Conclusions

This work was devoted to study the electrochemical behaviour of Cu / 0.5 mol L -1 + PDTC system to understand the interaction between PDTC and Cu-rich particles contained in some aluminium alloys. By combining different electrochemical methods, relevant results were obtained:

-PDTC is a good mixed inhibitor for Cu in 0.5 mol L -1 NaCl. PDTC acts by stabilizing the surface products that are formed on copper in chloride media i.e. CuCl and Cu 2 O. This stabilization is so strong that these products were not completely reduced under cathodic current.

-At 10 -4 mol L -1 , PDTC was found in the corrosion products (cf. Section 3.2) but this concentration is not sufficient to stabilize the products. However a slight decrease in the corrosion rate was observed after 12 h of immersion.

-At high concentrations, PDTC reduces considerably the copper surface area exposed to aggressive species and then hinders strongly the reactivity of the electrode, leading to high inhibitive efficiencies. This is more pronounced at 10 -2 mol L -1 .

-A quaint result was observed for the anodic behaviour of the copper at 10 -3 mol L -1 PDTC

showing that 1 h immersion seems to be a transition time to stabilize the corrosion products at this concentration.

These findings are highly useful for various applications of copper. Moreover, regarding our main objective and as shown in Fig. 6, the cathodic current density decreased dramatically in presence of sufficient concentration of PDTC. This compound decreases then the rate of ORR on pure copper. As ORR takes place on copper rich particles in 2xxx series aluminium alloys, this finding explains some results obtained with preliminary tests on Al [START_REF] Kendig | Nace Research in Progress Symposium, session: Coatings and Inhibitors, Corrosion/08[END_REF]. This is encouraging for further investigations.

Figure captions

e R f R t R F C f , n f C d , n d C F , n F

  Model FAS-1 or 300C. Polarization curves were plotted from two independent measurements in a new test solution for each run: one from the open circuit potential towards about -1.5 V after 1 or 24 hours immersion period, and another from the open circuit potential to about +1.5 V at the potential scan rate of 1 mV s -1 . 5 to 10 replica experiments were carried out for each experimental condition. In spite of careful experimental manipulations, the open circuit corrosion potentials were not as reproducible as expected, and changed by about ±50 mV. Therefore, instead of the mean value, the most representative one was presented in this paper.

  products covering the electrode surface protect the substrate copper. b a values decreased when the PDTC concentration increased as can be readily evaluated from the wide range polarization curves. b c , though scattered considerably, no clear tendency can be seen from these data. The value of the Stern -Geary coefficient B that links the corrosion rate and the slope of the polarization curves (R p ), B = 1/(b a -b c ) does not change significantly. Thus the mean B value of these data can be used to evaluate the corrosion rate (section 3.7) from the polarization resistance measurements R p (determined from EIS tests) with the following relationship

2 O

 2 much higher compared with CuCl. It should be noticed also that the redox potential of Cu  Cu 2 O is much less negative than that of Cu  CuCl and therefore, the reversible redox process should be more closely associated with Cu / Cu 2 O. In spite of the increase of the Cu 2 O amount with immersion period, C F decreased. This indicated that the active reactivity of Cu 2 O taking part in the redox process decreased with time. R F at 10 -2 mol L -1 PDTC increased very quickly with immersion time in agreement with the change in C d . The adsorption of PDTC likely decreased the redox reaction rate between Cu and Cu 2 O.

  39. C. Noguet, C. Schwab, C. Sennett, M. Sieskind, C. Viel, J. Phys. France 26 (1965) 317. 40. I. Epelboin, C. Gabrielli, M. Keddam, H. Takenouti, A-C impedance measurements applied to corrosion studies and corrosion rate determination, in "Electrochemical Corrosion Testing",F. Mansfeld, U. Bertocci editors, STP 727,150-192. American Society for Testing and Materials, Philadelphia, Pa (1981).

Figure 1 : 1 -

 11 Figure 1: 1-pyrrolidine dithiocarbamate (M=147.26 g mol -1 ).

Figure 2 :

 2 Figure 2: Gravimetric measurements on electrochemically deposited Cu in presence of different concentrations of PDTC (1 pyrrolidine dithiocarbamate) in 0.5 mol L -1 NaCl. The insert indicates an initial mass change in enlarged scale.

Figure 3 :

 3 Figure 3: SEM image of copper surface after 24 hours immersion in 0.5 mol L -1 NaCl + 10 -4 mol L -1 PDTC (a) and EDX analysis of the film (b).

Figure 4 :

 4 Figure 4: SEM image of copper surface after 24 hours immersion in 0.5 mol L -1 NaCl + 10 -2 mol L -1 PDTC (a) and EDX analysis of the ridge area (b) and the crystal (c).

Figure 5 :

 5 Figure 5: Galvanostatic reduction of surface layers formed on the Cu electrode in 0.5 mol L -1 NaCl in presence of PDTC at different concentrations after 1 (a) or 24 (b) hours immersion. The reduction was carried out at -50 µA cm -2 in 0.01 mol L -1 H 3 BO 3 + 0.01 mol L -1 Na 2 B 4 O 7 (pH 9) purged of dissolved oxygen by Ar.

Figure 6 :

 6 Figure 6: Effect of PDTC at different concentrations on the electrode kinetics of copper electrode in 0.5 mol L -1 NaCl solution at 20 °C.

Figure 7 :

 7 Figure 7: Two different polarization curves collected after one hour immersion in presence of 10 -3 mol L -1 PDTC; the presence or the absence of the anodic current peak at about 0 V SCE .

