F Halter

V Glasziou

H Keck

G Legros

C Chauveau

Energetic Materials Combustion

Keywords: Metal fuels, Aluminum particle, ignition temperature

Aluminum particles are widely used as an additive to increase energetic performances. Recently, Aluminum has also received a growing interest as a source of hydrogen for fuel cell applications, or in novel power generation concepts as it provides a carbon-free energy that meets current policies for reducing emissions of greenhouse gases. Our recent studies focused on the description of the combustion process varying ambient conditions (pressure and oxidizer). Only the second stage of the combustion process, after the establishment of a steady state, was scrutinized. The proposed work sheds light more specifically to the initial stages where the particle is warmed up from the initial state to a thermal runaway condition. Indeed, the question arises at what temperature do these condensed fuels burst into flames, knowing that the presence of a passive coat greatly influences the process.

For this study, a dedicated set-up has been improved. It consists of an electrodynamic levitator in which a single particle, trapped by electrostatic forces, burns in a controlled atmosphere. A two-color ratio pyrometer has been specifically assembled with photomultipliers (PM) to obtain the time evolution of mean surface temperature. The PM signals are synchronised witha particle visualization device that is composed of a high-speed camera combined with a longdistance microscope. Subsequently, the temperature measurement of high temporal resolution can then be associated for the first time at a rate of 250 000 fps with the fast-apparent phenomenology of the pre-ignition processes. In addition, the spatial resolution of the visualization (1.92 μm/pix) allows the interpretation of the PM signals to be commented.

Introduction

Knowledge about the specific ignition temperature is crucial to the accurate modelling of the performances of practical combustion systems using metal particles such as propulsion systems, explosives, and pyrotechnics. Yet efforts to characterize aluminum ignition have produced up to now results that are contradictory. A major difficulty comes from the presence of the passive natural oxide that covers the particle. This coating consists of a thin layer of amorphous alumina whose thickness has been evaluated to vary from about 0.5 nm to 4 nm. Most of the authors suggest that micrometric aluminum powders are coated with a 2 -3 nm thick layer [START_REF] Zhang | Reaction Interface for Heterogeneous Oxidation of Aluminum Powders[END_REF][START_REF] Sundaram | Combustion of nano aluminum particles (Review)[END_REF][START_REF] Rufino | Influence of particles size on thermal properties of aluminium powder[END_REF][START_REF] Jeurgens | Growth kinetics and mechanisms of aluminum-oxide films formed by thermal oxidation of aluminum[END_REF].

Friedman and Macek [START_REF] Friedman | Combustion studies of single aluminum particles[END_REF] speculated that ignition of aluminum occurs upon melting of the oxide film, leading to an ignition temperature of Al close to the melting point of Al 2 O 3 (i.e., 2345 K). However, ignition temperatures as low as 850 K and as high as 2400 K have been reported [START_REF] Bar-Ziv | The electrodynamic chamber: A tool for studying high temperature kinetics involving liquid and solid particles[END_REF][START_REF] Panagiotou | Measurements of Particle Flame Temperatures Using Three-Color Optical Pyrometry[END_REF][START_REF] Monier | Liquid metals surface temperature fields measurements with a two-colour pyrometer[END_REF][START_REF] Levendis | Development of multicolor pyrometers to monitor the transient response of burning carbonaceous particles[END_REF][START_REF] De Arrieta | Radiative properties of micron-sized Al/air premixed flames described by an effective medium core-shell formulation[END_REF][START_REF] Guibaud | Accessing the soot-related radiative heat feedback in a flame spreading in microgravity: optical designs and associated limitations[END_REF][START_REF] Guibaud | Broadband modulated absorption/emission technique to probe sooting flames: Implementation, validation, and limitations[END_REF][START_REF] Braconnier | Aluminum combustion in CO2-CO-N2 mixtures[END_REF][START_REF] Mohan | Experiment and modeling : ignition of aluminum particles with a CO2 laser[END_REF][START_REF] Qiu | Characteristics of Radiation Absorption in Metallic Particles[END_REF][START_REF] Halter | Peculiarities of aluminum particle combustion in steam[END_REF]. The reported range of temperature appears to correlate with the mean particle diameter [START_REF] Braconnier | Experimental investigation of the aluminum combustion in different O2 oxidizing mixtures: Effect of the diluent gases[END_REF]. The observed discrepancies may also be attributed to different experimental methods employed, which resulted in different conditions, such as surrounding gas compositions and heating rates. The main experimental techniques illustrating the different heating regimes leading to aluminum ignition are the Thermal Analysis [START_REF] Rufino | Influence of particles size on thermal properties of aluminium powder[END_REF], the Wire Ignition [START_REF] Merzhanov | Aluminium ignition[END_REF], the Shock Tube [START_REF] Bazyn | Combustion of nanoaluminum at elevated pressure and temperature behind reflected shock waves[END_REF], and the laser heating [START_REF] Mohan | Aluminum particle ignition in mixed environments[END_REF]. The kinetics of exothermic reactions is commonly characterized by thermal analysis, where heating rates are very low (lower than 10 1 K/s) in comparison with heating rates of practical interest (higher than 10 5 K/s), which are not readily controlled. For the simplest analysis of a metal particle ignition in a practical combustion system, the rate of particle heating by external sources as it enters the combustion system should be compared against the thermal losses and the rate of particle self-heating due to oxidation. This study, based on both detailed experiments and a simplified thermal model, brings new insights into the ignition sequence and addresses the relative importance of the phenomena at play.

Experimental set-up description

The experimental device is mainly composed of a high-pressure combustion chamber into which an electrostatic levitation system is incorporated. The particle, initially charged with triboelectric effect, is introduced in the electric field in the center of the levitator. Electric forces generated by the levitator allow the charged particle to remain motionless at the center of the chamber without any contact with any surface. The temperature increase of the particle due to the laser heating leads to thermionic losses. In the case of aluminum with a work function that ranges between 4 and 4.2 eV, the characteristic time for charge loss is shown to be around 1 ms at 1300K [START_REF] Bar-Ziv | The electrodynamic chamber: A tool for studying high temperature kinetics involving liquid and solid particles[END_REF]. This single particle is isolated in the center of the chamber by modulating the different voltages of the levitator. A CO 2 laser (power of 50 W, wavelength of 10.6 µm) is used to warm up the particle till ignition. Once the combustion is initiated, the laser is stopped. The particle stabilized in levitation is heated almost uniformly on both sides as the initial beam is carefully splitted into 2 ones of equivalent power. Opposed laser beams are focused in the center of the chamber leading to an estimated beam waist 𝐷 𝑏𝑒𝑎𝑚 = 193 𝜇𝑚 at the particle's location. The positioning of the particle relative to the laser beam will be discussed later (see Section 3). The high-pressure chamber (up to 12 MPa by design) allows combustion experiments to be conducted in a controlled gaseous atmosphere, and with a wide variety of gas composition. Particles with diameters ranging from 20 to 150 m can be stabilized with this system.

The time evolution tracking of the burning particle is ensured by the use of an imaging device combining a fast camera PHANTOM and a long-distance microscope QUESTAR QM100 focused on the particle in reaction. The camera used is the PHANTOM V2512, allowing 25000 fps at maximum resolution (1280x800 px) up to 39000 fps at the resolution used in this study (768x768 px). The magnification obtained with the long-distance microscope is 1.92 µm/px. The imaging obtained by the fast camera allows not only a phenomenological description of the combustion process, but also, after the application of an image processing, the extraction of quantitative information such as for example the diameter of the initial aluminum particle after melting, the temporal evolution of this diameter, the migration speed of the alumina particles towards the drop, etc.

The method selected to measure experimentally the temperature is the 2-color pyrometry. This method makes it possible to determine the temporal evolution of the average temperature of the condensed phase during the combustion of the aluminum particle [START_REF] Panagiotou | Measurements of Particle Flame Temperatures Using Three-Color Optical Pyrometry[END_REF][START_REF] Monier | Liquid metals surface temperature fields measurements with a two-colour pyrometer[END_REF][START_REF] Levendis | Development of multicolor pyrometers to monitor the transient response of burning carbonaceous particles[END_REF]. To achieve this, an optical system composed of 2 photomultipliers (Thorlabs PMT1002) is set up to follow the temporal evolution of the light signal emitted by the particle during its combustion. The PMTs are operating in a spectral range between 250 and 900 nm and at a frequency of 250 kHz. A USB-1808X acquisition card is used to digitize the signal. Each PMT is equipped with an interference filter ( 1 =720 nm and  2 =830 nm with a 10nm FWHM). This couple of spectral ranges has been selected to avoid the emission lines of the major gaseous species identified by the literature in the reaction zone. As a result, the signal captured by the PMs is supposed to be highly dominated by the emission of the condensed matter's surface.

A schematic view of the full set-up is proposed in Figure 1.

Figure 1 : Schematic view of the complete experimental set-up

The pyrometer is calibrated using a certificated 45 W Quartz Tungsten Halogen Lamp (Newport). We assume in the following that the emissivity of the condensed matter does not vary over the selected spectral range (i.e. 720 nm to 830 nm) which is consistent in view of our latest results [START_REF] De Arrieta | Radiative properties of micron-sized Al/air premixed flames described by an effective medium core-shell formulation[END_REF].

Evaluating the ratio of the radiative intensity, Γ = 𝐿 𝜆 1 /𝐿 𝜆 2 , as given by Planck's law and measured over both selected spectral rangesthe expression for the temperature is derived as follows:

𝑇 (𝛤, 𝜆 1 , 𝜆 2 ) = [ 𝑘 ℎ.𝑐 . 𝜆 1 .𝜆 2 𝜆 2 -𝜆 1 . (𝑙𝑛(𝛤) -5. 𝑙𝑛 ( 𝜆 1 𝜆 2 ))] -1 Equation 1
with k the Boltzmann constant, h the Planck constant, and c the speed of light in vaccuum.

A major assumption of the processing is that the signal's absorption along the line-of-sight leading to the PM's is neglected. Actually, this assumption may lead to significant uncertainties in the evaluation of the radiative heat feedback from the flame to the condensed matter [START_REF] Guibaud | Accessing the soot-related radiative heat feedback in a flame spreading in microgravity: optical designs and associated limitations[END_REF]. Still, the assumption skews at a similar level both signals as they are collected over close spectral ranges and the interpretation of their ratio delivers decent evaluation of the surface's temperature [START_REF] Guibaud | Broadband modulated absorption/emission technique to probe sooting flames: Implementation, validation, and limitations[END_REF]. More details about this device are given in Ref. [START_REF] Braconnier | Aluminum combustion in CO2-CO-N2 mixtures[END_REF].

