Baltzer
email: baltzer@geos.unicaen.fr

B Tessier

H Nouzé

R Bates

C Moore

D Menier

Seistec Seismic Profiles: A Tool to Differentiate Gas Signatures

The presence of gas is a common feature in many seismic sections. However, the origin of the gas is often difficult to determine. Recently acquired very high resolution seismic profiles using an IKB Seistec boomer provide useful insight to the understanding of the gas origins in a range of environmental settings including sea lochs and coastal lagoons. The gas features are described both from a qualitative point of view through their acoustic facies, and quantitatively through the associated seismic signal (polarisation, amplitude). Acoustic facies include acoustic turbidity, gas "curtains" and "blankets" as well as "white fringes" and "black shadows". All the features encountered have been related to specific gas nature generated by different sources (organic matter degradation in paleo-valley infillings, waste material effluent).

Introduction :

Gas in seafloor sediments has been recognized for a number of years [START_REF] Ko | Gases in marine sediments[END_REF][START_REF] Hovland | Seabed pockmarks and seepages, impact on geology[END_REF]) and continues to be actively studied [START_REF] Garcia-Gil | Shallow gas features in incised-valley fills Ria de Vigo, NW Spain †: a case study[END_REF][START_REF] Missiaen | Very high resolution seismic mapping of shallow gas in the Belgian coastal zone[END_REF] In shallow sediments, gas occurence is often associated with an acoustic turbidity phenomenon, particularly in association with soft sediments resulting in an almost total blanking of the underlying geological structure [START_REF] Taylor | Nearshore shallow gas around the U.K. coast[END_REF][START_REF] Missiaen | Very high resolution seismic mapping of shallow gas in the Belgian coastal zone[END_REF][START_REF] Gorgas | Evidence for gassy sediments on the inner shelf of SE Korea from geoacoustic properties[END_REF].

Gas charged sediments are manifest in seismic data by an abrupt decrease of acoustic velocity combined with a strong ringing or reverberation accompanied by a decrease in the amplitude of full waveforms that is not observed in gas-free sediments [START_REF] Gorgas | Evidence for gassy sediments on the inner shelf of SE Korea from geoacoustic properties[END_REF]. The abundance of free gas bubbles in the sediment is controlled by temperature, salinity and pressure which in turn control bioactivity and chemical processes within the water and sediment column [START_REF] Wever | Variability of acoustic in Eckernforde Bay (southwest Baltic Sea) related to the annual temperature cycle[END_REF]. Two potential origins have been postulated, namely gas derived from organic material, with the biogenic process relying on bacterial activity, or gas derived from thermogenic methane at depths greater than 1000 m [START_REF] Davis | Shallow gas † an overview[END_REF].Despite these observations on the occurrence of gas, the origin of the gas remains difficult to define on seismic data.

In the framework of several projects dealing with recent nearshore sediment Holocene infillings and wedges, very high resolution seismic data have been acquired in different environmental settings (fig. 1) such as lochs (e. g. Creran and Sunart Lochs, west coast of Scotland) and lagoons (e.g. Leucate and Thau Lagoons, Mediterranean Sea). In all cases, despite differing geological and climate context, seismic signatures reflecting the occurrence of gas in shallow sediment layers have been found. Typical gas-related seismic signatures, similar to those reported in the literature, include acoustic blankets, acoustic curtains, acoustic plumes † [START_REF] Taylor | Nearshore shallow gas around the U.K. coast[END_REF][START_REF] Garcia-Gil | Shallow gas features in incised-valley fills Ria de Vigo, NW Spain †: a case study[END_REF] and acoustic turbidity [START_REF] Hovland | Seabed pockmarks and seepages, impact on geology[END_REF] (fig. 2). Gas curtains and gas blancket are characterised by a similar dark and diffuse buried facies, topped by a clear horizontal limit and abrupt lateral facies changes (fig. 2). The main difference between these two facies consists in shorter extension for the curtain facies. Two additional signatures were also recognized: the "black shadow" and the "white fringe" facies. The black shadow facies differs from the gas curtain facies as it reaches the sea floor (not buried) (fig. 2). The white fringe facies appears very different, as its top limit follows the seafloor morphology (fig. 3C).

For each case study, the gas-related acoustic facies is characterized by a particular seismic signature.

This paper presents three of these characteristic seismic gas facies, the blanket facies, the white fringe and the black shadow facies, identified from signal analysis. We show that the seismic signal shape and geometry, can provide important clues about the gas origin.

Seismic data acquisition

All the data presented in this paper have been acquired using the boomer based IKB-Seistec profiler [START_REF] Simpkin | For Seismic profiling in very shallow water, a novel receiver[END_REF]. This profiler has been designed specifically for collecting very high-resolution data in shallow water environments but can be used as well in water depth > 200m [START_REF] Simpkin | For Seismic profiling in very shallow water, a novel receiver[END_REF][START_REF] Mosher | Status and Trends of Marine High-Resolution Seismic Reflection Profiling: Data Acquisition[END_REF]. The Seistec profiler comprises a 2.5 m long catamaran supporting both the boomer source and receiver. The source is an IKB model B3 wide band electrodynamic "Boomer" producing a single positive peak pressure impulse with a peak pressure of 5 x 10 10 μPa measured on-axis and referred to a distance of 1m (214 dB // 1 μPa @ 1 m ) with a pulse duration in the range from 90 to 200 μs. This is very important as the narrower a pulse is, the broader (or wider) the band of frequencies that exist. So to be able to resolve two close spaced reflectors, it is necessary to have a narrow pulse so that the echoes separate sufficiently. Thus a narrow pulse width which possesses a wide frequency band is essential. That means it is bandwidth and not frequency that determines resolution.The receiver is a line-in-cone receiver located adjacent to the boomer plate (70 cm). The source emits useful frequencies in the range 1-10 kHz and gives a resolution down to 25 cm. Penetration is up to 100 m in soft sediments, and 200 m in deep water soft sediments. As we used the signal processed outpout from the SPA3 (the signal conditioning unit) for digitization, the high cut is reduced to 10 Khz.

During the surveys, carried out on board of small boats at a speed of about 3 knots, a SIG Energos power supply was used with a power of 50, 100 or 200 J (mainly 100 J) and a shooting rate of 2 †shots/s. The seismic data were recorded with an Elics-Delph digital acquisition system with 4000 bits per sample, a sampling frequency of 20 kHz and a recording length of 200 ms. The position was determined by a differential GPS directly connected to the Delph.

The exceptional time resolution and the fixed source-receiver geometry of the Seistec profiling system, together with its high sub-bottom penetration, allow a quantitative analysis of the different seismic signature shapes and geometries and signal amplitudes.

Seismic processing

Signal processing, including signal polarity and relative amplitude analysis were performed using the SithËre-Sisbise software, developed at IFREMER, for all the main reflectors along the seismic lines.

Seismic polarisation analysis

A reflector with a reversed polarity (or phase inversion) indicates a sediment layer of higher acoustic impedance overlying a sediment layer of lower acoustic impedance. Since acoustic impedance is a product of sound speed and density, such a reflector implies a decrease in P-wave velocity or/and in bulk density which can generally be attributed to gas occurrence.

Since the data were digitally recorded, it was possible to study the raw signal (unfiltered or stacked) and to access the original phase of the signal. Thus, signal polarisation analysis of the recorded signals was conducted for several survey cases. This analysis involved comparing the phase of the sea-floor signal with the phase of the "selected" (presumably gassy) reflector signal.

As the seafloor signal is assumed to be characterised by a "normal polarisation", due to the positive water/sediment impedance contrast (fig. 4), a phase inversion (reversed polarity) reveals a negative contrast, which in turn indicates possible presence of gas. Numerous signal analyses were conducted along the profiles but for presentation reasons we will only discuss a single shot signal (fig. 4), chosen as the most representative sampled signal for the facies.

It must be noted that the source signal could be slightly different for each shot. However, for a single shot, it is considered that the comparison of the phase between two different reflectors is valid (Nouzé and Baltzer, 2003). As the frequency band of the data is between 2.5 and 6.0 kHz , at a 20 kHz sampling frequency, this means there are about 4 data points per cycle. Thus, a visual examination of the signal polarity seems valid (even though some signals are truncated).

The Sithère software enables the selection of the seafloor reflector signal, and the gas reflector signal, along a vertical axis (a single shot) together with a visualisation of the appropriate phases (fig. 4). The signal and phase inversion can be clearly observed for each case. Amplitudes given here are normalised and thus represent a relative amplitude value. To obtain a reliable amplitude value, additional software was developed (Sisbise) providing another amplitude analysis, based on extraction of the instantaneous amplitude.

Seismic amplitude analysis

The instantaneous amplitude of the signal was analysed for selected reflections on each seismic profile. After geometrical attenuation correction, the amplitudes provide information about the relative reflectivity of the interface responsible for each reflection. The relative reflectivity is then plotted along each profile.

Along any reflecting horizon, two analyses are performed:

1/ Amplitude analysis was made by extracting the maximum value of the instantaneous amplitude of the signal for a given reflector. Low sampling frequency with respect to the central frequency of the signal might lead to underestimation of the amplitude values: peaks of the signal might not be properly sampled. Computing the instantaneous amplitude of the signal lowers the frequency content of the signal and minimizes this problem. No examples of these analysis are presented here, as we were confronted to the irregular (non continuous) aspect of the gas reflector. Thus, we preferred the second method, described below.

2/ Where gas occurs in the sediments, the signal is diffracted and there is no single reflector that can be easily followed. In such a case, in order to get a representative value of the strength of a given reflector, an average of the instantaneous amplitude over a time window was computed.

This average can be regarded as a numerical value representative of the "darkness" of the seismic section on a narrow band along the reflector. This second method offered the possibility to determine the acoustic areas characterised by a facies of acoustic turbidity (no continous top reflector) and to quantify the signal amplitude (figs. 5 & 6). Thus, amplitude values were integrated between + 0.5 ms above the selected reflector to -2 ms below it in order to obtain an integrated instantaneous amplitude value . The Leucate and Thau lagoons are part of a lagoonal system located on the Rhone Delta, Western Gulf of Lions, near the border with Spain (fig. 1). These lagoons are separated and protected from the sea by sandy barriers, or lidos, and are characterised by very shallow waters from 2 to 5 m water depth. Most lagoons of the system started to infill around 6000 years BP as sea level stabilized and the lido could settle. The infilling of the lagoons reflect the progressive closure by the barriers and is typically composed of alternative deposits of lagoonal clays and marine sands.