Figure 8 :

 8 Figure 8: The polarization curves of narrow potential range to evaluate the corrosion parameters after 1 or 24 hours immersion in 0.5 mol L -1 NaCl in presence of different PDTC concentrations at 20 °C. Stationary Cu electrode.

Figure 9 :

 9 Figure 9: An example of the impedance spectra obtained with Cu electrode in 0.5 mol L -1 NaCl solution in presence of 10 -3 mol L -1 PDTC. Symbols () are experimental

Figure 10 :

 10 Figure 10: Electrical equivalent circuit to reproduce experimental impedance spectra for Cu electrode in NaCl solution with or without PDTC.

Figure 11 :

 11 Figure 11: R f and C f (highest frequency loop) change as a function of immersion period for Cu / 0.5 mol L -1 NaCl + PDTC at different concentrations; stationary electrode at 20 °C.

Figure 12 :

 12 Figure 12: R t and C d (medium frequency loop) change as a function of immersion time for Cu / 0.5 mol L -1 NaCl + PDTC at different concentrations; stationary electrode at 20 °C.

Figure 13 :

 13 Figure 13: R F and C F (lowest frequency loop) as a function of immersion time in Cu / 0.5 mol L -1 NaCl + PDTC at different concentrations; stationary electrode at 20 °C.

Figure 14 :

 14 Figure 14: Variation of the polarization resistance R p with respect to the immersion time. Cu / 0.5 mol L -1 NaCl + PDTC at different concentrations; stationary electrode at 20 °C.

  Figures

Figure 1 : 1 -Figure 2 :Figure 3 :

 1123 Figure 1: 1-pyrrolidine dithiocarbamate (M=147.26 g mol -1 ).

Figure 4 :Figure 5 :

 45 Figure 4: SEM image of copper surface after 24 hours immersion in 0.5 mol L -1 NaCl + 10 -2 mol L -1 PDTC (a) and EDX analysis of the ridge area (b) and the crystal (c).

Figure 6 :Figure 7 :Figure 8 :Figure 9 :

 6789 Figure 6: Effect of PDTC at different concentrations on the electrode kinetics of copper electrode in 0.5 mol L -1 NaCl solution at 20 °C.

R

Figure 10 :

 10 Figure 10: Electrical equivalent circuit to reproduce experimental impedance spectra

Figure 11 :

 11 Figure 11: R f and C f (highest frequency loop) change as a function of immersion period for Cu / 0.5 mol L -1 NaCl + PDTC at different concentrations; stationary electrode at 20 °C.

Table captions Table 1 :

 captions1 Rate of mass change at the beginning (1-4 hours) and the end (21-24 hours) of immersion test in absence and in presence of PDTC at different concentrations.

Table 2 :

 2 Potentials and charges corresponding to the plateaux obtained by reduction of corrosion products formed on Cu electrode in 0.5 mol L -1 NaCl in presence of different concentrations of PDTC, and then reduced in borate buffer solution purged of dissolved oxygen at -50 µA cm -2 .

Table 3 :

 3 Quantities of O and Cl involved in Cu 2 O and CuCl formation determined from the charges Q 1 and Q 2 corresponding to Cu 2 O and CuCl reduction.

Table 4 :

 4 Copper mass loss calculated from the EQCM and coulometric measurements at 1 and 24 hours immersion at different PDTC concentrations.

Table 5 :

 5 The corrosion kinetic parameters of Cu in 0.5 mol L -1 NaCl in absence and in presence of PDTC at different concentrations for two different immersion times.

Table 6 :

 6 Film capacitance C f at 1 and 24 hour immersion and the estimation of the CuCl film thickness for Cu electrode immersed in 0.5 mol L -1 NaCl in presence of different concentrations of PDTC. d(Q 2 ) is the thickness of CuCl film calculated from coulometric results.

Table 7 :

 7 The inhibitive effect of PDTC at different concentrations determined by various techniques used in this study. For the blank test solution, the corrosion current density estimated by each method is shown.

Table 1 :

 1 5 mol L -1 NaCl + PDTC at different concentrations; stationary electrode at 20 °C.Figure 13: R F and C F (lowest frequency loop) as a function of immersion time in Cu / 0.5 mol L -1 NaCl + PDTC at different concentrations; stationary electrode at 20 °C.Figure 14: Variation of the polarization resistance R p with respect to the immersion time. Cu / 0.5 mol L -1 NaCl + PDTC at different concentrations; stationary electrode at 20 °C. Rate of mass change at the beginning (1-4 hours) and the end (21-24 hours) of immersion test in absence and in presence of PDTC at different concentrations.

	t /  cm 2	10 6		(a) R t					10 mM
	Charge transfer resistance, R	10 3 10 4 10 5	0	6	12	18	24	30	36 Blank 42 1 mM 0.1 mM	48
					Immersion time, t / h

Table 2 :

 2 Potentials and charges corresponding to the plateaux obtained by reduction of corrosion products formed on Cu electrode in 0.5 mol L -1 NaCl in presence of different concentrations of PDTC, and then reduced in borate buffer solution purged of dissolved oxygen at -50 µA cm -2 .

