Cong Xin

Philippe Veber

Mael Guennou

Constance Toulouse

Nathalie Valle

Monica Ciomaga Hatnean

Ciomaga Hatnean

Geetha Balakrishnan

Raphael C Haumont

Romuald Saint

Matias Velázquez

Matias Velazquez

Alain Maillard

Daniel Rytz

Michael Josse

Mario Maglione

Jens Kreisel

°C using optical floating zone technique and growth prospects from BaB2O4 flux at 1350 °C

Single crystal growth of BaZrO3 from the melt at 2700

Introduction

Perovskites with general formula ABO 3 are an important family of multifunctional materials [START_REF] Mitchell | Perovskites: modern and ancient[END_REF] and exhibit a broad variety of outstanding properties, which are widely used in various technological applications such as Solid Oxide Fuel Cells (SOFC) [START_REF] Tao | A redox-stable efficient anode for solid-oxide fuel cells[END_REF][START_REF] Huang | Double perovskites as anode materials for solid-oxide fuel cells[END_REF], steam electrolysis [START_REF] Tsekouras | Step-change in high temperature steam electrolysis performance of perovskite oxide cathodes with exsolution of B-site dopants[END_REF], substrates for multiferroic materials [START_REF] Wang | Epitaxial BiFeO 3 multiferroic thin film heterostructures[END_REF][START_REF] Ramesh | Multiferroics: progress and prospects in thin films[END_REF], piezoelectricity [START_REF] Liu | High-performance piezoelectric (K,Na,Li)(Nb,Ta,Sb)O 3 single crystals by oxygen annealing[END_REF][START_REF] Veber | Growth and characterization of lead-free piezoelectric single crystals[END_REF][START_REF] Benabdallah | Continuous cross-over from ferroelectric to relaxor state and piezoelectric properties of BaTiO 3 -BaZrO 3 -CaTiO 3 single crystals[END_REF][START_REF] Liu | Influence of Ta 5+ content on the crystallographic structure and electrical properties of [001] PCoriented[END_REF][START_REF] Rödel | Perspective on the development of lead-free piezoceramics[END_REF][START_REF] Saito | Lead-free piezoceramics[END_REF][START_REF] Liu | Orientationdependent electromechanical properties of Mn-doped (Li,Na,K)(Nb,Ta)O 3 single crystals[END_REF] or catalysis [START_REF] Maeda | Water oxidation using a particulate BaZrO 3 -BaTaO 2 N solidsolution photocatalyst that operates under a wide range of visible light[END_REF][START_REF] Tsekouras | The role of defect chemistry in strontium titanates utilised for high temperature steam electrolysis[END_REF][START_REF] Varghese | Ferroelectric nanoparticles, wires and tubes: synthesis, characterisation and applications[END_REF]. Among different perovskite compounds, barium zirconate BaZrO 3 (BZO) has attracted renewed interests as a high-temperature proton conductor [START_REF] Kreuer | Proton-Conducting Oxides[END_REF], a dielectric material for wireless communication applications [START_REF] Sebastian | Dielectric materials for wireless communication[END_REF], a substrate for thin-film growth [START_REF] Azad | On the development of high density barium metazirconate (BaZrO 3 ) ceramics[END_REF][START_REF] Dobal | Micro-Raman scattering and dielectric investigations of phase transition behavior in the BaTiO 3 -BaZrO 3 system[END_REF] and as an inert crucible for superconductors crystal growth [START_REF] Celik | BaZrO 3 insulation coatings for HTS coils[END_REF]. Owing to its large lattice constant, high melting point, low thermal expansion coefficient, low dielectric loss, and low thermal conductivity [START_REF] Vassen | Zirconates as new materials for thermal barrier coatings[END_REF][START_REF] Davies | Effect of ordering-induced domain boundaries on low-loss Ba(Zn 1/3 Ta 2/3 )O 3 -BaZrO 3 perovskite microwave dielectrics[END_REF][START_REF] Akbarzadeh | Combined theoretical and experimental study of the low-temperature properties of BaZrO 3[END_REF], BaZrO 3 has become popular for both fundamental research and device applications. More particularly, BaZrO 3 is known to be used in lead-free BaTiO 3 -based (BCTZ) [START_REF] Acosta | BaTiO 3 -based piezoelectrics: Fundamentals, current status, and perspectives[END_REF] solid solutions which exhibits high piezoelectric response [START_REF] Liu | Large piezoelectric effect in Pb-free ceramics[END_REF], whereby its role in the polarization mechanisms at the phase convergence region of this system remains poorly understood [START_REF] Keeble | Revised structural phase diagram of (Ba 0.7 Ca 0.3 TiO 3 )-(BaZr 0.2 Ti 0.8 O 3 )[END_REF]. For a better understanding of the remarkable piezoelectric response of BCTZ solid solution, it is of great importance to measure the intrinsic properties [START_REF] Akbarzadeh | Combined theoretical and experimental study of the low-temperature properties of BaZrO 3[END_REF] of BaZrO 3 single crystals in order to comprehend and to model more accurately BaTiO 3 -CaTiO 3 -BaZrO 3 phase diagram as previously reported [START_REF] Amoroso | First-principles study of (Ba,Ca)TiO 3 and Ba(Ti,Zr)O 3 solid solutions[END_REF][START_REF] Nahas | Microscopic origins of the large piezoelectricity of leadfree (Ba,Ca)(Zr,Ti)O[END_REF][START_REF] Yang | Mechanisms Responsible for the Large Piezoelectricity at the Tetragonal-Orthorhombic Phase Boundary of (1-x)BaZr 0.2 Ti 0.8 O 3 -xBa 0.7 Ca 0.3 TiO 3 System[END_REF].

To the best of our knowledge, there are only a few publications available on the investigation of BaZrO 3 single crystal growth [START_REF] Helal | Terahertz time-domain spectroscopy and Raman scattering studies of incipient ferroelectric BaZrO 3[END_REF][START_REF] Helal | Softening of infrared-active mode of perovskite BaZrO 3 proved by terahertz time-domain spectroscopy[END_REF][START_REF] Babinskij | Growth and thermoluminescence of BaZrO 3 monocrystals[END_REF][START_REF] Paun | Single crystal growth of high melting oxide materials by means of induction skull melting[END_REF]. Single crystals are usually colored, small in size and their properties vary dramatically with the growth method. Because of the very high congruent melting point of BaZrO 3 in the range of 2500 ºC-2700 °C [START_REF] Gong | Thermodynamic investigation of ZrO 2 -BaO system[END_REF][START_REF] Paschol | Phase equilibria in the pseudoquaternary BaO-UO 2 -ZrO 2 -MoO 2 system[END_REF][START_REF] Shevchenko | Reactions in the system HfO 2 -SrO, HfO 2 -BaO, and ZrO 2 -BaO in high HfO 2 or ZrO 2 regions[END_REF], the growth of high-quality BaZrO 3 single crystal demands suitable techniques which avoid the contamination of single crystals by conventional container-using methods such as Czochralski and Bridgman methods. The skull-melting method has been used, but led to fractured millimeter-sized crystals featuring a strong orange color [START_REF] Paun | Single crystal growth of high melting oxide materials by means of induction skull melting[END_REF].

In this work, we present the first -to the best of our knowledge -crystal growth of BaZrO 3 by optical floating zone and flux technique. In the former case, millimeter-sized boules grown from BaZrO 3 melt at 2700 °C were obtained. A (100)-oriented single crystal plate was successfully extracted and its properties were compared to those of commercial crystals with the same orientation. Chemical properties have been investigated through GDMS and SIMS. The band gap energy has been investigated by optical spectroscopy on the different single crystals. On the other hand, by using BaB 2 O 4 flux, we were able to grow sub-millimetric single crystals at half the melting point of BaZrO 3 . Finally, Raman spectroscopy was performed on single crystals obtained by both the flux and the optical floating zone methods. Results are discussed by comparing the data from literature and those obtained on commercial samples.

Characterization methods

Three sets of BaZrO 3 single crystals are characterized in this paper: i) flux grown single crystals, ii) crystals obtained with the crucible-free optical floating zone technique using a mirror furnace, labelled BZO1, from which a (100)-oriented plate was extracted, and iii) two (100)-oriented commercial single crystals, labelled BZO2 and BZO3, purchased from Crystal Base Co.Ltd. and grown by tri-arc Czochralski method.

Laue back-scattering patterns were recorded using a CCD-camera device (Photonic Science dual lens coupled X-rays Laue system) after a 3-5 mn stationary crystal irradiation with polychromatic X-rays supplied by a molybdenum anode. The Orientexpress software [START_REF] Ouladdiaf | OrientExpress: A new system for Laue neutron diffraction[END_REF] was used for pattern indexing. Single crystals obtained with mirror furnace were cut along (100) direction with a diamond wire saw with an absolute accuracy less than 1º.