Case histories

Infilling sequences of

The acoustic turbidity reaches up to the sea floor but appears to be diffuse. The often sharp lateral contact with the gas-free sediments appears to correspond with a greater thickness of sediment infill of the lagoonal depression. The result is that gassy sediments predominantly occupy the deeper part of the channels. Signal analysis was unsuccessful on these data because of the diffuse aspect of the sea floor.

Loch Creran and Loch Sunart are situated on the west coast of Scotland, north of Oban (fig. 1).

They are typical of fjord style sea lochs with large water depth variations and narrow, steep sides.

The gas curtain facies observed in Loch Creran (fig. 3B) is marked by a diapiric shape. The occurrence of this facies coincides with the buried valley axis which is draped by a stratified sequence and buried below a thick transparent unit. Gas blanket facies is observed in many places at Loch Sunart (figs 3B, 4A, 6). It is similar to the gas curtain facies except that it extends laterally over a much greater distance. This acoustic facies was observed by [START_REF] Taylor | Nearshore shallow gas around the U.K. coast[END_REF] where very sharp contacts exist with the overlying sediments. Seismic polarisation analysis of the gas blanket facies in Loch Sunart shows a gas blanket with relative amplitudes varying from -500 to 1000 for the seafloor reflector (fig. 4A). Although this seafloor reflector appears discontinuous, the signal (note the phase) shows an important positive peak (amplitude of 950) between two negative peaks, which is typical of bottom reflectors (see also fig. 4C). The gas reflector shows very high amplitude values (from -1500 to 1500) and is characterised by a phase inversion confirming the presence of gas.

The signal of the gas reflector is diffracted and there is no single reflector that can be easily followed. In order to obtain a representative value of the strength of this reflector an average of the instantaneous amplitude over a time window (+ 0.5 ms above the selected reflector and 2 ms below it) was computed. Figures 5 &6 show the integrated amplitude data along two profiles in Loch Sunart and Loch Creran, revealing amplitude values for the gas reflector varying from 100 to 300. Note that in these examples seismic amplitude analysis was able to distinguish between the morainal seismic facies, characterised by high amplitude values (from 400 to 700), and the gas facies showing lower amplitude values (< 300).

White fringe facies in Loch Sunart

Seistec profiles acquired in the relatively deep parts of Loch Sunart between 40 and 50 m water depth, show a facies that is characterised by a transparent fringe overlying an acoustically turbid facies (fig. 3C). This facies is marked by a relatively constant thickness of about 1.3 m, except on its extremities . No clear reflector delineates the top of the acoustically turbid facies.

The polarisation analysis reveals a seafloor signal with relative amplitude values from -500 to 500 (fig. 4B). The continuous, dark bottom reflector corresponds to a signal with a first important negative peak, between two small positive peaks. This bottom signal appears different from the bottom signals observed in figs. 4A & 4C, which may suggest an important water content of the superficial sediments [START_REF] Nouzè | Shallow bottom-simulating reflectors on the Angola margin, in relation with gas and gas hydrate in the sediments[END_REF].

The gas reflector corresponding to the white fringe facies shows relative values ranging from -1200 to 700, with no clear phase inversion, but with a typical signature of acoustic turbidity (chaotic reflection) (fig. 4B). These relative amplitude values are two times less than the relative amplitudes observed in the gas blanket facies (fig. 4A).

No integrated instantaneous amplitude analysis was conducted in this case because of the irregular morphology of this facies which did not allow a clear definition of the time window.

Black shadow facies in Loch Creran

Loch Creran contains several fish farms that emit waste material into the loch close to the area where seismic profiles were acquired. At theses locations, a unique phenomenan is observed on the seismic profiles: a very dark facies with distinct multiple reflections (figs. 2, 4C and 5). The lateral extent of this facies is about 300 m and the lateral boundaries are abrupt.

The signal polarisation analysis shows very high relative amplitude values (from -60 000 to 60 000) with no underlying reflectors (fig. 4C). All of the signal energy seems to be reflected by the surficial sediments.

Taken from the same time window, the seismic amplitude analysis reveals extremely high values of the integrated instantaneous amplitudes ranging from 1000 to 1600 (fig. 5). Such large values are not observed in any of the previous cases, even for morainal facies or rocky outcrops.

Discussion

Acoustic turbidity, gas blanket and gas curtain facies in Lagoons of Leucate and Thau (Gulf of Lyon, France) and Lochs Creran and Sunart †(Scotland) Results of the signal analysis show that the facies of the gas blanket and gas curtain are characterised by: -a relative seafloor amplitude from -500 to 1000 with a first significant positive peak (fig. 4A) -a strong relative gas reflector amplitude from -1500 to 1500 (fig. 4A) -a phase inversion for the gas reflector (fig. 4A) -an integrated gas reflector amplitude varying from 100 to 400 (fig. 5 &6) Gas blanket and gas curtain facies were observed in Loch Creran and Loch Sunart (fig. 3B), only when a suffisant thickness of sediments is reached †above a rich organic sedimentary deposit usually in valley bottoms or depressions. Thus, blanket and curtain facies do appear only where the sediment infilling is the most important (pressure conditions), and usually not on the sides of the depressions. Similar observations were made for the acoustic turbidity facies in Thau and Leucate lagoons: acoustic turbidity facies occurs within the deeper part of the lagoons: they correspond to the thickest part of the infilling sequences.These observations agree with those made by [START_REF] Taylor | Nearshore shallow gas around the U.K. coast[END_REF] where the occurrence of gas curtains (100 to 500 m of lateral extension) and gas blankets (> 500 m of lateral extension) in the Firth of Forth was also linked to thicker accumulations of sediment in the centre of channels. But for Loch Sunart, the phase inversion observed at the top of the gas blanket seems to suggest that the quantity of gas contained in the sediments is larger compared to Taylor's observations. According to [START_REF] Taylor | Nearshore shallow gas around the U.K. coast[END_REF], the mode of formation of gas curtains could be a combination of biogenic factors (deposition of organic matter in the channels) and thermogenic factors (through structural joints and faults) as the Firth of Forth and the river Clyde are situated along a major fault.In most of our examples,Thau and Leucate lagoons, the eastern parts of Loch Creran and Sunart, the gas likely has a biogenic origin resulting from the decay of organic matter in shallow waters and confined areas. For the deeper part of Loch Sunart (still very confined and narrow sides), the seismic data do not seem to give any evidence of faulting (even on neighbour profiles) and the geological map of this area doesn't show any fault going along the lochs in these places.

Thus it could not be exluded as faults exist in all this area, but lochs are different from rivers systems, and so do not follow pre-existing fault pattern: the faults occuring in Loch Sunart and Loch Creran are faults perpendicular to the main lochs axis, and don't seem to influence the spatial repartition of the gas blanket and gas curtain facies.

It is postulated that the gas blanket at Loch Sunart is just another expression of acoustic turbidity, but here it is sealed by an impermeable layer. Most likely the acoustic turbidity, gas curtains and gas blankets observed here are a similar expression of the occurrence of biogenic gas, derived from the deposition and decay of organic material on the bottom of buried depressions or valleys.The difference between the acoustic turbidity facies on one hand and the curtain and blanket facies seems to be the amount of gas content. Gas curtains and gas blankets facies seem to be marked by a higher gas content. These facies represent the sub surface tracers, footprints of buried channels and depressions.

White fringe facies in Loch Sunart

This facies is characterised by: -a relative low amplitude of the sea floor reflector † (< 500) with a first negative peak (fig. 4B) -a relative amplitude value of the gas reflector from -1000 to 800 -a constant thickness (except at the edges) of non-gassy sediments at the top of the sedimentary column, whatever the morphology or water depth.

The white fringe facies seems to appear only in near-surface unconsolidated sediment bodies (the transparent body was sampled by a 2m length core) with high water content, such as valley deposits and drifts within the loch. This observation is in good correlation with the low amplitude of the bottom reflector and a first negative peak of the signal, typically for unconsolidated sediments. The acoustic facies appears like an "infilling" of the transparent body, but keeping a top white fringe. It is proposed that this facies is similar to the gas curtain facies but with "visible" chemical and bacterial reactions in the superficial sediments. These reactions would consume gas coming from below and thus reduce the gas proportion in the top transparent fringe, resulting in a gas content that is too low to be observed on the Seistec profiles. This assumption seems to be confirmed by the low values of the signal amplitude. Thus, this facies would be a non-mature version of the gas curtain or blanket: a curtain/blanket facies in formation within unconsolidated sediments.

Black shadow facies in Loch Creran

This facies is characterised by -a very high relative amplitude of the bottom reflector from-60 000 to 60 000 (fig. 4C) -a very high integrated instantaneous of the bottom reflector amplitude from 600 to 1800 (fig. 5) -the occurrence of several well-marked multiples (fig. 5) This facies remains restricted to the factory's vicinity. In the literature, similar strong sea floor reflectors and multiples have been related to waste material sometimes encountered in estuaries and harbours [START_REF] Missiaen | Very high resolution seismic mapping of shallow gas in the Belgian coastal zone[END_REF]McGee, 1989).

At Loch Creran, fish farm wastes are deposited in this area as confirmed by diving observations. This surface deposit generates gas directly by organic matter decomposition and this could explain the very high signal amplitudes and the ringing effect observed. McGee (1989) observed that the waterlayer multiples are decaying more slowly over the waste material than the over the adjacent sea floor. This indicates that the reflectivity of the waste is greater than that of either the natural sediments or bed outcrop. That seems to confirm what we oberved on figure 2. But also the role of air caught in these fresh deposits and the contribution of the surface rugosity could be of some importance to explain the very high amplitude of the gas signature.