Model description

The system we consider for this model is composed of an aluminum particle, covered with a thin layer of alumina and located in an oxidizing medium. The model is constructed to describe the average temperature of the particle surface until its ignition. In the case of our experimental setup, the particle's diameter is typically of the order of 20-100 m, and the combustion process happens at the latest 10 ms after the start of the heating process. As a first approximation, the ignition temperature in this model is considered to be the alumina melting temperature 𝑇 𝑚𝑒𝑙𝑡_𝐴𝑙 2 𝑂 3 = 2345 𝐾 . Within the range of the particle's size studied, heat conduction in the particle dominates the convective exchanges occurring at the surface, leading to a Biot number << 1. Consequently, we consider in the sequel that the temperature gradients are negligible throughout the particles.

As described in the experimental section, the particle is heated until its ignition with a 50 W CO 2 laser (λ =10.6 μm). An optical efficiency, due to the different optics met along the optical pathways (i.e. mirrors, windows, and lenses) has been experimentally evaluated to be 0.84. Thus, a power of 42 W distributed along a gaussian profile is considered to be delivered by the laser at the center of the combustion chamber. The experimental setup allows a symmetrical heating of the particle by the careful splitting of the laser illuminating the particle from both sides. However, for the sake of clarity, a single laser beam is considered in this model. Thermal losses by conduction and radiation towards the surroundings are considered while convection is neglected as the particle is considered spatially static and because of the absence of any flow induced within the very short process studied.

The numerical model is a sequence of three phases. The first one consists in the heating of the particle from room temperature to the melting point of aluminum. During this phase, the energy transmitted from the laser to the particle is used to heat up the particle up to 933 K. When this temperature is reached, the energy from the laser is mainly used to melt the aluminum, that the second phase of the model characterizes. This phase is assumed to take place at constant temperature. When the aluminum is fully melt, the last phase brings the temperature of the particle to its ignition temperature.

The heat delivered by the laser to the particle consists in the integration of the laser power density along the effective surface of the particle, weighed by its absorption efficiency 𝜂 𝑎𝑏𝑠 . Little information is available in the literature on the absorption efficiency and even less on its evolution with the diameter and the temperature of the particle. For this model, we relied on the work of Mohan [START_REF] Mohan | Experiment and modeling : ignition of aluminum particles with a CO2 laser[END_REF] as discussed later.

The power density of the laser being distributed along a gaussian profile, it can be expressed as follows:

𝐼(𝑦, 𝑧) = 𝑊 2 𝜋 𝜎 2 𝑒 -𝑦 2 2 𝜎 2 𝑒 -𝑧 2 2 𝜎 2 Equation 2
𝑊 is the total power delivered by the laser (i.e. 42 W), 𝜎 is the standard deviation along the gaussian profile and as described by Mohan [START_REF] Mohan | Experiment and modeling : ignition of aluminum particles with a CO2 laser[END_REF], it can be assumed that 6 𝜎 = 𝐷 𝑏𝑒𝑎𝑚 , where 𝐷 𝑏𝑒𝑎𝑚 = 193 𝜇𝑚.

In other words, it can be considered that 99.7% of the power lies within the beam diameter 𝐷 𝑏𝑒𝑎𝑚 . 𝑦 and 𝑧 are the coordinates from the center of the gaussian profile as indicated in Figure 2. The total power transmitted to the particle from the laser is then

𝑄 ̇𝑙𝑎𝑠𝑒𝑟 = 𝜂 𝑎𝑏𝑠 ∫ ∫ 𝐼(𝑦, 𝑧) 𝑑𝑦 𝑑𝑧 𝑧 𝑝 +𝑟 𝑝 -(𝑟 𝑝 2 -(𝑧-𝑧 𝑝 ) 2 ) 0.5 𝑧 𝑝 +𝑟 𝑝 𝑧 𝑝 -𝑟 𝑝 Equation 3
It is considered in this expression that the particle is centered along the 𝑦 axis with respect to the gaussian profile, and shifted from the center of the gaussian profile along the 𝑧 axis by 𝑧 𝑝 . The power density is integrated along the surface of a circle of radius 𝑟 𝑝 , that is the radius of the particle. Indeed, the effective surface illuminated from one side by the laser is 𝜋 𝑟 𝑝 2 . The laser lasts till the particle ignition following a top-hat function of time.

The absorption efficiency allows to quantify the amount of energy transmitted to the particle as an important part of the energy is reflected. This coefficient is theoretically derived in Ref. [START_REF] Qiu | Characteristics of Radiation Absorption in Metallic Particles[END_REF] and numerical values are given for aluminum in Ref. [START_REF] Mohan | Experiment and modeling : ignition of aluminum particles with a CO2 laser[END_REF]. This coefficient depends on the temperature, the size of the particle (considered constant during the heating process), the refractive index of the material (alumina in our case), and the wavelength of the laser. The numerical values chosen for the model come from Ref. [START_REF] Mohan | Experiment and modeling : ignition of aluminum particles with a CO2 laser[END_REF]. We consider the absorption efficiency to vary linearly from 0.018 to 0.08 between 300 and 2345K. The position of the particle with respect to the laser profile is represented in Figure 2. The particle is indicated by the grey circle. For simplicity, it is assumed to be spherical. The laser profile is represented by the blue Gaussian function. The particle position is vertically shifted by 𝑧 𝑝 with respect to the maximum of the Gaussian profile. No shift is considered in the 𝑦 direction (note that the laser beam profile is assumed to be asymmetrical).

Figure 2 : View of the position of the particle (grey sphere) with respect to the laser profile (blue shaded region)

Once the laser power transferred to the particle is determined, the power used to heat the particle and the heat losses may be evaluated.

The heat to raise the temperature of the particle up to its melting point is

𝑄 ̇ℎ𝑒𝑎𝑡 = 𝑚 𝑝 𝑐 𝑝 𝑑𝑇 𝑑𝑡 Equation 4
𝑚 𝑝 being the mass of the particle, 𝑐 𝑝 the heat capacity of aluminium, and 𝑇 the temperature of the particle. The heat lost by radiation to the surrounding follows the Stefan-Boltzmann's law and is expressed as 𝑄 ̇𝑟𝑎𝑑 = 𝜎 𝑆𝐵 𝜖 𝐴𝑙 2 𝑂 3 (𝑇 4 -𝑇 0 4 ) Equation 5With 𝜎 𝑆𝐵 the Stephan-Boltzmann constant, 𝜖 𝐴𝑙 2 𝑂 3 the emissivity of alumina as it is what the external layer of the particle is made of, and 𝑇 0 the room temperature. Then, heat losses by conduction through the particle's surface S 𝑝𝑎𝑟𝑡 follow the Fourier's law and can be expressed as 𝑄 ̇𝑑𝑖𝑓𝑓 = -𝜆 𝑆 𝑝𝑎𝑟𝑡 𝛻𝑇 Equation 6𝜆 being the thermal conductivity of the ambient gases. Expressing this term in spherical geometry, we get 𝑄 ̇𝑐𝑜𝑛𝑑 = -𝜆 4𝜋𝑟 2 𝑑𝑇 𝑑𝑟 Equation 7Then, integrating between the particle radius and an infinite distance 𝑄 ̇𝑐𝑜𝑛𝑑 = -𝜆 4𝜋𝑟 𝑝 (𝑇 -𝑇 0 ) Equation 8For a particle temperature below the melting temperature of aluminum, we can thus write 𝑄 ̇𝑙𝑎𝑠𝑒𝑟 = 𝑄 ̇ℎ𝑒𝑎𝑡 + 𝑄 ̇𝑟𝑎𝑑 + 𝑄 ̇𝑐𝑜𝑛𝑑 Equation 9When reaching the melting point of aluminum, the energy transmitted to the particle is used for melting at constant temperature (𝑇 𝑚𝑒𝑙𝑡_𝐴𝑙 = 933 𝐾).

𝑄 ̇𝑚𝑒𝑙𝑡𝑖𝑛𝑔 = 𝐿 𝐴𝑙 𝑑 𝑚 𝑙𝑖𝑞_𝐴𝑙 𝑑𝑡 Equation 10With 𝐿 𝐴𝑙 the latent heat of fusion and 𝑚 𝑙𝑖𝑞_𝐴𝑙 the mass of liquid aluminum. Thus, at the melting temperature of aluminum and while the aluminum is not fully molten, we write 𝑄 ̇𝑙𝑎𝑠𝑒𝑟 = 𝑄 ̇𝑚𝑒𝑙𝑡𝑖𝑛𝑔 + 𝑄 ̇𝑟𝑎𝑑 + 𝑄 ̇𝑐𝑜𝑛𝑑 Equation 11Once the aluminum only exists in its liquid phase and up to the ignition temperature, we use Eq. 11.

We thus have a system of equation that we can solve with respect to the temperature In this way, it is possible for a given particle radius and particle height above the peak of the laser gaussian profile 𝑧 𝑝 to compute the temperature of the particle with respect to time and the contribution of each term (laser and losses) to the particle. In the proposed model, the heat produced by surface reaction on the particle has been neglected as it is supposed that the thermal runaway occurs as soon as the oxide layer breaks. This hypothesis will be confirmed later.

Ignition process description

An aluminum particle is isolated and stabilized in the center of the levitator. The particles studied here have a size between 30 and 100 µm. The surrounding gas is composed of air at atmospheric conditions. Figure 3 displays the ignition process of an aluminium particle having an initial diameter of 88 µm. The initial morphology of the particle is not exactly spherical as seen in the early stages of the process (𝑡1 to 𝑡4). When subjected to CO 2 laser heating, we observe an increase in particle luminosity due to an increase in temperature. Unfortunately, the sensitivity of the camera and the selected exposure time (i.e. 10 s) do not allow to detect the particle before the initial time 𝑡1 = 2450 𝜇𝑠 of the sequence captured. Till 𝑡4 = 3116 𝜇𝑠, the morphology of the particle remains almost unchanged. Only slight modifications can be observed on smallest structures. These changes are hardly discernible with the current image display quality. More or less luminous zones are observed on the surface of the particle. These differences can be linked to the presence of an non-uniform oxide layer thickness on the surface or to non-uniform heating. As a reminder, the particle has an equivalent diameter of 88 µm while the diameter of the laser beam waist is approximately of 200 µm. In addition, the particle is simultaneously heated from both sides. However, it cannot be ruled out that the powers of the two laser beams are not perfectly balanced. At 𝑡5 = 3142 𝜇𝑠, the particle becomes spherical. It is still covered with alumina but the latter must have changed phase exposing, in certain areas, bare liquid aluminum to the surrounding oxidizer.

