G Tastevin

B Glowacz

P.-J Nacher

Using a cw 465nm laser to probe metastable He 2 molecules *

Keywords: numbers: 33.20.Kf, 42.65.Ky, 67.25.-k Keywords: He 2 metastable molecules, laser absorption, frequency mixing in PP-KTP, spectroscopy in dense He

A cw 465 nm laser has been built for sensitive detection of helium metastable molecules by resonant absorption on the a 3 Σ + u (v=0) -e 3 Πg(v=0) transition band. The frequency-mixing radiation is obtained from commercial laser diodes in a periodically-poled KTP non-linear crystal. The 1.3-nm tuning range includes the main rotational lines of 3 He2, 4 He2, and 3 He -4 He dimers. Measurements of absolute molecular densities down to a few 10 9 cm -3 are reported in low pressure (1-400 mbar) room temperature He gas excited by a weak rf discharge. Unsophisticated detection techniques provide signals with good signal-to-noise ratios thanks to the narrow absorption linewidths (a few GHz, due to Doppler and moderate collisional broadenings) in the fully resolved spectrum. Prospects for use or upgrade of this blue laser to probe the broadened and shifted molecular lines in condensed He are discussed.

I. INTRODUCTION

Metastable molecules (He *

2 ) are ubiquitous in helium systems. They are produced by any excitation process that feeds in enough energy to create excited, ionic or metastable atomic species (electric discharge, bombardment by charged particles, irradiation with ionizing radiation or intense laser pulses), provided that the density of ground state atoms is high enough to allow conversion by 3-body collisions. Extensive work has dealt with room temperature gas for basic studies of this unique dimer and of the complex processes involved in He plasmas, with a renewed interest driven by a variety of applications both at low and high pressures. Low temperatures provide an exceptionally clean environment where the lowest-energy triplet state a 3 Σ + u is very long-lived both in gas and liquid He [START_REF] Mckinsey | Radiative decay of the metastable He2(a 3 Σ + u ) molecule in liquid helium[END_REF]. Emission spectra bring a wealth of information about the processes of formation [START_REF] Tokaryk | Infrared emission spectra from cryogenic proton-irradiated helium gas[END_REF] and decay [START_REF] Benderskii | Photodynamics in superfluid helium: Femtosecond laser-induced ionization, charge recombination, and preparation of molecular Rydberg states[END_REF] of He * 2 . However, only absorption techniques can allow straightforward absolute density measurements. They also provide complementary spectroscopic data [START_REF] Hill | Evidence of metastable atomic and molecular bubble states in electron-bombarded superfluid liquid helium[END_REF][START_REF] Soley | Pressure shifts and quenching of atomic and molecular states produced in electronbombarded liquid helium[END_REF] for quantitative investigation of the structural changes induced in the fluid by the electronic excitation and of the de-excitation channels [START_REF] Benderskii | Photodynamics in superfluid helium: Femtosecond laser-induced ionization, charge recombination, and preparation of molecular Rydberg states[END_REF][START_REF] Kafanov | Structure and dynamics of the He[END_REF][START_REF] Eloranta | Structure and energetics of He2 * bubble-states in superfluid 4 He[END_REF]. Moreover, they potentially provide high sensitivity and time resolution for improved detection, imaging, or tracking of dissolved He * 2 (currently performed using light-induced fluorescence techniques [START_REF] Benderskii | A direct interrogation of superfluidity on molecular scales[END_REF][START_REF] Guo | Metastable helium molecules as tracers in superfluid 4 He[END_REF]).

So far, He * 2 absorption has been probed at low temperatures with fairly low resolution using broad lamps and monochromators [START_REF] Kafanov | Structure and dynamics of the He[END_REF][START_REF] Brooks | Absorption spectra from high vibrational levels of He2[END_REF] or with sophisticated lasers [START_REF] Rellergert | Detection and imaging of He2 molecules in superfluid helium[END_REF] and systematic studies involve only the most prominent lines [START_REF] Eloranta | Structure and energetics of He2 * bubble-states in superfluid 4 He[END_REF]. Here, a compact laser source designed for absolute measurements of molecular density by laser absorption on the a 3 Σ + u (v=0) -e 3 Π g (v=0) vibrational band at 465 nm is presented. Detailed description and characterisation will be provided elsewhere [12]. The present report focuses on the sensitivity of absorption measurements demonstrated in low pressure room temperature He gas. Prospects for upgrade of the laser and for use as a tool to probe He * 2 in cold and condensed He are discussed.

II. LASER SET-UP AND PERFORMANCES

Continuous-wave frequency up-conversion is performed with a commercial 2×1×10 mm 3 crystal of periodically poled KTiOPO 4 (PP-KTP) (Raicol Crystals Ltd.), selected for optimal performance [START_REF] Le Targat | 75%-efficiency blue generation from an intracavity PPKTP frequency doubler[END_REF]. The crystal is phase-matched for frequency doubling at 930 nm. Given the prohibitive cost-to-power ratio of a custom made laser diode (LD) at 930 nm (more than twice higher price per item with only 40-mW output), emission at 465 nm is actually obtained by frequency mixing from two 100-mW single-mode distributed feedback LDs at 923.6 and 937.6 nm available off-theshelf. The difference in wavelengths excludes the usual configuration with incident perpendicular beams collinearly combined using a dichroic mirror. The laser set-up (Fig. 1) therefore includes a set of three dispersive glass prisms that introduce a small individual difference in deflection angle (0.005 • nm -1 at minimum deviation) and several mirrors that provide incident beams with a matched small tilt angle. The power at the output of the crystal reaches 0.23 mW at 465 nm for 75 mW and 80 mW input at 923 and 937 nm, respectively. The effective conversion efficiency (3.9 %W -1 cm -1 ) is in line with expectations taking into account the elliptical shapes of the collimated LD beams (typically 1 mm×2 mm FWHM) [12]. This is slightly lower than the 4.6 %W -1 cm -1 maximum computed for perfect Gaussian beams and optimal beam waist [START_REF] Le Targat | 75%-efficiency blue generation from an intracavity PPKTP frequency doubler[END_REF]. The laser can be tuned over 1.3 nm at a rate of 39 GHz • C -1 by temperature control of the infrared LDs (operating range: 5-45 • C). The crystal temperature must be correlatively adjusted between 20 and 68 • C to preserve optimal phase matching, due to the measured 0.07 nm FWHM (97 GHz) tolerance at fixed PP-KTP temperature [START_REF] Risk | Phase-matching characteristics of periodically poled KTP[END_REF]. The laser is very stable in intensity and its spectral bandwidth, set by the input LDs, is on the order of a few MHz.

¡

III. ABSORPTION MEASUREMENTS

Molecular absorption on the a → e (0 -0) transition is probed in sealed glass cells (length 11 cm, inner diameter 1.6 cm) filled with He gas at pressures ranging between 8 and 400 mbar, in conditions typical for room temperature optical pumping: excitation by rf discharge at 3 MHz using external wire electrodes, with a density of metastable 2 3 S atoms in the 10 9 -10 12 cm -3 range (this density is simultaneously measured using the 2 3 S-2 3 P atomic line at 1083 nm with a commercial distributed Bragg reflector LD [START_REF] Courtade | Magnetic field effects on the 1083 nm atomic line of helium[END_REF][START_REF] Millard | Diode laser absorption measurements of metastable helium in glow discharges[END_REF]). The single-pass transmitted blue light is monitored by a silicon photodiode.

The measured absorption ranges from a few 10 -4 to 10%, depending on pressure and discharge intensity. This corresponds to a He * 2 density in the 10 8 -10 12 cm -3 range [START_REF] Neeser | Lifetime of the He2 (e), He2 (d) and He2 (f ) states[END_REF]. Fig. 2 shows a signal recorded with a signal-to-noise ratio (SNR) of order 2000 in pure 3 He at 96 mbar. The inset displays the set of rotational lines identified in the accessible part of the 3 He * 2 spectrum [START_REF] Dieke | The molecular spectrum of He 3[END_REF]. The intensity distribution corresponds to thermal equilibrium around 300K, with stronger even-numbered lines due to the nuclear spin 1/2 of 3 He [START_REF] Dieke | The molecular spectrum of He 3[END_REF]. The less numerous rotational lines of 4 He * 2 have been similarly probed (data not shown), its band head lying only 0.06 nm below that of 3 He * 2 [START_REF] Dieke | The molecular spectrum of He 3[END_REF][START_REF] Brown | Spectrum and structure of the He2 molecule VI. Characterization of the states associated with the UAO's 3pπ and 2s[END_REF]. Sensitivity easily reaches 10 -5 with a lock-in amplifier when either the gas excitation or the laser output are modulated. Molecular density is here varied by low frequency (a few Hz) modulation of the discharge amplitude, the He * 2 lifetime being ultimately limited to tens of milliseconds by diffusion to the walls (D=310 cm 2 s -1 for 4 He * 2 at 1 Torr [START_REF] Phelps | Absorption studies of helium metastable atoms and molecules[END_REF]). A double-modulation scheme can further improve the SNR. Signal modulation at significantly higher frequencies can be achieved using, e.g., a mechanical chopper to modulate the laser intensity. A cheaper alternative is to periodically drive one of the input LDs above and below laser threshold by strong modulation of its feeding current. For sharp resolved lines, the modulation of one feeding current may also be adjusted so as to bring the laser on-and off-resonance with He * 2 and to obtain a 100% modulation of the absorption rate with moderate impact on the laser power.