It has been shown by Anderson & Hampton (1980a, b) and [START_REF] Tuffin | Temporal variability of P-Wave attenuation due to gas bubbles in a marine sediment[END_REF] that gas bubbles in sediments are resonators and the seismic image strongly depends on the frequency spectrum of the source as well as on the physical bubbles parameters and bubble density. As we compare only Seistec subbottom profiles, we avoid any frequency misinterpretation caused by using different frequencies sources. [START_REF] Wilkens | The influence of gas bubbles on sediment acoustic properties: in situ, laboratory and theoretical results from Eckernforde Bay, Baltic sea[END_REF] show that between 1.5 kHz and 25 kHz, bubble resonance greatly affects seismic velocity and attenuation with the greatest reverberation occuring when the acoustic frequency matches the resonance of the bubbles. According to this diagram the resonance frequency window from 1 kHz to 10 kHz, which corresponds to the Seistec equipment, covers bubble sizes from 0.6 mm to 6 mm, the most frequent size of gas bubbles (in fact 0.5 to 5 mm) reported in literature (Anderson & Hampton, 1980a;[START_REF] Wilkens | The influence of gas bubbles on sediment acoustic properties: in situ, laboratory and theoretical results from Eckernforde Bay, Baltic sea[END_REF]. This observation could explain why the Seistec system is so sensitive to the presence of gas.

Furthermore, the bubble characteristics (depending on the grain size of the particles) could remain a controlling factor for the appearance of the acoustic facies. This hypothesis needs to be further investigated by comparing cores sampled in different gas facies areas.

The Seistec profiles clearly indicate the gas originated from superficial sediments or from deeper layers in the subsurface. The superficial factory wastes facies (fig. 5) shows very strong integrated instantaneous amplitude values (> 1000) and a strong ringing; the succession of strong multiples appear as a black shadow on the Seistec profiles. The same observation was made for air caught in soft sands [START_REF] Baltzer | Identification des sables vacuolaires ‡ terre et en mer[END_REF]. Gas generated from deeper sediments is characterised by lower instantaneous amplitude values (< 300) and shows no ringing effects. Acoustic turbidity, gas curtains, gas blankets or white fringes facies are characterised by blank "shadows" directly underneath the gas signatures, due to the absorption of the signal.

Conclusions

The exceptional resolution of the Seistec boomer coupled with a good penetration allows a close observation of the different shapes and geometries of some typical seismic gas facies such as acoustic turbidity, gas curtains and blankets, and white fringes. Examples from abundant literature on shallow gas occurrence often remain largely based on qualitative observations. The Seistec data discussed here allowed us to present a more quantitative approach, based on signal analysis using software developed at IFREMER. These results linked to the environmental conditions could provide some clues about the gas source for the different seismic signatures observed.

The Leucate and Thau lagoons on one hand, and the Lochs Creran and Sunart † on the other hand, show infilling of valleys or depressions. Acoustic turbidity, gas curtains and gas blankets constitute a similar expression of gas occurrence most likely related to biogenic gas (methane) originating from the deposition and maturation of organic material on the bottom of the depressions or valleys. The difference between the acoustic turbidity facies and the curtain or blanket facies is that the latter are generally sealed by a top layer or cap. These seismic facies therefore seems to represent the sub-surface footprints of buried channels and depressions.

The Loch Sunart †profile shows another example of infilling in its deeper parts. Associated facies shows gas curtain characteristics marked by a white fringe underlying the surface reflector. This facies most likely corresponds to a gas curtain in formation within unconsolidated sediments.

At Loch Creran a profile shows an example of fish farm wastes. In this surface †deposit facies, gas is generated directly at the sediment surface by the organic matter decomposition. The decaying waste accounts for the very strong signal amplitude †and the ringing effect observed.

The role of air caught in these deposits and the contribution of the roughness of the deposit may play an additional role in the abnormal amplitude values of the gas signature.

The broad frequency range of the Seistec boomer has proved to be extremely sensitive for gas detection. Combined with signal amplitude analysis, this has allowed discrimination of gas origin from superficial sediments (the fish farm wastes case, amplitude > 1000) or from deeper layers in the sedimentary column (infillings of valleys, amplitude < 300).

Aknowledgements

The writers are especially grateful to the different people who participated in the Seistec surveys: Creran. This amplitude analysis is realised on a time window (between 0.5 and -2 ms) for all the shots along the profile. Note the extreme high amplitude values (from 1000 to 1800) of the organic wastes from the fish farm factory compared with the other values, including gassy sediments (around 300). Due to the scale, differentiation between the gas and moraine facies is not so clear.

Figure 6. Seismic integrated amplitude analysis (instantaneous amplitude) of Seistec data acquired in Loch Sunart. This amplitude analysis is realised on a time window (between 0.5 and -2 ms) for all the shots along the profile. The signal analysis allowed to easily differentiate morainal facies (integrated amplitude values from 600 to 700) from gas blanket facies (amplitude values from 100 to 300).

  Figures 3A & 3B show different Seistec profiles marked by characteristic facies associated with gas occurrence. These facies include the following: -acoustic turbidity facies in Leucate and Thau lagoons, -gas curtain facies in Loch Creran, -gas blanket facies in Loch Sunart.

F

  . Lelong, J.M. Rousset, P. Bretel , B. Austin, J. & J-C Rousset, R. Certain. Special thanks

Figure 1 .

 1 Figure 1. Location of the different study areas and Seistec profiles acquired in Scotland (Loch Sunart and Loch Creran) and France (Lake Leucate and Thau Lagoon)

Figure 3 .

 3 Figure 3. Different Seistec profiles showing three characteristical gas facies. 3A) Acoustic turbidity facies facies in infillings of depressions in Leucate and Thau lagoons. 3B) Gas curtain facies in Loch Creran which coincides with the deepest part of the depression, and gas blanket (similar facies to curtain but with a greater extension) in Loch Sunart. 3C) White fringe facies, revealed by two profiles acquired in Loch Sunart. This facies is characterised by a white or transparent fringe, keeping the same thickness (around 1.3 m) all along the profile whatever the water depth and topography, except at the contact with the valley edges.

Figure 4 .

 4 Figure 4. Seismic polarisation analysis of three different cases of gas facies selected on Seistec sections. Single shot signal is shown on the right. Amplitude values shown are relative amplitudes. 4A) Profile acquired in Loch Sunart presenting a typical gas blanket facies. The signal polarisation analysis shows the signature of the seafloor reflector with a positive high first peak of the amplitude (1000), and the gas reflector showing a negative peak of the amplitude (-1500) with a phase inversion. 4B) Profile acquired in Loch Sunart presenting a very particular gas facies characterised by acoustic turbidity topped by a transparent white fringe. The amplitude of the seafloor reflector is negative (-500) and the amplitude of the gas reflector varies from -1000 to 1000. The signal polarisation analysis cannot indicate a clear proof of a phase inversion. 4C) A section of profile 10 (fig.2) acquired in loch Creran corresponding to the fish factory waste site. Two multiples occur, very well marked, masking all other sub reflectors. The amplitude of the bottom reflector reaches the huge value of 60 000 !


M L Hattali
email: hattali.lamine@gmail.com

N Mesrati

D Tréheux

Electric charge trapping, residual stresses and properties of ceramics after metal/ceramics bonding

Keywords: Joining, Toughness, Residual stress, Al 2 O 3, Dielectric properties

The use of ceramic components in electrical engineering and mechanical applications is rapidly increasing. Most of these applications require the use of ceramics bonded with metal. In this paper, we have studied the role of residual stresses occurring after joining between an industrial alumina ceramic (Al 2 O 3 ) and Ni-based super-alloy, on the dielectric behaviour of ceramics. The electric charging phenomenon i.e. trapping-detrapping or diffusion of electric charges is studied by Scanning Electron Microscope Mirror Effect (SEMME) coupled with the Induced Current Method (ICM). Knowing that localized trapped charges in ceramics is a source of damage, the correlation between residual stress intensity, apparent-toughness of ceramics and ability to trap charges near the interface was demonstrated: the SEMME and ICM measurements of the quantities of trapped charges near the interface, highlighted the changes in the ceramic properties related to residual stresses due to both thermo-mechanical effect and diffusion of metallic species in the ceramics, during the bonding process.

Introduction

Ceramics are potentially interesting for industrial application under severe conditions of service. Their insulating properties, wear resistance and high melting point make them attractive. But it is sometimes difficult to use ceramic materials as general structural materials because they have some fatal weakness in impact resistance, toughness, or during manufacturing process.