The evaporation of the liquid aluminum is no longer hindered by the presence of the oxide layer. An increase in luminosity is observed in the close vicinity of the drop. Vaporized aluminum reacts extremely fast with the oxygen of air to produce alumina. Since the local temperature is lower than alumina dissociation temperature (as confirmed later), alumina instantly condenses in the form of nanoparticles.

These nanoparticles are at very high temperature (~3500 K) and emit radiation. Thermal runaway is clearly observed as soon as the oxide layer breaks. Observations clearly indicate that for these heating rates, heterogenous reactions on the liquid aluminum surface play a very limited (if not zero) role in the ignition processes. After 𝑡5 = 3142 𝜇𝑠, the alumina recovering the surface retracts to form several lobes which eventually merge into a single one. This process lasts around 600 𝜇𝑠 from 𝑡6 = 3168 𝜇𝑠 till 𝑡10 = 3783 𝜇𝑠. The laser is stopped after 𝑡 𝐿𝐴𝑆𝐸𝑅 = 3600 𝜇𝑠, i.e. between 𝑡9 and 𝑡10. The energy provided by the laser after the thermal runaway induces an over-production of aluminum vapour, which accelerates the process to reach steady state. As soon as the laser is stopped, the dimension of the luminous halo (corresponding to the gas phase reaction zone) remains unchanged. Thermal equilibrium is reached almost immediately. Subsequently, the combustion of the aluminum drop continues following approximately a d 2 -law [START_REF] Halter | Peculiarities of aluminum particle combustion in steam[END_REF][START_REF] Braconnier | Experimental investigation of the aluminum combustion in different O2 oxidizing mixtures: Effect of the diluent gases[END_REF].

Temperature evolution during the ignition process

In this part, we compare the temperature evolution obtained with that measured experimentally. First, the relative contributions of the different terms involved in the energy equation (Eq. 13) (i.e. 𝑄 ℎ𝑒𝑎𝑡 , 𝑄 𝑟𝑎𝑑 , 𝑄 𝑑𝑖𝑓𝑓 ) are evaluated. The total laser power (𝑃 𝐿𝐴𝑆𝐸𝑅 ) is taken constant and equal to 42 W. The vertical shift (𝑧 𝑝 ) is adjusted to 42 µm. This value was calibrated by positioning the particle relative to the laser by adjusting the settings of the levitator (i.e. modulating the different voltages of the levitator). Figure 4 is limited to the first 3 ms after the start of the laser shot. The power transmitted by the laser to the particle (solid black line) is not constant, due to a variable absorption efficiency with the particle temperature. We clearly identify three successive phases corresponding to: i) heating of solid aluminum (till 1.2 ms) ii) aluminium fusion (till 1.7 ms) iii) heating of liquid aluminum (till the end). Over the entire heating, the power provided by the laser is much higher (at least two orders of magnitude) than losses by diffusion (solid red line) and radiation (solid blue line). We can therefore consider that the ignition process is almost adiabatic.

² Figure 4 : Contribution of the different terms of the energy equation (Eq. 13) for an Aluminum particle of 88 µm in diameter heated by a CO 2 laser beam (𝑃 𝐿𝐴𝑆𝐸𝑅 = 42 𝑊).

The measured and simulated temperature evolutions are shown in Figure 5. The raw (red solid line) and smoothed (black solid line) experimental traces are plotted with the modelled temperature evolution (dashed black line). The current system using photomultipliers is not sensitive enough to provide information on the evolution of the temperature below 1200 K. Aluminum melting can therefore not be detected. This particular point will be improved in the future. Once the aluminum is liquified, the temperature of the particle increases rapidly up to about 2500 K (around 3 ms). During this phase, the absorption efficiency increases and therefore more laser power is transmitted to the particle. According to Figure 3, alumina melting and thermal runaway occur shortly after 3 ms, once the passive oxide layer melts. Is it observed in Figure 5 that this moment occurs around 2500 K, i.e., slightly above the alumina melting temperature. The difference between the measured temperature and the expected one may be due to a non-uniform heating on either side of the particle due to a slight asymmetry in the power distribution of the two laser beams. Under these conditions, one side may be slightly overheated before complete fusion of the oxide.

Once ignition starts, the gas phase combustion process takes place with a maximum surface temperature slightly below 3500 K, which agrees with thermodynamic calculations.

The proposed model accurately predicts the characteristic times for the different phases (i.e. solid and liquid aluminum heating and aluminum melting), which confirms the hypothesis that no exothermic surface reaction occurs before thermal runaway.

Conclusions

This study focuses specifically on the ignition of a micrometric aluminum particle stabilized in a levitator. The particle is heated up using a laser which gives access to heating rates of relevance for industrial systems (i.e., in the order of 10 6 𝐾/𝑠). To complete the detailed visualizations of the combustion process, a two-color pyrometry diagnostics has been set up which allows following the time evolution of the particle's temperature. The experimental results are compared with a simplified heating model. In particular, the different heating phases are very decently reproduced. Thermal runaway is triggered by the rupture of the oxide layer which occurs after melting. At this level of heating rate, the process is shown to be almost adiabatic as radiation and diffusion heat losses from the particle to the surroundings are almost negligible. Moreover, surface reactions do not have time to take place on the surface once the aluminum is in contact with the oxidizing atmosphere. Thermal runaway is almost immediate, occurring at time scale shorter than 50 microseconds after the heating started.

{

  𝑄 ̇𝑙𝑎𝑠𝑒𝑟 = 𝑄 ̇ℎ𝑒𝑎𝑡 + 𝑄 ̇𝑟𝑎𝑑 + 𝑄 ̇𝑐𝑜𝑛𝑑 𝑤ℎ𝑒𝑛 𝑇 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑒 < 933 𝐾 𝑄 ̇𝑙𝑎𝑠𝑒𝑟 = 𝑄 ̇𝑚𝑒𝑙𝑡𝑖𝑛𝑔 + 𝑄 ̇𝑟𝑎𝑑 + 𝑄 ̇𝑐𝑜𝑛𝑑 𝑤ℎ𝑒𝑛 𝑇 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑒 = 933 𝐾 𝑄 ̇𝑙𝑎𝑠𝑒𝑟 = 𝑄 ̇ℎ𝑒𝑎𝑡 + 𝑄 ̇𝑟𝑎𝑑 + 𝑄 ̇𝑐𝑜𝑛𝑑 𝑤ℎ𝑒𝑛 𝑇 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑒 > 933 𝐾 Equation 12

Figure 3 :

 3 Figure 3 : Sequence of frames captured by the high-speed camera within the heating and ignition processes of an aluminium particle (initial diameter: 88 m).

Figure 5 :

 5 Figure 5 : Particle temperature evolution until ignition. Red solid line: raw experimental results; Black solid line: smoothed experimental results; Black dashed line: Model. Particle of aluminum in atmospheric air (𝑟 𝑝 = 88 𝜇𝑚)

Acknowledgements

This work was supported by the region of « Centre Val de Loire » and the European Regional Development Funds (FEDER).


Diane Dewez

Ludovic Hoyet

Anatole L Écuyer

Ferran Argelaguet 1 Fig

Anatole Lécuyer

Ferran Argelaguet
email: ferran.argelaguet@inria.fr

Do You Need Another Hand? Investigating Dual Body Representations During Anisomorphic 3D Manipulation

Keywords: Avatar, Sense of embodiment, Interaction, Virtual reality

Illustration of two types of dual body representation studied in this paper. On the left image, a ghost representation enables remote manipulation while a realistic co-located representation provides feedback with respect to the real user's position. On the right image, a ghost representation provides feedback with respect to the user's real position while a realistic representation enables precise manipulation with the environment thanks to slowed motion.

INTRODUCTION

Anisomorphic interaction techniques, i.e. techniques that alter users' input motion, are often used in Virtual Reality (VR). These techniques alter users' motion differently according to their own purpose. For example, the Go-Go technique [START_REF] Poupyrev | The go-go interaction technique: non-linear mapping for direct manipulation in VR[END_REF] applies a gain to amplify users' motion and enables reaching distant objects. The motion can also be decreased to gain precision during fine manipulation [START_REF] Frees | PRISM interaction for enhancing control in immersive virtual environments[END_REF], or sometimes even constrained to respect real-world physical constraints [START_REF] Prachyabrued | Visual feedback for virtual grasping[END_REF]. While these techniques have been widely used for several years with simple user representations, it remains unclear how the use of high-fidelity avatars would influence users' actions and body perception.

Nevertheless, representing users with high-fidelity avatars is common as they offer several advantages. Avatars were found to increase spatial awareness [START_REF] Draper | Exploring the influence of a virtual body on spatial awareness[END_REF], or even to impact perceived effort during a physical task [START_REF] Kocur | Flexing muscles in virtual reality: Effects of avatars' muscular appearance on physical performance[END_REF]. During cognitive tasks, the effect of seeing an avatar is not clear yet, as it was sometimes found helpful to reduce mental load [START_REF] Steed | The impact of a self-avatar on cognitive load in immersive virtual reality[END_REF] or having no effect on accuracy [START_REF] Peck | The impact of a self-avatar, hand collocation, and hand proximity on embodiment and stroop interference[END_REF] depending on the task. There are also an increasing number of applications using full-body avatars, especially social VR applications. Because they cannot see their real body anymore, users can feel that this virtual representation of themselves is their real body [START_REF] Kilteni | The sense of embodiment in virtual reality[END_REF]. This phenomenon is called the sense of embodiment, and VR developers usually want to maximize it to offer a better user experience. To maximize it, it was found that spatially and temporally congruent visuomotor stimulation, i.e. having an avatar moving according to users' movements, was an efficient method [START_REF] Kokkinara | Measuring the effects through time of the influence of visuomotor and visuotactile synchronous stimulation on a virtual body ownership illusion[END_REF]. If movements are distorted during interaction, this could be detrimental to the sense of embodiment, as spatial congruency would not be respected anymore. While the dominance of vision over proprioception makes small offsets acceptable [START_REF] Burns | The hand is slower than the eye: a quantitative exploration of visual dominance over proprioception[END_REF], bigger offsets can impact body perception [START_REF] Kalckert | The spatial distance rule in the moving and classical rubber hand illusions[END_REF]. In this case, the question arises as to which is the best visual feedback to apply motion distortion on users' avatar, to preserve both interaction efficiency and sense of embodiment [START_REF] Dewez | Towards "Avatar-Friendly" 3D Manipulation Techniques: Bridging the Gap Between Sense of Embodiment and Interaction in Virtual Reality[END_REF]. To start answering this question, Feuchtner and Müller [START_REF] Feuchtner | Extending the body for interaction with reality[END_REF] investigated different avatar appearances when using the Go-Go technique in augmented reality. The preferred condition was an extended arm, as participants appreciated that it preserved body continuity. However, this type of feedback deformed the avatar and made users lose the feedback of their real motion.