465.0 465.5 466.0

Q(2) Q(3) Q(4) Q(5) Q(6) Q(7) Q(8) R(0) Q(1) P (2

IV. DISCUSSION

Well resolved rotational lines are observed, with Voigt profiles that result from combined Doppler and collisional broadenings (Gaussian FWHM W D =3.25 GHz and Lorentzian FWHM W L =3.72 GHz , respectively, for Fig. 2 data). The peak heights can thus easily be related to the absorption rate A and transmission rate T = 1-A. Their reciprocals uniformly scale linearly with W L at small W L and this increase transiently slows down when W L becomes comparable to the average splitting, due the rising contribution of the neighbouring lines. For instance, the Q(2) peak in Fig. 2 is already 2.3 times smaller than it would be at W L = 0, and could be about 40 (resp. 64) times smaller for W L =100 (resp. 200) GHz. The He * 2 number density N m can be directly inferred from the measured optical density using the expressionln T =N m σl, where l is the optical path length and σ is the optical cross section. The cross section σ lies in the 3 -5 10 -15 cm 2 range for the main 4 He * 2 rotational lines at 465 nm [START_REF] Neeser | Lifetime of the He2 (e), He2 (d) and He2 (f ) states[END_REF]. This value is 100 times lower that of the atomic 1083 nm line [START_REF] Courtade | Magnetic field effects on the 1083 nm atomic line of helium[END_REF][START_REF] Millard | Diode laser absorption measurements of metastable helium in glow discharges[END_REF].

The 1.3-nm tuning range of the laser includes the main rotational lines of 3 He 2 , of 4 He 2 , and hence of 3 He -4 He dimers. It thus allows systematic studies of the a → e line that are useful for quantitative comparison with quantum chemistry computations and fundamental to understanding of many collisional and quenching processes. This provides, in particular, accurate tests of the interaction potentials between ground and excited states, especially relevant for investigations on low temperature systems such as quantum He nanodroplets and laser-cooled metastable He [START_REF] Vrinceanu | He(1 1 S)-He(2 3 S) collision and radiative transition at low temperatures[END_REF].

Finding a laser with suitable wavelength and characteristics is often difficult. Today, established up-and downfrequency conversion methods and high quality crystals released in the last 15 years allow experimentalists to build lasers tailored to their needs. Moreover, the limited time and effort required to put together an appropriate system is usually worth the cost difference between the set of optical components and a custom-made source. The cw 465nm laser described in this report has been specifically developed for He * 2 absorption measurements that require very little power and narrow spectral bandwidth. Its innovative design is relevant when available lasers have wavelengths close to, but not exactly at, twice the desired one. Various strategies exist for a straightforward increase of the laser performances. For higher laser output, one can use a ring or standing-wave enhancement cavity to boost up the conversion efficiency [START_REF] Le Targat | 75%-efficiency blue generation from an intracavity PPKTP frequency doubler[END_REF] and use tapered amplifiers [START_REF] Batchko | Continuous-wave quasiphase-matched generation of 60mw at 465nm by single-pass frequency doubling of a laser diode in backswitch-poled lithium niobate[END_REF] or other 930 nm lasers [START_REF] Le Targat | 75%-efficiency blue generation from an intracavity PPKTP frequency doubler[END_REF] to increase the input power. Alternatively, 465 nm light can be obtained by frequency mixing from very different wavelengths combining, e.g., a diode-pumped solid-state laser for high input power and a small power LD to retain the tuning and high-frequency modulation abilities [START_REF] Johansson | Generation of turquoise light by sum frequency mixing of a diode-pumped solid-state laser and a laser diode in periodically poled KTP[END_REF]. When optical saturation of the entire line profile is required, the LD can also be replaced by a diode array (nm-broad) or a fiber laser (bandwidths range from tens of kHz to nanometer) to have more output power and to optimally match the absorption linewidth, taking advantage of the wide phase-matching bandwidth for sum-frequency generation in PP-KTP [START_REF] Risk | Diode laser pumped blue-light source based on intracavity sum frequency generation[END_REF][START_REF] Sakata | Type II quasi-phase-matched sum-frequency mixing of diode lasers in KTiOPO4 with broad spectral and temperature acceptance bandwidths[END_REF]. Of course, different input wavelengths and PP-KTP grating period (or angle tuning) can be selected to probe other visible He * 2 or impurity lines. Following early studies of low pressure microwave gas discharges and applications in microelectronics, recent He work strongly focuses on high pressure plasmas (with or without selected impurities) for use in surface treatment, laser sources, plasma displays, and biology. With room temperature He * 2 densities up to 10 15 cm -3 [START_REF] Neeser | Lifetime of the He2 (e), He2 (d) and He2 (f ) states[END_REF][START_REF] Myers | Rate measurements of reactions of helium metastable species at atmospheric pressures. II -He2 (2 3 Σ + u ) in pure afterglows[END_REF][START_REF] Takao | Optical absorption spectra and kinetic behavior of helium excited diatomic molecule (a 3 Σ + u )[END_REF][START_REF] Emmert | Reaction kinetics of the He(2P) and the He2*(a,v) states in high-density helium[END_REF], a compact 465 nm laser may be a useful high resolution dedicated probe for plasma diagnosis or study of rotational relaxation and collisional transfer rates [START_REF] Takao | Optical absorption spectra and kinetic behavior of helium excited diatomic molecule (a 3 Σ + u )[END_REF][START_REF] Nakata | Collisonal electronic and rotational relaxation of molecular helium in an afterglow of pulsed discharge[END_REF], providing higher sensitivity and resolution than He or Xe excitation lamps [START_REF] Phelps | Absorption studies of helium metastable atoms and molecules[END_REF][START_REF] Takao | Optical absorption spectra and kinetic behavior of helium excited diatomic molecule (a 3 Σ + u )[END_REF] and tunable dye lasers [START_REF] Neeser | Lifetime of the He2 (e), He2 (d) and He2 (f ) states[END_REF][START_REF] Collins | Study of excitation transfer in a flowing helium afterglow pumped with a tuneable dye laser. II. Measurement of the rate coefficient for the rotational relaxation of He2(3p 3 Πg)[END_REF]. The laser can also be used for new investigations in cold He gas where unusual spectral features, differences between isotopomers, and specific molecular dynamics are observed [START_REF] Brooks | Absorption spectra from high vibrational levels of He2[END_REF][START_REF] Kimura | New emission bands at high-density excitation of dense gas and liquid of helium irradiated by nitrogen ions[END_REF][START_REF] Noma | Gas-temperature-dependent generation of cryoplasma jet under atmospheric pressure[END_REF]. Sensitive measurements are expected thanks to the still fairly high He * 2 densities (10 12 cm -3 at 4.2K) and to the nicely resolved absorption lines [START_REF] Brooks | Absorption spectra from high vibrational levels of He2[END_REF][START_REF] Dennis | Spectroscopic identification of excited atomic and molecular states in electron-bombarded liquid helium[END_REF][START_REF] Brooks | Helium emission spectra from doped samples of solid hydrogen and deuterium[END_REF]. For light-induced fluorescence studies, one advantage of the 465 nm line potentially lies in the proximity between the e, d, and f states of He * 2 [START_REF] Neeser | Lifetime of the He2 (e), He2 (d) and He2 (f ) states[END_REF]. In its current configuration, the laser may also be used to probe He * 2 with high spectral and temporal resolution in He clusters or nanodroplets and to get insight on their dynamics inside the fluid. In both cases, sharp and unshifted lines with well resolved rotational structures have been observed, due to the presence of free molecules expelled from the surface [START_REF] Von Haeften | Bubble Formation and Decay in 3 He and 4 He Clusters[END_REF][START_REF] Yurgenson | Detachment of metastable helium molecules from helium nanodroplets[END_REF]. Direct observation is potentially more challenging for He * 2 floating at the surface of superfluid He [START_REF] Hu | Spectroscopy of He * 2 floating on the liquid helium surface[END_REF] or dissolved in bulk liquid, despite the very long lifetimes (up to 13 s [START_REF] Mckinsey | Radiative decay of the metastable He2(a 3 Σ + u ) molecule in liquid helium[END_REF]) and high densities (10 12 -10 14 cm -3 ) reported in this latter system. Significant line shifts and broadenings result from the repulsive interaction between He * 2 and the surrounding liquid, leading to partly resolved rotational spectra [START_REF] Hill | Evidence of metastable atomic and molecular bubble states in electron-bombarded superfluid liquid helium[END_REF][START_REF] Soley | Pressure shifts and quenching of atomic and molecular states produced in electronbombarded liquid helium[END_REF]. A good quantitative description of both systems is required to account for the non trivial observed changes, that strongly depend on initial and final electronic states [START_REF] Kafanov | Structure and dynamics of the He[END_REF][START_REF] Eloranta | Structure and energetics of He2 * bubble-states in superfluid 4 He[END_REF]. Emission and absorption lines shift at markedly different rates with increasing pressure [START_REF] Hill | Evidence of metastable atomic and molecular bubble states in electron-bombarded superfluid liquid helium[END_REF][START_REF] Soley | Pressure shifts and quenching of atomic and molecular states produced in electronbombarded liquid helium[END_REF], in a way that also differs from that observed in He gas [START_REF] Li | Luminescence from liquid helium excited by corona discharges[END_REF]. If most lines experience a blue shift on the order of several tenths of nanometer per bar, the triplet a → b absorption line, for instance, does not shift at all [START_REF] Soley | Pressure shifts and quenching of atomic and molecular states produced in electronbombarded liquid helium[END_REF][START_REF] Kafanov | Structure and dynamics of the He[END_REF]. To our best knowledge, no computations and very few data are available for the a → e (0 -0) line at 465nm. This line is weak but clearly present in fluorescence spectra of liquid He at saturated vapour pressure [START_REF] Benderskii | Photodynamics in superfluid helium: Femtosecond laser-induced ionization, charge recombination, and preparation of molecular Rydberg states[END_REF] and appears quite sharp in high-resolution 4.2K studies [START_REF] Li | Spectroscopic investigation of liquid helium excited by a corona discharge: evidence for bubbles and "red satellites[END_REF][START_REF] Li | Décharge couronne dans l'hélium liquide et gaz dense sous champ électrique intense : pré-claquage, transport de charge, spectroscopie d'émission[END_REF]. So far, no significant shift has been measured for light emission induced by a corona discharge in liquid 4 He (464.95 ± 0.10 nm [START_REF] Li | Décharge couronne dans l'hélium liquide et gaz dense sous champ électrique intense : pré-claquage, transport de charge, spectroscopie d'émission[END_REF]).