However, in order to be used more effectively, ceramics can be joined to metals. In this context, joining of ceramic has been the subject of numerous researches, essentially focused on solid state bonding, [START_REF] Hattali | Calculation and experimental determinations of the residual stress distribution in alumina/Ni/alumina and alumina/Ni/nickel alloy systems[END_REF][START_REF] Hattali | Study of SiC-nickel alloy bonding for high temperature applications[END_REF][START_REF] Hattali | Effect of thermal residual stresses on the strength for both alumina/Ni/alumina and alumina/Ni/nickel alloy bimaterials[END_REF] direct copper bonding technique (DCB) [START_REF] Beraud | Study of copper-alumina bonding[END_REF][START_REF] Ning | Preoxidation of the Cu layer in direct bonding technology[END_REF] and active metal brazing, whose feasibility has been demonstrated. [START_REF] Burgess | Direct bonding of metals to ceramics by the gas-metal eutectic method[END_REF][START_REF] Sun | A new hybrid power technique utilizing a direct copper to ceramic bond[END_REF] The reader is referred to the numerous references reported in our previous papers. [START_REF] Hattali | Calculation and experimental determinations of the residual stress distribution in alumina/Ni/alumina and alumina/Ni/nickel alloy systems[END_REF][START_REF] Hattali | Study of SiC-nickel alloy bonding for high temperature applications[END_REF][START_REF] Hattali | Effect of thermal residual stresses on the strength for both alumina/Ni/alumina and alumina/Ni/nickel alloy bimaterials[END_REF][START_REF] Hattali | Interfacial behavior on Al 2 O 3 /Haynes ® 214 TM joints fabricated by solid state bonding technique with Ni or Cu-Ni-Cu interlayers[END_REF] In general, ceramics and metals are joined at high temperature. Residual stresses appear in the joint during the cooling process according to the plastic and elastic properties of both materials. These stresses influence the strength and fracture toughness of the bond. [START_REF] Hattali | Calculation and experimental determinations of the residual stress distribution in alumina/Ni/alumina and alumina/Ni/nickel alloy systems[END_REF][START_REF] Hattali | Study of SiC-nickel alloy bonding for high temperature applications[END_REF][START_REF] Hattali | Effect of thermal residual stresses on the strength for both alumina/Ni/alumina and alumina/Ni/nickel alloy bimaterials[END_REF] The stress-field depends of many factors, e.g. thermo-mechanical properties of both material (temperature dependence of mechanical characteristics, thermal expansion coefficients), thermal cycle used, geometric design [START_REF] Hattali | Calculation and experimental determinations of the residual stress distribution in alumina/Ni/alumina and alumina/Ni/nickel alloy systems[END_REF][START_REF] Hattali | Effect of thermal residual stresses on the strength for both alumina/Ni/alumina and alumina/Ni/nickel alloy bimaterials[END_REF][START_REF] Hattali | Interfacial behavior on Al 2 O 3 /Haynes ® 214 TM joints fabricated by solid state bonding technique with Ni or Cu-Ni-Cu interlayers[END_REF][START_REF] Tréheux | editors. Interfacial science in ceramic joining[END_REF] and also electrical charge generation occurring during contact, diffusion and friction. [START_REF] Fayeulle | Role of dielectric properties in the tribological behaviour of insulators[END_REF][START_REF] Guerret-Piecourt | Limitation induced by electrical charges effects on micromechanisms[END_REF] The breakdown of insulators, [START_REF] Bonnelle | Les isolants, physique de la localisation des porteurs de charge. Application aux phénomènes d'endommagement[END_REF][START_REF] Gressus | Breakdown phenomena related to trapping/detrapping processes in wide band-gap insulators[END_REF] also the tribology (friction coefficient, wear) [START_REF] Fayeulle | Role of dielectric properties in the tribological behaviour of insulators[END_REF][START_REF] Guerret-Piecourt | Limitation induced by electrical charges effects on micromechanisms[END_REF] and the fracture of ceramics [START_REF] Haddour | Relationships between microstructure, mechanical and dielectric properties of different alumina materials[END_REF] were explained on the basis of the concept of the space charge physics. [START_REF] Bonnelle | Les isolants, physique de la localisation des porteurs de charge. Application aux phénomènes d'endommagement[END_REF][START_REF] Blaise | Mise en évidence d'un claquage des isolants associé à la déstabilisation d'une charge d'espace localisée[END_REF] According to this approach and knowing that a localized trapping of charges on structural defect (point defects, dislocations, grain boundaries, and more generally interfaces) is a source of damage, the properties (toughness, friction, wear mechanical, breakdown strength) of insulating materials (ceramics, polymers, composites) can be explained in a new way. [START_REF] Bonnelle | Les isolants, physique de la localisation des porteurs de charge. Application aux phénomènes d'endommagement[END_REF] In order to understand the reasons of ceramic damage when there are joined to metal, we proposed to study the trapping and/or diffusion of charges i.e. the storage and/or dissipation of polarization energy into alumina materials, close to the metal-ceramic interface, according to the concept of the space charge physics. Previously, the residual stresses expected in Al 2 O 3 /Ni/HAYNES ® 214 TM joints were characterized mechanically and simulated using Vickers indentation method (VIF), X-ray diffraction (XRD) and Finite Element Analysis (FEA). [START_REF] Hattali | Calculation and experimental determinations of the residual stress distribution in alumina/Ni/alumina and alumina/Ni/nickel alloy systems[END_REF] It was proved that fracture strength of the joint and apparenttoughness of alumina decrease near the metal-ceramic interface due to tensile residual stresses developed close to the interface. (The term apparent-toughness has been used rather than toughness, to show that the measures correspond to the superposition of several effects.)

In this study, the Scanning Electron Microscope Mirror Effect (SEMME) [START_REF] Vallayer | Space charge measurement in a dielectric material after irradiation with a 30 kV electron beam application to single-crystal oxide trapping properties[END_REF] and Induced Current Method (ICM) [START_REF] Temga | Conduction and trapping of electric charges in an anisotropic material after irradiation with an e-beam: application to TiO 2 single crystal[END_REF] were used to determine the quantity of charges trapped in ceramics. This work was the first attempt to apply the SEMME techniques to metal-ceramic joints. The results show the correlation between ceramic apparent-toughness, intensity of residual stresses and the capacity of charges to trap near the metal-ceramic interface. Al 2 O 3 /Ni/HAYNES ® 214 TM and Al 2 O 3 /Cu/Ni/Cu/HAYNES ® 214 TM systems were considered as model for metal-ceramic joint.

Experimental procedure

Materials

As in the previous study, [START_REF] Hattali | Calculation and experimental determinations of the residual stress distribution in alumina/Ni/alumina and alumina/Ni/nickel alloy systems[END_REF][START_REF] Hattali | Effect of thermal residual stresses on the strength for both alumina/Ni/alumina and alumina/Ni/nickel alloy bimaterials[END_REF][START_REF] Hattali | Interfacial behavior on Al 2 O 3 /Haynes ® 214 TM joints fabricated by solid state bonding technique with Ni or Cu-Ni-Cu interlayers[END_REF] the ceramic material used in this work is a commercial polycrystalline ␣-Al 2 O 3 (AL23 alumina), from UMICORE Marketing services. The metal is the nickel super alloy (HAYNES ® 214 TM ), which is widely used in technological applications at high temperatures and in severe chemical atmosphere such as that encountered in industrial heating market (petrochemical, etc.). For bonding, both base material blocks were cut into small pieces with the dimension 15 mm × 5 mm × 5 mm. The pure metal foils used as filler materials were commercially available and of high purity. Both Ni and Cu foils were prepared by cold rolling down to a thickness of 200 m and 100 m respectively and had the same dimensions as the ceramics and alloy blocks (Fig. 1).

Prior to bonding, the faying surfaces of HAYNES ® 214 TM and alumina were well polished. The R a surface roughness of HAYNES ® 214 TM and alumina was equal respectively to 0.08 and 0.48 m. All surfaces of base materials were cleaned by immersion in acetone with ultrasonic agitation for 0.5 h. The Al 2 O 3 /Ni/HAYNES ® 214 TM system was fabricated by solid state bonding 1,3 at 1150 • C for 1 h. A pressure of 16 MPa was applied through a pneumatic piston. Heating speed was 150 • C/h and cooling speed was 200 • C/h. The solid state bonding was carried out in primary dynamic vacuum (10 -3 Pa) during all the thermal cycles. The optimized process was selected taking into account previous studies. [START_REF] Hattali | Calculation and experimental determinations of the residual stress distribution in alumina/Ni/alumina and alumina/Ni/nickel alloy systems[END_REF][START_REF] Lourdin | Nickel-alumina bonds: mechanical properties related to interfacial chemistry[END_REF] For the Al 2 O 3 /Cu/Ni/Cu/HAYNES ® 214 TM system, the process was as follows [START_REF] Hattali | Interfacial behavior on Al 2 O 3 /Haynes ® 214 TM joints fabricated by solid state bonding technique with Ni or Cu-Ni-Cu interlayers[END_REF] : firstly, copper foils (100 m thick) are polished with a 6 m diamond paste, and oxidized under low oxygen pressure (10 -3 Pa) at 950 • C for 0.5 h to obtain a superficial Cu 2 O film. [START_REF] Beraud | Study of copper-alumina bonding[END_REF] Cu 2 O is removed on one of the copper surfaces in contact with nickel. The Cu 2 O-Cu/Ni/Cu multilayer, was placed between alumina and HAYNES ® 214 TM substrates, for good metal-ceramic contact. Secondly, the sandwiches were heated (200 • C/h) and maintained at 1075 • C for 10 min, and then cooled to room temperature (150 • C/h). During the entire cycle a minimum low pressure of 1 MPa was applied on the sandwiches and the vacuum in the furnace was maintained at 10 -3 Pa. At 1075 • C, a eutectic liquid appears at the Cu 2 O-Cu interface; its amount can be controlled by the oxide growth conditions of the copper foil. This thin molten film produces an intimate contact between copper and alumina and generates, after cooling, a strong bond between metal and ceramic materials. [START_REF] Beraud | Study of copper-alumina bonding[END_REF][START_REF] Burgess | Direct bonding of metals to ceramics by the gas-metal eutectic method[END_REF][START_REF] Hattali | Interfacial behavior on Al 2 O 3 /Haynes ® 214 TM joints fabricated by solid state bonding technique with Ni or Cu-Ni-Cu interlayers[END_REF]

Characterization techniques

When an insulating material such as alumina ceramics is bombarded by electrons with sufficiently high energy, electrons can penetrate few atomic layers and generate excitons (electron-hole pairs). These electrons and hole can be recombined or can be trapped on structural defects. In this way, the use of Scanning Electron Microscope (SEM) is interesting to simulate the injection of charges (electrons) and then to observe the trapping of charges. [START_REF] Vallayer | Space charge measurement in a dielectric material after irradiation with a 30 kV electron beam application to single-crystal oxide trapping properties[END_REF] The Induced Current Method (ICM) [START_REF] Temga | Conduction and trapping of electric charges in an anisotropic material after irradiation with an e-beam: application to TiO 2 single crystal[END_REF] gives the dynamic aspects during the charge injection. On the other hand, the SEMME method gives the capacity of charges to be locally trapped in the insulating materials. The use of both techniques can be applied to any insulating material and allows us to determine the ability of the insulator to trap or to diffuse electric charges.

Induced Current Method (ICM)

In our study, the alumina ceramics is bombarded at 25 • C by electrons, in focused mode, at high voltage (30 KeV) in SEM (LEO 440, Electron Microscopy Ltd., Cambridge, UK) equipped with a beam blanking device to select the irradiation duration (injection time t m = 100 ms), in order to know the total quantity of injected charges (Q i = 100 pC) (Fig. 2a).