To keep the advantages of both anisomorphic interaction and spatial reference offered by the avatar, we propose to provide the visual feedback of both distorted and real motions by displaying two avatars.

This type of feedback is referred hereinafter as dual representation, as opposed to single representations where only one avatar shows one type of feedback (real or distorted). The aim of this dual representation is to offer both the feedback of real users' motions as well as distorted motions during anisomorphic manipulation. In the context of this study, we define the dual representation as made of an interactive representation, used for interaction and displaying altered movements, and a co-located representation displaying real movements. This type of feedback has already been explored in the context of finger collisions with objects for example [START_REF] Canales | Virtual grasping feedback and virtual hand ownership[END_REF][START_REF] Prachyabrued | Visual feedback for virtual grasping[END_REF], and we believe that it could be valuable to all interaction techniques involving motion distortions to support both the sense of embodiment and interaction capabilities.

In order to evaluate the impact of dual representations on the sense of embodiment and interaction performance, we conducted two withinsubject experiments to compare different user representations (both single and dual) during anisomorphic manipulation. To this end, two common interaction methods were considered: remote manipulation with a gain that increases movement [START_REF] Poupyrev | The go-go interaction technique: non-linear mapping for direct manipulation in VR[END_REF], and precise manipulation with a gain that decreases movement [START_REF] Frees | PRISM interaction for enhancing control in immersive virtual environments[END_REF]. Both manipulation techniques create offsets between virtual and real hands, by either amplifying or decreasing users' movements. The interactive and co-located representations had either a realistic visual appearance or a transparent/ghost appearance. This paper provides insight on which user representation to choose during 3D anisomorphic manipulation.

The structure of the paper is as follows. Section 2 presents the main related work on anisomorphic techniques, their impact on the sense of embodiment, and multiple body representations. The context and an overview of our experiments are presented in Section 3. Section 4 and 5 present the two conducted experiments focusing on amplified motion and decreased motion respectively. Section 6 concludes the paper with a global discussion on the use of dual representations, as well as perspectives for future research.

RELATED WORK

In this section, we will present common techniques involving motion alteration, then the impact they can have on the sense of embodiment. Finally, we will present related work on extended avatars and dual representations. These are an open lead to explore towards reconciling embodiment and motion alteration.

Interaction Techniques Using Motion Alteration

Motion data captured from hand-held devices (e.g. VR controllers) can be modulated by the interaction technique in order to alter the user's interaction capabilities [START_REF] Frees | PRISM interaction for enhancing control in immersive virtual environments[END_REF][START_REF] Poupyrev | The go-go interaction technique: non-linear mapping for direct manipulation in VR[END_REF]. Precisely, the Control-Display gain (CD gain) defines how users' motions are mapped to their virtual counterparts. The CD gain is a factor that will either increase (if it is greater than one) or decrease (if it is lower than one) users' motion. Only a CD gain of one will result in a total co-location of the real and virtual counterparts. For example, the Go-Go technique [START_REF] Poupyrev | The go-go interaction technique: non-linear mapping for direct manipulation in VR[END_REF] enables remote object selection and manipulation by increasing the motion performed by the user, therefore increasing unrealistically the distance between the user's physical hand and the virtual hand. The gain can also be decreased, such as in the PRISM technique [START_REF] Frees | PRISM interaction for enhancing control in immersive virtual environments[END_REF], to increase precision. If the displayed motion is slower than the real performed motion, it is possible to move objects more slowly, to translate or rotate them precisely. Motion distortion can also be used to provide haptic feedback by leading users to a physical prop [START_REF] Azmandian | Haptic retargeting: Dynamic repurposing of passive haptics for enhanced virtual reality experiences[END_REF]. This technique applies an offset to the virtual hand so users will reach the real tangible object. Movements can be distorted to assist users during a manipulation task by leading the virtual hand towards the virtual target [START_REF] Porssut | Reconciling being in-control vs. being helped for the execution of complex movements in vr[END_REF], or even to reduce arm fatigue [START_REF] Feuchtner | Ownershift: Facilitating overhead interaction in virtual reality with an ownership-preserving hand space shift[END_REF]. Globally, at a given time, motion can be either amplified or decreased during anisomorphic interaction.

Motion Alterations and the Sense of Embodiment

The medium through which motions are displayed to users is their virtual representation. While we start being able to represent users using high-quality full-body virtual humans, this type of representation might not be compatible with motion alteration as it imposes physical constraints (arm length, co-location with the real body). Using a highfidelity avatar raises the question of which feedback should be used, for example when using increased motion with the Go-Go technique. Should we use a floating hand, or an elongated arm [START_REF] Feuchtner | Extending the body for interaction with reality[END_REF]? Depending on the feedback chosen, this might disturb users which could feel like their avatar is not their body. Nevertheless, it would be beneficial for user experience to keep the illusion that this virtual body, replacing users' real body not visible anymore because of the Head-Mounted Display (HMD), is their real body. This illusion, the sense of embodiment [START_REF] Kilteni | The sense of embodiment in virtual reality[END_REF], is similar to the rubber hand illusion [START_REF] Botvinick | Rubber hands 'feel' touch that eyes see[END_REF] in which participants can integrate a rubber hand as part of their body thanks to visuotactile stimulation. It is similar in VR but with a 3D virtual body [START_REF] Slater | Towards a digital body: the virtual arm illusion[END_REF], often seen from a first-person perspective. Users can feel that the avatar is part of their own body (sense of ownership), that they can control the avatar's movements (sense of agency) and that the avatar is colocated with their real body (sense of self-location) [START_REF] Kilteni | The sense of embodiment in virtual reality[END_REF]. The most effective way to elicit embodiment is to use multisensory stimulation, such as congruent visuomotor or visuotactile stimulation [START_REF] Kokkinara | Measuring the effects through time of the influence of visuomotor and visuotactile synchronous stimulation on a virtual body ownership illusion[END_REF]. The problem is that motion alterations break spatial congruency, which can potentially impact the sense of embodiment, especially the sense of agency [START_REF] Farrer | Effect of distorted visual feedback on the sense of agency[END_REF][START_REF] Kalckert | The spatial distance rule in the moving and classical rubber hand illusions[END_REF][START_REF] Kokkinara | The effects of visuomotor calibration to the perceived space and body, through embodiment in immersive virtual reality[END_REF]. In some cases where the alteration is done slowly, the impact can be limited on the sense of embodiment, for example the reaction to a threat can be as high as when there is no distortion [START_REF] Yuan | Is the rubber hand illusion induced by immersive virtual reality?[END_REF]. It was also found that it can change body perception, for example predicting future motion and displaying it can give the illusion of having a lighter body [START_REF] Kasahara | Malleable embodiment: changing sense of embodiment by spatial-temporal deformation of virtual human body[END_REF], which can be wanted in some applications but not in others where motion is just altered for interaction.

Despite a growing literature on the relation between sense of embodiment and interaction, the impact of motion distortion has not been deeply investigated. Some studies tried to measure the noticeable perception thresholds for motion alterations. It was found to be influenced by the motion direction [START_REF] Esmaeili | Detection of scaled hand interactions in virtual reality: The effects of motion direction and task complexity[END_REF] or by avatar anthropomorphism [START_REF] Ogawa | Effect of avatar appearance on detection thresholds for remapped hand movements[END_REF]. The impact on virtual embodiment is however unclear. Additionally, these studies focused on cases where users saw only one virtual representation of their movements. We can wonder what would happen if users could see multiple representations of their movements during interaction.

Extended and Multiple Body Representations

Few studies have investigated virtual representations with additional body parts or full bodies. In this part, we will present these studies which investigated either extended or multiple body representations.

Extended Body Representations

Avatar extensions have been explored, such as human tails [START_REF] Steptoe | Human tails: Ownership and control Computer Graphics. The final version of this record is available at: 10.1109/TVCG.2022.3150501 of extended humanoid avatars[END_REF] or third arm [START_REF] Laha | Evaluating control schemes for the third arm of an avatar[END_REF]. These studies used a material reminiscent of a ghost to distinguish the additional part from the rest of the body. They found that people could experience a sense of embodiment towards these additional limbs. Elongated arms have also been studied [START_REF] Kilteni | Extending body space in immersive virtual reality: A very long arm illusion[END_REF], one study investigating its use in augmented reality while keeping a representation of the real arm [START_REF] Feuchtner | Extending the body for interaction with reality[END_REF]. This feedback was less appreciated than only an extended arm, without the real arm. More recently, an appended limb has been used as a spatial reference to extend proprioception and increase target selection [START_REF] Tian | Virtuallyextended proprioception: Providing spatial reference in vr through an appended virtual limb[END_REF]. This study found that people could feel a sense of ownership towards this limb, especially when they could control and move it prior to the selection task. The authors also tested different transparency levels for the additional limb, and found no impact on the selection performance.