Exhaustive trace element chemical analysis (75 elements except C, H, O, N) was performed by Glow Discharge Mass Spectrometry (GDMS) on an as-grown single crystal grown by mirror furnace (BZO 1). GDMS model VG9000 was used with the instrumental uncertainty in between about 20 % from traces (ppb wt) to 0.1 wt%. Measurements were performed under a pressure of 10 -4 -10 -5 mbar at liquid nitrogen temperature.

Samples BZO#1, #2, and #3 were analyzed using a CAMECA SC Ultra SIMS instrument with O 2 + primary ion beam to enhance the ionization of the impurities studied. Impact energy for O 2 + was 1 keV with incidence angle of 64º. A beam current of 50 nA was set over a 250 μm 2 area and ions detected from a 30 μm diameter region at the center of the crater. Analyses were performed under high vacuum conditions with chamber pressure of approximately 1 × 10 -8 mbar. They were carried out at low mass resolution. Then, the contribution of 48 Ca on mass 48 was deduced from the intensity measured for 40 Ca. Samples were sputtered up to a 1 μm deep. Sputtering rate was determined by sputtering a region for an extended span of time so that the crater was sufficient to be measured despite the surface roughness. The sputtering rate obtained was 0.08 nm.s -1 .

Optical transmittance of BaZrO 3 (100)-oriented single crystals from 200 nm to 3000 nm was recorded with a Cary 5000-UV-vis-NIR spectrometer and a resolution of 1 nm. Micro Raman spectra were recorded in a backscattering geometry on a Renishaw in Via micro-Raman spectrometer. Two excitation laser lines were used: the 442 nm line of a He-Cd laser with an output power of 58 mW, and the 785 nm line of a Renishaw laser diode with an output power of 9 mW.

Results and discussion

Crystal growth by the optical floating zone technique

Feed rods were prepared with BaZrO 3 commercial powder (Fox-Chemicals GmbH, 99.9 %) with impurity contents less than 0.006 at.% for Al 2 O 3, , 0.0025 at.% for SiO 2 , 0.0021 at.% for SrO, 0.008 at.% for Fe 2 O 3 and 0.007 at.% for Na 2 O+K 2 O. The powder was sealed in a rubber tube and pressed under hydrostatic pressure around 50 MPa. Compacted cylindrical rods were then sintered in a horizontal furnace at 1200 ºC for 24 hours under air atmosphere leading to sintered feed rods about 6.5 mm in diameter and 65 mm in length. X-ray diffraction performed on crushed pieces of sintered rods confirmed the presence of the cubic perovskite BaZrO 3 phase without any detectable secondary phases. BaZrO 3 was grown using a four-mirror furnace equipped with xenon arc lamp (3 kW) (CSI FZ-T-12000-X_VI-VP, Crystal Systems Incorporated, Japan). The crystal growths were carried out under dynamic argon atmosphere with pressure in the range of 5-6 bar. A polycrystalline BaZrO 3 sintered rod was used as a seed. Once a good quality boule was obtained, a crystal seed was used for subsequent growths. The crystal boules were obtained with a growth rate of 18-25 mm.h -1 . The two rods, the feed rod and the seed, were counter rotated at a rate of 5-15 rpm. Despite the use of a high gas-pressure, a white colored deposition was observed on the quartz tube surrounding the feed rod and the seed rods, indicating the evaporation of BaO during the growth process as usually observed in zirconia-based systems [START_REF] Jacobson | Thermodynamic properties of some metal oxide-zirconia systems[END_REF][START_REF] Fisher | Reactive sintering of BaY 0.1 Zr 0.9 O 3 -δ proton conducting ceramics with CuO liquid phase sintering aid[END_REF][START_REF] Gonçalves | Enthalpy of formation and thermodynamic insights into yttrium doped BaZrO 3[END_REF] at high temperature. The as-grown BaZrO 3 boules were polycrystalline and mostly opaque and white colored (Fig. 1(a)) at the ends close to the seed, at the beginning of the growth. As the growth progressed, the boules produced single crystals and developed facets. Without any post-treatment annealing, as-grown single crystals are transparent to visible light, colorless and size up to 6 mm long (Fig. 1(b)). A single crystal, namely BZO#1, about 1.99 × 1.80 × 0.6 mm 3 (Fig. 1(c)) oriented along (100) (Fig. 1(d))

was extracted from the boule and optically polished for further investigations. Its properties were compared to those of two commercial samples, BZO2 and BZO3, (Fig. 1 (e) and (f)) purchased from Crystal Base company (Japan).

Chemical analysis

The concentration of impurities in a 0.269 g-BZO1 single crystal was measured by GDMS. The result of main impurities contents is summarized in Table 1. Dominant impurities in BZO1 were found to be Ca, Sr, Ti, and Hf and belong to the IIA and IVB columns of the periodic table of the elements, like Ba and Zr, respectively. Solid solutions in between these impurities and BaZrO 3 are easily formed [START_REF] Rödel | Perspective on the development of lead-free piezoceramics[END_REF][START_REF] Acosta | BaTiO 3 -based piezoelectrics: Fundamentals, current status, and perspectives[END_REF] so that they can be incorporated in the perovskite matrix with ease. On the contrary, we note that Al, Si, K and Fe impurities, as foreign elements from those of the IIA and IVB columns, present very low contents compared to those in the raw material (see 3.1). The SIMS intensities are not proportional to the elemental concentrations but depend on factors like ionization yields and matrix effects. In particular, alkali and alkaline-earths ionize very easily so that Ca and Sr produce a more intense signal than Hf, even at lower concentrations. It is therefore not possible to be quantitative with SIMS alone, but we can combine SIMS with the GDMS results known for BZO#1 in order to derive the main impurity concentrations for BZO#2 and #3. Under the assumption that the concentration of the main elements Ba, Zr and O is identical between all samples (identical matrices for the three samples studied), the secondary ion intensity for impurities scales linearly with concentration. Therefore, we can use the GDMS/SIMS measurements on BZO#1 as a calibration to determine the concentrations in BZO2 and BZO3.

Hence, we determined the concentrations for the main impurities (Hf, Sr, Ca and Ti) by rescaling their SIMS intensity with respect to the Zr intensity. The results are reported in Effective segregation coefficients of impurities depend on the growth rate as theoretically predicted by Burton et al. [START_REF] Burton | The distribution of solute in crystals grown from the melt. Part I. Theoretical[END_REF]. Here, we can assume that the growth velocities in the mirror furnace (18-25 mm.h -1 ) are substantially larger than the pulling velocities used for Czochralski growth of oxides, which commonly range within a few mm.h -1 . As experimentally referenced by Fukuda et al. [START_REF] Fukuda | Solid-solution LiTa x Nb 1-x O 3 single crystal growth by Czochralski and edge-defined film-fed growth technique[END_REF], effective segregation coefficients tend to be 1 for high pulling speeds. They are usually lower than 1 for foreign ions [START_REF] Burton | The distribution of solute in crystals grown from the melt. Part I. Theoretical[END_REF], so that impurities are rejected at the liquid-solid interface during the growth and their contents increase progressively into the liquid during the growth. In the present work, owing to the high pulling velocity employed, impurities with effective segregation coefficient lower than 1 have been incorporated in a larger amount within the crystal grown with a mirror furnace than those grown in a tri-arc Czochralski furnace. On the contrary, impurities with effective segregation coefficients higher than 1, are incorporated in a lower amount with the mirror furnace grown crystal. Considering that the raw materials are of the same minimum purity (3N),

we can reasonably conclude that the effective segregation coefficient of Sr in the A site of BaZrO 3 perovskite is higher than 1, similarly to what can be deduced from the sign of the solidus and liquidus curves of the BaTiO 3 -SrTiO 3 perovskite system [START_REF] Basmajian | Phase Equilibria in the System BaTiO 3 -SrTiO 3[END_REF]. In the same way, effective segregation coefficients of Hf, Ca and Ti are lower than 1, as previously referenced for Ca and Ti in BaZrO 3 -based perovskite solid solutions [START_REF] Buse | Spinodal decomposition in lead-free piezoelectric BaTiO 3 -CaTiO 3 -BaZrO 3 crystals[END_REF].