			First plateau	Second plateau	Total charge
			(Cu 2 O/Cu)	(CuCl/Cu)	(Cu + /Cu)
	t	PDTC	E	Q 1	E	Q 2	Q t
	(hours)	(mol L -1 )	(SCE/V)	(mC cm -2 )	(SCE/V)	(mC cm -2 )	(mC cm -2 )
	1	0	-0.575	-6.90	-0.679	-12.32	-18.9
		10 -4	-0.515	-15.68	-0.676	-5.02	-20.6
		10 -3	-0.673	-1.31	-0.736	-3.78	-4.99
		10 -2	-0.761	-8.94	-0.937	-8.54	-17.3
	24	0	-0.596	-205.5	-0.784	-26.37	-231
		10 -4	-0.523	-162.9	-0.941	-21.13	-184
		10 -3	-0.757	-17.2	-0.987	-26.16	-42.7
		10 -2	-0.935	-6.28	-1.035	-3.07	-9.27

Table 3 :

 3 Quantities of O and Cl involved in Cu 2 O and CuCl formation determined from the charges Q 1 and Q 2 corresponding to Cu 2 O and CuCl reduction.

	t	PDTC	Q 1	m(O)	Q 2	m(Cl)	m total
	(hour)	(mol L -1 )	(mC cm -2 )	(µg cm -2 )	(mC cm -2 )	(µg cm -2 )	(µg cm -2 )
		0	-6.90	0.572	-12.32	4.53	5.10
		10 -4	-15.7	1.30	-5.02	1.84	3.14
	1
		10 -3	-1.31	0.109	-3.78	1.39	1.50
		10 -2	-8.94	0.741	-8.54	3.14	3.88
		0	-206	17.0	-26.4	9.69	26.7
		10 -4	-163	13.5	-21.1	7.76	21.3
	24
		10 -3	-17.2	1.43	-26.2	9.61	11.0
		10 -2	-6.28	0.521	-3.07	1.13	1.65

Table 4 :

 4 Copper mass loss calculated from the EQCM and coulometric measurements at 1 and 24 hours immersion at different PDTC concentrations.

	t	PDTC	EQCM	Coulometry	m Cu	<j corr >
	(hour)	(mol L -1 )	(µg cm -2 )	(µg cm -2 )	(µg cm -2 )	(µA cm -2 )
		0	-5.83	5.10	-10.9	4.81
		10 -4	-9.68	3.14	-12.8	5.40
	1
		10 -3	14.3	1.50	12.8	-5.40
		10 -2	14.8	3.88	10.9	-4.60
		0	-242	26.7	-268.7	4.73
		10 -4	-150	21.3	-171.3	3.01
	24
		10 -3	92.6	11.0	81.6	-1.43
		10 -2	20.1	1.65	18.4	-0.324

Table 5 :

 5 The corrosion kinetic parameters of Cu in 0.5 mol L -1 NaCl in absence and in presence of PDTC at different concentrations for two different immersion times.

	PDTC (mol L -1 )		1 h immersion			24 h immersion
		Blank	0.1	1	10	Blank	0.1	1	10
	E corr (V / SCE)	-0.271 -0.272 -0.214 -0.225 -0.226 -0.228 -0.155 -0.179
	j corr (µA cm -2 )	6.14	5.22	1.23	0.0274	3.47	4.61	0.755	0.00433
	b a (V -1 )	31.6	30.7	20.4	15.5	32.6	20.1	17.9	16.8
	b c (V -1 )	-27.7	-18.3	-14.0	-20.6	-49.0	-5.18 -30.8	-18.2
	B (mV)	16.9	20.4	29.1	27.7	12.3	39.6	20.5	28.6

Table 6 :

 6 Film capacitance C f at 1 and 24 hour immersion and the estimation of the CuCl film thickness for Cu electrode immersed in 0.5 mol L -1 NaCl in presence of different concentrations of PDTC.

			1 h immersion			24 h immersion
	PDTC	C f		d(Q 2 )	C f		d(Q 2 )
			d (nm)			d (nm)
	(mol L -1 )	(µF cm -2 )		(nm)	(µF cm -2 )		(nm)
	0	8.65	0.77	30.5	0.731	9.1	65.2
	10 -4	9.05	0.73	12.4	56.4	0.12	52.2
	10 -3	0.324	20.5	9.35	0.085	78.3	64.7
	10 -2	0.159	41.7	21.1	0.060	110	7.6

Table 7 :

 7 The inhibitive effect of PDTC at different concentrations determined by various techniques used in this study. For the blank test solution, the corrosion current density estimated by each method is shown. d(Q 2 ) is the thickness of CuCl film calculated from coulometric results.

	ImmersionTime PDTC (mol L -1 ) EQCM EQCM+Q* Voltammetry	EIS
	Blank (µA cm -2 )	4.15	4.81	6.14	3.43
	10 -4 (%)	3.13	-12.3	15.0	-9.33
	1 hour
	10 -3 (%)	-143	212	80.0	61.7
	10 -2 (%)	88	196	99.6	96.0
	Blank (µA cm -2 )	6.08	4.73	3.47	2.98
	10 -4 (%)	75.2	36.4	-32.9	23.5
	24 hours
	10 -3 (%)	94.3	130	78.2	96.8
	10 -2 (%)	99.0	107	99.9	99.7

* j corr calculated by coupling EQCM and Coulometric methods

Acknowledgements

The University of El Jadida who authorized Pr. W. Qafsaoui to attend the LISE before the end of University year is gratefully acknowledged.