Multiple Body Representations

A study investigated the experience of having two bodies in VR, but neither spatially co-located nor interacting with the Virtual Environment (VE) [START_REF] Heydrich | Visual capture and the experience of having two bodiesevidence from two different virtual reality techniques[END_REF]. Similar studies with additional body parts were conducted in real life with the rubber hand illusion [START_REF] Ehrsson | How many arms make a pair? Perceptual illusion of having an additional limb[END_REF], using video stream [START_REF] Chen | Body ownership and the four-hand illusion[END_REF][START_REF] Guterstam | Duplication of the bodily self: a perceptual illusion of dual full-body ownership and dual self-location[END_REF] or augmented reality [START_REF] Rosa | The supernumerary hand illusion in augmented reality[END_REF]. More recently, a study investigated "distributed embodiment", i.e. embodiment towards up to four bodies [START_REF] Miura | Multisoma: Distributed embodiment with synchronized behavior and perception[END_REF]. In this study, participants had split views over four first-person perspectives. The results tended to show that subjects switched attention between bodies, but kept a global parallel sense of embodiment towards all bodies. Another study explored the use of additional hands for interaction, to decrease object selection time but without investigating embodiment [START_REF] Schjerlund | Ninja hands: Using many hands to improve target selection in vr[END_REF]. Some VR applications have two superimposed representations, but they are usually used for guidance [START_REF] Yang | Implementation and Evaluation of "Just Follow Me": An Immersive, VR-Based, Motion-Training System[END_REF] so one representation is not controlled by users. While studies using several controlled bodies often used similar textures for all representations, these guidance applications used a "ghost" metaphor [START_REF] Yang | Implementation and Evaluation of "Just Follow Me": An Immersive, VR-Based, Motion-Training System[END_REF], where the second representation used for guidance is more transparent than the main one [START_REF] Chinthammit | Ghostman: augmented reality application for telerehabilitation and remote instruction of a novel motor skill[END_REF][START_REF] Han | Ar-arm: Augmented visualization for guiding arm movement in the first-person perspective[END_REF].

While people replicated the RHI with two arms or bodies, or used a second representation for guiding users, to our knowledge no study investigated the use of a second representation to show real movements during anisomorphic manipulation. For this reason, we conducted two experiments to explore the embodiment of a dual representation, with two different anisomorphic interaction techniques, involving either amplified or decreased movements.

CONTEXT AND EXPERIMENTS OVERVIEW

In this study, we were interested in the potential use of dual representations for interaction in VR. More precisely, the context of this paper is the use of dual body representations when there is an offset created between users' real position and the virtual representation position during anisomorphic manipulation. In this context, we consider that dual representations are made of an interactive representation used for interaction with the environment, and a co-located representation showing users' real movements. Therefore, users have control over two avatars simultaneously, but the mapping and visual appearance differ. More precisely, when a motion distortion is applied to a part of the avatar, the hand in our experiments, the two avatars will dissociate, providing visual feedback of both the co-located and distorted motions. When no distortion is applied, a single default representation can be used (a realistic full-body avatar) to maximise embodiment [START_REF] Latoschik | The effect of avatar realism in immersive social virtual realities[END_REF][START_REF] Martini | The body fades away: investigating the effects of transparency of an embodied virtual body on pain threshold and body ownership[END_REF] and avoid a visual overlap between the two representations. In this paper, when talking about our different experimental conditions, we will always indicate the visual appearance of the interactive representation first (letter on the left), then the co-located representation one (letter on the right). The context of this paper is close to the ghost metaphor [START_REF] Yang | Implementation and Evaluation of "Just Follow Me": An Immersive, VR-Based, Motion-Training System[END_REF] in which the trainer's movements are superimposed to users' one. However, in our context, users are in control of both representations at the same time.

Dual body representations could be adapted to other situations, but in this study we focus on anisomorphic manipulation techniques. These techniques usually increase or decrease users' motion. In this paper, we therefore conducted two experiments investigating either amplified motions or decreased motions. The goal was to have two main types of motion alteration. We chose two very well-known manipulation techniques: the Go-Go technique which amplifies movements to reach remote objects [START_REF] Poupyrev | The go-go interaction technique: non-linear mapping for direct manipulation in VR[END_REF], and the PRISM technique which decreases movements to gain precision [START_REF] Frees | PRISM interaction for enhancing control in immersive virtual environments[END_REF].

EXPERIMENT 1: DUAL REPRESENTATIONS FOR INCREASED MOTIONS DURING OUT-OF-REACH MANIPULATION

The first within-subject experiment investigated the use of a dual representation when interacting with remote objects using the well-known Go-Go technique [START_REF] Poupyrev | The go-go interaction technique: non-linear mapping for direct manipulation in VR[END_REF]. When using this technique, users' movements above a certain threshold are amplified, enabling both close and remote object manipulation.

Participants and Apparatus

Twenty-four participants (age min=20, max=36, avg=25.0±4.1, 11 women and 13 men) took part in this experiment. On a 7-point Likert scale, 7 of them were VR experts (score equal to or greater than 6), 6 were knowledgeable (score between 3 and 5) and 11 were beginners (score equal to or lower than 2). The majority of participants were students and staff recruited on our campus. All participants gave written and informed consent. The study conformed to the declaration of Helsinki, and was approved by the local ethical committee. Participants did not receive any compensation. They all had a normal or corrected vision.

Participants were equipped with a Valve Index Head-Mounted Display and Knuckles enabling finger tracking, tracked by two base stations. The experiment was performed using a desktop computer (Nvidia GeForce RTX 2080 Super, 32GB RAM, Intel(R) Xeon(R) W-2125 CPU), ensuring a minimum frame rate of 90fps under all conditions. The experiment was developed using Unity 2019.4.12f1. We used gender-matched avatars based on users' self-identification from the Rocketbox library. Avatars were not race-matched. They were animated using the RootMotion Final IK plugin. For object interaction (selection and hand poses), we used the SteamVR interaction system.

Go-Go Technique Implementation

Our implementation is based on the original Go-Go implementation [START_REF] Poupyrev | The go-go interaction technique: non-linear mapping for direct manipulation in VR[END_REF]. A one-to-one mapping is applied until the real hand position R r reaches a threshold value D, then it becomes non-linear. The virtual hand position R v is determined as follows:

R v = R r i f R r < D R r + k(R r -D) 2 otherwise
The threshold D was 2/3 of the real arm length. The factor k was computed to ensure that the maximum arm length was four times the real arm length. Therefore, when users had their real arm fully extended, the virtual arm was four times longer than their real arm. This k value enabled users to perform the task by reaching all the targets, while maintaining a relatively accurate interaction. To extend the arm, users' motion direction needed to be known. There are several ways of determining it, depending on whether the head, the shoulder or the controller positions are used to compute the direction vector. Different ways were tested, and finally the normalised vector linking the shoulder position to the controller position was kept, similarly to the implementation by Feuchtner and Müller [START_REF] Feuchtner | Extending the body for interaction with reality[END_REF]. This was the most stable solution, as it did not depend on the head orientation.

Tasks

Main Experimental Task: Picking and Sorting Apples

The Go-Go technique uses an adaptative gain, enabling both close and remote interaction. An ecological task involving both interaction in the peripersonal (close to the body) and the extrapersonal (far from the body) space was necessary. We chose the ecological task of picking apples. Participants were facing a virtual tree containing 20 apples (see Figure 1). They had to extend their arm to reach the apples placed in their extrapersonal space and pick them. Then, there was a sorting task. Participants had to bring back the apple to their peripersonal space to inspect it. They needed to look at them to see if they were flawless, or rotten, i.e. with stains on them. The goal was to make participants bring the apples back to them, to interact closer to their body and be aware of their dual representation. After looking at the apples, they had to put them in the appropriate basket situated just outside their peripersonal space beside the tree (flawless apples in the left basket, or rotten apples in the right basket). To pick/release the apples, subjects had to grab/release the Knuckles' handle. When they grabbed the handle, this would snap the apple to the virtual hand and trigger a hand pose.

Additional Task: Maximum Reach Estimation Task

Interacting with remote objects can alter the mental arm representation, and give a sensation of real elongated arm [START_REF] Feuchtner | Extending the body for interaction with reality[END_REF]. We therefore decided to add a task to estimate whether the perceived maximum reach distance was influenced by the user representation. For this additional task, participants were immersed in an empty scene, with only the Unity default skybox. They had no virtual body during this task and only saw a red virtual sphere in front of them. They were instructed to move this virtual sphere using the Knuckles controllers' pad to place them at the distance where they thought they could reach it with their arm fully extended. They had to keep their arms at their sides during the measure. This measure was repeated six times, three times with the sphere coming from away (at a distance of twice users' real arm length), and three times from near participants (at a distance of 0.3 times the real arm length), alternatively.

Experimental Design and Protocol

This experiment had a within-subject design with Representation as the independent variable. For each level of Representation, distorted (interaction) motions and real motions could be represented either by a realistic body (R), a ghost body (G) or nothing (Ø). For this experiment we chose three levels of Representation, defined as : R -Ø, R -G and G -R (see Figure 2). In R -Ø, only distorted motions were represented with a realistic elongated arm. In R -G, a realistic elongated arm was used for interaction, while real motions were represented by a ghost transparent arm. G -R corresponded to the opposite combination. We did not use other possible combinations, as the goal was to always have a different visual appearance for interactive and co-located representations (so for example the R -R combination was excluded), and only a realistic appearance when using a single representation (therefore no single ghost extended arm). Also, when using the Go-Go technique, the user must see where the virtual hand is to be able to interact during motion distortion, therefore combination Ø -R could not be used in this experiment. The order of the different Representations was fully counterbalanced. Participants were first welcomed, the experiment was explained to them and they had to read and sign the consent form. Before starting the main experiment, users were first immersed in the experimental scene but without any task to perform. They were represented by an avatar with a global scale calibrated to match their height. They were immersed 45 seconds to look at the scene and their avatar. The goal was to have a first perception of the virtual scene, with a normal calibrated avatar without extendable arms. After the 45 seconds, they performed the maximum reach estimation task. After this first exposure, the main experiment began, which was divided into three blocks, one for each Representation. Before the experimental task began, they had 20 seconds of free exploration of their avatar and the environment. Then, they had to press the controller's trigger to start a training with only five apples. After pressing the trigger again, they could start the main task, with 20 apples. The apples were placed similarly in all conditions to be able to compare results between the different representations. At the end of each block, participants were placed in another scene to do the maximum reach estimation task. In total, the experiment session lasted around 50 minutes.

Experimental Data

Subjective Measures

Participants' sense of embodiment was measured using the questionnaires from Peck and Gonzalez-Franco [START_REF] Peck | Avatar embodiment. a standardized questionnaire[END_REF] (mentioned as PGF questionnaire hereafter) and from Roth and Latoschik [START_REF] Roth | Construction of the Virtual Embodiment Questionnaire (VEQ)[END_REF] (mentioned as RL questionnaire). We decided to use both questionnaires because they have complementary questions. The questionnaire items were evaluated on a 7-point Likert Scale from 1 (Strongly disagree) to 7 (Strongly agree). For the PGF questionnaire, we customised question R7 as recommended in their paper. We used the question "I felt as if my real arm were becoming longer". For question R8, we asked "I felt a realistic sensation in my body when I saw my virtual body" as recommended. For R14 to R16, we used apples as the source of touch. We thought Table 1. Additional embodiment questions asked after conditions involving a dual representation (in R -G and G -R). it would be clearer for participants than asking about the touch of the ground as recommended in the questionnaire paper. It is possible to consider that the passive haptic feedback provided by the Knuckles grip could elicit a sensation of touch, even though the shape was different from an apple. In addition, six questions specific to this study were added. For the two conditions with a dual representation, questions related to senses of ownership and agency (from the RL questionnaire) were added for both virtual representations, the realistic (opaque) and the ghost (transparent) representations (see Table 1). We used the words opaque and transparent to have neutral words describing the representations that would not bias participants' responses.