Physical properties

Optical properties

Figure 3 shows the transmittance of the three samples BZO1, 2 and 3. BaZrO 3 single crystals are essentially transparent in the visible and NIR regions and exhibit a sharp absorption edge in the near-UV. BZO2 is remarkably different from the other crystals in that it exhibits a strong absorption band in the lower part of the visible spectrum, with a main peak around 800 nm, which is consistent with the blueish color of the crystal (see photo Fig. 1(e)). This blueish color and the absorption band can likely be attributed to differences in oxygen stoichiometry, as previously

shown for example on La 0.5 Na 0.5 TiO 3 [START_REF] Haumont | Growth of quantum paraelectric La 1/2 Na 1/2 TiO 3 single crystals using optical floating zone technique[END_REF]. BaZrO 3 is expected to have an indirect gap according to most electronic structure calculations reported in the literature [START_REF] Kouchaksaraie | Theoretical calculation of electrical and optical properties of BaZrO 3[END_REF][START_REF] Yang | Pressure effects on structural, electronic, elastic, and optical properties of cubic and tetragonal phases of BaZrO 3[END_REF][START_REF] Cavalcante | Intense violetblue photoluminescence in BaZrO 3 powders: A theoretical and experimental investigation of structural order-disorder[END_REF][START_REF] Terki | Full potential calculation of structural, elastic and electronic properties of BaZrO 3 and SrZrO 3[END_REF], with a conduction band minimum at the Γ point (0,0,0) and a valence band maximum at the R point (½,½,½), where the phonon energies can be as high as 100

meV [START_REF] Akbarzadeh | Combined theoretical and experimental study of the low-temperature properties of BaZrO 3[END_REF][START_REF] Bennett | Effect of symmetry lowering on the dielectric response of BaZrO 3[END_REF]. These computations also suggest a relatively flat valence band, with a local maximum at Γ only 250 meV lower than the absolute maximum, where direct transitions would be allowed. One study even reports a direct band gap at Γ [START_REF] Alay-E-Abbas | Formation energies and electronic structure of intrinsic vacancy defects and oxygen vacancy clustering in BaZrO 3[END_REF].

Experimentally, the nature and the value of the band gap can in principle be determined by the Tauc plot, i.e. plotting (α.hν) 1/r as the function of the energy hν, where α is the absorption coefficient and r equals ½ or 2 for, respectively, a direct or indirect gap. The absorption can be determined from the transmittance T, reflectivity R, and thickness d of the samples using the relation T ≈ (1-R) 2 exp(-αd) [START_REF] Bhatt | Urbach tail and bandgap analysis in near stoichiometric LiNbO 3 crystals[END_REF]. We neglect reflectivity in the following. The absorption edge is expected to exhibit a linear regime for r=1/2 or r=2, depending on the character of the gap. Fig. 3 (b) and (c) show the Tauc plots for the three crystals, whereby the blueish BZO#2 appears very different from the two others. Colorless BZO#1 and BZO#3 exhibit a linear region in both plots, and extrapolations of this linear region gives gap values that are very similar for the two samples:

4.89/4.86 eV for r=1/2 (hypothesis of the direct gap) and 4.76/4.74 eV for r=2 (hypothesis of the indirect band gap) for BZO#1/#3 respectively. Those values, consistent between the two samples and very close to each other, seem to confirm the indirect character of the gap, but also the existence of direct transitions allowed at a hardly higher energy, as suggested by computations. In addition, both crystals show a pronounced Urbach tail linked to near-edge defects states. In BZO#2, this appears so dominant that no linear region can be identified.

In the present work, the value of ~4.8 eV matches the 4.1-4.8 eV reported in a study of BaZrO 3 powders prepared with various degrees of disorder [START_REF] Cavalcante | Intense violetblue photoluminescence in BaZrO 3 powders: A theoretical and experimental investigation of structural order-disorder[END_REF], and the 5.0 eV found on a similar study with powder prepared by solid state reactions [START_REF] Volkova | Ferroelectric perovskite oxides for photovoltaics and photocatalysis[END_REF]. On the other hand, it is much higher than the 3.8 eV reported in [START_REF] Lee | Band-gap nonlinearity in perovskite structured solid solutions[END_REF] a study of powders obtained by the sol-gel route. As commonly observed, many computations underestimate the band gap value (3.2 eV in Refs. [START_REF] Yang | Pressure effects on structural, electronic, elastic, and optical properties of cubic and tetragonal phases of BaZrO 3[END_REF][START_REF] Terki | Full potential calculation of structural, elastic and electronic properties of BaZrO 3 and SrZrO 3[END_REF][START_REF] You | Temperaturedependent Raman spectra and microstructure of barium metaborate crystals and its melts[END_REF]).

Raman spectroscopy analysis

BaZrO 3 is cubic with the space group and has no first-order Raman-active phonon mode. 𝑃𝑚3𝑚

In spite of this, an intense Raman spectrum is usually observed [START_REF] Helal | Terahertz time-domain spectroscopy and Raman scattering studies of incipient ferroelectric BaZrO 3[END_REF][START_REF] Chemarin | A high-pressure Raman study of mixed perovskites BaCe x Zr 1-x O 3 (0 ≤ x ≤ 1)[END_REF][START_REF] Colomban | Raman intensity: an important tool in the study of nanomaterials and nanostructures[END_REF], in a way that is reminiscent of other cubic perovskites (SrTiO 3 [START_REF] Nilsen | Raman spectrum of strontium titanate[END_REF], KTaO 3 [START_REF] Nilsen | Raman spectrum of potassium tantalate[END_REF]). Second-order scattering processes involving combinations of two phonons usually explains it.

Figure 4 (a) shows the Raman spectra of BZO1, BZO2 and BZO3 single crystals measured at ambient conditions in parallel scattering geometry. The spectra present a comparable signature, which are similar in their main features to the few spectra reported in the literature [START_REF] Helal | Terahertz time-domain spectroscopy and Raman scattering studies of incipient ferroelectric BaZrO 3[END_REF][START_REF] Chemarin | A high-pressure Raman study of mixed perovskites BaCe x Zr 1-x O 3 (0 ≤ x ≤ 1)[END_REF][START_REF] Mączka | Growth and characterization of acentric BaHf(BO 3 ) 2 and BaZr(BO 3 ) 2[END_REF].

Raman spectra were collected upon cooling down to 4.2 K (Fig. 4 (b)). The comparison of the spectra at low and room temperatures does not show any indication for a phase transition. Slight changes, such as a subtle sharpening of the bands, very small shifts in their positions, or the general weakening of the low frequency bands can all be attributed to thermal effects. The emergence of a couple of very weak and thin peaks can be noticed, but have to be attributed to extrinsic effects and defects. This is demonstrated by the observation that they are found at different positions in BZO1 grown in the present work and in the commercial BZO2 and BZO3 samples. Such spurious Raman lines are not uncommon in perovskites [START_REF] Saha | Magnetic modes in rare earth perovskites: A magnetic-field-dependent inelastic light scattering study[END_REF], but cannot be assigned conclusively here.

Investigation on BaZrO 3 growth with BaB 2 O 4 as a potential flux

The flux method from high temperature solution is investigated in order to decrease the crystallization temperature of BaZrO 3 . As suggested by the BaTiO 3 -BaB 2 O 4 phase diagram referenced by Goto et al . [START_REF] Goto | Phase diagram of the BaTiO 3 -BaB 2 O 4 system and growth of BaTiO 3 crystals in the melt[END_REF] and because B ions cannot be inserted into the BaTiO 3 lattice as a foreign element, BaB 2 O 4 (BBO) could be used as a suitable self-flux [START_REF] Azad | On the development of high density barium metazirconate (BaZrO 3 ) ceramics[END_REF] to grow BaTiO 3 at temperatures as low as 942 °C, which is more than 650 °C below the BaTiO 3 melting point (T f = 1618 °C). Although the detailed BaZrO 3 -BBO phase diagram is unknown, we assumed that the BaZrO 3 -BBO system exhibits the same phase diagram feature than that of BaTiO 3 -BBO because of the chemical similarity of Ti 4+ and Zr 4+ cations. Therefore, BaZrO 3 :BBO mixtures were investigated for different molar ratios such as 10:90, 20:80, 30:70, 35:65, 40:60 and 50:50, in order to characterize the solubility of BaZrO 3 in BBO. The BBO flux was prepared by solid state reaction [START_REF] Deyra | Impact of BaB 2 O 4 growth method on frequency conversion to the deep ultraviolet[END_REF] by mixing BaCO 3 (Alfa Aesar, 99.95 %) and B 2 O 3 (Alfa Aesar, 99.98 %). Then, BBO was mixed with BaZrO 3 commercial powder (Fox-Chemicals GmbH, 99.9 %). 10 g ball-milled mixtures were placed into a platinum crucible tightly covered with a platinum lid and placed in a large alumina crucible. A two-heating-resistive-zone furnace was used to achieve a 1°C.cm -1 longitudinal thermal gradient. The process was performed in the following steps: i) an increasing temperature ramp of 120 °C.h -1 up to 1350 °C, with a 1-hour dwelling time at this temperature, ii) a decreasing ramp with 1 °C.h -1 down to 1250 °C, and finally, iii) the mixtures were cooled down at a rate of 30 °C.h -1 down to room temperature. The mass of the crucible containing the mixtures was monitored all along the thermal process.

