K H Tran

M Kühni

C Morin

P Guibert

Anisole laser induced fluorescence (LIF) for imaging local heterogeneities in temperature in a rapid combustion machine

The aim of the study is to characterize the local temperature heterogeneities for HCCI combustion mode in a rapid compression machine (RCM), where temperature and pressure vary simultaneously. The technique of Laser Induced Fluorescence (LIF) is used with one excitation wavelength and two integration spectral bands. Fluorescence signal depends on several parameters such as temperature, pressure, tracer concentration, bath gas and laser energy. Firstly it is necessary to perform a parametric analysis and a calibration in a high pressure-high temperature (HP-HT) vessel, before investigation in controlled conditions such as in a RCM.

Introduction

A precise knowledge of thermodynamic conditions such as the concentration, the temperature and the pressure is essential for present combustion and pollutant formation in IC engine. The technique of laser induced fluorescence (LIF) was developed for the temperature measurement in controlled conditions such as in a Rapid Compression Machine (RCM), engine, shock tube or heated flow cell [1][2][3]. Most of tracers commonly used in the literature are, aromatics, polycyclic aromatic hydrocarbons (PAHs) and ketones which have a high fluorescence quantum yield (toluene 17 . 0 f , benzene 22 . 0 f ) [4].

In a global and macroscopic balance formulation, the fluorescence signal depends for weak excitation on thermodynamics parameters according to the following equation: (1) with E the laser fluence, (hc/λ) the energy of a photon at the excitation wavelength λ, opt the overall efficiency of the collection optics, dV c the collection volume, the molecular absorption cross section of the tracer, Φ the fluorescence quantum yield, X the mole fraction, P the total pressure, k the Boltzmann constant and T the temperature.

Different LIF techniques exist to measure the temperature based on Eq. (1), the selection of one of them is dictated by thermodynamic quantities to study, the selected tracer, thus the experimental conditions.

The first one is the single color detection technique used for homogeneously seeded flows. This technique is restrictive as it applies in constant total pressure conditions, with homogeneous concentration distribution at constant volume.

Proceedings of the European Combustion Meeting 2011

Experimental conditions have to be set such as laser intensity, excitation wavelength, optical yield and volume. Thus the fluorescence signal is proportional to the tracer absorption cross section and the fluorescence quantum yield. The fluorescence signal evolves in T 1 :

P T T T S f , , , 1 (2)
The second technique is the single excitation twocolor detection technique, it enables to measure fluorescence signal in inhomogeneously seeded flow. In fact, we take away from homogeneously tracer concentration condition thanks to the two integration bands ratio. The detected fluorescence signal ratio of spectral bands 1 and 2 is presented below:

y x T f P y x T P y x T P T y x S P T y x S f f , , , , , , , , , , , 2 1 1 1 2
(3) The fluorescence signal ratio is proportional to the temperature.

The third technique is the dual-excitation wavelength technique. It also enables to measure fluorescence signal in inhomogeneously seeded flow. This technique is applied in case where the tracer partial pressure varies and the tracer concentration is inhomogeneous. The fluorescence signal ratio from two excitation wavelengths 1 and 2 gives:

P T T P T T E S E S f f , , , , , , 1 1 2 2 , , 21 21 1 1 (4)
This last technique enables to obtain a better signal to noise ratio than the second technique, whereas it requires an important experimental set up and a high cost because it involves two lasers with pulses slightly delayed to ensure separate detection of the two fluorescence signals in the same wavelength band.

In our study, the second technique is selected, taking account on experimental conditions noted later, and satisfactory fluorescence signal compared to the first technique.

The single excitation two-color detection technique was used by Luong et al. [3] with the excitation wavelength of 248 nm and toluene in air over a temperature range of 300-675 K in shock tube and heated flow-cell for in-cylinder temperature distribution measurements. Kaiser et al. [5] also used this technique with the excitation wavelength of 266 nm and naphthalene as tracer in nitrogen and oxygen over a temperature range of 400-928 K at 0.1 MPa for equivalent ratio and temperature imaging.