Five questions on the global opinion regarding the virtual representation as well as usability were asked (see Table 2). Subjects were also asked to rank the conditions in order of preference after the experiment, and to explain why they chose this order.

Objective Measures

Because previous studies showed a possible impact of the virtual representation on performance [START_REF] Tran | Effects of virtual arm representations on interaction in virtual environments[END_REF], we included a performance measure in this experiment by measuring the time to pick the 20 apples. Similarly, related work showed that having a virtual extended arm can impact arm length perception, because it alters body schema [START_REF] Kilteni | Extending body space in immersive virtual reality: A very long arm illusion[END_REF]. We therefore included the maximum reach estimation task described in Section 4.3.2 after each main task, to evaluate whether the virtual representation would impact maximum reach perception.

Hypotheses

Previous studies showed that from a certain length (four times the real arm length), an extended arm do not feel like the users' own arm anymore [START_REF] Kilteni | Extending body space in immersive virtual reality: A very long arm illusion[END_REF]. Reminding users of their real arm might help to create a global sense of embodiment towards their representation, which could be higher than having only one visible arm. Also, having the constant feedback of their own arm might be good for the sense of agency, as they can always see a congruent feedback of their real actions, which is an important factor in eliciting a sense of agency [START_REF] David | The "sense of agency" and its underlying cognitive and neural mechanisms[END_REF]. Therefore we propose as hypothesis that dual body representations influence the sense of embodiment, especially the sense of agency (H1.1).

While some studies suggested that additional visible information can impact performance [START_REF] Tran | Effects of virtual arm representations on interaction in virtual environments[END_REF], we hypothesised that seeing real movements could help to understand the transformation applied in the case of anisomorphic manipulation. This should not be enough to increase performance in the case of the Go-Go technique, as the technique is simple and the task we used does not necessitate precision. We hypothesised that we should not observe differences in terms of performance between the conditions (H1.2), as dual representations should not hinder the task.

When using a single representation, an offset is created between the remote virtual hand interacting with objects and the real hand. Because of this mismatch between real and virtual movements, users might get frustrated. On the contrary, having a dual representation lets users see both their real and virtual movements, we hypothesised that this type of representation would be preferred. Also, it could be perceived as fun to embody two different virtual representations. Our third hypothesis was therefore that people prefer having a dual representation (H3.1).

Results

Considering the ordinal nature and the non-normal distribution of subjective questionnaires scores, we performed Friedman non-parametric tests (with α = 0.05) on the different embodiment components and the user experience questions, followed by post-hoc pairwise Wilcoxon signed-rank tests with Bonferroni correction. For objective measures, we used a one-way repeated measures ANOVA and Tukey's post-hoc tests when the normality assumption was verified by the Shapiro-Wilk test (for the estimated maximum reach distance), and Friedman tests followed by Wilcoxon tests when the normality assumption was not verified (for the time completion). For the additional questions on realistic and ghost representations in R -G and G -R, we performed Wilcoxon signed-rank tests.

Sense of Embodiment and Preference

Embodiment questionnaires For the RL questionnaire, we computed the global scores for the three components (Ownership, Agency and Change) as recommended. Globally, the Ownership (Mdn = 5, IQR = 3.438 -5.812) and Agency (Mdn = 6, IQR = 5.75 -6.5) scores were high in all conditions while the Change (Mdn = 2.25, IQR = 1.25 -3.75) scores were very low. For all components, we did not find any significant difference.

For the PGF questionnaire, we also computed global scores for each component (Ownership, Appearance, Multi-Sensory and Response), as well as the Multi-Sensory sub-score, the Agency score, using the two questions directly related to the sense of agency. Components scores were globally low (Mdn as if it was my own body or I felt as if my body was located where I saw the virtual body) while questions not adapted to the experimental protocol gave very low scores (such as questions on tactile sensations). For the Response component, there was a main effect of Representation (χ 2 = 7.525, p < 0.05). The Kendall's W value showed a small effect (W = 0.157). Pairwise tests showed that the score for R -Ø was significantly higher than for G -R (p = 0.01). This seemed mostly influenced by the question I felt a realistic sensation in my body when I saw my virtual hand. For the Multi-Sensory component, Friedman's test showed a main effect of the Representation (χ 2 = 6.1, p < 0.05), but the pairwise test did not reveal any significant difference between the conditions. The results of these two components are visible in Figure 3.

For all components, we did not find any order effect.

Additional questions For the additional ownership and agency questions about the realistic and the ghost representations (G -R and R -G only), there were significant differences in agency questions scores between R -G and G -R (see Figure 5). For the question controllingOpaque, the scores were significantly higher in R -G than G -R (p < 0.05). Participants felt more in control of the realistic arm when it was used for interaction than when it was displaying real motions. For the question controllingTransparent, the scores were significantly higher in G -R than R -G (p < 0.01). Similarly to the realistic arm, control was higher when interacting with the ghost arm. For the question causingTransparent, the scores were significantly higher in G -R than R -G (p < 0.01). No significant difference was found for the questions on ownership. We compared scores for similar types of representation (interactive or co-located) depending on the visual appearance. The scores for the interactive representations were not significantly different between the two appearances, neither for the co-located appearance. For each dual representation condition, we also looked at the potential difference in the scores between the interactive and the co-located representation. For the controlling and causing questions, scores were significantly higher for the realistic extended arm than for the ghost short arm in R -G (p < 0.05).

Preference R -Ø was the preferred condition, with 10 participants ranking it as their most preferred condition (see Figure 4). Its average rank was 1.67. R -G and G -R had the same average rank equal to 2.17. There was no significant difference on the user experience questions. The scores were average for the question liked (Mdn = 4, IQR = 2.75 -6), low for disturbing (Mdn = 2, IQR = 1 -5), high for questions easy (Mdn = 6, IQR =5.75 -7), clear (Mdn =6, IQR =6 -7) and exciting (Mdn =6, IQR =6 -7).

Objective Measures

For the mean estimated maximum reach distance, corresponding to the average of the six measures from the maximum reach estimation task, the sphericity was violated so a Greenhouse-Geisser correction was applied. We found a main effect of Representation (F 2.37,54.52 = 10.48, p < 0.001, η 2 p = 0.313). Post-hoc test showed that the baseline (the measure prior to the experiment) was significantly lower than measures after the different levels of Representation, as all the pairs with the baseline in them had a p-value lower than 0.001. However, there were no differences between the three levels of Representation. In the baseline, the estimated maximum reach was equal to 83.4 ± 15.2% of the real arm length. After the different conditions, it was respectively equal to 91.7 ± 16.2% of the arm length after R -Ø, 89.9 ± 16.9% after R -G and 90.6 ± 16.9% after G -R. We did not observe any order effect.

There was also no significant difference in the time completion between the different blocks. There was a learning effect, with participants performing better at the end of the experiment than at the beginning. People took 105.7 ± 40.7 seconds in average in condition R -Ø, 96.6 ± 25.3 seconds in R -G and 99.3 ± 22.8 seconds in G -R.

Discussion

The main goal of this experiment was to explore the impact of dual representations on the sense of embodiment and user performance. Except for a difference on the Response component between R -Ø and G -R, embodiment scores did not highlight differences between the conditions, which does not support H1.1. Additional questions showed that for both visual appearances, people had a higher sense of agency towards the arm when they were interacting with it (even if it was sometimes extended), than when the arm was showing their real movements. Also, the agency scores for the ghost arm were significantly lower than the realistic arm in R -G. The opposite was not true in G -R. This means than participants could feel control over a ghost extended arm because they were interacting with it, but they felt less control over a co-located ghost arm while interacting with a realistic extended arm. These results seem to suggest that the interaction capabilities of the avatar play an important role on the sense of agency, and that the interaction capabilities of the avatar could compensate for deformations of the shape and the motions of the avatar.

Additionally, the maximum reach estimation task revealed an increased perceived maximum reach distance after all the conditions, compared to the measure prior to the experiment. This shows that interacting with an extended arm changed users' mental representations of themselves, and they perceived their real arm to be longer. This is similar to results found in other studies [START_REF] Feuchtner | Extending the body for interaction with reality[END_REF][START_REF] Lin | The influence of perceptual-motor variability on the perception of action boundaries for reaching[END_REF].

Regarding user performance, we did not observe any significant difference in task completion time, which does not contradict H1.2. Participants performed similarly under all conditions. This is probably due to the fact that the co-located representation did not interfere during the task, as it was only an additional information not located where the manipulation was happening.

Almost half of participants preferred using only one representation, which does not support H1.3. They found it more "playful", or that knowing only where the interactive hand is was "the most important". Several participants also reported it as more "realistic" because they only had one arm, even though this arm was extended. This is in line with the study by Feuchtner and Müller [START_REF] Feuchtner | Extending the body for interaction with reality[END_REF] that also considered an extendable arm in augmented reality, in which 67% of participants preferred not seeing their real arm, and could embody the extended arm when interacting with it. Still, some participants preferred having a dual representation, to have the information of their real arm position. Some of them preferred their real body represented by a realistic body. These participants considered the ghost arm as an "extension", a superpower they had in the VE. Other people preferred interacting with a realistic arm, as they felt they were not able to pick apples with an "intangible" arm, and were more precise with a realistic arm. These two opposite opinions explain the similar mean ranking for R -G and G -R. Even though the single representation was preferred, there seems to be some individual differences.

The results could also be greatly influenced by the type of mapping. For this reason, we conducted a second study investigating another type of motion distortion.

EXPERIMENT 2: DUAL REPRESENTATIONS FOR DE-CREASED MOTIONS DURING PRECISE MANIPULATION

The goal of the second within-subject experiment was to investigate the use of a dual representation when forced to be always in sight during a task in the peripersonal space. We also wanted to test a technique with a different type of gain, i.e. decreasing users' movements. The PRISM technique was chosen as it scales down users' movements to increase precision.