In all cases, a strong volatilization as well as flux creeping outside of the crucible were observed with a loss of approximatively one half of the total weight of the loads.

In the lower-BaZrO 3 -content solutions corresponding to 10:90, 20:80 and 30:70 BaZrO 3 -BBO mixtures, no BaZrO 3 crystallites were detected. In the particular case of 10:90 BaZrO 3 -BBO solution, millimeter-sized yellow single crystals (Fig. 5(a)), with (111)-natural facets (Fig. 5(b)). corresponding to rhombohedral BaZr(BO 3 ) 2 (BZB) phase, were extracted from the solidified solution.

Higher-BaZrO 3 -content solutions corresponding to 35:65, 40:60 and 50:50 BaZrO 3 -BBO mixtures displayed two solidified zones with white and brown colors (Fig. 5(c)). In the brown zone, BaZrO 3 single crystals were successfully obtained (Table 3) and their structure has been confirmed by single crystal XRD with space group and a=4.18 Å. 𝑃𝑚3𝑚

The largest BaZrO 3 crystals that can show a rectangular-like face with a size up to 150-200 µm (Fig. 5(d)) were found in the brown zone from the 35:65 BaZrO 3 -BBO solution where yellow BZB micrometric crystals with triangle-like shape are also detected. Crystallites with micrometric-sized were detected in 40:60 and 50:50 BaZrO 3 -BBO solutions.

XRD diffractograms (Fig. 6) display BaZrO 3 , Ba 2 B 2 O 5 and BZB phases in both white and brown zones with a higher proportion of BaZrO 3 in the brown zone where rectangular single crystals were detected and collected. Micro-Raman spectroscopy was performed on the as-grown crystals from the 35:65 BaZrO 3 -BBO solution (Fig. 7). Crystals with large flat facets exhibit a spectrum that matches previously reported Raman spectra of BaZrO 3 [START_REF] Helal | Terahertz time-domain spectroscopy and Raman scattering studies of incipient ferroelectric BaZrO 3[END_REF][START_REF] Chemarin | A high-pressure Raman study of mixed perovskites BaCe x Zr 1-x O 3 (0 ≤ x ≤ 1)[END_REF][START_REF] Mączka | Growth and characterization of acentric BaHf(BO 3 ) 2 and BaZr(BO 3 ) 2[END_REF] as well as the spectra shown in Fig. 4. Finally, as we obtained only BZO, BZB and Ba 2 B 2 O 5 phases in the brown zone where, respectively, their structures are cubic (ICDD N°006-0399), rhombohedral (ICDD N°056-0239) and monoclinic (ICDD N°024-0087), it is worth noticing that crystals with natural rectangular faces observed on figure 1b features typically the habit of crystals belonging to orthogonal coordinate systems, so that brown rectangular single crystal shown in figure 5(d) can only correspond to cubic BaZrO 3 . In the same way, yellow triangular crystal shown in figure 5(d) corresponds to rhombohedral BZB.

The spectrum of BZB [START_REF] Mączka | Growth and characterization of acentric BaHf(BO 3 ) 2 and BaZr(BO 3 ) 2[END_REF] was also confirmed, consistently with the XRD analysis. Some additional spectral features are observed but cannot not be conclusively identified. In particular, they do not match the Raman spectrum expected from the BBO flux [START_REF] You | Temperaturedependent Raman spectra and microstructure of barium metaborate crystals and its melts[END_REF] or ZrO 2 [START_REF] Ishigame | Temperature dependence of the Raman spectra of ZrO 2[END_REF]. We may tentatively attribute it to the Ba 2 B 2 O 5 detected in XRD, but no reference Raman spectrum is available in the literature to confirm this hypothesis. We conclude that a chemical reaction occurs between BZO and BBO, which leads to the formation of BZB and Ba 2 B 2 O 5 at 1350 °C, and it is described by the following equation:

  1350 2 4 3 3 2 2 5 2 2 C BaB O BaZrO BaZr BO Ba B O     
Furthermore, in an attempt to increase the crystal size by avoiding the flux creeping and its volatilization from the solution, a crystal growth was performed with BaZrO 3 -BBO 35:65 molar ratio under controlled atmosphere in a sealed and gas-proof platinum assembly for saturating the vapor pressure as previously reported by Albino et al. [START_REF] Albino | Growth and characterization of centimeter-sized Ba 2 LaFeNb 4 O 15 crystals from high-temperature solution under a controlled atmosphere[END_REF]. The same thermal protocol was performed as above-mentioned. After thermal process, XRD analysis revealed the presence of BaZrO 3 , BBO and BZB compounds in the solidified solution (see insert of Fig. 8) without any BaZrO 3 single crystal visible under a microscope (X64). This confirms that the use of a sealed atmosphere prevents the decomposition of BBO into Ba 2 B 2 O 5 but not the formation of BZB, and this points out that the solubility of BaZrO 3 in BBO is very low and below the XRD detection limit. In addition, this highlights that the flux volatilization was the driving force governing the growth of BaZrO 3 obtained in the tightly lid-covered crucible.

Finally, we noted that the formation of BZB is systematically observed, whatever the platinum assembly used. The chemical reaction in between BBO and BaZrO 3 decreases the amount of the latter in the BBO-based solution, partly impeding then its growth at 1350 °C. Hence, a growth attempt was performed with BZB as a self-flux with BaZrO 3 -BZB 50:50 molar ratio. BZB was synthesized by solid state reaction [START_REF] Hornebecq | BaZr(BO 3 ) 2 : a noncentrosymmetric dolomite-type superstructure[END_REF] and mixed to BaZrO 3 commercial powder. The load was

held in an open iridium crucible under argon atmosphere for 12 h at a temperature in a 1550 °C-1640 °C range, as determined through pyrometric measurement. A strong volatilization of the solution was observed, prohibiting the growth of BaZrO 3 single crystal from BZB flux. Indeed, XRD analysis (Fig. 9) revealed the presence of only BaZrO 3 and ZrO 2 phases resulting likely from the decomposition of BZB at high temperature. We infer that the high thermal chemical stability of BaZrO 3 makes it difficult to dissolve in the investigated Ba-based flux of the present work. A yellow BZB crystal with a triangle-like shape is detected.

Conclusion

The growth of BaZrO 3 by the optical floating zone technique and its investigation by the flux method are reported for the first time. BaZrO 3 boules have been successfully grown by optical floating zone technique with single crystal sizes up to 6 mm long. They are transparent, colorless and have been shaped as millimeter-sized plate oriented along (100) direction. Four dominant elements that can make easily solid solutions with BaZrO 3 were observed through GDMS and SIMS analysis: Ca, Ti, Sr and Hf, whereas other foreign elements contents are drastically decreased compared to their initial content in the raw material. This highlights that optical floating zone method is a suitable technique for removing foreign impurities from oxide crystals contrary to oxide crystals classically grown with a highly polluting environment, such as refractory ceramics and metallic crucibles, inducing a higher impurities contents with the same or a higher raw material purity [START_REF] Velázquez | Spectroscopic properties of newly flux grown and highly Yb 3+ -doped cubic RE 2 O 3 (RE=Y, Gd, Lu) laser crystals[END_REF][START_REF] Veber | Flux growth at 1230 °C of cubic Tb 2 O 3 single crystals and characterization of their optical and magnetic properties[END_REF][START_REF] Veber | Flux growth of Yb 3+doped RE 2 O 3 (RE=Y, Lu) single crystals at half their melting point temperature[END_REF]. Optical measurements performed on BaZrO 3 single crystal exhibit a high optical band gap energy of ~4.8 eV, most probably indirect, although direct electronic transitions are only slightly higher in energy. Raman study revealed a second-order spectrum, with sharp features. This Raman spectrum does not change significantly down to the lowest temperature measured (4.2 K). The availability of large single crystals opens the possibility for fundamental studies of BaZrO 3 , notably its dynamics at the macroscopic and local scales. While it is demonstrated that BaZrO 3 solubility is poor in BaB 2 O 4 flux, this self-flux growth approach makes possible to crystallize this highly refractory material at low temperatures.