Any evolution of the total electric charges in the SEM/insulating material system involves a flow of induced charges towards the ground of the SEM. [START_REF] Temga | Conduction and trapping of electric charges in an anisotropic material after irradiation with an e-beam: application to TiO 2 single crystal[END_REF] This evolution is recorded in the form of induced current I inf . This current is measured using a pico-ammeter (Keithley type). The curve I inf = f(injection time) permits to obtain the total quantity of charges Q m , distributed in the samples (bulk and surface) during the injection:

Q m = I inf dt (1)
The typical evolution of I inf = f(t) during injection (Fig. 2b) gives information on the different steps of diffusion or trapping of charges present in the insulating sample:

-Step 1 -unsteady step: a decrease in I inf is observed as a result of the increase in the surface potential, according to trapped charge stabilized just beneath the beam impact point.

The incident electrons are slowed down and then I inf decreases continuously. -Step 2 -steady step: an asymptotic profile is observed. It corresponds to a self-regulating process between slowing of incident electrons, secondary emission, diffusion and trapping of charges. -Step 3 -after the stop of the electron injection, a depolarization-detrapping step, can be observed. This phenomenon leads to a brutal fall of the current curve and a part of the trapped charges can be brutally re-emitted.

Sometimes this phenomenon of charge relaxations can appear during the electron injection: the influence current (I inf ) falls brutally, associated with secondary emission.

Scanning Electron Microscope Mirror Effect (SEMME) method

The SEMME method [START_REF] Vallayer | Space charge measurement in a dielectric material after irradiation with a 30 kV electron beam application to single-crystal oxide trapping properties[END_REF] will be performed after the injection step. The observation at lower energy (V = 100-3000 V) of the irradiated zone makes it possible to put in evidence the mirror effect. [START_REF] Vallayer | Space charge measurement in a dielectric material after irradiation with a 30 kV electron beam application to single-crystal oxide trapping properties[END_REF][START_REF] Temga | Conduction and trapping of electric charges in an anisotropic material after irradiation with an e-beam: application to TiO 2 single crystal[END_REF] In fact negative charges Q p locally trapped and stabilized near to the injection point can induce the deflection of incident electrons: the mirror image is a view of the SEM chamber (Fig. 2c). The quantity Q p of trapped charges can be deduced from the slope of the linear part of the "mirror" curve 1/d = f(V) (Fig. 2d) according to an electrostatic law, established by Vallayer, [START_REF] Vallayer | Space charge measurement in a dielectric material after irradiation with a 30 kV electron beam application to single-crystal oxide trapping properties[END_REF] relating the real diameter d of the last output diaphragm and the apparent one, d, measured on the mirror image:

1 d = 4L d 2πε 0 (ε r + 1) Q p V ( 2
)
where L is the working distance of the SEM, fixed at 10 mm; d , the diameter of the last output diaphragm (5.3 mm); d, the apparent diameter measured on the mirror image; V, the acceleration potential of the electron beam; ε r , the permittivity of alumina. The comparison between Q m , measured by the ICM and Q p , measured by the mirror method, gives information of the charge state in the sample. In fact, if Q p /Q m = 1, the charges are stabilized and trapped near the injection point. However, if Q p /Q m tends towards 0, the charges diffuse in the sample from the injection point.

If the electron beam energy is increased, the mirror curve (Fig. 2d) departs from a straight line beyond a certain tension V lim . Several authors have shown that the mirror curve curvature is due to a non-punctual charge distribution. [START_REF] Vallayer | Space charge measurement in a dielectric material after irradiation with a 30 kV electron beam application to single-crystal oxide trapping properties[END_REF][START_REF] Temga | Conduction and trapping of electric charges in an anisotropic material after irradiation with an e-beam: application to TiO 2 single crystal[END_REF][START_REF] Ghorbel | Analytical description of mirror plot in insulating target[END_REF][START_REF] Damamme | Space charge characterization for the 21st century[END_REF] Before testing, each sample i.e. alumina ceramics, Al 2 O 3 /Ni/ HAYNES ® 214 TM or Al 2 O 3 /Cu 2 O-Cu/Ni/Cu/HAYNES ® 214 TM joint is annealed (700 • C for 3 h), in vacuum to avoid the surface contamination. The aim of this annealing is to limit, for the dielectric study, the defects (dislocations [START_REF] Hockey | Plastic deformation of aluminium oxide by indentation and abrasion[END_REF] ) and stresses produced by the polishing of the assembly. This annealing also leads to neutralization, total or partial, of the oxygen vacancies by the monovalent or divalent impurities. Four samples of alumina ceramics and multi-materials were analyzed using Induced Current Method (ICM) method and Scanning Electron Microscope Mirror Effect (SEMME) method.

Various parameters can be deduced from ICM and SEMME method (Fig. 2b andd), we selected:

-I max : initial current that informs on the response of material without any perturbation; -Q m : quantity of charges distributed in the sample (ICM); -Q p : quantity of charges locally trapped near the injection point (SEMME method).

Results and discussion

Al 2 O 3 before joining

As reference, alumina before bonding was studied. Alumina sample received a thermal cycle identical to that used for the assemblies. As reported in Fig. 3a, induced current curves I ind = f(t), measured for alumina sample before joining, present a high I max (785 ± 55 pA) and then, a strongdecrease. The presence of mirrors (Fig. 3b) confirms that the charges remain concentrated (Q p = 46.1 pC for 100 pC injected electrons and Q m = 47 ± 4 pC), moreover they are stable (tension of disappearance of the mirror for approximately 2200 V).

Al 2 O 3 after joining

The dielectric measurements take place on the free edge of ceramics, close to the interface (L1; Fig. 4). To highlight the influence of the residual stresses (primarily σ yy ) and the diffusion of metallic species on the phenomenon of trapping charges, we choose to carry out a filiation of electron injection, in staggered, from the metal-ceramic interface to ceramic bulk. The results of these injections appear in Table 1 for the Al 2 O 3 /Ni/HAYNES ® 214 TM system and in Table 2 for the Al 2 O 3 /Cu 2 O-Cu/Ni/Cu/HAYNES ® 214 TM system.

Two typical examples of I ind curves are given in Figs. 5 and6. The curve of Fig. 6a presents a very fast decrease in the current at the first moments of the injection. The post-injection observation shows a mirror effect due to a quantity of trapped charges Q p localized close to the electronic beam injection point. The asymptotic value of the current is low, corresponding to behaviour of trapping-localization of the charges. The curve of Fig. 5a is very different from the first. During the electron injection, a phenomenon of charge relaxation occurs, accompanied by a brutal emission of secondary electrons, generally due to the local saturation of the traps. After each injection, an observation with low tension (starting from 200 V) of the irradiated zone is carried out. The observation of mirrors is frequent and indicates that the charges are localized in the vicinity of the injection point. Far from the interface (>100 m), the majority of the electron injections lead to circular mirrors. In certain cases, in the vicinity of the interface (50 m), they take an elliptic form (Figs. 5b and6b), with the large axis of the ellipse forming with the interface an angle of 35-40 • . Let us note that this angle corresponds to the angle observed for the propagation of cracks (Fig. 7) related to the residual stress field close to the metal/Al 2 O 3 interface. Consequently, in the vicinity of the interface, an anisotropic trapped charge distribution is obvious. [START_REF] Temga | Conduction and trapping of electric charges in an anisotropic material after irradiation with an e-beam: application to TiO 2 single crystal[END_REF] These mirrors could be measured what made it possible to determine the quantities Q p of charges remained locally trapped after injection. The mirrors were stable until a tension from 2 to 2.4 kV.

Until a lecture potential V lim of approximately ∼600 V, the 1/d ratio follows a law of proportionality with the lecture potential (Fig. 8) and allows the calculation of the quantity Q p of trapped charges. Then, the mirror is stretched and becomes more or less deformed; the slope of the 1/d = f(V) curve decreases, indicating that the charges spread out, laterally. [START_REF] Vallayer | Space charge measurement in a dielectric material after irradiation with a 30 kV electron beam application to single-crystal oxide trapping properties[END_REF][START_REF] Temga | Conduction and trapping of electric charges in an anisotropic material after irradiation with an e-beam: application to TiO 2 single crystal[END_REF][START_REF] Ghorbel | Analytical description of mirror plot in insulating target[END_REF]

Correlation between trapping of electric charges and apparent-toughness

Fig. 9 shows the evolution of alumina toughness from the metal-ceramic interface. Apparent-toughness is very weak level near the interface and then increases to be maxima to 150-200 m. It then tends to the toughness of the starting alumina (5.8 MPa m 1/2 ), away from the interface.

Three zones can be highlighted (Fig. 9), related to trapping of charges (Tables 1 and2 relatively constant (Tables 1 and2). So Q p (not Q m ) is the quantity most appropriate for understanding the evolution of toughness.

However, a question arises: how to explain the large increase in toughness (reinforcing effects) associated exactly to the nullification of Q p (zone 2)?

The most likely hypothesis, usually advanced for this type of reinforcement, [START_REF] Simpson | Effect of microstructure on measurement of grain size dependence of fracture energy in Al 2 O 3[END_REF] is the formation of a micro-cracking due to residual stresses developed in the assembly. These microcracks are favourable to the diffusion of charges proved by the Q p decrease. [START_REF] Meyza | Thermal shock on an alumina: influence on mechanical and dielectric properties[END_REF] Beyond this zone 2, the micro-cracking disappears and the energetic potential increases gradually leading to a new decrease in toughness (zone 3). Q p , Q m and K IC tend far from the interface to their values before bonding.

Correlation between trapping of electric charges and residual stresses

The results of the quantities of influence charges Q m and charges locally trapped Q p for the Al 2 O 3 /Ni/nickel alloy system, were compared with those of the residual stresses determined by Finite Element Analysis (FEA) [START_REF] Hattali | Calculation and experimental determinations of the residual stress distribution in alumina/Ni/alumina and alumina/Ni/nickel alloy systems[END_REF][START_REF] Hattali | Effect of thermal residual stresses on the strength for both alumina/Ni/alumina and alumina/Ni/nickel alloy bimaterials[END_REF] (Fig. 10).