Participants and Apparatus

In this second experiment, we also had 24 participants (age min=22, max=36, avg=26.8±3.3, 12 women and 12 men). None of them did the first experiment. 4 of them were VR experts, 13 were knowledgeable and 7 were beginners.

The apparatus was the same to that in the first experiment.

PRISM Technique Implementation

The PRISM technique (see implementation in [START_REF] Frees | PRISM interaction for enhancing control in immersive virtual environments[END_REF]) adjusts the virtual hand motion (translation and rotation) depending on users' hand speed.

For translation, we scaled users' motion depending on the speed measured in m/s. For rotation, the scale depends on the angular speed in degree/s. When implementing PRISM, the gains can be applied on global translation/rotation or independently on each axis. In our experiment, we used global scaling as it was found more intuitive after several tests. The different cases determining the gains applied are described here. The threshold values MinV , SC and MaxV were empirically adjusted to our task and its difficulty. Considering the chosen thresholds, and thanks to the instantaneous offset recovery when exceeding MaxV , the offsets stayed reasonable and the virtual hand could not drift away.

• HandSpeed < MinV . In this case, the motion is considered as noise and not intentional motion. We used MinV = 0.01m/s for translation and MinV = 5 degree/s for rotation. • MinV < HandSpeed < SC. Users are performing a slow motion in order to be precise. The function determining the gain applied is linear (going from 0 for MinV to 1 for SC). We used SC = 0.1m/s for translation, and SC = 50 degree/s for rotation. • SC < HandSpeed < MaxV . In this case we use a one-to-one mapping. We chose MaxV = 0.2m/s for translation and MaxV = 60 degree/s for rotation. • HandSpeed > MaxV . This case shows an intention to remove the current offset created between the virtual hand and the real hand. When the speed is greater than MaxV , the offset is instantly recovered.

Task

The task was similar to the 6-DOF task considered by Frees et al. [START_REF] Frees | PRISM interaction for enhancing control in immersive virtual environments[END_REF]. Subjects were instructed to grab a 3D object and place it at a given target position. The object was a cylinder, to have a simple hand pose to grab it, with an antenna on top of it to constrain its rotation on the three axes. The target consisted of the same object but transparent and red (see Figure 1). When the object was correctly placed, it turned green. The global object was considered well placed when its centre was within 3 millimetres of the target's centre and its three axes of rotation were aligned with those of the target (with a tolerance of 2.25 degrees per axis). The objects were always shown in the same order across the different conditions.

Experimental Protocol and Design

Similarly to Experiment 1, the experiment followed a within-subject design with Representation as the independent variable. This variable had four levels (see Figure 6). We added a fourth condition, Ø -R, because the PRISM technique is compatible with having only the real movements displayed and the object slowed down, contrary to the Go-Go technique. The order of the different Representation conditions was counterbalanced using a 4 × 4 Latin square design. Participants were first welcomed, the experiment was explained to them and they had to read and sign the consent form. Then they were immersed in the VE where they first had 20 seconds to observe their virtual body and the environment. Then, they had to press the controller's trigger to start a training with only two objects to place. They had a maximum of 30 seconds to place each object after which they could start the main experiment by pressing again the trigger. They had three minutes to place as many objects as possible. In order to keep the exposure time the same for each condition, we preferred to fix the total exposure time because we expected stronger user variability in this task than in the first experiment task. The next object (and the associated target) would appear when the previous one was correctly placed. In total, the experiment session lasted about an hour.

Experimental Data

We used the same subjective questionnaire as in Experiment 1. Only the following questions were changed, in the PGF questionnaire:

• R7: I felt as if my body had changed • R14: It seemed as if I felt the touch of the virtual objects in the location where I saw the virtual body touched • R15: It seemed as if the touch I felt was caused by the virtual objects touching the virtual body • R16: It seemed as if my real body was touching the virtual objects As objective data, the number of completed placements was counted. The times when each object was displayed, picked, released and correctly placed were logged. The hand velocity (in m/s) as well as the offset between the virtual and the real hands were also saved at each frame, for both left and right hands.

Hypotheses

We globally had the same hypotheses as in the first experiment. This experiment was slightly different from the first one, both because the technique decreased users' movements and the interaction was happening in users' peripersonal space. The PRISM technique alters users' movements which creates an offset between the real and the virtual hands [START_REF] Farrer | Effect of distorted visual feedback on the sense of agency[END_REF]. Yet, compared to the Go-Go technique, the offset introduced will tend to be smaller and will also increase progressively during the manipulation process. On the one hand, as the offset increases progressively and will tend to be small, the chances of noticing the offset are reduced. On the other hand, as the offset is dependent of the motion speed, the user might perceive the virtual hand motion as unpredictable. The dual representation could enable to decrease this effect of control loss. However, having two bodies with different mappings in the peripersonal space could be disturbing for users. One hypothesis was that dual representations influence the sense of embodiment, especially the sense of agency (H2.1).

We were still expecting similar performance between the different conditions. Because in all the dual representation conditions one representation is semi-transparent, we hypothesised that it would not add too much visual information to affect performance, so that we should not observe differences in terms of performance between the conditions (H2.2).

Finally, because with dual representations there would be two virtual hands very close to each other, we were expecting a less pronounced preference for dual representations compared to the Go-Go technique, but still a preference because it would not result in any loss of information about users' movements. The third hypothesis was that people would slightly prefer having a dual representation (H2.3).

Results

We performed a similar analysis to Experiment 1. Subjective data were analysed using Friedman's tests and objective measures using either one-way repeated measured ANOVA or Friedman's test depending on the Shapiro-Wilk test significance.

Sense of Embodiment and Preference

Embodiment questionnaires No significant effect of Representation on the components from the RL questionnaire was found. Ownership (Mdn = 4.625, IQR = 3 -5.5) and Agency (Mdn = 5.125, IQR = 4 -6.062) were above average. The Change component scores were low (Mdn = 2, IQR = 1 -3.062). Interestingly, the question It felt like the virtual body was my body was influenced by Representation (p < 0.05), with single representations having better scores than dual representations, and the pairwise tests showing that Ø -R was significantly higher than G -R. Other questions had similar results between Representations, resulting in no significant difference in the Ownership component.

For the PGF questionnaire, components for which we found a main effect of Representation are shown in Figure 7. Components scores were globally average (Mdn = 3.833, IQR = 2.667 -4.667 for Ownership, Mdn = 3.125, IQR = 2.125 -3.5 for Appearance, Mdn = 3.5, IQR = 2.958 -4.375 for Multi-Sensory, Mdn = 3, IQR = 2 -3.667 for Response and Mdn = 4, IQR = 3 -5 for Agency). For both the Response (χ 2 = 8.85, p < 0.05) and the Ownership (χ 2 = 8.38, p < 0.05) components, Friedman's test showed a main effect of the Representation. R -Ø had higher scores but the pairwise tests did not reveal any significant difference between the conditions. However, the question I felt as if my body was located where I saw the virtual body was significantly higher in R -Ø compared to R -G and G -R. Even though there was an offset with their real hand, participants felt co-located with their avatar. There was also an effect of Representation on the question I felt that my own body could be affected by the virtual world, with R -Ø having significantly higher scores than R -G. Friedman's test showed a main effect of the Representation on the global embodiment score (χ 2 = 9.63, p < 0.05). Embodiment score in R -Ø was significantly higher than in R -G (p < 0.05). There was no other significant difference.

For all components, we did not find any order effect.

Additional questions For the additional questions, we did not find any significant difference (see Figure 8). But, we also compared the scores between each similar question (for example, controllingOpaque and controllingTransparent). The scores for the question on controlling the interactive representation (i.e. showing distorted motions) were higher with the realistic appearance (in R -G) than with the ghost appearance (in G -R) (p < 0.05). The scores for the question wasMyBody for the interactive representation were also higher with the realistic texture (in R -G) than with the ghost texture (in G -R) (p < 0.05). In G -R, the scores for wasMyBody and controlling were significantly higher for the co-located realistic arm than for the ghost interactive arm (p < 0.01).

Preference R -Ø was the preferred condition, with 13 participants ranking it as their most preferred condition (see Figure 9). Its average rank was 1.92. R -G came second with an average rank of 2.38, then Ø -R (2.58) and G -R (3.13) as the least preferred. There was no significant difference in the responses to user experience questions, similarly to Experiment 1. The scores were average for the question liked (Mdn = 4, IQR = 2.75 -5), between low and average for disturbing (Mdn = 3, IQR = 2 -5), high for questions easy (Mdn = 5, IQR = 3 -6), clear (Mdn = 6, IQR = 5 -7) and exciting (Mdn = 6, IQR = 5 -7).

Objective Measures

The number of completions was slightly higher in R -Ø but no significant difference was found between the four conditions (F 2.76,60.63 = 2.43, p = 0.079, η 2 p = 0.100). There was a learning effect, with participants performing better at the end of the experiment than at the beginning. People correctly placed 15.71 ± 4.91 objects in average in R -Ø, 13.38 ± 6.02 in G -R, 13.46 ± 5.08 in R -G and 13.33 ± 5.31 in Ø -R. We also counted the number of adjustment releases (when objects were released without being correctly placed), for which there was a significant effect of Representation (p < 0.01). Pairwise showed that the number of adjustment releases was significantly higher in Ø -R than in R -Ø. The average number of adjustment releases was equal to 15.79 ± 7.04 in R -Ø, 16.42 ± 9.49 in G -R, 15.29 ± 7.77 in R -G and 19.71 ± 7.37 in Ø -R. Also, neither the offsets created nor the hand velocity were significantly different between the conditions. The average offset was 3.39 ± 0.97cm.

Discussion

The goal of this second experiment was to study the influence of dual representations on embodiment and performance when the interaction took place in the peripersonal space. We expected embodiment scores to be higher with the dual representations. On the contrary, there was a tendency of higher scores in R -Ø but pairwise tests could not show significant results. Only the global embodiment score showed that embodiment in R -Ø was higher than in R -G. Indeed, participants reported being disturbed by the presence of two hands, which could explain this result. Between the two dual representations, the results on the sense of embodiment tended to show that in this experiment, the hand appearance (realistic versus transparent) had an impact. The scores for the questions about control and ownership were higher with the realistic hand appearance than with the ghost one.

We did not find a significant difference between the conditions for the performance measure (number of completions), which does not contradict H2.2. However, performance tended to be higher for R -Ø, and equivalent for dual representations and the single representation Ø -R. The absence of a significant difference may be due to the number of participants. We also reported that the number of adjustment releases was high in Ø -R. It was indeed observed during the experiment that in this condition, where the hand would cross the object when there was an offset, people would tend to release and grab again the object to remove the offset. This is in line with papers showing that users are disturbed by interpenetrations [START_REF] Canales | Virtual grasping feedback and virtual hand ownership[END_REF].