Antonio Franchi
email: antonio.franchi@laas.fr

Antonio Petitti
email: petitti@ba.issia.cnr.it

Alessandro Rizzo
email: alessandro.rizzo@nyu.edu

Distributed Estimation of the Inertial Parameters of an Unknown Load via Multi-Robot Manipulation

In this paper, we propose a distributed strategy for the estimation of the kinematic and inertial parameters of an unknown body manipulated by a team of mobile robots. We assume that each robot can measure its own velocity, as well as the contact forces exerted during the body manipulation, but neither the accelerations nor the positions of the contact points are directly accessible. Through kinematics and dynamics arguments, the relative positions of the contact points are estimated in a distributed fashion, and an observability condition is defined. Then, the inertial parameters (i.e., mass, relative position of the center of mass and moment of inertia) are estimated using distributed estimation filters and a nonlinear observer in cooperation with suitable control actions that ensure the observability of the parameters. Finally, we provide numerical simulations that corroborate our theoretical analysis.

I. INTRODUCTION

Last decades have seen raising attention to cooperative manipulation by teams of robotic agents, due to its relevant applications in several fields, such as search and rescue and disaster recovering, cooperative transportation, and service robotics. In [START_REF] Yamashita | Motion planning of multiple mobile robots for cooperative manipulation and transportation[END_REF], for example, a path planner for cooperative manipulation is proposed. It consists of two components; a global path planner, responsible of obstacle avoidance, and a local manipulation planner, responsible of the object's manipulation via the control of its position. In [START_REF] Sieber | Formation-based approach for multi-robot cooperative manipulation based on optimal control design[END_REF], the control of robotic teams with the aim of combining optimal goal regulation and relaxation of the formation rigidity constraint is carried out. Cooperative manipulation has also been dealt with in aerial applications [START_REF] Gioioso | The Flying Hand: a formation of uavs for cooperative aerial tele-manipulation[END_REF], [START_REF] Lee | Geometric control of cooperating multiple quadrotor UAVs with a suspended payload[END_REF], using cables, or other interaction tools. However, the majority of the works does not focus on an accurate dynamic modeling of the payload, which is either modeled as a point mass or as a rigid body with unknown dynamical parameters. In [START_REF] Satici | Nonholonomic cooperative manipulation of polygonal objects in the plane[END_REF], the problem of cooperatively manipulating an object on a plane by a team of non-holonomic wheeled robots is dealt with. The problem is solved by defining the dynamics of the whole system (i.e., the robot team and the object) and through a decomposition technique into a task and a null space. The dynamic model of the object is accounted for, but its inertial parameters are assumed to be known, as well as the positions of the contact points between the robots and the object.

An accurate and on-line estimation of the inertial parameters of the payload is very useful for at least two reasons: first, it allows to manipulate unknown objects with effective techniques once the inertial parameters are estimated, e.g., by exploiting force control and pose estimation algorithms [START_REF] Yong | Inertia parameter estimation of planar object in pushing operation[END_REF], [START_REF] Kubus | On-line estimation of inertial parameters using a recursive total least-squares approach[END_REF]. Second, it is very important when time-varying inertial parameters are present, such as, e.g., when part of the load on a cart is dropped or an additional load is added by an external agent. In the case of time-varying parameters, their on-line estimation will allow to implement effective adaptive control techniques, as well as to carry out fault detection (e.g., raise an alarm when part of the load is dropped during manipulation), or event-based control (e.g., change the controller's behavior when the load changes one or more of its characteristics). Moreover, since cooperative tasks exhibit great advantages when carried out via decentralized strategies [START_REF] Lynch | Distributed Algorithms[END_REF], decentralized inertial parameter's estimation algorithms are highly desirable, to maintain the features of robustness, flexibility, and fault tolerance of the cooperative manipulation system as a whole.

Some effort has been made in the estimation of inertial parameters. In [START_REF] Yong | Inertia parameter estimation of planar object in pushing operation[END_REF], an algorithm to estimate the inertial parameters of an unknown planar object is proposed. The object is pushed by robot fingers and the estimate is carried out on the basis of the fingertip velocities and accelerations, and of the exchanged forces. In [START_REF] Kubus | On-line estimation of inertial parameters using a recursive total least-squares approach[END_REF], an approach to estimate the inertial parameters of rigid loads attached to a manipulator, is proposed. The proposed algorithm estimates the inertial parameters on-line by a total least-squares approach. The cited papers rely on acceleration measurements and are based on centralized approaches.

In this paper, we consider a team of Autonomous Ground Vehicles (AGVs) that manipulate an unknown rigid body by applying forces to contact points whose relative (and, therefore, also absolute) position is not known a priori. We propose a distributed algorithm for the estimation of the relative positions of the contact points and of the inertial parameters of the rigid body based on velocity measurements only (neither absolute/relative positions nor the accelerations of the contact points are needed). This is the first step in the design and implementation of advanced cooperative control strategies, based on the knowledge of the position of the contact points and on the estimate of the dynamical model of the payload.

The paper is structured as follows. In Sec. II we formally define the model and the estimation problem. A general overview of the estimation algorithm is given in Sec. III. In Sec. IV, we deal with the distributed estimation of the relative positions of the contact points, which is instrumental to estimate the payload's inertial parameters. In Sec. V, we propose a distributed algorithm for the estimation of the inertial parameters whose main part consists of the design of a nonlinear observer to track the relative position of the center of mass. Finally, in Sec. VI we validate the proposed approach by means of numerical simulations, and draw our conclusions in Sec. VII.

II. PROBLEM STATEMENT

We consider a load modeled as a planar rigid body B, whose center of mass is denoted with C. The load is manipulated by a group of n AGVs, where each AGV can exert a force on a contact point C i , i = 1, . . . , n, of B. We assume that the number n is constant and known to all robots 1 . Consider a planar reference inertial frame W = {O W -x W y W } and denote with p C ∈ R2 the position of C in W, with p Ci the position of C i in W and with f i the force applied by the i-th robot at C i and expressed in W. Figure 1 schematizes the problem setting. We also assume that friction on the load is negligible with respect to the total force exerted by the robots 2 .