Recently, visualisation of He * 2 clouds and measurements of normal-fluid velocity in superfluid He using optically tagged molecules have been demonstrated [START_REF] Guo | Metastable helium molecules as tracers in superfluid 4 He[END_REF]. The detection scheme involves 2-photon excitations of a → d transitions and monitoring of the d → c spontaneous emission. This scheme has also be proposed for applications such as detection of ionizing radiation [START_REF] Mckinsey | Trace detection of metastable helium molecules in superfluid helium by laser-induced fluorescence[END_REF] and imaging of Compton gamma rays [START_REF] Rellergert | Detection and imaging of He2 molecules in superfluid helium[END_REF]. In addition to the two pulsed lasers used for pump and probe excitation, two powerful infrared lasers are required to recover the molecules that end up in off-resonant vibrational levels of the a 3 Σ + u state. The intrinsic loss of sensitivity introduced by the fluorescence detection (1% solid angle) is also partly compensated by the number of cycles each molecule can make during its lifetime (10-100 emitted photons, ultimately limited by the b state decay). Detection by absorption on a single resonant dipolar transition potentially provides a high sensitivity without need of high laser power, sophisticated photodetectors, or large optical access inside the cryostat. The main advantage of this technique is to allow absolute density measurements.

FIG. 2 :

 2 FIG. 2: Resonant laser absorption signal obtained on the Q(2) line in 96 mbar 3 He gas at room temperature. Inset: Observed and identified lines in the 3 He * 2 rotational spectrum. Line intensities have been normalised to the highest peak, Q(2).


prof A Antonello
email: antonello.barresi@polito.it

Barresi

 Extended Kalman filter

Keywords:

This paper is focused on the monitoring of the primary drying phase of the lyophilisation process of pharmaceuticals in vial. Monitoring is required to ensure that the maximum temperature of the product is maintained at a safe value in order to avoid denaturation, and the position of the moving front of sublimation has to be monitored since its evolution gives the state of progression of the primary drying. Furthermore, the information coming from the monitoring system, which include the estimation of the transport coefficients, can be used in a control loop designed to minimise the drying time beside ensuring product quality.

To this purpose, a soft-sensors (observer) has been developed, based on the extended Kalman filter algorithm: it requires a model of the process (a simplified model is used in order to reduce the computational load) and some physical measurements (in this case the temperature of the product at the bottom of the vial, that can be measured by a thermocouple). The main issues arising in the design of this observer have been discussed. A detailed mono-dimensional model experimentally validated has been used at first to compare the results provided by the observer by means of numerical simulations, and then the results obtained in a pilot freeze-dryer are shown.

Introduction

Freeze-drying, or lyophilisation, is the process where water or another solvent is removed from a frozen product by sublimation. The lyophilization technique consists of three main steps:

1.

Freezing: the product to be dried is frozen at low temperature;

2. Primary drying: in this phase the ice is sublimated, generally operating under reduced pressure. Vapour originated at the moving sublimation front flows through the dried material into the lyophilization chamber and a refrigerated trap connected to the chamber -the ice condenser -continuously removes it.

In vial freeze-drying heat is continuously supplied to the product through an heating shelf; this is required in order to compensate for the energy required by the endothermic sublimation process.

3.

Secondary drying: the last stage of the freeze-drying process is a desorption step where the residual moisture, which is strongly bounded by adsorption phenomena to the partially dried cake, is reduced to a low level ensuring long term product preservation at room temperature. This step is usually carried out at high vacuum and moderate temperature ( +20 to +60°C).

In most of the industrial freeze-dryers control actions are often based on empirical information obtained in previous experimental runs carried out with the product of interest. Nevertheless, Guidance for Industry PAT (Process Analytical Technology)

issued by the US Food and Drug Administration in September 2004, encourage to develop in-line monitoring and control tools in order to improve the manufacturing process. The most important parameter to be monitored and to be controlled during primary drying is the temperature of the product. In fact primary drying should be carried on at a temperature below the eutectic point of the crystalline solute, to avoid formation of liquid. Many solutes, such as proteins, do not crystallise during freezing, remaining amorphous. The formation of a glass can be beneficial for the protein activity preservation; but, it makes the drying process more demanding in terms of process time and physical conditions, because the primary drying must be carried out below the glass transition temperature to avoid collapse of the cake structure. Another important variable that would be useful to monitor is the position of the sublimating interface, the evolution of which gives the state of progression of the primary drying. Mass transfer across the porous matrix of the product and heat transfer between the shelf and the vial are two other critical aspects that influence the drying time. Mass transfer is related to the temperature and pressure conditions and to the resistance of the dried cake to the vapour flow. Heat transfer is mainly linked to the composition and pressure of the gas phase in the lyophilisation chamber and to the geometry of the vial in the region of contact with the heating shelf.

The insertion of thin thermocouples in the vials is a widely used method to measure the product temperature. Moreover, if multiple thermocouples are inserted the position of the moving front can be monitored. In fact, as the sublimating interface passes in correspondence of the sensor, a change in the slope of the temperature profile is observed since the thermal conductivity of the frozen and dried layer is different.

However, for practical reasons, only one thermocouple is generally used placed in close contact with the internal bottom of the vial, whose temperature is assumed representative of the whole product. The method proposed in this paper, based on the Extended Kalman Filter (EKF) concept, is able to determine the temperature of the product at any axial position, as well as the dynamic evolution of the sublimating interface, only exploiting the single point measure of the temperature at the bottom of the product. Some parameters related to mass and heat transfer are also estimated, that can be fed to an in-line predictive control system [START_REF] Fissore | In-line control of a freeze-drying process in vials[END_REF]. Furthermore, differently from manometric temperature measurement (MTM) like methods (for a review, see Velardi et al., 2008) , this approach allows continuous monitoring either in one or several single vials placed in different positions, giving a direct in-line measure of the batch variance.

Most of the modern theories on the control of dynamic systems are actually based on a state-space representation. This representation allows for the description of the system behaviour through the variation of its state. The state of a system can be defined as the minimum set of information needed to completely describe the system at a given time instant. The knowledge of the vector of the state is needed in order to apply the command law. This can be achieved by using physical sensors. However, in many cases due to cost consideration and physical constraints, the number and type of sensors could be very limited. An observer, or soft-sensor, combines a priori knowledge about the physical system -the model -with experimental data -the on-line measurementsto provide an on-line estimation of states and/or parameters.

The synthesis of observers for non-linear systems is generally a difficult task and many types of observers have been proposed in the literature. All the approaches proposed are based on the analysis of the observability (sensitivity of the output measurements with respect to state). The observability analysis yields to special forms of systems (called normal form, or canonical form). Among these canonical forms, linear systems up to output injection are systems whose dynamics can be splitted into two parts: a linear one with respect to state and a nonlinear one which depends only on the output measurements [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF]. For these systems, it is possible to design a Luenberger observer. Using a Kalman-like observer, this result has been extended to state affine systems up to output injection [START_REF] Hammouri | Global time-varying linearization up to output injection[END_REF].

Kalman Filter also requires some observability hypothesis in order to guaranty the stability of the filter. For the time-variant linear systems [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF] showed that if the system is completely uniformly observable then the filter becomes stable. Similar hypothesis can be assumed in order to design an extended Kalman observer. A peculiarity of the Extended Kalman Filters is that they can be used without transforming the system on a particular form. However, the convergence is guaranteed only if the initialization of the observer is chosen in a local domain of the unknown state.

In practice this domain may be large. The calibration of the constant matrix gain which appears in the Riccati equation depends on the dimension of the system and the output sensitivity. For small dimension and in the single output case, it is generally easy to obtain this calibration.