Fig. 1 :

 1 Fig. 1: (a) Portion of a BaZrO 3 boule grown by the floating zone method. (b) 6-mm long BaZrO 3 single crystal extracted at the end of the boule. (c) Cut and (100)-oriented BaZrO 3 single crystal (BZO#1) ~ 1.99 × 1.80 × 0.6 mm 3 (d) Back-scattered Laue diffraction pattern of BaZrO 3 single crystal with (100) orientation. Red dots correspond to the calculated Laue pattern. (e) BZO2 commercial sample. (f) BZO3 commercial sample.

Figure 2 Fig. 2 :

 22 Figure2shows the secondary ion yield variations observed as a function of sputtering time t s under oxygen bombardment into BaZrO 3 samples BZO1, BZO2 and BZO3. t s is directly proportional to the distance z normal to the samples surface (about 120 nm in Fig.2). From the results shown in figure2, the intensities of most ions stabilize quite rapidly, except Ti and Ca in BZO2 and BZO3 single crystals. Comparison between crystals shows a higher Ca and Ti secondary ion intensity in BZO#1 than in BZO#2 and BZO#3, and an opposite trend for the Sr concentration.The prominence of Sr, Ca, Ti and Hf impurity elements observed by GDMS is thus confirmed by SIMS analysis

Fig. 3 :

 3 Fig. 3: (a) Optical transmission spectra of BaZrO 3 single crystal of BZO1, BZO 2, BZO3 in a wide wavelength range. (b) and (c) Tauc plots with, respectively, r=1/2 and r=2.

Fig. 4 :

 4 Fig. 4: Raman scattering spectra of BaZrO 3 single crystals measured with a 442 nm excitation at (a) ambient temperatures and (b) at 4.2 K. The positions of the main peaks are indicated. The stars mark the weak lines emerging at low temperature, as discussed in the text.

Fig. 5 :

 5 Fig. 5: Flux growth with 35:65 BaZrO 3 -BBO solution: (a) BaZr(BO 3 ) 2 (BZB) single crystals grown from 10:90 BaZrO 3 -BBO solution with (b) (111)-natural faces highlighted in a Laue pattern (red dots correspond to theoretical pattern whereas white dots correspond to experimental pattern). (111) rhombohedral faces correspond to (001) hexagonal face, noted (H001). (c) Crucible containing the 35:65 BaZrO 3 -BBO solution exhibiting the BBO flux (white zone) and the BZO crystals (brown zone). (d) Zoom on BZO single crystals (brown zone) with a rectangle-like shape.

Fig. 6 :

 6 Fig. 6: XRD patterns of crushed sample extracted from the brown zone (red pattern) and the white zone (black patterns) obtained from the flux growth attempt into a tightly lid-covered platinum crucible with 35:65 BZO-BBO solution.

Fig. 7 .

 7 Fig. 7. Selected Raman spectra recorded in different parts of the flux grown crystals with a 785 nm excitation line. (a) and (b) show two representative spots where the spectra of BaZrO 3 and BZB can be observed as indicated in (c), together with other spectra showing mixed cases and Raman bands (marked with a star) that are tentatively assigned to Ba 2 B 2 O 5 .

8 :

 8 XRD pattern of crushed sample from the flux growth attempt into a sealed platinum crucible with 35:65 BZO-BBO solution. In insert: view of the load after the thermal process. No crystal was detected.

Fig. 9 :

 9 Fig. 9: XRD pattern of crushed sample from the flux growth attempt into an open iridium crucible with 50:50 BZB-BZO solution.

Finally, the flux

  method using BaB 2 O 4 solvent enables to grow 150-200 µm-sized single crystals at half the melting point (1350 °C) of BaZrO 3 . Crystals size and quality are restricted by the formation of BaZr(BO 3 ) 3 , which has been revealed by Raman spectroscopy and XRD diffraction.

Table 1 :

 1

	Element	B	Mg	Al	Si	P	S	Cl	K	Ca	Fe	Ti	Sr	Hf
	BZO1	0.8	3.4	6.2	47.3 1.5	7 15.6 4.5	179.4	1.3	370	1420	3001

GDMS analysis results of the impurity content in a 0.269 g BZO1 single crystal (ppm at.).

Table 2 .

 2

	8

Table 2 :

 2 Summary of the main impurity contents (ppm at.) for the three single crystals derived from GDMS and SIMS measurements.

Table 3 :

 3 Mixtures of BaZrO 3 -BaB 2 O 4 with different molar ratio together with the typical size of BZO crystals grown in the tightly lid-covered crucible.

Acknowledgments

The authors would like to thank Dr Stanislas Pechev and M Alexandre Fargues for, respectively, BaZrO 3 single crystal XRD and optical transmission analysis performed at ICMCB. This work was supported by the Innovative Training Networks (ITN) -Marie Skłodowska-Curie Actions-European Joint Doctorate in Functional Material Research (EJD-FunMat) project (nº 641640). The work at the University of Warwick was supported by the EPSRC, UK, Grant EP/M028771/1.

Cong Xin, Dr. Mael Guennou, Dr Constance Toulouse and Prof. Jens Kreisel acknowledge support from the National Research Fund Luxembourg through a Pearl Grant (FNR/P12/4853155).


This paper deals with the global stabilization of a rigid body by means of a bounded quaternion-based feedback. In addition to input bounds, the nonlinear control takes into account the slew rate limits. The proposed control scheme is generic and can be applied to all systems falling in the framework of rigid bodies. Furthermore, its extreme simplicity rendering it suitable for embedded implementation. It is also shown that the control strategy can be applied to attitude stabilization of a four-rotor mini-helicopter despite gyroscopic effects appear that are not present in the rigid body framework. Several real-time experiments have been performed on a real mini-helicopter in order to show the effectiveness of the proposed control approach.

INTRODUCTION

The problem of attitude stabilization of a rigid body has been a subject that has attracted considerable amount of interest over the past 30 years. Several approaches have been applied such as feedback linearizing control, [START_REF] Wie | Quaternion feedback regulator for spacecraft eigenaxis rotations[END_REF], [START_REF] Fjellstad | Quaternion feedback regulation of underwater vehicles[END_REF], feedback proportional-derivative control, [START_REF] Wen | The attitude control problem[END_REF], [START_REF] Joshi | Robust attitude stabilization of spacecraft using nonlinear quaternion feedback[END_REF], predictive control in [START_REF] Wen | Nonlinear predictive control applied to spacecraft attitude control[END_REF] and [START_REF] Hegrenas | Attitude control by means of explicit model predictive control, via multi-parametric quadratic programming[END_REF]), integral backstepping in [START_REF] Kristiansen | Satellite attitude control by quaternion-based backstepping[END_REF] and robust control applied to tactical missiles, [START_REF] Song | Robust control of the missile attitude based on quaternion feedback[END_REF]. This list is not exhaustive. Within these mentioned approaches, the feedback linearizing control and the feedback proportionalderivative control are widely applied. This is due to the fact that these control laws achieve global stabilization and they are simples to implement. The major criticism of the two last approaches is that for large attitude or angular velocity error, the control law may yield a control input that the actuators are unable to produce. This fact becomes apparent due to their linear nature. In practice, limitations on available energy impose bounded input signal. Moreover, it is common that due to sensors limitation the output of the system are bounded. Actually, the above cited attitude control approaches do not consider the problem which takes the input constraints into account. Few publications have dealt with this problem. In [START_REF] Tsiotras | Control of underactuated spacecraft with bounded inputs[END_REF], a saturating control law for the kinematic system of an under-actuated rigid spacecraft is studied. This control law uses a non-standard attitude representation, which allows the decomposition of general motion into two rotations. In [START_REF] Belta | On controlling aircraft and underwater vehicles[END_REF], a control law that drives a rigid underwater vehicle between arbitrary initial and final region of the state space while satisfying bounds on control and state is proposed. The approach is based in a control of multi-affine systems.

⋆ This work was supported in part by CONACYT-Mexico The authors in [START_REF] Boskovic | Robust stabilization of spacecraft in the presence of control input saturation using sliding mode control[END_REF] have studied the robust sliding mode stabilization of the spacecraft attitude dynamics in presence of control input saturation based on the variable structure control (VSC) approach. The application of optimal control to attitude problem of a rigid body has been also the interest of many research, (see [START_REF] Scrivener | Survey of timeoptimal attitude maneuvers[END_REF] and the references therein). In order to circumvent the solution of Hamilton-Jacobi equations, different works have used the so-called inverse optimal method, [START_REF] Osipchuck | Achieving good performance in global attitude stabilization[END_REF], [START_REF] Krstić | Inverse optimal stabilization of a rigid spaceraft[END_REF]. Unfortunately, the above cited methods are either non smooth or computationally expensive (optimal control). This renders difficult their practical implementation.