Also, the preference for condition R -Ø was clearer than in the first experiment. But interestingly, R -G was often ranked as second, showing a good acceptance of the ghost hand to show real movements. Some participants reported that it was easier to understand their virtual movements when they had two representations. The dual representation also enables keeping the contact between a virtual hand and the virtual object. This is different from the original PRISM implementation [START_REF] Frees | PRISM interaction for enhancing control in immersive virtual environments[END_REF] where the virtual representation moves away from the object (like Ø -R in our experiment), and the offset is represented by either a line between the hand and the object for translations, or two sets of 3D axes for rotations. Our visual feedback is more compatible with avatars, as it leverages avatars to represent the offset.

GENERAL DISCUSSION

This section gives global discussion on the two experiments with some leads of future studies.

Impact of Dual Body Representations on Embodiment and Performance

Embodiment scores for dual representations were globally similar to the scores for single representations, especially when using the Go-Go technique, which is in line with previous studies [START_REF] Chen | Body ownership and the four-hand illusion[END_REF][START_REF] Miura | Multisoma: Distributed embodiment with synchronized behavior and perception[END_REF]. While we expected that showing real movements would increase the sense of agency, it was not higher compared to conditions when people could only see their distorted movements. This is in accordance with other studies which showed that people are not disturbed by small offsets [START_REF] Kokkinara | The effects of visuomotor calibration to the perceived space and body, through embodiment in immersive virtual reality[END_REF][START_REF] Porssut | Reconciling being in-control vs. being helped for the execution of complex movements in vr[END_REF] or can embody an extended arm until a certain length [START_REF] Feuchtner | Extending the body for interaction with reality[END_REF][START_REF] Kilteni | Extending body space in immersive virtual reality: A very long arm illusion[END_REF]. People even felt more in control of the extended arm than of the arm showing their real movements. But in the second experiment when the interaction happened in the peripersonal space, the avatar visual appearance seemed to have a higher impact on agency. Both virtual representations were visible and closer during object manipulation, which may have increased the comparison between both hands and the difference in embodiment scores. This may also be due to people focusing more on the objects than on their representation, and not being disturbed by the offset which was never big enough to be distracting. We also compared embodiment scores of the two experiments using a Student's t-test. For all conditions, the agency component score from the RL questionnaire was higher in the first experiment than in the second (p < 0.01 for G -R and R -G, and p < 0.05 for R -Ø). People therefore felt a lower sense of agency using the PRISM technique than using the Go-Go technique. The other components scores were similar in both experiments.

As expected, we could not find a significant impact of dual representations on performance. Contrary to papers showing that additional visual information can impact performance [START_REF] Tran | Effects of virtual arm representations on interaction in virtual environments[END_REF], in our study dual representations were neither beneficial nor detrimental to the task. It therefore seems possible to use dual representations in such anisomorphic manipulation tasks.

User Preference and Recommendations

The two experiments showed that people globally preferred having a single representation over a dual representation for manipulation. Still, results showed a good acceptance of all representations, dual or single. The questions on user experience had relatively high scores, demonstrating that people liked the different representations and found the interaction easy and clear with them. However, while the general preference was for displaying only the interactive representation, some people found dual representations useful. We hypothesise that individual differences might exist regarding the perception of such a dual representation. One potential explanation could be related to people's ability to estimate their real body position or whether they are task-oriented. Depending on users' preferences, they could choose the representation that better suits them.

The results from both experiments give us some insight on how to choose the best virtual representation during 3D anisomorphic manipulation. In general, we suggest to use a single realistic arm as the interactive representation, and to let users the possibility to activate a ghost co-located arm. However, when using a technique in the peripersonal space with small offsets, only keeping a single realistic interactive representation seems sufficient and preferred.

Measuring Embodiment

One of the challenges that we faced in both experiments was how to measure embodiment. While the most common method is the use of subjective questionnaires, it has several limitations as it relies on the participants' understanding of the questions. Moreover, existing questionnaires have not been designed to assess dual representations. To cope with these limitations, we added specific questions to address dual representations. Furthermore, we considered it a good opportunity to use two questionnaires [START_REF] Peck | Avatar embodiment. a standardized questionnaire[END_REF][START_REF] Roth | Construction of the Virtual Embodiment Questionnaire (VEQ)[END_REF], namely PGF and RL, which could be more sensitive to different aspects of embodiment in such a situation. The two questionnaires have different components, but still some comparisons could be made. The RL Ownership and Agency scores were high, and the Change scores low, while the PGF questionnaire scores all tended to be below average.

The RL Ownership and Agency questions are commonly used in the literature and tended to be high in all conditions, while the Change component questions were found irrelevant by the participants. They reported that they had difficulties answering the questions, suggesting that this component may be used in experiments investigating morphological differences, but probably not in experiments with calibrated avatars. For the PGF questionnaires, all components scores tended to be below average, as some questions (e.g. about tactile sensations) not adapted to the experiment seemed to have lowered global scores and added noise to the results. The questions in this questionnaire are more varied, suggesting a better adaptability to experiments with rich sensory stimulation, as there are questions asking specifically about it.

Overall, the two questionnaires provided different and complementary results. More questionnaires could be designed, as VR setups and experiments are various, and having only one standardised measure of embodiment seems like a difficult goal [START_REF] Skarbez | Mixed reality doesn't need standardized evaluation methods[END_REF].

Limitations and Future Work

Our experiments were designed to study two common types of motion alteration, either increasing or decreasing motion. However, other types of distortion could be influenced by the representation. For instance, another type of common distortion is collision handling. Dual representations of the virtual hand have been studied for fine manipulation involving collision handling [START_REF] Canales | Virtual grasping feedback and virtual hand ownership[END_REF], where a dual representation provided a good trade-off between performance and preference. It would be interesting to explore the use of dual representations when a collision happen after a larger arm movement, and not only finger movements. While not studied here, dual representations could be appropriate as collisions can create huge offsets between the virtual and the real hand. Dual representations could also be studied when using haptic retargeting [START_REF] Azmandian | Haptic retargeting: Dynamic repurposing of passive haptics for enhanced virtual reality experiences[END_REF], but the results might be similar to the results in our experiment with PRISM as it usually does not create big offsets. Moreover, we decided to display the co-located representation as soon as there was a gain applied. It would also be possible to display the co-located representation only if the offset is greater than a certain threshold. It would avoid having an additional information for precise manipulation like PRISM, when the offset is not disturbing for users, and only display the real hand when users can notice the mismatch and feel a loss of control. In this case, dual representations could be preferred by users as it would be more useful.

Multiple bodies have not been investigated a lot in VR studies for the moment, but they might be more explored in the future. It would be interesting to propose a definition and a taxonomy of such multiple representations. Several factors already appear differentiating in such representations: visual appearance, perspective (first-person versus third-person), control (isomorphic versus anisomorphic; synchronised versus alternate), location (co-located bodies or not). Dual representations could represent co-located bodies seen from a first-person perspective, while multiple bodies seen from different perspectives like in the study by Miura et al. [START_REF] Miura | Multisoma: Distributed embodiment with synchronized behavior and perception[END_REF] could be called duplicated representations. We hope this work will steer new studies and open more questions on the topics of embodiment and control of multiple representations.

Fig. 2 .

 2 Fig. 2. The three levels of Representation in the first experiment with amplified motion. The interactive arm is either realistic (in R -Ø or R -G) or transparent (in G -R). When a co-located arm is displayed, a ghost representation (in R -G) or a realistic one (in G -R) is used.

  ID Question wasMyBodyOpaque I felt like my opaque representation was my body. wasMyBodyTransparent I felt like my transparent representation was my body. controllingOpaque I felt like I was controlling the movements of my opaque representation. controllingTransparent I felt like I was controlling the movements of my transparent representation. causingOpaque I felt like I was causing the movements of my opaque representation. causingTransparent I felt like I was causing the movements of my transparent representation.

  = 3.75, IQR = 3 -4.667 for Ownership, Mdn = 2.625, IQR = 1.75 -3.406 for Appearance, Mdn = 3.42, IQR = 2.667 -4.167 for Multi-Sensory, Mdn = 2.75, IQR = 2 -3.833 for Response and Mdn = 4, IQR = 3.5 -5 for the Agency sub-component), with common questions getting relatively high scores (such as I felt like I could control the virtual body

Fig. 3 .

 3 Fig. 3. Results for the two components Response and Multi-Sensory from the PGF questionnaire in Experiment 1, for which a main effect of Representation was found. Only the pairwise test on the Response component showed a significant difference between R -Ø and G -R (p < 0.05). There was no main effect for other components.

Fig. 4 .

 4 Fig. 4. Ranking for the different conditions in the Go-Go experiment. R -Ø was preferred with 10 participants ranking it as first.

Fig. 5 .

 5 Fig. 5. Results of additional questions on the senses of ownership and agency for R -G and G -R in the experiment with the Go-Go technique. * p < 0.05, ** p < 0.01

Fig. 6 .

 6 Fig. 6. Different levels of Representation in the second experiment with amplified motion. The interactive arm is either realistic (in R -Ø or R -G), transparent (in G -R) or not displayed (in Ø -R). A co-located arm is displayed, either a transparent one (in R -G) or a realistic one (in G -R and Ø -R).

Fig. 7 .

 7 Fig. 7. Results for the two components Response and Ownership, as well as the global Embodiment score (from the PGF questionnaire) in Experiment 2, for which a main effect of Representation was found. Only the pairwise test on Embodiment scores showed a significant difference between R -Ø and R -G (p < 0.05). There was no main effect for other components.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Results of additional questions on the senses of ownership and agency for G -R and R -G in the experiment with the PRISM technique. * p < 0.05, ** p < 0.01

Table 2 .

 2 Questions on user experience asked after all the conditions.

	ID	Question
	liked	I liked my virtual representation(s).
	disturbing	My virtual representation(s) was/were disturbing.
	easy	It was easy to interact.
	clear	It was clear how I could interact with the environment.
	exciting	It was exciting to interact with the environment.

ACKNOWLEDGMENTS

We wish to thank all the participants who took part in these two experiments. This work was sponsored by the Inria Research Challenge Avatar, and from the French National Research Agency under the Investments for the Future program (PIA) grant (ANR-21-ESRE-0030 CONTINUUM).