The EKF has been chosen for this work as it is well known in the process industries and the most frequently used state estimator [START_REF] Cutler | An industrial perspective on advanced control[END_REF][START_REF] Richalet | Industrial applications of Model-Based Predictive Control[END_REF][START_REF] Qin | An overview of industrial model predictive control technology[END_REF]. In spite of its use, relatively few papers have been published on industrial application of Kalman filtering: some examples can be found in the works by [START_REF] Dimitratos | An experimental study of adaptive Kalman filtering in emulsion copolymerization[END_REF], [START_REF] Krämer | Simultaneous estimation of heat transfer coefficient and reaction heat in semi-batch processes[END_REF], Alvares and Simutis (2004), Bentes Freire and Giudici (2004). In the authors knowledge this is the first published application referring to freeze-drying of pharmaceuticals.

The structure of the paper is the following: in the first section the simplified model used to design the EKF is introduced; then, the observer is given and the results obtained are discussed. Basic concepts concerning the EKF and details on the calculations are summarised in Appendix A and Appendix B.

The simplified model

The simplified model that is used to design the EKF was previously shown by [START_REF] Velardi | Freeze-drying: Modelling, monitoring and control[END_REF] and Velardi and Barresi (2008). It is based on the energy balance for the frozen product and the mass balance for the water vapour inside the dried product, both taken in pseudo-stationary conditions because of the slow dynamics of the process. The presence of an inert gas is neglected as well as heat transfer in the vial glass. The temperature at the bottom of the product is assumed to be the physically measured variable of the process. The reference geometry is shown in Figure 1. The final form of the simplified mono-dimensional model is given by:

      1 , , 1
w i i w c II Ie i k dH M p T p dt RT H    (1)       1 1 , , 1
s shelf i w i i w c v II i H M k L H T T p T p K k RT H                    (2)     1 1 1 II shelf shelf i v v II z H T z T T T K K k                   (3)
The key parameters of the model are the mass transfer coefficient in the dried layer ( 1 k )

and the heat transfer coefficient between the shelf and the bottom of the product ( v K ).

The non-linear equation (2) provides a relationship between the interface temperature, i T , and the position of the interface between the frozen and the dried region, H , while Eq. (3) gives the temperature profile of the product along the axial coordinate z . Thus, the value of the product temperature at the bottom of the sample ( B T ), that is the measured variable of the process, can be determined setting z L  in Eq. (3). Eq. (1) describes the dynamics of the interface, relating it to the sublimation flux.

The proposed model is a quite simplified one, since heat transfer in the dried layer is not considered and pseudo-steady state is assumed in the frozen layer, leading to the approximation that all the energy transferred to the product is used for sublimation.

Furthermore the vial sides are assumed to be insulated from radiation and this makes this model not suitable for the vials located near the edge of the tray or in proximity of the walls of the chamber.

The adequacy of this model to describe the dynamics of the primary drying of a freeze-drying process has been proven by means of comparison with the results obtained using a detailed one-dimensional model experimentally validated (Velardi and Barresi, 2008).

Kalman filter design and validation

In this section an Extended Kalman Filter type observer is synthesised using the simplified model previously developed. The target observer should estimate the dynamics of the interface temperature at any time during primary drying, by using the temperature at the bottom of the frozen product as measured variable. The effective diffusion coefficient in the dried layer, that affects mass transfer, and the heat transfer coefficient between the shelf and the vial bottom, are considered as unknown parameters and are estimated too.

In order to develop an observer with the required structure [that will be shown by Eq. ( 19)-( 22)], the system must be put in the general state-based form:

    ,u t   x f x v  (4)     , y h u t   x w (5)
where u and y are the scalar input and output of the system ( 4) respectively (because a single manipulated input and a single measure are adopted), and   Since it is proposed to estimate the dynamics of the interface temperature i T , and the values of the diffusion coefficient 1 k and of the heat transfer coefficient v K , the state vector of the process can be defined as:

    1 2 3 1 T T i v x x x T K k   x and shelf u T  (6)
while the state equation of the measure, giving the frozen product bottom temperature B T , can be expressed as:

    1 2 3 B 1 , , , , , , i v shelf y h x x x u T T K k T   (7)
where the shelf temperature shelf T is taken as process input. Thus, the dynamic system to be solved can be written as:

    1 2 3 1 1 1 2 3 2 3 , , , 0 , 0 dx dt dx dt dx dt x f x x x u x u x                                                                  x f x     (8)   , y h u  x (9)
where the first derivatives of 2 v x K  and 3 1 x k  are set equal to zero because they are considered constant parameters of the process. Of course this is an approximation, but it is correct if we assume that the observer can almost compensate for the variation of v K and 1

k through the effect of the term    19) in the followings], that corrects the dynamics of the state estimates x  . The dynamic equation ( 1) of the simplified model only provides the evolution of the position of the moving front H, but the form of system ( 8)-( 9) requires an expression for the derivative of 1 i

t y y  K [see Eq. (
x T  with respect to time. This can be calculated by differentiating with respect to time the function

    1 , , , , i v shelf H H T K k T H u   x , obtaining: 1 1 1 dx dH H du H dt dt u dt x                             (10)
It follows from Eq. ( 10) that in order to determine the expression of 1 dx dt , the partial derivative of H with respect to 1 x has to be calculated; thus the expression of H as a function of the state is needed and it can be determined by rearranging equation (2), finding:

H      (11)
where the following substitutions have been made:

      3 1 3 , 1 , 1 , s w i w c H M x x x p x p R x       (12)     1 1 , II x u k u x      (13)   2 2 II k x L x      , ( 14
)
defining in addition:

  1 s II Ie H       (15)
Adopting the substitutions ( 12)-( 15), the expressions for dH dt , 1 H x   and H u   are obtained, the last one determined by derivation of Eq. ( 11):

dH dt       (16)   2 H d u du                     (17)   2 1 1 1 H d x x dx                           (18)
The combination of the three previous equations according to Eq. ( 10), finally yields the dynamic evolution of the interface temperature 1 i x T

 . Thus, with the previous notation the estimate state

  1 2 3 ˆˆˆ, , x x x x  of (x1, x2, x3
) is given by the EKF dynamic system:

                      1 1 3 2 ˆ1 ˆˆˆ0 0 ˆ, d x dx d du du dt t r t y t y t u r t y t y t                                                                     x x x K f x K  (19)   2 ˆˆ, ˆŷ u h u x        x (20)     1 ˆ, T u h t t                 x K S x (21)           ˆˆˆ, , , ,
T T u u u u t t t t t r                                       x x x x f f h h S S S S S x x x x  (22)
where Eq. (12,13,14) for these expressions),

  1 3 ˆ, x x    ,   1 ˆˆ, x u    ,   2 x    (see

 

y t is the on-line physical output measurement,  is a symmetric positive definite matrix, and r is a positive tuning parameter. Once the state of the system is known, the other variables of interest for process monitoring and control can be easily evaluated: in fact, the position of the interface, which gives the progress of the drying operation can be calculated by Eq. ( 11), while the sublimation flux can be obtained by the differentiation of the previous variable, or better from Eq. (11,12), noting that it is simply given by        s H [compare also Eq. ( 1)]. Further details concerning the expression of the Jacobian matrix   f x and of the gradient vector   h x are provided in Appendix B.

The proposed observer has been validated both by means of simulations and through experimental data obtained in a pilot-scale freeze-dryer. In the first case the detailed mono-dimensional model developed by Velardi and Barresi (2008) equal to 20 mm.

Figure 2 shows the observer predictions compared to the simulated temperature at the interface and at the bottom of the product; in Figure 3 the time evolution of the moving front position, calculated from Eq. ( 11), is plotted. The agreement between the estimation and the real original values is generally good, except for the first part of the drying period, where the performance of the observer are poorer.

Poor initial performances are mainly related to the goodness of the initial approximation of the state of the system, and in part are due to the simplifications introduced by the reduced model. In fact, it should be pointed out that the observer was developed starting from a model that, among the other simplifications, does not take into account the accumulation term in the product; this can play an important role in the first part of the drying cycle, when the temperature decreases due to sublimation at the very low pressure used in simulations and the heat supplied from the shelf is not able to balance the temperature diminution. When the transient effects are no longer important and the estimated dynamics approaches the actual one, the observer can provide a good estimate of the temperature at the interface and of the front position as well all along the rest of the process, giving practically coincident final drying times.

Concerning the parameters v K and 1 k , some considerations can be made by looking at Figure 4, that shows the estimates of the two parameters along the drying cycle. Comparing the values of Table 2 with the estimated ones, it comes out that the effective diffusivity k1 is well estimated in all the four cases considered, except for the initial part of the drying cycle. Instead, the heat transfer coefficient v K is slightly overestimated by 15-20%. This is due to the simplifications introduced in developing the simplified model: in fact, when considering the complete transient model, the effect of heat transfer in the vial glass is taken into account; thus, the energy coming from the shelf is provided to the product mainly at the bottom of the sample, but to some extent is transferred to the product from the vial sides too as a consequence of conduction through the glass. In the simplified model the presence of the vial is not taken into account and the energy coming from the shelf reaches the product only through the bottom. This results in an effective value of v K , higher than the actual one, weighing up the additional heat input due to heat transfer from the vial sides. The way to calculate this effective v K has been shown in Velardi and Barresi (2008). It must be evidenced that the estimated effective values agree with the expected ones, calculated as said above; in addition, the fact that the observer estimates an effective Kv is not a limitation in model-based control applications, because in this case a simplified model is generally adopted, that uses this effective coefficient.