A strong regular decrease of tensile stresses was observed (let us recall that in fact tensile stresses are harmful) and, in parallel, Q m and Q p decrease until reaching a minimum, then increase towards their values for alumina before junction. This difference in evolution can be explained by the fact why FEA simulation does not take into account the evolution of the properties of ceramics in the vicinity of the interface, as shown previously (Fig. 9).

Otherwise, Vickers indentation technique makes possible to consider the residual stresses in the same area as that corresponding to dielectric measurements. [START_REF] Hattali | Effect of thermal residual stresses on the strength for both alumina/Ni/alumina and alumina/Ni/nickel alloy bimaterials[END_REF][START_REF] Hattali | Interfacial behavior on Al 2 O 3 /Haynes ® 214 TM joints fabricated by solid state bonding technique with Ni or Cu-Ni-Cu interlayers[END_REF] For the two systems, the evolution of the quantity of charges Q m vs the distance to the interface (Fig. 11 and Tables 1 and2) shows:

-A similar evolution for Q m and tensile residual stresses measured by Vickers indentation. -The quantities of charges (Q m or Q p ) stored in alumina are very strongly increased close to the interface (Tables 1 and2). -The strong fall of Q m and Q p , below approximately 100 m far from the interface, is related to the fall of stresses observed by indentation. -For the Al 2 O 3 /Ni/nickel alloy system, Q m becomes again higher at long distance (1 mm) while tending towards the value of alumina, before bonding. This tendency is slower for the system Al 2 O 3 /Cu 2 O-Cu/Ni/Cu/HAYNES ® 214 TM . -The Q p /Q m ratio, close to 1 for alumina before bonding, is low or zero in areas of high tensile stresses (Tables 1 and2). This shows that tensile stresses favour the mobility of the charges starting from the injection point of electrons.

These results are to be compared to those obtained by in situ electron injection on alumina subjected to bending stresses [START_REF] Juvé | CSC'3. Le Vide: science[END_REF] : in fact mechanical load influences the capacity of material to trap and/or to diffuse electric charges. The interpretation which can be given was developed in [24]. Under mechanical loading, the distortions of the lattice can allow a trapping of charges leading to storage of polarization energy. This energy can be sufficient to create dislocations which appear in the alumina lattice, thus creating new sites of trapping. [START_REF] Zong | Model for radiation induced degradation of ␣Al 2 O 3 crystals[END_REF] These created dislocations are mobile and subjected to an electric field. Consequently, an increase in the trapping of the charges occurs but these charges do not remain localized (mirror not very stable and Q p /Q m < 1).

Let us note that these measurements are sensitive as well to the stress field as with the effect of diffusion of nickel (Ni) or copper (Cu) and then, these two factors favour the trapping and/or the mobility of the charges. Lastly, according to our results it is obvious that the Al 2 O 3 /Cu/Ni/Cu/HAYNES ® 214 TM system is less stressed than the Al 2 O 3 /Ni/HAYNES ® 214 TM system (weaker Q m ), which induces mechanical properties of the assembly more favourable (shearing, bending, interfacial fracture energy, toughness). [START_REF] Hattali | Effect of thermal residual stresses on the strength for both alumina/Ni/alumina and alumina/Ni/nickel alloy bimaterials[END_REF][START_REF] Hattali | Interfacial behavior on Al 2 O 3 /Haynes ® 214 TM joints fabricated by solid state bonding technique with Ni or Cu-Ni-Cu interlayers[END_REF]

Conclusions

The two complementary techniques: Induced current Method (ICM) and Scanning Electron Microscope Mirror Effect (SEMME) method are the non-destructive technique developed for insulator characterization for the study of charging properties of ceramics. Knowing that localized trapped charges in ceramics is a source of damage, the correlation between residual stress intensity, ceramic apparent-toughness and ability to trap charges near the interface was demonstrated. In summary, the following points can be retained:

1. A residual stress field appears close to the metal-ceramic interface, in particular, due to thermal expansion mismatch between metal and ceramics. The comparison between the various methods (XRD, 1,8 FEA, [START_REF] Hattali | Calculation and experimental determinations of the residual stress distribution in alumina/Ni/alumina and alumina/Ni/nickel alloy systems[END_REF] and VIF [START_REF] Hattali | Effect of thermal residual stresses on the strength for both alumina/Ni/alumina and alumina/Ni/nickel alloy bimaterials[END_REF][START_REF] Hattali | Interfacial behavior on Al 2 O 3 /Haynes ® 214 TM joints fabricated by solid state bonding technique with Ni or Cu-Ni-Cu interlayers[END_REF] ) and estimations of the stress field in ceramics, reveals a global tendency, to pass from traction (near interface) towards compression (medium of ceramics), before cancelling itself (far from the interface); but only VIF and SEMME, by their good spatial resolution, demonstrate wide variations of stresses in sub-layer. 2. It is confirmed that the mechanical and dielectric properties of ceramics are significantly altered in the vicinity of metal-ceramic interface. These changes, due to the process, are related both to thermo-mechanical stresses and diffusion in ceramics of metallic species. 3. The results of Vickers indentation mainly revealed us that the apparent-toughness of ceramics falls in the vicinity of the metal-ceramic interface what leads to the embrittlement of ceramics. This fall can be explained by the existence of the stress field and/or the diffusion of the metals (Ni, Cu) in alumina. Indeed, the developed stresses will create a distortion of the crystal lattice, being able to play the role of sites of charge trapping, then creation of dislocations. 12,25 4. The mirror method made it possible to highlight the role played by the residual stresses in diffusion of electric charges in metal-ceramic assemblies. The residual stress field close to the interface modifies significantly the response of ceramics to the charge injection. 5. Toughness and breakdown being dependent, [START_REF] Haddour | Relationships between microstructure, mechanical and dielectric properties of different alumina materials[END_REF][START_REF] Fothergill | Filamentary electromechanical breakdown[END_REF] a fall of resistance to the breakdown of ceramics in the vicinity of the junction can be expected. 6. Finally, we show that SEMME characterization can be a good method for study of residual stress and changes in properties of materials developed in metal/ceramic joints. However, for a careful estimation of the residual stresses alone, SEMME method and ICM should be calibrated using previously described protocols [START_REF] Juvé | CSC'3. Le Vide: science[END_REF] to avoid the superposition of several phenomena (diffusion, stresses, micro-cracking, etc.).

Fig. 1 .

 1 Fig. 1. Schematic illustration of a Al 2 O 3 /HAYNES ® 214 TM joint (a) with Ni interlayer; (b) with Cu 2 O-Cu/Ni/Cu interlayer.

Fig. 2 .

 2 Fig. 2. (a) Schematic experimental set-up for measuring the induced and surface currents; (b) schematic ground current curve; (c) mirror image; (d) mirror curve 1/d = f(V).

Fig. 3 .

 3 Fig. 3. (a) Evolution of induced currents I ind during electron irradiation for Al 2 O 3 ceramics before joining; (b) the mirror image of Al 2 O 3 ceramics, before joining.

Fig. 4 .

 4 Fig. 4. Schematic representation of the specimen for dielectric measurements.

Fig. 5 .

 5 Fig. 5. (a) Typical I ind (t) plots for injections at the Ni/Al 2 O 3 interface: behaviour of periodic secondary electronic relaxation on the level of the interface. (b) The corresponding mirror image (Al 2 O 3 /Ni/HAYNES ® 214 TM system).

  Fig. 6. (a) Typical I ind (t) plots for injections at ∼50 m from the Ni/Al 2 O 3 interface: behaviour of trapping-localization of charges; (b) the corresponding mirror image (Al 2 O 3 /Ni/HAYNES ® 214 TM system).

Fig. 7 .

 7 Fig. 7. Crack propagation in alumina ceramic in the Al 2 O 3 /Ni/HAYNES ® 214 TM system after joining process. It will be noted that the angle of the crack with the interface is close (35-45 • ) to that observed for the elliptic mirror, following in that the stress field lines. 1,3 (a) Optical image of polished cross-section perpendicular to the interfaces; (b) typical elliptic mirror observed after injection near to the interface (∼<50 m).

Fig. 8 .

 8 Fig. 8. Evolution of the ratio 1/d ratio vs lecture potential, after 100 pC injected electrons on the surface of alumina near the interface (∼90 m) for the Al 2 O 3 /Ni/nickel alloy system.

Fig. 10 .

 10 Fig. 10. (a) Evolution of the quantities of charges Q m and Q p and normal stress σ yy measured on the free edge (L1), vs distance y from the metal-ceramic interface. (b) Residual stresses determined by Finite Element Analysis (FEA) 1,3 (Al 2 O 3 /Ni/HAYNES ® 214 TM system).

Fig. 11 .

 11 Fig. 11. Comparing the quantity of influence charges Q m and residual stress estimated by Vickers indentation in both types of assemblies: (a) Al 2 O 3 /Ni/HAYNES ® 214 TM , (b) Al 2 O 3 /Cu 2 O-Cu/Ni/Cu/HAYNES ® 214 TM .

Table 1

 1 Characteristic parameters deduced from ICM for Al 2 O 3 /Ni/HAYNES ® 214 TM joint.

	Distance from interface (m)	I max (pA)	Q m (pC)	Q p (pC)	Q p /Q m	Relaxations
	Before joining	942.187	47.65	46.1	0.967	No
	After joining
	0	879.37	69.7	35.5	0.509	Yes
	50	750.34	26.16	2.96	0.113	No
	100	832.44	29.9	0	-	-
	140	856.34	26.8	0	-	-
	250	829.43	26.1	6.9	0.264	No
	500	789.34	21.2	10.5	0.495	No
	1000	709.4	38.9	15.2	0.391	No
	Q p = 0 corresponds to the absence of mirror image due to the diffusion of charges.
	Table 2
	Characteristic parameters deduced from ICM for Al 2 O 3 /Cu 2 O-Cu/Ni/Cu/HAYNES ® 214 TM joint.
	Distance from interface (m)	I max (pA)	Q m (pC)	Q p (pC)	Q p /Q m	Relaxations
	Before joining	942.187	47.65	46.1	0.967	No
	After joining
	20	822.43	58.2	37.5	0.644	Yes
	45	788.33	29.12	0	-	-
	80	790.45	28.3	0	-	-
	120	844.21	26.1	0	-	-
	250	756.21	26.4	12.3	0.465	No
	1000	734.12	22.1	14.45	0.653	No
	Q

p = 0 corresponds to the absence of mirror image due to the diffusion of charges.