The dynamical model of the load B is therefore that of a rigid body subjected to the n forces f 1 , . . . ,

f n pC = 1 m n i=1 f i (1) ω = 1 J n i=1 (p Ci -p C ) ⊥ T f i , (2)
where m > 0 is the mass of B, ω ∈ R is its angular velocity, J > 0 is its moment of inertia, and the operator (•) ⊥ is the linear operator that, given a vector v ∈ R 2 , v = (v x v y ) T , provides the perpendicular vector to v

v ⊥ = Qv = 0 -1 1 0 =Q v x v y = -v y v x . (3)
From an actuation and sensing point of view, we assume that the i-th robot can control the exerted force f i and can measure the velocity of its contact point ṗCi . Note that we do not assume that the robot can measure the position of the contact point, nor its acceleration. 3 Furthermore, we do not assume kinematically controlled robots, i.e., robots cannot set the speed of the contact point at will.

From a communication point of view, we assume that the robotic agents can transmit data through a network modeled by an undirected graph G = (I, E), where I = {1, . . . , n}, E ⊂ I × I is a set of unordered pairs of nodes representing the one-hop communication channels between agents. The set N i = {j ∈ I : (i, j) ∈ E} indicates the communication neighborhood of agent i. We also assume that the communication graph is connected, and that the link set E does not change in time.

Problem II.1 (Inertial Parameters' Distributed Estimation). Define a distributed algorithm such that each robot i is able to compute an estimate of 1) the constant parameters m and J, and 2) the time-varying vector p Ci -p C ∈ R 2 , only controlling the local force f i (to be computed by the algorithm), and only resorting to local measurement ṗCi and to local information coming from the communication neighbors.

In this work, we are interested in the fundamental problem of letting a multi-robot system estimate the inertial parameters in a distributed way. Thus, we consider a deterministic setting, i.e., the noise affecting measurements, control inputs, and communication inputs is negligible. The noisy case will be the subject of a future work.

III. ESTIMATION ALGORITHM

Define p G = 1 n n i=1 p Ci as the geometric center of all the contact points and define also

z ij = p Ci -p Cj , z i = p Ci -p G , and z C = p G -p C . Note that p Ci -p C = z i +z C .
We first recall a simple fact.

Fact III.1. Denote with z 1 and z 2 the relative positions between two pairs of points of B expressed in W.

Then, consider two time instants t and t . The following relation, based on the rigid-body constraint, provides a straightforward way to compute z 2 (t ) from z 1 (t ), z 2 (t ), and z 1 (t ):

z 2 (t ) = Γ(z 1 (t ), z 2 (t ), z 1 (t )) = (4) = z 2 (t ) T z 1 (t ) z 1 (t ) + z 2 (t ) T z ⊥ 1 (t ) z ⊥ 1 (t ) z 1 (t )
.

The proposed distributed estimation algorithm follows a multi-step approach whose main parts are described in the next sections. Before going into the details, in order to facilitate the comprehension of the whole algorithm, we sketch all the steps in a chronological order (refer to Fig. 2 for a block diagram representation of the interconnections among the several steps).

The algorithm starts at time t = t 0 and passes through four key time instants t 0 ≤ t 1 ≤ t 2 ≤ t 3 ≤ t 4 that define when some crucial quantities become available due to the convergence time of some of the estimation steps. After t 4 , the algorithm has identified the two constant parameters m and J and can observe the time-varying parameter p Ci -p C .

Step 1: z ij (t) becomes available after t 1 .

Robot i employs the velocity measurement ṗCi and the velocities of its neighbors ṗCj , with j ∈ N i , in order to obtain, in a distributed fashion, an estimate of the relative positions z ij (t), j ∈ N i and sign(ω). This estimation process needs a short convergence time to retrieve the inter-distance between contact points. Therefore, z ij (t) will be available for any time t ≥ t 1 , where t 1 ≥ t 0 . This step is detailed in Sec. IV;

Step 2: z i (t) becomes available after t 1 . For any time t ≥ t 1 , robot i uses the relative position measurements z ij (t) (available thanks to Step 1) to compute, using the algorithm in [START_REF] Aragues | Distributed centroid estimation from noisy relative measurements[END_REF], the vector z i (t);

Step 3: ω(t) becomes available after t 1 . For any time t ≥ t 1 , robot i uses the following formula to compute, locally, the angular velocity of B:

ω(t) = sign(ω) żT ij z ⊥ ij (t) z ij 2 , (5)
where j can be any possible neighbor in N i ;

Step 4: J becomes available after t 2 . For all t ∈ (t 1 , t 2 ), each robot i applies a force f i (t) = z ⊥ i (t) (available thanks to Step 2) and observes ω(t) (available thanks to (5)). This allows robot i to reach, at t = t 2 , an estimate of the constant parameter J. Therefore, for any t ≥ t 2 , J is known by all the robots. This step is detailed in Sec. V-A;

Step 5: p Ci -p C becomes available after t 3 . For any t ∈ (t 2 , t 3 ), each robot i applies the same constant nonzero force f i (t) = f and measures ω(t). Using the nonlinear observer detailed in Sec. V-B together with the estimated value of J (known thanks to Step 4), each robot is able to obtain an estimate that eventually converges to the time-varying vector z C . For any t ≥ t 3 , each robot is then able to compute the sought vector p Ci -p C using

p Ci (t) -p C (t) = z i (t) + z C (t) = (6) = z i (t) + Γ(z C (t 3 ), z ij (t 3 ), z ij (t)),
where Γ is defined in (4) and j is any neighbor in N i ;