Figure 5 shows the predictions of the observer when a noisy measurement of B T is simulated. For this purpose, a normal white noise characterised by a standard deviation of 1 K was added to the output signal simulated through Eq. (3). It can be seen that the interface temperature (l.h.s) continues to be well estimated, while a different behaviour is observed concerning the position of the ice front (r.h.s). The noise of the measure is in this case amplified, making the H provided by the EKF rather noisy, even if the global trend of the curve is followed.

As discussed before, the tuning of the EKF requires to select the terms of the matrix  , which must be a semi-positive definite matrix, and the value of the tuning parameter r . A possible choice consists in assuming a diagonal form for  , with the positive diagonal terms to be selected by trial and error. In the case of our system,  is a 3 3  matrix and the selection of the three diagonal terms was made carrying out different simulations in the conditions of case 1 of Table 2, while r was chosen equal to 1. The selected  was then used for the other simulated cases, showing that the observer robustness is good, when a suitable  has been defined.

Preliminary experimental results obtained in a Telstar Lyobeta25 freeze-dryer (chamber volume = 0.2 3 m ) confirm the goodness of the state estimation and the influence of the measurement noise on the evaluation of the front position. The apparatus used is a special prototype that allows to monitor the product temperature and to estimate the process parameters through the DPE method (Velardi et al., 2008;Barresi et al., 2009a). The DPE operating principle is based on the pressure increase measured during the so-called pressure rise test (PRT) timely run during primary drying, isolating for a few seconds the dryer chamber from the condenser. Thus, DPE provides at discrete times average estimations of the state of the system, since all the vials of the batch contribute to the pressure increase.

The product used in the experimental cycles consisted of a 10% w/w sucrose solution, distributed in 636 tubing type vials (internal diameter = 14.25 mm, filling volume = 1 mL). Four vials located in different zones of the tray were equipped with a thermocouple placed in contact with the vial bottom. The product, initially frozen at -42°C, was maintained at a total pressure of 10 Pa during the primary drying.

It has been observed that the performance of the observer can be improved if it is coupled with the DPE tool. In fact, especially at the start-up, the error of the estimation requires a certain time to converge to zero; but if the values of the interface temperature, the heat transfer coefficient and the mass transfer resistance calculated with the pressure rise test carried out at the beginning of the of primary drying are used to initialise the observer equations, the time requested by the Kalman filter estimation to converge to the actual solution is minimised.

In Figure 6a the measured bottom temperature is compared with the estimated values of B T and i T . It must be pointed out that the temperature of the moving interface can not be physically measured; thus, the performances of the observer were judged comparing the values of the physical measure B T with the corresponding estimation, provided by Eq. ( 20), verifying that the dynamics of the measured variable was retrieved satisfactorily.

As anticipated in the Introduction, the state estimation obtained with the method presented in this paper is limited to a single vial. Nevertheless, several vials of the batch can be monitored. Figure 6b shows the evolution of the interface temperature estimated in four different vials, compared with the batch average front temperature determined through the DPE method described above. The figure shows that the results of the two methods are consistent, and highlights at the same time the difference in the type of information that can be obtained. The use of several observers can allow the estimation of the variance of the batch; in fact, ice nucleation is a stochastic process, and thus different ice structures and different sublimation rates can results. In addition, the effect of temperature differences in the shelf, and eventually a different contribution from wall radiation, can results in different thermal histories for vials in different positions [see for example Barresi et al. (2008a) and [START_REF] Pisano | Modelling and experimental investigation of radiation effects in a freeze-drying process[END_REF], for an experimental evidence of this]. In this case vial #2 has a behaviour very close to the average of the batch; this is reasonable as it was placed in the centre of the tray and, apart for intra-batch intrinsic variability, this makes it the most representative for the average batch. This is confirmed in Figure 6d, where the moving front positions obtained with the two methods are compared and good agreement can be observed too.

The estimation of the front position deserves some additional comments concerning the effect of the noise in the temperature measurements; the simulation work had already highlighted this problem, evidencing that is was affecting more seriously the estimation of the front position, while the interface temperature is only slightly affected (see Fig. 5). The experimental results confirmed this conclusion, evidencing that the noise was amplified up to the point that only the general trend of the interface evolution could be catch adopting Eq. ( 11) for the direct evaluation of the interface position. A much better results was obtained on the other hand calculating the sublimation rate first, using Eq. ( 12) and the estimated state of the system, and then the position of the interface by integrating it over time; the results shown in Fig. 6d have been obtained by this way.

Finally, Figure 6c shows the temperature profile of the frozen product in the four monitored vials, taken after about 1.5 h of drying. In fact, it must be pointed out that the proposed observer is able to estimate the entire temperature profile of the product in the vial and not only the temperature in a particular point, as commonly obtained using a thermocouple.

As concerns the value of the other two state variables, the heat and mass transport coefficients, an almost constant value v K = 7

-2 -1 -1 J m s K is estimated by the EKF, which is consistent with that calculated by means of literature correlations [START_REF] Pikal | Mass and heat transfer in vial freeze-drying of pharmaceuticals: role of the vial[END_REF];

the relatively low value depends on the fact that the product was placed on a tray, that strongly increases the resistance to heat transfer. An higher variability through the monitored vials is observed in the estimated value of mass diffusivity 1 k : in particular a 3 1.1 0.3 10   

2 -1 m s average value is obtained for vials #1 to #3, while an higher value is obtained for vial #4 (7.2•10 -3 m 2 s -1 ). The values obtained are within the range generally reported in literature, and the results evidence a large intra-batch variance, that can have also been favoured by inserting the thermocouple in the product; in fact, during freezing the thermocouple can favour nucleation at lower supercooling, generating larger ice crystal and thus a final matrix with higher conductivity. It is also possible that some compensation between the parameters occurs in the optimisation procedure that leads to the state estimation; this is not a very serious drawback for control applications, if a similar model is used for prediction and state estimate.

Conclusions and final remarks

In-line monitoring of the primary drying phase of a freeze-drying process in vial has been addressed in this paper. It has been pointed out that the most important parameters to be monitored during primary drying are the temperature of the product, in order to avoid collapse, and the position of the moving front of sublimation, that gives the progress of the primary drying phase. The use of a completely predictive approach through mathematical simulation may not be feasible, since mathematical models of freeze-drying need many parameters, primarily the values of the heat and mass transfer coefficients, that often are not known a priori. Thus, from the simplified model showed in the first part of the work a non-linear observer was synthesised, capable to infer inline the evolution of the moving front temperature and position, without any knowledge of the heat and mass transfer related parameters. The performances of the developed observer, based on the Extended Kalman Filter concept, were validated with a detailed mono-dimensional model and tested with experimental data.

The proposed observer, based on the single point measure of the product temperature at the bottom of the frozen layer, has proven to be able to estimate the state of the system even starting from not perfectly known initial conditions. To this end, it is important to note that the EKF is an intrinsically time-variant non-linear estimator for which there is generally no guarantee to work properly like in the linear case.

Nevertheless the filter works well for many application, as for the one presented in this paper, once a proper tuning of the observer has been made.

Differently from methods based on PRT, that provide an average estimation of the state of the system, the estimations of the developed EKF are limited to a single vial.

Nonetheless, several vials placed in different position in the vacuum chamber could be monitored contemporaneously and independently to allow the evaluation of the heterogeneity of the batch.

Poor process control is a major limitation of freeze-drying where control actions are often based on empirical laws and data collected off-line. Besides monitoring, the method proposed in this paper can be employed to realize an efficient control of the process. In fact, the parameters estimated by the observer could be used to close the loop, enabling an in-line control system capable to manipulate the temperature of the heating shelf in order to maintain the product at a safe temperature level, avoiding product denaturation. Examples of possible application, also in conjunction with the PRT methods, and in particular with the DPE method, are given in [START_REF] Fissore | In-line control of a freeze-drying process in vials[END_REF] and [START_REF] Barresi | In-line control of the lyophilization process. A gentle PAT approach using software sensors[END_REF]. It can be noted that coupling with DPE could be particularly advantageous even just for initialization; in fact, it has been shown experimentally that this can guarantee faster convergence to the exact solution, and thus may significantly improve performance and convergence of the observer.

The use of this type of observer for industrial applications is potentially very interesting; in fact it is possible to realize a wireless system to measure the temperature of several vials and send the signal in real-time to an acquisition system placed outside the freeze-dryer [START_REF] Barresi | Understanding, monitoring and control the freeze drying process by means of model based tools and Computational Fluid Dynamics[END_REF][START_REF] Barresi | In-line control of the lyophilization process. A gentle PAT approach using software sensors[END_REF][START_REF] Fissore | Applying Process Analytical Technology (PAT) to the lyophilization process[END_REF]. Thus, this apparatus is suitable to be employed even in very large industrial freeze-dryers equipped with automatic loading and unloading systems. The hypothesis given by [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF] for time variant linear systems consists in assuming that the following artificial linearized system along      

1 ˆ, ˆˆ, , T meas u t t t r t t t t t                    x h x f x u S h x u y v x  (A3)           ˆˆˆ, , , ,
T T t t t t t r                                       x u x u x u x u f f h h S S
                2 1 ˆˆ, , , , T u u u t r t t t t t                         x x x f h h e S
ˆ, t t x u :     ˆ,u t t    x f ξ ξ x  (A6)     ˆ,u t t     x h Y ξ x (A7)
is completely uniformly observable. Under this hypothesis, there exist two constants

1 0 c  and 2 0 c  such that       2 2 1 2 T c t t c t   e
e S e e

. Set         T t t t t  V e S e ,
after some calculations, we can show that there exist some constants 0 c  , 0 c   , and 0 c   such that:

                                                      2 , T T T T T t t
; if     0 t t   v w , then   t   e
, where  is a constant which depends on the upper bounds of v and w . If v and w are small, so is for  .