Specific objectives

The single-excitation two-colour detection technique for the temperature measurement depends on pressure and temperature which have an influence on the variations of the absorption cross section and the fluorescence quantum yield. Preliminary it is necessary to investigate a fluorescence signal calibration in a set up where thermodynamic conditions are perfectly known. So a prior study has been realised in a high pressure and high temperature (HP-HT) cell over a large range of temperature (473-773 K) and pressure (0.2-4 MPa) to determine the selection of the tracer, the optimal concentration, the excitation wavelength and the laser intensity in order to obtain a satisfactory fluorescence signal with different sensibilities according to the temperature and the integration bands selected.

Experimental set-up

The fluorescence signal calibration of tracer is realised in a HP-HT cell, as shown in Fig. 1 and described in Ref [6]. The inside volume is 7.6969 x 10 -4 m 3 , heated and maintained at constant temperature by eight electrical resistance (230 W each). A system composed of three thermocouples inside the cell enables to regulate it and ensures the cell temperature homogeneously. The liquid fuel/tracer blend is vaporized by reaching the boiling point and the saturate vapour pressure thanks to vacuum pump in the inside volume. An admission circuit composed of pressure sensors and flood gates enables to up the cell under pressure with different bath gases. The optical system is composed of a frequency-quadrupled and pulsed Nd:Yag laser (266 nm, 90 mJ/pulse at 10 Hz). The pulse width fluctuation is 3-4 nm with an energetic stability of ± 8 %. The laser beam crosses over the centre of the cell and it is recovered by a beam trap and a fluxmeter that enables to know the temporal laser fluctuations and the laser intensity. The fluorescence signal is collected at 90° from the incident beam by a high sensitivity spectrometer supplied with an UV lens (motorized slit, three diffracting gratings of 150, 600 and 1200 grooves per mm, integration time of 150 μs), and an ICCD camera supplied with an UV lens. The filters are placed at front of measurement devices.

Several tracers were tested in order to select the more appropriate. Fluorescence spectra were measured for polycyclic aromatic hydrocarbons These different tracers present low fluorescence signal in the explored experimental conditions, except anisole. This tracer is then selected for its satisfactory spectroscopy proprieties [7] and its physical proprieties close to these of isooctane (C 8 H 18 ), the traced fuel. The anisole is a methyl oxide-ether of phenol (cf. Fig. 2), it presents under liquid form. Anisole is a very volatile compound (100% at 294 K), its saturate vapour pressure is 0.0013 MPa at 315 K and its boiling point is equal to 429 K [8] that enables a good vaporization in our experimental conditions.

Results and discussion

Anisole absorption and fluorescence

For different LIF techniques, the absorption cross section is taken into account to calculate the fluorescence signal which varies with the temperature and the excitation wavelength. At a constant excitation wavelength, the absorption cross section depends only on the temperature. It is deduced from Lambert-Beer's law: (5) with I 0 the incident intensity, I the transmitted intensity, l the path length and C the anisole concentration. As presented in Fig. 3, the absorption spectrum of anisole is located between 240-290 nm, as underlined in Ref [9] too. The absorption spectra comparison shows a maximum difference of absorption cross sections of 12% for different temperatures at 266 nm. It will be then necessary to normalize the fluorescence signal to the absorption cross section corresponding to a given temperature for the used excitation wavelength of 266 nm. The anisole fluorescence spectrum in isooctane is included between 270-400 nm, the peak is centred at 295 nm (cf. Fig. 4 and Ref. [START_REF] Matsumoto | Teijiro Ichimaru[END_REF][11]. This figure shows that fluorescence spectrum strongly decreases with increasing of the temperature, which underlines the interest of anisole as tracer for the measurement of temperature by LIF. As pictured in Fig. 5, the fluorescence signal evolution versus molecular density is represented for 2% anisole/isooctane blend. This concentration is selected to stay in the linear regime. The signal increases up to reach a saturation level which corresponds to for 773 K. After these respective maxima, the fluorescence signal decreases due to two quenching phenomena, the reabsorption and the auto-quenching (collision with anisole molecules in the ground state). Therefore to avoid these quenching phenomena, the experimental molecular density is fixed at (cf. Fig. 5). Similarly, all experiments are investigated in the linear fluorescence regime, with a laser energy laser fixed at 90 mJ/pulse.