In the present work a bounded control law for the global attitude stabilization of a rigid body is proposed. With the proposed approach, the control bounds along the three axis can be completely freely chosen. The bounded control law can force the closed-loop angular velocity to any a priori specified region of the state space and maintains them in it thereafter. Therefore, in addition to input constraints, the control law enforces angular velocity constraints, i.e. slew rate limits. Hence, the control law is robust towards bounded angular velocities measurements as it happens with classical rate gyros. Furthermore, the control law is robust towards modelling errors because it does not depend on the knowledge of the inertial parameters of the system. The proposed control is inspired by the nested saturation functions approach proposed by [START_REF] Teel | Global stabilization and restricted tracking for multiple integrators with bounded controls[END_REF] for the problem of stabilizing linear systems with bounded controls. The attitude is parameterized using the unit quaternion, which allows nonlinear large angles attitude manoeuvres. This makes a distinct contrast from the case using three parameters representation, where an inherent geometric singularity appear (see [START_REF] Shuster | A survey of attitude representations[END_REF] for more details on attitude representations). The principal property of proposed control law is its extreme simplicity com-pared to the other existing approaches, rendering it very suitable for embedded implementation, where on-board computational resources are challenging. Furthermore, the control signal computed with this control law is smooth. In recent years, significant research effort has been directed toward the control, modelling and design of flying robots. This interest arise principally of military purposes but also of civilian application such as disaster monitoring, environmental surveillance or even cinematography. Within flying robots, the four-rotor helicopter has aroused great interest because of its highly manoeuvrability and the ability to hover. This Vertical Take-Off and Landing (VTOL) vehicle has some advantages over conventional helicopters: owing to symmetry, this vehicle is simple to design and construct. The low moment of inertia of this aerial robot makes it vulnerable to large angular acceleration. Therefore, for many potential missions an efficient attitude control is crucial, since it allows the vehicle to maintain a desired orientation. Several control techniques have been used for the attitude stabilization of the four-rotor helicopter, for instance, PID and LQ control in [START_REF] Bouabdallah | PID vs LQ control techniques applied to an indoor micro quadrotor[END_REF], sliding mode control in [START_REF] Bouabdallah | Backstepping and sliding-mode techniques applied to an indoor micro quadrotor[END_REF] and Backstepping approach applied in [START_REF] Tayebi | Attitude stabilization of a vtol quadrotor aircraft[END_REF]. Although the results presented in the above cited works are nice and founding, the boundedness of the control input is not considered. One particular aspect in the four-rotor helicopter dynamics, is that the dynamical model for the attitude may be reduced to this one of a rigid body. Therefore, the proposed approach for the stabilization of a rigid body, is naturally extended to the attitude stabilization of the four-rotor mini helicopter. Owing to the extreme simplicity and the smoothness of the control approach, a real-time stabilization was performed successfully on the four-rotor mini helicopter, developed at gipsa-lab control system department (former LAG) in Grenoble. The control capabilities were tested for different scenarios, showing a good performance in terms of settling time stabilization, robustness towards disturbance rejection and tracking trajectories capability. This paper extends the initial work proposed in [START_REF] Guerrero-Castellanos | Bounded attitude stabilization: Application on four rotor helicopter[END_REF] by proposing a control law with more tuning parameters, established robustness properties and a real-time implementation. The present paper is organized as follows. In section 2, a rigid body quaternion-based orientation is given. Then the main problem is formulated in section 3. The section 4 is devoted to the control law design, where its stability is proved and its properties are presented. The extension of the proposed control law for the attitude stabilization of the four-rotor helicopter is demonstrated in section 5. Then, the implementation and experimental results are presented and discussed in section 6.

MATHEMATICAL BACKGROUND

The attitude of a rigid body can be represented by a quaternion, consisting of a unit vector e, known as the Euler axis, and a rotation angle β about this axis. The quaternion q is then defined as follows

q =    cos β 2 e sin β 2    = q 0 q ∈ H (1)
where H = {q | q 2 0 + q T q = 1, q = [q 0 q] T , q 0 ∈ R, q ∈ R 3 } (2) q = [q 1 q 2 q 3 ] T and q 0 are known as the vector and scalar parts of the quaternion respectively. In attitude control applications, the unit quaternion represents the rotation from an inertial coordinate system N (x n , y n , z n ) located at some point in the space (for instance, the earth NED frame), to the body coordinate system B(x b , y b , z b ) located at the center of mass of the rigid body. The rotation matrix C(q) corresponding to the attitude quaternion q, is computed as C(q) = (q 2 0q T q)I 3 + 2(qq Tq 0 [q × ]). where I 3 is the identity matrix and [ξ × ] is the skew symmetric tensor associated with the axial vector ξ.

Denoting by ω = [ω 1 ω 2 ω 3 ] T the angular velocity vector of the body frame B relative to the inertial frame N , expressed in B, the kinematics equation is given by

q0 q = 1 2 -q T I 3 q 0 + [q × ] ω = 1 2 Ξ(q)ω (3)
The attitude error is used to quantify the mismatch between two attitudes. If q defines the current attitude quaternion and q d is the reference quaternion, i.e. the desired orientation, then the error quaternion that represents the attitude error between the current orientation and the desired one is given by

q e = q ⊗ q -1 d (4)
⊗ denotes the quaternion multiplication and q -1 d is the complementary rotation of the quaternion q d , which is the quaternion conjugate (see [START_REF] Shuster | A survey of attitude representations[END_REF]) for more details). The attitude dynamics of a rigid body is described by J ω = -ω × Jω + Γ (5) where J ∈ R 3×3 is the symmetric positive definite constant inertial matrix of the rigid body expressed in the B frame and Γ ∈ R 3 is the vector of control torques. Note that these torques also depend on the environmental disturbance torques (aerodynamic, gravity gradient, etc.).

PROBLEM STATEMENT

The objective is to design a control law that drives the rigid body attitude to a specified constant orientation and maintains this orientation. Let q d denote the desired constant rigid body orientation, then the control objective is described by the following asymptotic conditions q → q d , ω → 0 as t → ∞ (6) The quaternion error that represents the attitude error between the current orientation and the desired one is given by equation ( 4). If the the desired attitude is q d = [±1 0 0 0] T (inertial coordinate frame), the quaternion error (4) coincides with the current attitude quaternion, that is, q e = q. This control objective is then q → [±1 0 0 0] T , ω → 0 as t → ∞ (7) In this study the case q d = [1 0 0 0] T that represents the attitude aligned up with the inertial coordinate system axes is considered. Nevertheless, the results can be applied to another desired orientation. It is well known that actuator saturation reduces the benefits of the feedback. When the controller continuously outputs infeasible control signals that saturate the actuators, system instability may follow. Therefore, besides the asymptotic stability, the objective of the control law is to take into account the physical constraints of the control system, in order to apply only feasible control signals to the actuators.

BOUNDED ATTITUDE CONTROL FORMULATION

In this section, a control law that stabilizes the system described by ( 3) and ( 5) is proposed. The goal is to design a control torque that is bounded. Definition 1. Given a positive constant M, a continuous, nondecreasing function σ M : R → R is defined by

(1) σ M (s) = s if |s| < M ;
(2) σ M (s) = sign(s)M elsewhere;

Theorem 1. Consider the rigid body rotational dynamics described by ( 3) and ( 5) with the following bounded

control inputs Γ = [Γ 1 , Γ 2 , Γ 3 ] T such that Γ i = -α i σ Mi (λ i [ω i + ρ i q i ]) (8
) with i ∈ {1, 2, 3} and where σ Mi are saturation functions with M i ≥ 3λ i ρ i . α i , λ i and ρ i are positive parameters. Then the inputs (8) globally asymptotically stabilize the rigid body to the origin (q 0 = 1, q = 0 and ω = 0).