Appendix B -Details of Jacobian and gradients calculations.

The Jacobian matrix   f x and the gradient vector h  x , needed to compute the equations of the dynamic system ( 19)-( 22), have the following form:

ˆ, 1 2 3 ˆ, u u h h h h x x x                      x x x (B1) 1 1 1 1 2 3 0 0 0 ˆ, 0 0 0 ˆ, f f f x x x u u                            x x f x (B2)
The components of the gradient of   , h u x can be calculated by derivation of equation ( 20), obtaining:

2 1 1 1 2 2 2 2 3 3 3 1 1 1 1 T T h x x x x h h x x x x h x x x                                                                                                                                                              x (B3)
while the components of the first row of the Jacobian of  

,u f x are derived from equation ( 19), getting:

  2 2 2 2 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 2 1 1 1 1 1 1 3 1 x x d d x dx x dx x x d d x dx x dx f d x x dx d du du dt x                                                                                     (B4)   1 1 2 1 2 2 2 1 d x dx f x x                                           (B5)   2 1 3 1 3 1 1 1 1 2 1 3 1 3 1 1 1 1 2 1 3 3 3 1 3 1 x x x x d d x dx x dx x x x x d d x dx x dx f x x d du du dt x                                                                 (B6)
where the derivatives of the functions  

1 3 , x x  ,   1 , x u  , and   2 x 
can be determined by the trivial derivation of Eq. ( 12)-( 14):

  w,i 1 c w,i 3 1 1 1 1 s p x p dp H M x x R x x dx                     (B7)   w,i 1 c 3 1 s p x p H M x R x       (B8) 2 2 w,i 3 2 2 1 1 1 1 1 2 s d p H M x x x x R x dx           (B9) 2 1 3 3 1 1 x x x x         (B10) II 1 k x      (B11) II k u     (B12) 2 2 1 0 x     (B13) II 2 2 2 k x x      (B14) Notation   t e
error of estimation of the observer f vectorial function giving the derivatives of the state h vector of equations giving the state space equations of the measures

H moving front position, m s H  enthalpy of sublimation, -1 J kg k thermal conductivity, -1 -1 -1 J m s K 1 k effective diffusivity coefficient, 2 -1 m s   t K observer gain v K overall heat transfer coefficient, -2 -1 -1 J m s K L total

  on the state dynamics and on the output measurements. Details about the EKF and its convergence are summarised in Appendix A.

  bounded functions on any bounded set of e , v , and w .

Figure 5 T

 5 Figure 5Kalman filter predictions when a noisy measure is used. Condition of Table2, case 1. Left: comparison between true and estimated temperatures. Solid line (  ):

Figure 6

 6 Figure 6Freeze-drying of a 10% w/w sucrose solution at 10 Pa. (a) Comparison between experimental and estimated temperatures. Solid line (  ): estimated i T ; dashed line ( ): estimated B T ; open circles (  ): experimental B T . (b) Solid line (  ): estimated front temperature in four vials placed in different positions on the tray; solid circles (  ): i T determined through DPE method. (c) Temperature profile along the frozen product in the vials monitored as for graph (b) at t=5000s. (d) solid line (  ): front position estimated from temperature measurements of graph (a); solid circles (  ): front position determined through DPE.

  Figure1

  has been used as a source of simulated experimental data. We remind that this model is fully transient and takes into account the effect of accumulation in the vial glass. The main parameters used in the simulations are given in Table 1. The detailed model has been used for simulations in four different conditions, corresponding to the couples of values of v K and 1 k reported in Table 2, thus obtaining the set of simulated measured values of the temperature at the bottom of the product. The sample thickness has been taken

  Comparison between true and estimated moving front position for the case studies of Table 2. Solid line (  ): H Kalman filter estimation; solid circles (  ): H detailed model. Values of the parameters v K and i k for the case studies of Table 2, estimated by Kalman filter.

	 List of Figures	mass density,	kg m	-3
	Figure 1				Schematic of the vial.
		Subscripts
	Figure 2	I			layer I, dried layer Comparison between true and estimated temperatures for the case studies of
		II	layer II, frozen layer Table 2. Solid line (  ): i T Kalman filter estimation; solid circles ( ): i T detailed
		c e			chamber effective model; open circles (  ): B T detailed model.
	Figure 3	i meas	interface measured
		shelf	heating shelf
		w	water vapour
	Figure 4				product thickness, m
		M Superscripts molecular weight,	kmol kg	-1
		p ^observer estimate pressure, Pa
		R 		ideal gas constant, first time derivative J kmol K -1
		r			tuning parameter
		S	  t	matrix giving the solution of the dynamic Riccati equation
		t			time, s
		T			Temperature, K
		B T	frozen layer temperature at z L  , K
		u		vector of the control variables
		v	    , t t w white noises
		x		state space vector
		y		vector of the measured outputs of the system
		z			axial coordinate, m
		Greeks
		, , ,     variables defined by equations (12)-(15)



matrix of tuning parameters for the Kalman observer

Table 1

 1

	Table1
	Parameter	Value	Unit
	p		1.08	Pa
	shelf T	263.15	K
	k	II		2.56	W m -1 K -1
	Ie 		328	kg m -3
	II 		1030	kg m -3
				exp[-20.9470(273.156/T-1)
	  i p T	-3.56654 ln(273.156/T) +2.0189(1-273.156/T)	Pa
				-5.0983]
	H 	s	2687.4•10 -3	J kg -1
	L		2•10 -3	m

Acknowledgement

The authors would like to acknowledge E.U. for financial support in the framework of the research project LYO-PRO: Optimization and control of the freeze-drying of pharmaceutical proteins (GROWTH Project GRD1-2001-40259-RTD).

The authors like to thank Dr R. Pisano for partecipating to the experimental validation of the observer and Dr. D. Fissore (Politecnico of Torino) for the contribution to the revision of the manuscript.

Appendix A -EKF generalities and convergence conditions.

A dynamic system is here defined by the following set of differential equations:

 giving the derivatives of the state depending on the state itself and on the control law u applied to the system, and v is a noise on the state.

In order to build an observer for a given system we need to add to the system (A1) some information concerning the physically measured variables:

where the components of the vector q  y  are the output signal, and w is a noise on the output. The Kalman Filter is essentially a set of mathematical equations that implement a predictor-corrector type estimator that is optimal in the sense that it minimizes the estimated error covariance when some presumed conditions are met; the extended Kalman filter (EKF) is based on the linearization of the non-linear functions of the process. We recall here the condition which guaranties the convergence of the filter. The EKF is given by the following dynamical system: Table 1 Values of the parameters used in the simulations.

List of Tables

Table 2

Values of the transport coefficients used in detailed model simulations.


Is there an Intermediate Phrase in French?

Amandine Michelas and Mariapaola D' Imperio Aix-Marseille I et Laboratoire Parole et Langage, CNRS, Aix-en-Provence, France michelas@lpl-aix.fr, mariapaola.dimperio@lpl-aix.fr

The existence of an intermediate level of phrasing (ip) has been shown for several Germanic as well as Romance languages, such as English [START_REF] Beckman | Intonational Structure in Japanese and English[END_REF], Italian (D'Imperio, 2002), Catalan [START_REF] Feldhausen | The Prosody-Syntax Interface in Catalan[END_REF], as well as in non-Indoeuropean languages such as Cairene Arabic (Helmuth, 2007).

Within the autosegmental-metrical framework of intonation [START_REF] Pierrehumbert | The phonetics and phonology of English intonation[END_REF][START_REF] Ladd | Intonational Phonology[END_REF], Jun & Fougeron have proposed a model of French intonation based on two units: the Intonation phrase (IP) and the Accentual Phrase (AP; [START_REF] Jun | The accentual phrase and the Prosodic structure of French[END_REF][START_REF] Selkirk | The interaction of constraints on prosodic phrasing[END_REF]. Within this model, the existence of an ip, a prosodic unit whose rank in the prosodic hierarchy is lower than the IP and higher than the AP, has also been proposed, though its status and effects on prosodic structure are still controversial. According to [START_REF] Fougeron | A Phonological model of French intonation[END_REF] this constituent is marked by a L-or a H-edge (depending on the illocutionary value of the utterance) tone, though its distribution seems to be mainly restricted to the intonation contour of early focus utterances. Further evidence for an ip in French has been found in tagquestions, dislocated theme/rheme structures and wh-questions. [START_REF] Fougeron | A Phonological model of French intonation[END_REF]. New empirical data collected on French seems to support the existence of the ip in French. (Di Cristo, to appear;D'Imperio et al., 2007;[START_REF] Portes | Acoustic properties of prenuclear Accentual Phrases in French: evidence from the initial rise[END_REF].