C l I I * ) ( / ) ( ln ) ( 0

Pressure and temperature dependences of the anisole fluorescence signal

The anisole fluorescence signal depends on thermodynamics parameters such as temperature and pressure, as seen in Eq. (1). Therefore it is necessary to study independently the pressure influence at fixed temperature and the temperature influence at fixed pressure, to build a cartographic of signal intensity in function of pressure and temperature. When normalized to the molecular density, the fluorescence signal noted * f S is proportional to the fluorescence quantum yield at fixed temperature and wavelength, as expressed in Eq. ( 6). The fluorescence signal decreases with the pressure as presented in Fig. 6. Indeed, it decreases by factor 2.5 (a decrease by a factor larger than two for 1,2,4-trimethylbenzene, [12]) when the pressure increases from 0.2 to 4 MPa. This decrease with the pressure depends on the temperature, whereas, it seems to have a low influence from 673 K. This important decrease could be due to high collision rate leading to more non-radiative collisional deexcitation. It is noted that, for the ketones, the fluorescence signal increases with pressure [2], a similar evolution is observed for PAHs (naphthalene [13] and fluoranthene [6]). The anisole fluorescence quantum yield decreases with increasing temperature between 473-773 K, as observed in Fig. 7. The sensitivity of the fluorescence signal is stronger to the temperature than the pressure. In fact, the fluorescence signal decreases with a factor of 5 for the explored temperature range. This fluorescence decrease is explained by the radiative mechanisms of intersystem crossing more efficient when the excited levels are energetic. A similar tends are observed in the literature for others tracers as acetone [2], 3-pentanone [1], for which the fluorescence quantum yield decreases with increasing temperature. As presented in Fig. 7, the fluorescence quantum yield decreases significantly, with a more marked decrease at high pressure. At 0.2 MPa, the fluorescence quantum yield decreases exponentially by one order of magnitude within the studied temperature range, the single exponential fit is given by: For imaging temperature by fluorescence, it is interesting to normalize the signal to the molar fraction and the absorption cross section for a given temperature. The fluorescence signal noted f S depends then on the inverse of the temperature and can be used to convert the fluorescence signal into temperature from the dual-colour detection and one excitation wavelength technique for the temperature measurement: [START_REF] Matsumoto | Teijiro Ichimaru[END_REF] As pictured in Fig. 8, the fluorescence quantum yield decreases significantly in our temperature range, this evolution is similar for all the studied pressure. The fluorescence decrease evolves as an exponential. For 0.2 MPa, within the studied temperature range, the exponential fit is written as:

) , , ( ) , , ( ) ( )

Influence of the ambient gas

The influence of gas on anisole fluorescence is observed for nitrogen, carbon dioxide and argon. The evolution of the fluorescence signal versus pressure is similar for nitrogen and carbon dioxide, whereas for argon, the fluorescence signal decreases lower than in the two other gases, as shown in Fig. 9.

Implication for temperature measurement by the single-excitation two-color detection technique

From the previous results where the influence of pressure and temperature are characterized, temperature can be inferred from the ratio of measured fluorescence signals for two integration bands at the excitation wavelength of 266 nm. As pictured in Fig. 10, this ratio is a single function of temperature. For the studied pressures, the ratio of fluorescence signal shows a satisfactory sensitivity versus temperature. Different integration bands were tested to optimize the ratio value. We have to take into account on the signal/noise ratio, concerning the integration band 2 located at the fluorescence spectral queue. In fact, we take care to select only 2 which present a signal/noise ratio with a factor of 2 or more at 773 K, as shown in Fig. 11. Integration bands selected are then 1 = 291-312 nm and 2 = 374-395 nm. The band integration 1 corresponds to the central eak of the fluorescence spectrum. For the studied temperature range, the ratio value reaches 3.45, the ratio of fluorescence signal increases of 70% in nitrogen. The signal/noise ratio for 1 and 2 are equal respectively to 51.6 and 2.9.