Step 6: ṗC (t) becomes available after t 3 . For any t ≥ t 3 , the velocity of the center of mass is computed locally by any robot i using

ṗC (t) = ṗCi (t) -ω(t)(p Ci (t) -p C (t)) ⊥ , (7)
where all the quantities in the right hand side of ( 7) are known at any t ≥ t 3 thanks to all the previous steps;

Step 7: m becomes available after t 4 . For all t ∈ (t 3 , t 4 ), each robot i applies the same constant nonzero force f i (t) = f m and measures ṗC (t) thanks to [START_REF] Kubus | On-line estimation of inertial parameters using a recursive total least-squares approach[END_REF]. This allows robot i to reach, at t = t 4 , an estimate of the constant parameter m, as detailed in Sec. V-C. Thus, at any t ≥ t 4 the mass m is known by all the robots. For t ≥ t 4 , the constant parameters m and J are known by every robot (Steps 7 and 4). Furthermore, each robot i can instantaneously compute p Ci (t)-p C (t) (Step 5). Therefore, Problem II.1 is solved.

In the following sections we detail the algorithms used in Steps 1, 4, 5, and 7.

IV. DISTRIBUTED ESTIMATION OF RELATIVE POSITIONS

OF THE CONTACT POINTS In this section, we propose an algorithm to compute the relative positions z ij , for each j ∈ N i , only resorting to local sensing and 1-hop communication.

The fact that C i and C j belong to the same rigid body constrains their inter-distance to be constant over time, i.e.,

z T ij z ij = const. (8)
Taking the time derivative of both sides of (8), we obtain a constraint on the difference between the velocities of C i and C j which, in fact, has to be perpendicular to z ij , i.e.,

żT ij z ij = 0. (9)
Noting that żij > 0 and z ij > 0, constraint (9) can be rewritten as

z ij z ij = sign(ω) ż⊥ ij ż⊥ ij i.e., z ij = sign(ω) z ij y ij , (10)
where we compactly recast

ż⊥ ij ż⊥ ij
as y ij to emphasize that quantity y ij is available to robot i resorting to sensing and one-hop communication. Thus, the constant distance between the two points z ij represents the only unknown toward the computation of the time-varying vector z ij .

Differentiating both sides of (10), we obtain

żij = sign(ω) z ij ẏij , (11)
which cannot be directly used to compute z ij , since only żij and y ij are measured, but ẏij is not. Therefore, we apply the technique described in the Appendix using żf ij and y f ij instead of żij and y ij , respectively. We observe that the sign of (11) depends on sign(ω). Therefore, in order to estimate a positive quantity, we use the squared norm of the filtered quantities. We can then summarize the previous derivations in the following result:

Proposition IV.1 (Relative position estimation from velocity measures). For each (i, j) ∈ E, if z ij > 0 and żij > 0, then z ij can be computed using only the velocities of C i and C j by means of the following estimator

z ij = żf ij 2 k 2 f y ij -y f ij 2 y ij (12)
where (•) f is a first-order low-pass filter and k f is its gain.

Finally, we obtain sign(ω) as sign żf ij T (y ij -y f ij ) .

V. ESTIMATION OF THE INERTIAL PARAMETERS

In this section we describe in detail the estimation algorithms for the constant quantities J and m, and a nonlinear observer for z C , which can be used in [START_REF] Yong | Inertia parameter estimation of planar object in pushing operation[END_REF] in order to have an estimate of the sought parameter p Ci -p C .

Since p Ci -p C = z i + z C , we rewrite (2) as

ω = 1 J n i=1 z ⊥ i T f i + 1 J z ⊥ C T n i=1 f i . (13)
Each vector z i can be computed in a distributed way by the i-th robot resorting to the distributed algorithm presented in [START_REF] Aragues | Distributed centroid estimation from noisy relative measurements[END_REF], which can be applied since the communication graph is connected by assumption.

A. Estimation of the moment of inertia J It is easy to verify that

n i=1 z ⊥ i = 0. Thus, if f i = z ⊥ i , ∀i = 1 . . . n, ∀t ≥ 0 (14)
then [START_REF] Fagiolini | On the robust synthesis of logical consensus algorithms for distributed intrusion detection[END_REF] simplifies in

ω = 1 J n i=1 z ⊥ i T z ⊥ i = 1 J n i=1 z i 2 . ( 15
)
Thus, J can be computed by applying the following distributed algorithm: 1) Distributively compute the value w = n i=1 z i 2 using an average consensus algorithm [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF], i.e., each robot sets a local state variable ξ i | t=t0 = z i 2 and applies the local update rule

ξi = j∈Ni (ξ j -ξ i ), (16)
which, since the communication graph is connected by assumption, leads ξ i to be asymptotically equal to w/n. Thus, after consensus is reached, each robot can compute w = nξ i ; 2) Each robot applies a constant force f i = z ⊥ i ; 3) Locally estimate J by using the filtering approach described in the Appendix, substituting u with w, y with ω, which is known locally thanks to (5), and θ with 1 J . Remark V.1. No time synchronization is needed for the starting time of applications of forces in [START_REF] Hermann | Nonlinear controllability and observability[END_REF]. The only requirement is that, eventually, all robots are applying the required forces.