In the last decade, a large body of evidence has shown that the placement of intonational boundaries in various languages is not exclusively dependent on syntactic constraints, since factors such as information structure, constituent weight and speech rate play a major role in phrasing decisions [START_REF] Nespor | Prosodic Phonology[END_REF][START_REF] Ghini | Phi-formation in Italian: a New Proposal[END_REF][START_REF] Selkirk | The interaction of constraints on prosodic phrasing[END_REF]. Within Romance languages, the role of prosodic branchingness has been underlined [START_REF] D'imperio | Intonational phrasing and constituent length in Romance[END_REF]. Finally, the role of alignment constraints on the placement of prosodic boundaries has been underlined [START_REF] Selkirk | Sentence prosody: intonation, stress, and phrasing[END_REF][START_REF] Truckenbrodt | On the relation between syntactic phrases and phonological phrases[END_REF].

Our assumption is that the emergence of an intermediate prosodic level (ip) in French is not simply linked to a specific focus or marked syntactic structure. Our hypothesis is that an ip boundary might appear within a broad focus utterance when the syntactic structure allows it. We specifically predict that both ALIGN-XP,R and WRAP-XP conspire to place an ip boundary in correspondence with a major syntactic phrase boundary (Fig. 1). In other words, the boundary between NP and VP might be signaled by prosodic cues that are stronger than the ones associated to an AP boundary which is not associated with a major syntactic break (Fig. 2). This alignment between syntactic and prosodic structure could involve the emergence of an ip boundary marked by mainly by a H-edge tone as well as preboundary lengthening.

We predicted that the higher the unit in the prosodic hierarchy, the stronger the prosodic cues at its boundary. 100 experimental sentences were presented to 10 French listeners in a reading task (100*4*10=4000 sentences). The sentences were read in both normal and fast speech rates, for a total of 8000 utterances. We measured several target syllables in 4 contexts: (1) within a prosodic word, (2) at an AP (ip-internal) boundary, (3) at a potential ip boundary and (4) at an IP boundary.

In line with our predictions, the results show that preboundary syllable length increases with prosodic boundary strength, in that longer syllables were measured at IP boundaries (Fig 3), and an intermediate degree of lengthening was found at ip boundaries. Our results suggest that prosodic cues are reinforced when there is an alignment between prosodic and syntactic boundaries and support the existence of an intermediate prosodic level in French.

FFig

  Fig 1: Noun ph edge is associate

Figures

  Figures


Thierry Duval
email: thierry.duval@irisa.fr

Anatole Lécuyer
email: anatole.lecuyer]@irisa.fr

Sébastien Thomas

SkeweR: a 3D Interaction Technique for 2-User Collaborative Manipulation of Objects in Virtual Environments

Keywords: CR Categories: H.5.2 [Information Interfaces and Presentation]: User Interfaces -Evaluation/methodology, Input devices and strategies, Interaction styles, User-centered design, H.5.1 [Information Interfaces and Presentation]: Multimedia Information Systems -Artificial, augmented, and virtual realities, Evaluation/methodology, H.1.2 [Information Systems]: User/Machine Systems -Human factors, Human information processing, I.3.6 [Computer Graphics]: Methodology and Techniques -Interaction techniques, Ergonomics 3D interaction, Multiple Users, Collaborative Work, Manipulation, Skewer

This paper describes a novel 3D interaction technique called the "SkeweR", dedicated to the 2-user collaborative manipulation of objects in virtual environments. This technique enables two users to move simultaneously the same virtual object in 3D. For this aim, each user manipulates the object by one crushing point, like handling the extremity of a skewer. The SkeweR uses only translation information from the users' motions to change both the position and orientation of the manipulated object. By using more crushing points, this technique could easily be extended to 3 or more users. Thus, the SkeweR technique could be used to improve the collaborative manipulation of objects in numerous applications of Virtual Reality, such as: virtual prototyping, maintenance and training simulations, architectural mock-up reviews, etc.

INTRODUCTION

Manipulation of objects is one of the most fundamental tasks of 3D interaction in Virtual Reality (VR) [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF]. In addition, the collaborative manipulation of virtual objects by multiple users seems to be a very promising area for Collaborative Virtual Environments (CVE) [START_REF] Benford | Collaborative Virtual Environments[END_REF]. The collaborative manipulation of objects seems indeed necessary in many different applications of VR such as for virtual prototyping, training simulations, assembly and maintenance simulations, etc [START_REF] Ruddle | Symmetric and Asymetric Action Integration During Cooperative Object Manipulation in Virtual Environments[END_REF].

In such collaborative tasks, all the users participate to the motion applied to the object manipulated in the VE. The separate motions of the different users are used to define the final motion of the virtual object. However, probably due to the complexity of the current VR interfaces, no universal solution has already been proposed for two distinct users to apply a motion to a unique manipulated object.

Therefore, in this paper we introduce a novel interaction technique for two users to move the same virtual object simultaneously in a VE. The technique is called the "SkeweR" technique since each user manipulates the object by one crushing point, like handling one extremity of a skewer. When using the SkeweR technique, the final motion of the manipulated virtual object is based on a combination of the two translation motions of the users.

The following paper begins with a description of related work in the field of collaborative manipulation of objects in virtual environments. Then it describes the concept of the SkeweR technique and its implementation. The paper ends with a conclusion and a description of potential perspectives.

RELATED WORK

The most frequent technique used in virtual environments for two users to interact with the same shared object remains unfortunately to allow only one user to manipulate the object at a time. However, recent studies upon highly collaborative manipulations have proposed novel 3D interaction techniques, which enable several users to interact with a shared object simultaneously [8] [9].

Figure 1: principle of the average technique of Ruddle et al. [START_REF] Ruddle | Symmetric and Asymetric Action Integration During Cooperative Object Manipulation in Virtual Environments[END_REF] (P stands for position, O stands for orientation)

Ruddle et al. [START_REF] Ruddle | Symmetric and Asymetric Action Integration During Cooperative Object Manipulation in Virtual Environments[END_REF] have proposed several solutions to combine the movements of both users in order to obtain the final movement of the virtual object during a collaborative task in VR. First, they proposed to add the two motions (asymmetric integration of movements). Second, they proposed to compute the average of the two motions (see Figure 1), which basically results in dividing by two the amplitude of the addition motion aforementioned. Third, they proposed to keep only the common part (intersection) of the two motions (symmetric integration of movements). In this third case, the signed minimum value on each axis is kept. Thus, two identical motions along the same axis but with opposite directions will end into a null common part and a null motion.

The results of Ruddle et al. [START_REF] Pinho | Cooperative Object Manipulation in Immersive Virtual Environments: Framework and Techniques[END_REF] showed that the average technique and the intersection technique were globally equivalent in terms of task completion time. However, the intersection technique was found better when the two users had to perform a similar action, for instance when moving a virtual piano in translation in a corridor. Conversely, the average technique was better when users had to perform different tasks, for instance turning the virtual piano in one corner of the corridor.

Figure 2: principle of a DOF separation technique (translations vs. rotations) [8]

Another way for two users to collaborate when moving a virtual object is to split the parts of the task among the users, as described by Pinho et al. [START_REF] Pinho | Cooperative Object Manipulation in Immersive Virtual Environments: Framework and Techniques[END_REF]. In this case, the number of DOF (Dregrees Of Freedom) that each user can access and control is limited. The architecture of such an implementation (described in [START_REF] Pinho | Cooperative Object Manipulation in Immersive Virtual Environments: Framework and Techniques[END_REF]) is displayed on Figure 2. In this case, one user controls the object's rotations while the other user controls the object's translations. Partial evaluations of this technique showed that the separation of DOF could be useful when users do not have the same view of the virtual scene, i.e. when they have two complementary views [START_REF] Pinho | Cooperative Object Manipulation in Immersive Virtual Environments: Framework and Techniques[END_REF]. Thus, the separation of DOF seems useful when each user interacts with the DOF he/she can best observe.

A lot of innovative 3D interaction techniques have also been proposed in the field of manipulation of virtual objects with the two hands of a single user [5] [7]. Cutler et al. [START_REF] Cutler | Two-Handed Direct Manipulation on the Responsive Workbench[END_REF] gave a taxonomy of two-hand tools for 3D interaction. Among others, they described the "grab-an-carry" technique, which is a bimanual symmetric tool that allows to carry and turn an object around with both hands. Each hand can also be used independently as a one-hand grab tool which allows to pick-up a single object an move it freely. In their taxonomy, Cutler et al.

[5] also described the "trackball" technique which can be considered as the equivalent (in the field of 2-hand 3D interaction techniques) of the 2-User DOF separation technique described above and displayed on Figure 2. The "trackball" technique is a bimanual asymmetric tool which allows to use the left hand to position a virtual object while the right hand rotates this object around its center.

In the field of haptic interaction, researchers have also developed some haptic techniques for 2 users to interact simultaneously with the same object using force-feedback devices. In such studies, each user is generally clutched to the manipulated object [START_REF] Basdogan | An Experimental Study on the Role of Touch in Shared Virtual Environments[END_REF]. The force-feedback of the haptic devices is directly used to constrain the motions of the users and to apply commands on the manipulated object using simple physically based model. For instance, in the study of Basdogan et al. [START_REF] Basdogan | An Experimental Study on the Role of Touch in Shared Virtual Environments[END_REF], the haptic interaction model simply simulates the translational movements of a ring on a wire and is not used to change the orientation of this ring.