Acknowledgements

Special thanks are due to the IFP Energies nouvelles, for providing the financial resources. The authors thank Denyse Juvé from École Centrale de Lyon for technical collaboration for the SEM equipment and for helpful discussions.


Anh Duc
email: duc-anh.ta@hds.utc.fr

Ta

Isabelle Fantoni
email: isabelle.fantoni@hds.utc.fr

Rogelio Lozano
email: rogelio.lozano@hds.utc.fr

Modeling and Control of a Convertible Mini-UAV

Keywords: Convertible Plane, Modelling, Control, Stabilization and Autonomous Transition

The aim of this paper is to present a UAV's configuration that combines the manoeuvrability of a rotary wing vehicle (helicopter) such as slow forward displacement, vertical take off and landing, and the performance of a fixed wing vehicle (plane) such as fast forward movement, long reach and superior endurance.

INTRODUCTION

In recent years, the interest for Unmanned Aerial Vehicle (UAV) has steadily increased. Aircrafts have been used at first mainly in the military, in reconnaissance, surveillance and also in offensive missions. Nowadays, we can find civilian applications such as traffic monitoring, surveillance and environmental protection, the search and rescue of injured persons or the management of major infrastructure. The purpose of this work is to develop a miniature aircraft which has the performance of a plane for the horizontal flight and the manoeuvrability of a helicopter for the vertical flight. In the past, this research has generated a considerable interest because this type of aircraft doesn't require a runway and the ability of hover makes it very useful for aerial surveillance missions. The most common configuration of convertible aircrafts is the Tail-sitter thanks to its ability of vertical takeoff and landing and the ability of horizontal flight like a conventional plane. Among convertible UAVs in the past, we have been interested in the Heliwing of Boeing and the T-Wing in [START_REF] Stone | Control architecture for a tail-sitter unmanned air vehicle[END_REF] of the University of Sydney. The Heliwing was developed in 1995, the T-wing in 2003 and they are both Tail-sitters. Using linear control techniques to stabilize and control the linearized dynamics, the T-Wing successfully performed the transition from vertical to horizontal and vice versa. However, the Heliwing was lost in an accident during its first transition from the horizontal flight to the vertical flight. Another interesting configuration found in [START_REF] Green | A Hybrid MAV for Ingress and Egress of Urban Environments[END_REF] which presents a small conventional plane with a capacity of hovering can make the transition from horizontal to vertical. However, this configuration presents a problem related to gyroscopic effect and then the yaw control is not assured when the plane flies vertically. In addition, this configuration can not take off and land vertically. Other simulation results for the transition of a miniature convertible tailsitter UAV using an adaptive quaternion controller and a quaternion gain sequencing feedback controller are presented in [START_REF] Knoebel | Adaptive Quaternion Control of a Miniature Tailsitter UAV[END_REF] and [START_REF] Bataillé | Development of a VTOL Mini UAV for Multi-Tasking Missions[END_REF]. However these con- This paper contributes to a proposed configuration of the convertible aircraft capable of performing the hovering, transition, vertical take off and landing (figure 1). The detailed mathematical model is obtained by the equations of Newton-Euler and the aerodynamic formulas. In terms of control, we propose a control strategy using the quaternion in order to avoid the singularity problem when the plane is vertical and a motion profile generator to realize the phases of take-off, landing and transition. The paper is organized in five sections. The second section is devoted to the representation of the attitude of the plane. The mechanical structure and the aerodynamic model of convertible aircraft will be presented in the third section. The fourth section contains our contributions, namely the presentation of a control strategy to achieve a longitudinal flight path. Simulation and experimental results are presented in the fifth section. Finally, the paper ends with some conclusions.

REPRESENTATION OF THE ATTITUDE

In our implementation, the quaternion was chosen for the presentation of the attitude as it presents no problem of singularity with respect to Euler angles when the aircraft is vertical (the pitch angle is equal to -π/2). The unit quaternion is composed of a unit vector e, called the Euler axis and a rotation angle β around this axis:

q = [ cos β 2 e sin β 2 ] T = [ q 0 q ] T (1)
When the system of coordinates E b attached to the rigid body rotates relatively to the inertial coordinate E f with an angular velocity ω = [ ω x ω y ω z ] T , kinematic equations in terms of quaternion are given by: q = ( q0 q ) = 1 2

( -q T I 3 q 0 + [ q × ] ) ω = 1 2 Ξ(q)ω (2)
In the attitude control, note that q d is the desired attitude and q the real attitude of the rigid body, the error attitude is represented by: q e = q -1 d ⊗ q

(3) where ⊗ is the multiplication of the quaternion and the conjugate quaternion q -1 is the complement rotation of quaternion q. In the case where the attitude error is zero, the error quaternion has two possible values: q e = [ ±1 0 T ] T . This is due to the non bijection of the quaternion with the group SO(3).

CONVERTIBLE PLANE

In the configuration shown in figure 1, two counter rotating rotors are placed side by side in the vehicle and inside the wing. The aircraft was properly calibrated so that the center of gravity coincides with the aerodynamic center of the wing and it is at a quarter of the wing's cord. From the model of the convertible plane described above and the kinematic equations (2), the Newton-Euler equations of motion describing the six degrees of freedom of the system can be separated in translational motion (Σ P ) and rotational motion (Σ A ):

Σ P : { ṗf = Rv b vb = R T g f + 1 m F b A,T -ω × v b (4) Σ A : { q = 1 2 Ξ(q)ω J ω = -ω × Jω + Γ A,T (5)
where: q = [ q 0 q ] T is the quaternion vector; R is the

rotation matrix from E b to E f ; p f represents the position of the plane in E f ; v b is the linear velocity vector presented in E b ; m = diag(m) ∈ R 3×3 where m (≃ 900g
) is the weight of the plane; g f is the vector of the gravity acceleration in E f ; ω is the angular velocity vector; J ∈ R 3×3 represents the inertial matrix of the plane;

F b A,T = F b A + F b
T is the control vector of the aerodynamic forces and the thrusts expressed in the coordinate E b (see figures 2 and 3); Γ A,T = Γ A + Γ T is the control torque vector. The forces F b T and the torques Γ T generated by the actuators are identical in both the horizontal and vertical flight. As shown in figures 2 and 3, the left motor rotates in the clockwise direction while the right rotates in anticlockwise direction. Each motor produces a force T i parallel to its axis of rotation and a reactive torque Q i opposite to the direction of rotation. The combination of the forces T i and the reactive torques Q i is given by: Since the rotor speed reaches very high values (more than 200 rad/sec), we can approximate the forces T i and the reactive torques Q i generated by the motors:

{ F b T = [ T 1 + T 2 0 0 ] T Γ T = [ Q 2 -Q 1 0 (T 2 -T 1 )l m ] T (6)
T i = b i s 2 i Q i = k i s 2 i (7)
where s i is the rotational speed of rotor; b i and k i are two positive parameters depending on the density of air, the radius, the shape, the pitch angle of the blade and other factors [START_REF] Fay | Derivation of the aerodynamic forces for the mesicopter simulation[END_REF]).

The aerodynamic forces F b A and torques Γ A depend on the working mode of the plane. In the helicopter mode (see figure 2), the aerodynamic forces and torques are simply calculated:

{ F b A = [ 0 -P s r P s al + P s ar ] T ≈ [ 0 0 0 ] T Γ A = [ (P s al -P s ar ) l m (P s al + P s ar ) l s P s r l s ] T ( 8
)
where

P s i = 1 2 ρv 2 in S s i C P δ i
with i = al, ar, r (left aileron, right aileron, and rudder); ρ is the density of air; C P δ i = C P i δ δ i ; δ i and S i are the deflection angle and the area of control surface; v in = √ 2Ti ρAprop is the air inflow velocity created by the propeller [START_REF] Fay | Derivation of the aerodynamic forces for the mesicopter simulation[END_REF]) and A prop = πR 2 prop where R prop is the propeller's radius.

In horizontal flight, we assume that there is no wind. So the air velocity is equal to the relative velocity of the UAV, but in the opposite direction : V air = -V . The vector of the air velocity creates with the plane an angle of attack α (figure 3). In this mode, the aerodynamic forces and torques are given by:

{ F b A = [ -T b -P s r P b + P s ar + P s al ] T ≈ [ -T b 0 P b ] T Γ A = [ (P s al -P s ar ) l m (P s al + P s ar ) l s P s r l s ] T (9)
The longitudinal dynamic model of the wing is shown in figure 4, the aerodynamic lift force P b and drag force T b in the frame E b are calculated from:

P b = P cos α + T sin α T b = -P sin α + T cos α (10)
where: α is the angle of attack, P , T are calculated from the aerodynamic formulas in table 1 (see [START_REF] Mc Cormick | Aerodynamics Aeronautics and Flight Mechanics[END_REF]), S is the wing's area, the functions f P (α) and f T (α) are the nonlinear functions of the lift and drag coefficients depending on the angle of attack and the wing's profile (figures 5(a) and 5(b)) and C T0 is the constant coefficient of the parasite drag.

C P = f P (α) T = 1 2 ρV 2 SC T C T = C Tα + C T 0 C Tα = f T (α)

CONTROL STRATEGY

Control Law

Altitude Control.