Proof. Consider the candidate Lyapunov function V , which is positive definite, radially unbounded and which belongs to the class C 2 . V represents the total energy of the system

V = 1 2 ω T Jω + κ((1 -q 0 ) 2 + q T q) = 1 2 ω T Jω + 2κ(1 -q 0 ) (9)
where J is defined as before, and κ > 0 must be determined. The derivative of (9) after using (3) and ( 5) is given by

V = ω T J ω -2κ q0 = ω T (-ω × Jω + Γ) + κq T ω = ω 1 Γ 1 + κq 1 ω 1 V1 + ω 2 Γ 2 + κq 2 ω 2 V2 + ω 3 Γ 3 + κq 3 ω 3 V3 (10)
V is the sum of three terms ( V1 , V2 , V3 ). First V1 is analyzed. From Γ 1 in (8) and equation ( 10), one gets

V1 = -α 1 ω 1 σ M1 (λ 1 [ω 1 + ρ 1 q 1 ]) + κq 1 ω 1 (11) Assume that |ω 1 | > 2ρ 1 , that is |ω 1 | ∈ ]2ρ 1 , +∞[. Since |q 1 | ≤ 1, it follows that |ω 1 + ρ 1 q 1 | ≥ ρ 1 + ǫ
for any ǫ > 0 sufficiently small. Therefore, ω 1 + ρ 1 q 1 has the same sign as ω 1 . From equation ( 11) and the norm condition on the quaternion, V1 takes the following form

V1 = -α 1 ω 1 σ M1 (λ 1 [ω 1 + ρ 1 q 1 ]) + κω 1 q 1 ≤ -α 1 |ω 1 |σ M1 (λ 1 (ρ 1 + ǫ)) + κ|ω 1 | (12) Taking κ < α 1 min(M 1 , λ 1 ρ 1 + ǫ) (13)
one can assure the decrease of V 1 i.e. V1 < 0. Consequently,

ω 1 enters Φ 1 = {ω 1 : |ω 1 | ≤ 2ρ 1 } in finite time t 1
and remains in it thereafter. In this case, (ω

1 + ρ 1 q 1 ) ∈ [-3ρ 1 , 3ρ 1 ].
Let M 1 verify the following inequality M 1 ≥ 3λ 1 ρ 1 , equation ( 13) then becomes: κ < α 1 λ 1 ρ 1 + ǫ (14) For t 2 > t 1 , the argument of σ M1 will be bounded as follows

|λ 1 (ω 1 + ρ 1 q 1 )| ≤ 3λ 1 ρ 1 ≤ M 1 (15) Consequently, σ M1 operates in a linear region Γ 1 = -α 1 λ 1 [ω 1 + ρ 1 q 1 ] (16) As a result, (11) becomes V1 = -α 1 λ 1 ω 2 1 -α 1 λ 1 ρ 1 ω 1 q 1 + κω 1 q 1 (17) Choosing κ = α 1 λ 1 ρ 1 which satisfies inequality (14), one obtains V1 = -α 1 λ 1 ω 2 1 ≤ 0 (18)
The same argument is applied to V2 and V3 , (10) becomes

V = V1 + V2 + V3 (19) = -(α 1 λ 1 ω 2 1 + α 2 λ 2 ω 2 2 + α 3 λ 3 ω 2 3 ) ≤ 0 (20)
In order to complete the proof, the LaSalle Invariance Principle is invoked. All the trajectories converge to the largest invariant set Ω in Ω = {(q, ω) : V = 0} = {(q, ω) : ω = 0}.

In the invariant set, J ω = -[α1λ1ρ1q1 α2λ2ρ2q2 α3λ3ρ3q3] T = 0 that is, Ω is reduced to the origin. This ends the proof of the asymptotic stability of the closed loop system.

Remark 1. Since a quaternion and its negative represent the same physical angular position, there exist two equilibrium points: (q 0 = ±1, q = 0, ω = 0). The equilibrium point (q 0 = -1, q = 0, ω = 0) can be considered a repeller point (see [START_REF] Joshi | Robust attitude stabilization of spacecraft using nonlinear quaternion feedback[END_REF]). However, it can be reached using the control law Γ i = -α i σ Mi (λ i [ω i -ρ i q i ]) instead of (8). Therefore, applying Γ i = -α i σ Mi (λ i [ω i + sign(q 0 )ρ i q j ]) (21) ensures that, of the two rotations of angle β and 2π-β, the one of smaller angle is chosen. This can be demonstrated by adapting the previous proof. Remark 2. Note that the stability analysis has been carried out considering the asymptotic condition (7). In the case that the asymptotic condition ( 6) is considered, the control law to be applied becomes

Γ i = -α i σ Mi (λ i [ω i + sign(q e0 )ρ i q ej ]) (22)
where q e represents the attitude error between the current orientation and the desired one.

Some properties of the control law

In practice, the measure of the angular velocity is generally obtained by rate gyros. These sensors can measure the angular velocity in an a priori specified range depending on the sensors and its technology. Therefore, the ability to maintain rotation about a rotational axis without exceeding a certain limit in the angular velocity arises essential. The control (8) allows to take into account the angular velocity constraints, i.e. slew rate limits.

Let ω imax denote the maximum magnitude angular velocity allowed for the rigid body manoeuvre about the i axis. Let us consider the following state constraint |ω i | ≤ ω imax (23) then, one has the following result: Property 1. (Slew rate constraints). The control law (8) globally asymptotically stabilizes the rigid body described by ( 3) and ( 5) to the origin. Moreover, the state constraint |ω i | ≤ ω imax is guaranteed to remain satisfied, for any

t 1 > t 0 if |ω i (t 0 )| > ω imax and for all t if |ω i (t 0 )| < ω imax , setting ρ i = ωi max 2 .
Proof. Assume that the system's initial condition lies anywhere in the state space where ω i belongs to the ensemble Φ i with Φ i = {ω i :

|ω i | ≤ 2ρ i }.
From the proof of Theorem 1, one observes that if ω i belongs to Φ i , ω i remains in Φ i for all time t > 0. Therefore, if ρ i = ωi max 2 , one concludes that |ω i | ≤ 2ρ i = ω imax , for all time t > 0.

In several situations, the body angular velocities exceed the capability of the rate gyros (for instance, in the case that an external disturbance is present). The following propriety establish the robustness of the proposed control law towards bounded angular velocities measurements.

Property 2. (Robustness ). Let ωimax represent the maximum magnitude angular velocity that can be measured by the rate gyro along the i axis. Let ρ i be chosen accordingly to property 1, that is, ρ i = ωimax 2 . Assume that the angular velocity exceeds the rate gyro limits, such as ω i > ωimax . The control inputs (8) globally asymptotically stabilize the rigid body to the origin, in spite of measure saturation. 9). The derivative of (9) after using (3) and ( 5) and the control law ( 8) is given by (10). Analyzing for V1 one gets,

V1 = -α 1 ω 1 σ M1 (λ 1 [ω 1max + ρ 1 q 1 ]) + κq 1 ω 1 (24) Since ρ 1 = ω1max 2
, it follows that |ω 1max + ρ 1 q 1 | ≥ 1 2 ω1max . Because sign(ω 1 ) = sign(ω 1max ), from equation ( 24) and the norm condition on the quaternion, V1 takes the following form

V1 = -α 1 ω 1 σ M1 (λ 1 [ω 1max + ρ 1 q 1 ]) + κω 1 q 1 ≤ -α 1 |ω 1 |σ M1 (λ 1 ( 1 2 ω1max )) + κ|ω 1 | (25) Taking κ < α 1 min(M 1 , λ 1 1 2 ω1max ) (26) one can assure the decrease of V 1 . Consequently, ω 1 enters Φ 1 = {ω 1 : |ω 1 | ≤ 2ρ 1 }, that means |ω 1 | < |ω 1max |.
Hence, the rate gyro is not saturated and the system evolves in the same way that the nominal system (no rate gyro saturation). The same argument is applied to V2 and V3 .

Note that in addition to control torque constraints, the slew rate constraints and the robustness towards bounded angular velocities measurements are established.

APPLICATION TO FOUR-ROTOR HELICOPTER

In this section one shows that the control attitude strategy proposed previously can be stabilize in attitude a fourrotor helicopter. This mini helicopter (see Fig. 1) has four fixed-pitch rotors mounted at the four ends of a simple cross frame. Given that the front and rear motors rotate counter-clockwise while the other two rotate clockwise, gyroscopic effects and aerodynamic torques tend to cancel in trimmed flight. The collective input (or throttle input) is the sum of the thrusts of each rotor (f 1 + f 2 + f 3 + f 4 ). Pitch movement (θ) is obtained by increasing (reducing) the speed of the rear motor while reducing (increasing) the speed of the front motor. The roll movement (φ) is obtained similarly using the lateral motors. The yaw movement (ψ) is obtained by increasing (decreasing) the speed of the front and rear motors while decreasing (increasing) the speed of the lateral motors. This should be done while keeping the total thrust constant. In order to model the system dynamics, two frames are defined: the inertial frame N (x n , y n , z n ) and the body-fixed frame B(x b , y b , z b ) as shown in Fig. 1. For more details the reader can refer to [START_REF] Pounds | Design of a four-rotor aerial robot[END_REF].