THE SKEWER TECHNIQUE

In this section, we describe a novel 3D interaction technique called the "SkeweR" technique, which allows multiple users to select and manipulate virtual objects simultaneously in a VE. Most of the previous interaction techniques were designed for the multiple users to manipulate the virtual object by translating or rotating its geometrical center (or gravity center). On the contrary, we propose to take into account the size and geometry of the object, in order to obtain a more natural interaction. Indeed, we propose to move the virtual object by grabbing any part of it, at any location.

The SkeweR technique can be first considered as an equivalent (in the field of 2-user collaborative manipulation) of the "graban-carry" tool described by Cutler et al. [START_REF] Cutler | Two-Handed Direct Manipulation on the Responsive Workbench[END_REF] which was designed for 2-hand manipulation. However, the principle of the SkeweR technique can also be extended, in order to be used by 3 (or more) users. The SkeweR technique can also be compared with the techniques developed for shared haptic interaction [START_REF] Basdogan | An Experimental Study on the Role of Touch in Shared Virtual Environments[END_REF]. However, the SkeweR technique does not use force-feedback and physical models, but simply uses kinematics and information of motions of the users. The SkeweR technique can tolerate large offsets between the position of the user's hand (extremity of the device) and the position of the shared object in the simulation.

Concept

The SkeweR technique uses only the translation motion of the users. These motions are measured by position sensors. They are used to move one virtual 3D cursor per user in the VE. These 3D cursors are used to activate some "crushing points" at the surface of the manipulated object (see Figure 3). We will assume that a virtual 3D cursor stays at the same position than its associated crushing point.

Figure 3: two users reach and crush a parallelepiped object with their virtual 3D cursors

When associated with the crushing points, the 3D cursors of the two users become the extremities of a virtual skewer which is used to hold and move the interactive object. As displayed on Figure 4, a virtual object (parallelepiped) can be turned and rotated easily in a VE without using any rotation motion from the users.

Figure 4: 2-user collaborative manipulation of an object with the SkeweR technique in one turn of a virtual maze

The SkeweR technique uses only the translation motions of the two users to move the objects within a complex 6 DOF

Implementation of the SkeweR technique

In this section, we will first present how it is possible to rotate and translate a virtual object thanks to only one crushing point. Then we will show how two users can use two crushing points. Last, we will describe how it is possible to extend the technique for 3 or more users.

Changing the orientation of the manipulated object with one crushing point

Once a crushing point (associated to a 3D cursor) is selected and activated at the surface of the manipulated object, the translation motions of the user are used to move the cursor and then to compute a new rotation increment for the object. At each time step, we consider an elementary straight motion of the cursor, and we compute an elementary rotation to apply to the object. Let us assume that Po is the geometrical center of the object, Pc t the position of the 3D cursor at time t, Pc t+1 the position of the 3D cursor at time t+1. Then, the elementary rotation to apply to the interactive object will be a rotation of an angle θ around the ω axis, i.e. the axis which is passing by Po and orthogonal to the plane defined by Po, Pc t and Pc t+1 . The angle θ and axis ω are computed as follows:

) cos( 1 1 + + × ⋅ = t t t t PoPc PoPc PoPc PoPc ar θ (1) 1 + ∧ = t t PoPc PoPc ω r (2) 3.2.2
Changing the position of the manipulated object with one crushing point

The change in position of the manipulated object is achieved by constraining the position of the crushing point (point located at the surface of the object) to follow the position of the cursor. Let us assume that R is the rotation used to determine the absolute orientation of the manipulated object. R is obtained by the accumulation of the elementary rotations defined previously with Equations 1 and 2. The new position of the object is then obtained using Equation 3. In this equation, (Pc/Po) represents the local position of the 3D cursor relatively to the center (Po) of the manipulated object, in the relative frame of this object. This position is the same than the position of its associated crushing point, and it remains constant as long as the user does not choose to manipulate the object thanks to another crushing point.

( )

Po Pc R Pc Po ⋅ - = (3)
As a result, this technique gives the user the impression that he/she is pulling the object with a virtual cord.

Extension to 2 crushing points

With two crushing points, each user may control the motion of the manipulated object as if he/she was holding one extremity of a virtual skewer. The final position of the interactive object is computed using Equation 4, in which Pc1 and Pc2 are the absolute positions of the cursors of the 2 users, and (Pc1/Po) and (Pc2/Po) are the local positions of these 3D cursor relatively to the center of the manipulated object, in the relative frame of this object. The final position reached by the object corresponds to the middle of the two positions obtained when using the two crushing points alone.

) 2 1 ( 2 1 ) 2 1 ( 2 1 Po Pc Po Pc R Pc Pc Po + ⋅ - + = (4)
Here again, the expression ((Pc1/Po + Pc1/Po)/2) remains constant as long as one user does not choose to manipulate the object thanks to another crushing point.

Since the 4 points Pc1 t , Pc1 t+1 , Pc2 t and Pc2 t+1 do not always remain in the same plane, the final rotation applied to the interactive object now uses the vector v defined by Equation 5. This vector is defined using the absolute position of the two cursors. The new rotation angle θ and the new rotation axis ω are then given respectively by Equation 6 and 7.

2 1Pc Pc v = (5) ) cos( 1 1 +
+ × ⋅ = t t t t v v v v ar θ (6) 1 + ∧ = t t v v ω (7) 3.2.4
With only 2 crushing points: one DOF is missing… It is not possible for the two users to rotate the virtual object around the axis of the skewer, since the rotation axis remains always orthogonal to the skewer's axis. To do so, the users have to stop their current motions and change the positions of their crushing points (=change the axis of the skewer). However, we can imagine complementary techniques to make up for the lack of this sixth degree of freedom, and apply rotations around the skewer axis. First, we could use the rotation motions of the users around the skewer's axis straightforwardly, as measured by orientation trackers (similar to the "grab-andtwirl" tool referenced in [START_REF] Cutler | Two-Handed Direct Manipulation on the Responsive Workbench[END_REF]). However, this solution can not be implemented with 3 DOF trackers. Second, when pressing a button, users could be enabled to move their 3D trackers along the skewer's axis to define the sign and angle of a rotation as a function of the translation motion of each user along the axis. For example when the 3D trackers would get closer the object would rotate positively around this axis, and it would rotate negatively otherwise.

Extension to 3 crushing points, or more…

The proposed technique could be extended to the use of more crushing points. This could indeed allow a better control of the motion of the interactive object.

To control the orientation of an object, the best solution is to use 3 crushing points (not aligned), as these 3 points can easily be used to define exactly the orientation of the manipulated object. Indeed, these 3 crushing points define a plane embedded in the manipulated object and we can easily compute one vector orthogonal to this plane. Each time one crushing point (i.e. one 3D cursor) is moved, it is possible to compute a new orthogonal vector and to determine the exact rotation that transforms the initial vector into this new one. Furthermore, the position of this object could be computed as the average of the positions of the 3 crushing points. In the case of a manipulation of three crushing points by one user, one or two crushing points could be used to fix temporarily some position constraints, and the remaining crushing points would be used to move the object and change its orientation.

With four crushing points or more, the orientation of the interactive object might become over-constrained. Thus, with more than three crushing points, one must find arbitrary simplifications to apply changes in orientation to the object. Future work must be done to manage the computation of the orientation in this over-constrained case. However, the position of the interactive object still remains easy to compute, using the average position of all the crushing points.

Preliminary experimental setup

This technique has been implemented thanks to our OpenMASK platform 1 within our Reality Center, with magnetic trackers (see 6). We have made some preliminary tests, asking two users to move a parallelepiped object inside a 3D maze. Our first observations indicate that the SkeweR technique seems very intuitive and natural to use. However, we must now perform a series of experiments to measure if this technique is easy and efficient enough for collaborative manipulations of 3D virtual objects.

GENERAL CONCLUSION AND PERSPECTIVES

The SkeweR technique is a 3D interaction technique for 2-user collaborative manipulation of virtual objects. This technique enables two users to move simultaneously the same virtual object in a 3D environment. The technique uses and combines the translation motions of the two users, as measured by 3 DOF position trackers. Each user manipulates the object by one "crushing point", like handling the extremity of a skewer. This technique could also be used with only one crushing point, or also with 3 or more crushing points.

The SkeweR technique could be used to improve the manipulation of virtual objects in several applications of Collaborative Virtual Environments such as virtual prototyping, maintenance or training simulations.

Future work. Future work deals mainly with an evaluation of the SkeweR technique, as compared to the previous techniques described in the literature for the 2-user collaborative manipulation of virtual objects, i.e. the separation of DOF [START_REF] Pinho | Cooperative Object Manipulation in Immersive Virtual Environments: Framework and Techniques[END_REF] and the average or intersection of the user's motions [START_REF] Ruddle | Symmetric and Asymetric Action Integration During Cooperative Object Manipulation in Virtual Environments[END_REF]. We would also like to study the use of three crushing points, with one, two or three users.

  (P 1 + P 2 ) / 2 O r = (O 1 + O 2 ) / 2

  scheme is thus different from the techniques described in section 2, making the SkeweR technique a hybrid technique, as displayed on Figure5.

Figure 5 :

 5 Figure 5: principle of the SkeweR technique

The

  f p (P 1, P 2 ) O r = f o (P 1, P 2 )

Figure 6 :

 6 Figure 6: snapshots of the experimental setup. Each user holds 2 input devices: one 3D tracker (dominant hand) and one "trigger" device with buttons (non-dominant hand)

Acknowledgements. The authors would like to thank Dr. Jean-Marie Burkhart for his valuable remarks.