Remark V.2. When applying the constant forces detailed in the previous algorithm, two cases can occur. The first case is that the body moves of pure rotation, therefore, the center of mass does not moves. This is, in any case, adequate to our purposes. The second case is that the body moves with constant angular acceleration. In this case, the forces can be safely applied only for a limited time, after which the movement of the body must be stopped, e.g., with a pure damping force based on velocity feedback. However, should the time be not enough for estimation purposes, the process can be repeated several times after each stop, to ensure the acquisition of the necessary measurements.

B. Observer for relative position of the Center-of-Mass z C

Denoting with f ∈ R 2 any nonzero constant vector, if

f i = f , ∀i = 1, . . . , n, ∀t ≥ 0, (17)
then [START_REF] Fagiolini | On the robust synthesis of logical consensus algorithms for distributed intrusion detection[END_REF] becomes

ω = 1 J n i=1 z ⊥ i T f + n J z ⊥ C T f = n J z ⊥ C T f , ( 18
)
where we rely, as before, on the fact that

n i=1 z ⊥ i T = 0.
Letting f = (n/J)f we have

ω = z ⊥ C T f . (19)
It is clear that, to keep the distributed character of the algorithm the value of f must be agreed on in a distributed way. This can be easily achieved by means of standard consensus algorithms [START_REF] Giannini | On the convergence of the max-consensus protocol with asynchronous updates[END_REF], [START_REF] Fagiolini | On the robust synthesis of logical consensus algorithms for distributed intrusion detection[END_REF]. Without loss of generality, up to a suitable coordinate change, we assume f = ( 0 fy ) T . Considering that the vector z C has constant length and is rigidly attached to the object, we also have

ż⊥ C = -z C ω. ( 20
)
We can write ( 19) and (20) as the following autonomous nonlinear system

         ẋ1 = -x 2 x 3 ẋ2 = x 1 x 3 ẋ3 = x 1 fy y = x 3 , (21)
where we set

z x C = x 1 , z y C = x 2 , ω = x 3 .
We observe that the only available output is y = x 3 , since the angular velocity is locally measurable by each robot using [START_REF] Satici | Nonholonomic cooperative manipulation of polygonal objects in the plane[END_REF]. Thus, the problem of estimating z C is tackled as the following nonlinear observability problem: Problem V.1. Distributively estimate z C is equivalent to observe the state of the autonomous nonlinear system (21), with output y = x 3 = ω, where fy = 0, is a given constant value.

In order to be able to design a suitable nonlinear observer we have first to study the observability of system (21). Proposition V.1. If x 3 = 0 and fy = 0, then system (21) is locally observable in the sense of [START_REF] Hermann | Nonlinear controllability and observability[END_REF].

Proof. The proof, omitted due to space constraints, follows the line of arguments in [START_REF] Hermann | Nonlinear controllability and observability[END_REF] and is based on the computation of the nonlinear observability matrix. Thus, z C is observable if and only if the angular velocity of the object and the constant force applied by each robot are both nonzero.

Since observability depends on the angular velocity ω = x 3 , it is important to analyze the trajectories of system (21). In particular, trajectories for which the angular velocity is constantly zero or asymptotically converges to zero would lead to an unobservable system. On the other hand, trajectories for which the angular velocity grows unbounded are practically unfeasible. 4The following result shows that, apart from a zero measure set, the trajectories of system (21) yield angular velocities that are suitable regarding both the observability and the practical feasibility issues.

Proposition V.2. The following facts hold for system (21):

1) the origin is a stable equilibrium point; 2) the angular velocity x 3 is bounded, and in particular:

|x 3 | ≤ x 2 3 (0) + 4 fy z C (22)
3) ∃T ≥ 0 such that x 3 (t) = 0 ∀t ≥ T if and only if the initial angular velocity x 3 (0) is such that:

x 2 3 (0) = 2 fy (x 2 (0) ± z C ). (23)
Proof. Omitted due to space constraints. It is based on Lyapunov arguments. Proposition V.2 guarantees that the angular velocity x 3 does not vanish, and therefore the system remains observable for all time, except for at most four initial condition points, which constitute a zero measure set that has probability zero to happen in real-world robotics applications. Moreover, the Proposition shows that the angular velocity x 3 and its derivatives remain bounded and gives an upper bound that can be decreased by acting on the input. This last feature is very important for the real applicability of the proposed method.

Theorem V.1. The dynamical system ẋ1 = -x 2 x 3 + fy (x 3 -x3 )

ẋ2 = x1 x 3 ẋ3 = x1 fy + k e (x 3 -x3 ), (24)
where k e > 0, is an asymptotic observer for system (21), i.e., defining x = (x 1 x2 x3 ) T and x = (x 1 x 2 x 3 ) T , x → x asymptotically, except for the zero measure initial condition set defined in (23).

Proof. Omitted due to space constraints. It is based on Lyapunov-LaSalle arguments.

Note that observer (24) can be implemented in a distributed fashion by resorting only to local information.

C. Estimation of the mass m

After having estimated J and observed z C , the quantity p iC is known by each robot i using [START_REF] Yong | Inertia parameter estimation of planar object in pushing operation[END_REF], and therefore each robot can estimate ṗC using [START_REF] Kubus | On-line estimation of inertial parameters using a recursive total least-squares approach[END_REF]. Thus, if each robot applies the same force vector as in Sec. V-B, Eq. ( 1) of the dynamical model of the payload becomes

pC = n m f , (25)
i.e., a linear system with measured output ṗC . Thus, an approach similar to the one used to estimate J, relying on the technique recalled in the Appendix, can be applied again to estimate m. We omit here the details for brevity. We remark that this solution is again distributed, since each robot needs only to receive velocity vectors from its neighbors.