The control of the altitude is used when the plane flies vertically but it is not used in the transition and horizontal flight. Extracting the translational motion on the z axis of E f in equation ( 4) yields:

mz = T cos ( π 2 -θ ) cos ϕ -mg ( 11
)
where T is the total thrust force of both motors. A simple PD control can guarantee the convergence of the altitude to the desired altitude z d :

T = T 1 + T 2 = -k p (z -z d ) -k d ż + mg cos ( π 2 -θ ) cos ϕ (12)
But for security reasons and in order to improve the performance, a PID control with saturation is used:

T = sat T ( -k p (z -z d ) -k d ż -k i ∫ (z -z d ) ) + mg cos ( π 2 -θ ) cos ϕ (13) where sat M (x) = {x if |x| M ; M sign(x) if |x| > M }
is the classical saturation function, M is the boundary saturation and sign(x) is the signum function defined as sign

(x) = {1 if x 0; -1 if x < 0}.
Attitude Control. In order to control the attitude and to avoid the singularity when the plane flies vertically, the bounded control law with the attitude error calculated by the quaternion error is used. First, we define a scalar function H(q 0 ): Definition 1. H(q 0 ) is a non-negative function on the interval q 0 ∈ [-1, 1] and it vanishes only at q 0 = -1 and/or q 0 = 1. H(q 0 ) also satisfies the Lipschitz condition on the interval

q 0 ∈ [-1, 1], that means ∂H ∂q0 ∞ γ.
Theorem 1. For any function H(q 0 ) defined above, the following bounded control law globally asymptotically converges the plane to the desired attitude:

Γ x = (P s al -P s ar ) l m = -sat Mx ( λ x ω x -ρ ∂H ∂q e0 q e1 ) (14)
Γ y = (P s al + P s ar )

l s = -sat My ( λ y ω y -ρ ∂H ∂q e0 q e2 ) (15) Γ z = (T 2 -T 1 ) l m = -sat Mz ( λ z ω z -ρ ∂H ∂q e0 q e3 ) ( 16
)
where

λ i , ρ > 0; M i 3ργ; i ∈ {x, y, z}
This theorem is proved in appendix A. Table 2 presents some functions H(q 0 ) with their asymptotic stable equilibrium points (a.s.e.p) and unstable equilibrium points (u.e.p) [START_REF] Fjellstad | Quaternion Feedback Regulation of Underwater Vehicles[END_REF]).

Table 2. Examples of function H(q 0 ) H(q 0 ) a.s.e.p u.e.p 1 -q 0 q 0 = 1 q 0 = -1 1 + q 0 q 0 = -1

q 0 = 1 1 -|q 0 | q 0 = ±1 1 -|q 0 | n+1 q 0 = ±1 cos ( πq 0 2 ) q 0 = ±1 cos n ( πq 0 2 ) q 0 = ±1
Since |q 0 | 1 and for simplicity in the embedded control, the function H(q 0 ) = 1 -|q 0 | was chosen. Its derivative is ∂H ∂q0 = -sign (q 0 ). Note that this function does not have unstable equilibrium points, and the control law of the attitude is given by (as in [START_REF] Guerrero | Bounded attitude stabilization : Application on fourrotor helicopter[END_REF]):

Γ x = -sat Mx (λ x ω x + ρsign (q e0 ) q e1 ) (17) Γ y = -sat My (λ y ω y + ρsign (q e0 ) q e2 ) (18) Γ z = -sat Mz (λ z ω z + ρsign (q e0 ) q e3 ) (19)
In order to optimize and improve the performance of the control law, the positive parameters λ i and ρ i were calculated according to the pole placement method of bounded control in [START_REF] Johnson | Nested saturation with guaranteed real poles[END_REF]. The function sign (q e0 ) ensures that between the two rotations of error angle β e and (2π -β e ), the smallest angle will be chosen.

Trapezoidal Trajectory of Velocity -Servo Control

In the case when the convertible plane autonomously takes off, lands and makes a transition, the controller requires the next desired position of the altitude or the attitude (pitch angle in our case). Then a trajectory generation algorithm must be used for optimum motion control. The Fig. 6. Trapezoidal Trajectory of Velocity trajectory generation may not be used by simply forcing the final desired position. However when the final desired position is reached, the required deceleration to stop the vehicle will be very high. Since the vehicle can not achieve such a high deceleration because it flies in the very low friction environment (air), it will certainly overpass the final position. Therefore the vehicle oscillates around the final position before reaching it. The overshooting and oscillation will slow the final objective and have dangerous consequences in the case of autonomous transition and landing. In order to produce a smooth motion, we need a motion profile algorithm that controls the speed and acceleration. A linear piecewise velocity trajectory is implemented: the velocity is incremented by a constant acceleration value until a specified maximum velocity is reached. The maximum velocity is maintained for a required amount of time and then decreased by the same acceleration (deceleration) value until zero velocity is attained (see Figure 6). A movement distance, velocity limit, and acceleration value are required to execute the profile. From these data, the trajectory generator will produce trapezoidal shaped velocity curves for a long movement and triangular curves for a short movement where maximum velocity was not reached. The trapezoidal profile of velocity must be within the velocity profile of ν max (maximum physical velocity) and νmax (maximum acceleration). The control law of altitude and attitude above will force the vehicle to track the new controlled positions created by the trajectory generator in order to realize the process of take off, landing and transition.

RESULTS

Simulation Results

In this section, some simulation results from Matlab-Simulink simulator are presented to demonstrate the performance of the control strategy. Initially, the plane takes off automatically up to 2m (figure 7), the altitude velocity has a trapezoidal form, it accelerates and reduces the speed when the vehicle reaches 2m (figure 8, period (1)). Then the plane makes the transition (period (2)) at second 3.5, the pitch angle varies from -90 • to -2 • . After the transition, the plane accelerates to reach a certain speed. Its velocity v x increases rapidly and converges to a constant value due to the friction drag. That means at this time, the longitudinal acceleration of the plane is zero. In figure 11, we can notice that the angle of attack also converges to a constant value. Despite this small value (approximately 1.5 • ), it is enough to create the lift force thanks to the high speed of the plane (note that nominal airspeed is about 13m/s). It flies horizontally about 5s (period (3)) and then makes the transition to the vertical at second 11 4)). After the transition, it descends and lands at second 21 (period ( 5) and ( 6)).

Note that the control law of the altitude is inactivated during the horizontal flight and during the transition phase. The profile of the altitude velocity v z is trapezoidal in the periods of take off (1) and landing (6) as much as the profile of the pitch rotational velocity ω y has a form of a triangle (figure 8). There is no overshooting in both cases. On the longitudinal flight path (figure 7), the plane loses the altitude in two cases: the first after the transition to the horizontal because its slow speed does not allow to create the needed lift force; the second before the transition to the vertical due to the deceleration of the plane.

Experimental Results

Platform -Embedded Control System. The convertible airplane has been completely designed and constructed at Heudiasyc laboratory (see Figure 1). It has a total weight The measurement used for the altitude is obtained from the low cost ultrasonic sensor (SRF10 ) which has a range of 10cm to 3m. Figure 12 shows the prototype during a vertical flight when the attitude and the altitude are controlled autonomously.

Vertical Flight Mode Test. In this test, the inputs of the system are roll, pitch and yaw angles given by the opera- tors R/C transmitter. The desired altitude in take off and landing phases is given by the trajectory generator and the global thrust is generated by the altitude control. Figure 13 represents the results of the attitude and altitude control algorithm during a vertical flight. The presented attitude is recalculated from the current quaternion vector. The pitch angle varies around 87.5 • because of the mechanical calibration error of the plane. However, the vehicle remains stable and follows the desired attitude. The altitude varies respect to the desired one, automatic take off and landing have been tested with success. Note that 19cm is the initial altitude of the ultrasonic sensor with respect to the ground since the ultrasonic sensor is placed at this distance from the bottom of the plane. One complete video of vertical take off and landing and vertical flight (including "handsoff", torque perturbation, 360 • yaw rotation tests) was uploaded on the site: http://www.youtube.com/watch? v=7D0qSMTFqWc

Transition and Horizontal Flight Mode Test. The tests for transition and horizontal flights are in progress. The results will be obtained soon.

CONCLUSIONS AND FUTURE WORKS

In this work, we have presented the dynamic model of a convertible plane and a simple control strategy to achieve a longitudinal flight path. The obtained simulation results demonstrate that the proposed control strategy could be implemented in real experiments. The control law is simple and suitable for embedded applications, it does not require a high computational cost for control loop. The proposed approach is currently implemented on the platform to verify the practical applicability of the control strategy. The vertical flight mode has been validated and the other experimental phases are in progress.

Fig. 1 .

 1 Fig. 1. Prototype of the Convertible Plane trol laws are high time-consuming and not suitable for embedded control.

Fig. 2 .

 2 Fig. 2. Vertical flight (helicopter mode)

Fig. 3 .

 3 Fig. 3. Horizontal flight (airplane mode)

Fig. 9 .

 9 Fig. 7. Longitudinal flight path

  Fig. 10. Speed of each rotor

  Fig. 13. Vertical Take Off and Landing Flight Test

Table 1 .

 1 Aerodynamic forces

	Lift force	Drag force
	P =	1 2	ρV 2 SC P

ACKNOWLEDGEMENTS

This work has been supported by the French Armament Procurement Agency (DGA), the French National Center for Scientific Research (CNRS) and the project HORUS from the ANR ARPEGE.

Appendix A. PROOF OF THEOREM 1

Proof. Let us define the Lyapunov function:

where κ > 0. Differentiating V with respect to time yields:

Because V is the sum of three terms

) , then we first analyse the term:

Applying the bounded control law ( 14), we have:

We assume that:

) has the same sign as ω x . Then:

By choosing:

λx } in finite time t 1 and remains in it thereafter. In this case, ( -3ργ, 3ργ]. Let's choose M x which satisfies the following condition: M x 3ργ. Then for time t 2 > t 1 , the argument sat Mx will be bounded as follows:

3ργ M x and then we have a condition for κ from A.10:

Consequently, sat Mx operates in a linear region: .12) and:

By choosing κ = ρ satisfying condition A.11, we have:

The same argument is applied to V2 and V3 , and V becomes:

In order to complete the proof, LaSalle Invariance Principle is invoked. All the trajectories converge to the largest invariant set Ω in Ω = { (q e , ω) : V = 0 } = {(q e , ω) : ω = 0}. In the invariant set, J ω = -ω × Jω + Γ = ρ ∂H ∂qe 0 [ q e1 q e2 q e3 ] = 0. That is, Ω is reduced to the origin: q e = ω = 0. This ends the demonstration of the asymptotic stability of the closed-loop system.