Conclusion

This study has investigated the influence of pressure, temperature and gas composition on anisole fluorescence in high-pressure and high-temperature device for the excitation wavelength of 266 nm. The aim of this parametric study is to calibrate the singleexcitation two color detection technique for heterogeneity temperature measurements in a RCM using anisole as a new molecule tracer. This LIF technique is preferred from others techniques introduced in this work, because it is able to measure temperature for inhomogeneously seeded flow, and for its low facility set up compare to the dualexcitation wavelength technique.

Anisole mixed with isooctane is selected as fuel tracer, for its satisfactory physical and spectroscopy properties at 266 nm. Anisole absorption and fluorescence are studied for a pressure range of 0.2-4 MPa, a temperature range of 473-773 K, a constant concentration of Anisole presents a fluorescence emission band between 270-400 nm and the peak is located at 295 nm. For the studied temperatures, the fluorescence signal decreases with increasing pressure from 0.2 to 4 MPa. This decrease could be due to high collision rate leading to more non-radiative collisional desexcitation.

Regarding the influence of temperature, the fluorescence quantum yield decreases significantly with increasing temperature for the studied pressure range. This important decrease is due to a vibrational energy increase, with more efficient radiative mechanisms of intersystem crossing.

From these experimental results, the temperature can be inferred from the fluorescence ratio for the excitation wavelength of 266 nm and the detection band. An optimised couple of detection band is fund for 1 = 291-312 nm and 2 = 374-395 nm. This combination corresponds to commercial bandpass filters called BrightLine FF01-292/27-50 and BrightLine FF01-386/23-50 (Semrock). The comparison between the calculating and experimental fitting from the filtered gives a good agreement. Calibration curves are presented in a large range of temperature, pressure and bath gases, in conditions close to those encountered in RCM. These calibrations are correct for the assumption where quasi-static results present a similar behaviour to dynamic measurements in RCM.

  (fluoranthene C 16 H 10 ,), aromatics (toluene C 7 H 8 , anisole C 6 H 5 OCH 3 ), ketones (3-pentanone C 5 H 10 O, acetone C 3 H 6 O) at 573 K and 0.2 MPa, for the excitation wavelength of 266 nm in nitrogen.

Figure 1 :

 1 Figure 1: High pressure and high temperature device with optical system

CH 3 OFigure 2 :

 32 Figure 2: Anisole chemical structure The HP-HT cell experiments are realised in different bath gases (nitrogen, argon, carbon dioxide and air), for a pressure range of 0.2-4 MPa and temperature range of 473-773 K. For given experimental thermodynamics conditions, the fluorescence signals are averaged from 900 emissions spectra. Every experiment is reproduced three times. The normal standard deviations are included between 1.5-15% for the different bath gases.

Figure 3 :

 3 Figure 3: Absorption cross section of anisole for different temperatures in nitrogen at 0.2 MPa

Figure 4 :

 4 Figure 4: Normalized fluorescence spectra of anisole for different temperatures and a pressure of 0.8 MPa in nitrogen (λ exc = 266 nm). The fluorescence spectrum is normalized to the maximum value recorded at 473 K

Figure 5 :

 5 Figure 5: Evolution of normalized anisole fluorescence signal versus molecular density in nitrogen at 0.2 MPa , 473 K and 773 K for the excitation wavelength of 266 nm. The signal is normalized to the reference value obtained for the highest fluorescence signal. Dash line represents the experimental molecular density value.