1 1 According to [START_REF] Pounds | Design of a four-rotor aerial robot[END_REF] and section 2, the four-rotor helicopter model may be expressed in terms of quaternions as

1 2 1 3 1 4 1 5 1 2 3
ṗ = v (27) v = g N - 1 m C T (q)T (28) q = 1 2 Ξ(q)ω (29) J h ω = -ω × J h ω -Γ G + Γ ( 30
) m denotes the mass of the helicopter, g N is the vector of the gravity acceleration expressed in the N -frame. p = (x, y, z) T and v = (v x , v y , v z ) T represent the position and the linear velocity, respectively, of the origin of the Bframe expressed in the N -frame and ω denotes the angular velocity of the helicopter expressed in the B-frame. Γ ∈ R 3 is the vector of the control torques and T = [0 0 T ] T is the total thrust expressed in the B-frame. The attitude model of the four rotor aircraft differs from the general model (3)-( 5) in the gyroscopic torques Γ G . Γ G ∈ R 3 is the result of the rotational motion of the mini helicopter and the speed of the four rotors. Equations ( 27)-( 30) describe the 6 degrees of freedom of the system and can be separated into translational ( 27)-( 28) and rotational ( 29)-(30) motions.

Four-rotor Torque Control Design

In order to stabilize the attitude of the four-rotor helicopter, equations ( 29)-( 30) are used. The angular motion of the helicopter responds to control torques arising from the linear mapping of the rotation speed of the rotors. Hence, the maximum airframe control torque depends of the much higher rotation speed capability of the motors that are used. In this application, the rotors are driven by DC permanent magnet motors which support a maximum voltage of 9 V. When this voltage is applied to the motor the rotation speed reaches s i,max = 260 rad/sec. Consequently, from the motor characteristics mentioned above and the from aerodynamical characteristics of the propeller, the maximum torque that is applied to influence the helicopter rotational motion is given by Γ1 = 0.40

N m Γ2 = 0.40 N m Γ3 = 0.15 N m
In order to avoid unwanted damages in the actuators and to maximize its effectiveness, the bounded attitude control presented in the previous section is applied to the subsystem ( 29)-( 30). Lemma 1. Consider the four-rotor helicopter rotational dynamics described by ( 29) and ( 30) with the following bounded control inputs

Γ 1 = -α 1 σ M φ (λ 1 [ω 1 + ρ 1 q 1 ]) Γ 2 = -α 2 σ M θ (λ 2 [ω 2 + ρ 2 q 2 ]) Γ 3 = -α 3 σ M ψ (λ 3 [ω 3 + ρ 3 q 3 ]) (31)
where σ M φ,θ,ψ are saturation functions such as M φ ≥ 3λ 1 ρ 1 , M θ ≥ 3λ 2 ρ 2 , M ψ ≥ 3λ 3 ρ 3 with α i , λ i and ρ i positive parameters. Then the inputs (31) globally stabilize the four-rotor helicopter to the origin (q 0 = 1, q = 0 and ω = 0). M φ,θ,ψ and α i are chosen to satisfy the following equations

Γ1 = α 1 M φ Γ2 = α 2 M θ Γ3 = α 3 M ψ
Proof. The steps of the proof are identical to the ones of Theorem 1. Indeed, the only difference lies in the vector Γ G that adds a term canceled because of the relation:

ω T Γ G = ω T (ω × z b ) 4 i=1 J r (-1) i+1 s i = 0
where J r represents the inertia of the rotor.

EXPERIMENTAL RESULTS

The aim of this section is to show the effectiveness of the proposed control approach. For this purpose, an attitude stabilization in real-time was performed on the four-rotor mini helicopter prototype of "gipsa-lab". This prototype is based in the mechanical structure (airframe, motors and blades) of the Draganflyer III developed by RC Toys.

For the elaboration of real-time processing board, it was used an embedded microcontroller (µC) C8051F022 manufactured by Silicon Laboratories, an inertial measurement unit (3DM-G from Microstrain), a bi-directional communication unit (SPM2-433-28 from Radiometrix), a power module to drive the motors (MOSFET transistors) and four optical sensors attached to the ends of the mechanical frame to measure the rotor speed. The power is supplied by a 9 Volts Lithium Polymer battery. On the other hand, a communication unit (SPM2-433-28 from Radiometrix) is linked to a PC, in order to provide and to obtain the process data. The µC acquires the angular velocity and attitude (quaternion) provided by the IMU and it obtains the desired attitude incoming from the ground station. Thus, the µC executes the attitude control law and computes the PWM level to control the four motors. Optionally, the embedded system provides the process data to the ground station, in order to monitor the experiment . The attitude control loop runs at f s = 76Hz, this is due to IMU constraints. Three experiments have been accomplished. At the first and second case the goal was the attitude stabilization at the origin (

q d = [1 0 0 0]. i.e. φ d = θ d = ψ d = 0).
In the third case a command following was performed.

The desired thrust is fixed to T = mg = 4.59N and the maximum torque frame that can be applied is Γ = [0.40 0.40 0.15] T N m. The tuning parameters of the control input are as follows: α i = 1, ρ 1 = ρ 2 = 2.1, ρ 3 = 0.87. The Euler angles are shown, nevertheless, the algorithm is carried out in the quaternion formalism.

• Stabilization

The first experiment tests the control capabilities to stabilize the system at the origin. The obtained results are depicted in Fig. 2 -Fig. 4. According to the angles trajectories, the stabilization is achieved in 1.5 sec, which is a very suitable time. Moreover, the constraints in the control signal are largely satisfied.

• Disturbance rejection

For the second experiment, the robustness of the proposed controller towards disturbance rejection has been tested. The disturbances along each axis (in both directions) are introduced into the system after to achieve the attitude stabilization. As can be seen in the Fig. 5 and Fig. 6, the disturbance produces a large error in the yaw angle as well as in angular velocity ω z . Consequently, the control signal Γ 3 reaches its limit (±0.15N • m) (see Fig. 7) and takes action on the system to overcome the disturbances. A similar rejection disturbance is observed for the pitch and the roll angle, while only feasible control signals are applied to the system. This study case shows that the controller proposed in this paper is robust with respect to external disturbances, and the more important, the control law maximize the effectiveness of the actuators without endanger the system stability. The robustness property is essential in real missions where aerodynamic forces and others factors are presents.

• Command following

The third experiment deals with the attitude tracking. It is observed (see Fig. 8) that the controller is able to following the reference signal r t = 9 • + 9 • sin( 15 π t) along the axis z b . In the case that step signals are considered to be tracked, the problem results in an stabilization problem (see remark 2). This experiment enables to show the performance of the proposed control scheme to tracking a reference attitude, even thought the controller was designed for stabilization.

CONCLUSIONS

In this paper, a new bounded control law for the global stabilization of a rigid body is proposed. The objective of this control approach is to maximize the effectiveness of the actuators without taking a risk in the system stability.

The control design takes into account the slew rate limits, avoiding possible rate gyro saturation. The presented scheme control is especially simple. It is based on nested saturation approach and the attitude is parameterized by the unit quaternion. The proposed approach is extended to the stabilization of a four-rotor helicopter. Thus, the implementation in real-time is achieved. Several tests are performed showing the performance in term of settling time, disturbance rejection and trajectory tracking. Remain to compare the proposed approach with other control schemes. However, owing to simplicity, the proposed control law is suitable for application where on-board computational resources are challenging.

Proof.

  Assume that |ω i | > |ω imax | and sign(ω i ) = sign(ω imax ) because to gyro rate saturation. Consider the candidate Lyapunov function (

Fig. 1 .

 1 Fig. 1. Four-Rotor helicopter configuration: the inertial frame N (x n , y n , z n ) and the body-fixed frame B(x b , y b , z b )

Fig. 2 .Fig. 4 .

 24 Fig. 2. Stabilization: The convergence of the roll, pitch and yaw angles with initial conditions φ = -28 • , θ = 13 • , ψ = -23 • .

Fig. 5 .Fig. 6 .

 56 Fig. 5. Disturbance rejection: The convergence of the roll, pitch and yaw angles, while the system is subject to external disturbances.

17th IFAC World Congress (IFAC'08) Seoul,Korea, July 6-11, 2008

0.1 0.15 0.2 N*m Time (sec)Γ 1 Γ 2 Γ 3 Γ 2 Γ 3 Γ 1 Fig. 7. Disturbance rejection: The bounded control torque signal, while the system is subject to external disturbances. The control constraints are satisfied.17th IFAC World Congress (IFAC'08) Seoul,Korea, July 6-11, 2008