VI. NUMERICAL SIMULATIONS

A network of n = 4 robots manipulates a C-shaped unknown planar object B, see Fig. 3. The object has a uniformly distributed mass m = 5 kg, and its inertia moment is J = 12.3795 kg m 2 . The robot's communication network is a line topology, i.e., E = {(1, 2), (2, 3), [START_REF] Gioioso | The Flying Hand: a formation of uavs for cooperative aerial tele-manipulation[END_REF][START_REF] Lee | Geometric control of cooperating multiple quadrotor UAVs with a suspended payload[END_REF]}.

First, each robot executes Step 1 to estimate the relative positions of its neighbors by setting a random force f i . The results of such estimates are illustrated in Figs. 4(a)-4(f). Then, once Steps 2-3 are performed, Step 4 is executed, in which each robot i sets its input force as f i = z ⊥ i in order to estimate the inertia moment J. The initial guess of J is set to 1 kg m 2 . The evolution of the estimate Ĵ is illustrated in Fig. 5(a). In Step 5, the input force is set as

f i = f = (0 f y ) T ,
for all i ∈ I, in order to estimate the vector z C . The convergence of the observer to z C is illustrated in Fig. 6(a), where the value f x = J/n is used by each agent so that fy = 1. Figures 6(b) and 6(c) show the trend of the error e and of the Lyapunov function V (e) used in the proof of the observer, respectively. Both converge quickly to zero, as predicted by our theory. At this point, each robot is able to measure the velocity of the center of mass, ṗC , in order to estimate the mass m, whose initial guess is set at m = 1 kg. Convergence of the estimated parameter is illustrated in Fig. 5(b).

VII. CONCLUSION

In this paper, we have proposed a distributed strategy for the estimation of the inertial parameters of an unknown body manipulated by a team of networked mobile robots. (f) Fig. 4: Estimated coordinates of the displacements between the contact points (continuous lines) vs. ones (dashed lines) for (a)

pC 1 -pC 2 (b) pC 2 -pC 1 (c) pC 2 -pC 3 (d) pC 3 -pC 2 (e) pC 3 -pC 4 (f) pC 4 -pC 3 .
The estimation is performed through a series of steps, that eventually yields a complete estimation. All the assumptions made are realistic. In particular, we do not assume that the robots' velocity can be controlled, yet applied forces have to be measured and controlled. Moreover, only the velocity of contact points should be measured, whereas their positions and accelerations are not needed. The proposed strategy involves low computational burden, simulation results are very satisfactory and confirm the effectiveness of our approach.

APPENDIX

We briefly recall a well known parameter estimation procedure based on filtering linear dynamics [START_REF] Slotine | Applied nonlinear control[END_REF], which is used in our algorithm during the estimation of constant parameters such as the inter-distance between contact points, the moment of inertia J and the mass m. Notably, given the linear system ẏ = θu, the problem is to estimate the unknown constant parameter θ ∈ R on the basis of measurements of the quantities u ∈ R l and y ∈ R l only, where l ≥ 1. To this aim, a low-pass filtered versions of the measured signals u and y, namely u f and y f , is computed as uf = k f (u -u f ), ẏf = k f (y -y f ), with k f > 0. Due to the linearity, ẏf = θu f holds, i.e.,

k f (y -y f ) = θu f , (26)
which relates known quantities, except for θ. We remark that (26) holds after a transient phase due to the filter initial conditions.

Fig. 1 :

 1 Fig. 1: Top view of the problem setting. A manipulation task has to be performed by a team of n = 4 AGVs endowed with a planar manipulator each. The unknown inertial parameters of the load need to be estimated using local action, sensing and communication.

Fig. 2 :

 2 Fig. 2: A block diagram showing the sequence of estimators applied in order to solve Problem II.1.

Fig. 3 :

 3 Fig. 3: The simulated payload is an eight-sides polygon, obtained from a rectangular plate of sides 2 m×4 m, where a smaller rectangular portion of 0.5 m×2.5 m sides has been cut off from the longer side. The n = 4 contact points, C1, C2, C3, C4, the position of the center of mass, C, and the position of the geometric centroid, G, are illustrated.

Fig. 5 :

 5 Fig. 5: Estimation of the inertia moment J (a) and of the mass m (b) of the unknown payload B made by the 4 robotic agents.

Fig. 6 :

 6 Fig. 6: Observation of the center-of-mass position: (a) Cartesian coordinates of the displacement zC = pG -pC and of the angular velocity ω (dashed lines) versus their real values (continuos lines), (b) estimation errors, and (c) plot of the Lyapunov function V (e).

This assumption, that is the only source of centralized information needed, can be easily relaxed by implementing one of the several algorithms for the distributed estimation of a graph size before the algorithm starts[START_REF] Tel | Introduction to Distributed Algorithms[END_REF].

This can be ensured, e.g., by endowing the load with wheels.

Measuring the absolute position would require an additional sensing system, e.g., a GPS. Measuring the acceleration, on the other hand, is typically noisy. Measuring the instantaneous velocity can be instead reliably done on board by odometry, optical flow, etc... Preprint version, final version at http://ieeexplore.ieee.org/ 2

Preprint version, final version at http://ieeexplore.ieee.org/

53rd IEEE CDC

One cannot, in this case, stop the motion and restart it, as suggested in Remark V.2, since we are estimating a time-varying quantity and therefore we cannot restart the process each time with an improved initial estimate.