Figure 6 :

 6 Figure 6: Influence of pressure on anisole fluorescence in nitrogen for different temperatures at 266 nm. The signal * f S is normalized to the molecular density and the reference value at 0.2 MPa. It is necessary to take into account of the anisole absorption cross section to see the influence of temperature on the fluorescence quantum yield. Therefore the fluorescence signal * f S is normalized to the corresponding cross section value for a given temperature and excitation wavelength and is noted * * f S , as presented in Eq. (7):

2 RFigure 7 :

 27 Figure 7: Influence of temperature on anisole fluorescence in nitrogen for several pressures. The signal is normalized to the molecular density, the absorption cross section and the reference value obtained at the temperature of 473K.

Figure. 8 :

 8 Figure.8: Influence of temperature on anisole fluorescence excited at 266 nm in nitrogen for different pressures. The signal f S is normalized to the molar fraction, the absorption cross section and the reference value at 473 K.

Figure 9 :

 9 Figure 9: Influence of pressure on anisole fluorescence for different gases at 266 nm. The signal * f S is normalized to the molecular density and the reference value at 0.2 MPa.

Figure 10 :

 10 Figure 10: Evolution of the fluorescence signal ratio versus temperature in nitrogen at 1.2 MPa for the excitation wavelength of 266 nm.

Figure 11 :

 11 Figure 11: Anisole fluorescence and noise spectra with integration bands respectively located at 1 = 291-312 nm and 2 = 374-395 nm, for 1.2 MPa and 773 K in nitrogen.Experimental fluorescence signal ratios plotted in Fig.12decrease in our temperature range for different pressures in nitrogen. The fluorescence decreasing evolves as an exponential function.

Figure. 12 :

 12 Figure.12: Evolution of the experimental fluorescence signal ratio versus temperature for different pressures in nitrogen, with1 = 291-312 nm and 2 = 374-395 nm. For a pressure of 3 MPa, the comparison between the calculating and experimental fitting gives a good agreement, as seen below.

  intensity of 90 mJ/pulse in different bath gases as nitrogen, carbon dioxide, argon and air.

Acknowledgements This work was performed during the PhD thesis of K.H. Tran supported by the FUI (French Fond Unique Interministériel) in the framework of the MODELESSAIS Pôles de Compétitivité MOVE'O project.


M Filliung

J Drupt

C Peraud

C Dune

N Boizot

A Comport

C Anthierens

V Hugel

An Augmented Catenary Model for Underwater Tethered Robots

This paper examines the relevance of using catenary-based curves to model cables in underwater tethered robotic applications in order to take into account the influence of hydrodynamic damping. To this end, an augmented catenarybased model is introduced to deal with the dynamical effects of surge motion, sway motion or a combination of both on a cable. Experimental studies are carried out with eight cables of varying stiffness, weight and buoyancy. One end of the cable is fixed, while the other end is moved by the underwater robot. The obtained results help to determine which cables and which dynamics are compatible with a fair estimation of the cable shape through the proposed models.

I. INTRODUCTION

Remotely Operated Vehicles (ROVs) are used to explore unknown areas. Their tether is the only way to provide direct feedback to the pilot, as electromagnetic waves are absorbed in the first few centimetres of water. However, the tether can interfere with the robot's movement and, if its shape is not controlled, it can become entangled in obstacles. This is even more critical when exploring confined karstic terrain, wrecks or offshore installations with small observation ROVs. This study focuses on shape estimation of tethers of a few metres in length, e.g. the portion of cable connecting two mini-ROVs of a chain [START_REF] Laranjeira | Catenary-based visual servoing for tether shape control between underwater vehicles[END_REF] or a mini-ROV to an Unmanned Surface Vehicle (USV) [START_REF] Tortorici | Towards active self-management of umbilical linking rov and usv for safer submarine missions[END_REF]. To be used in robot control for path following and obstacle avoidance, the chosen geometric model must be parametrised and fast to compute.

The standard catenary model, bound to a vertical plane, is defined as the shape of an idealized homogeneous hanging cable with fixed length and fixed ends, only subject to its own weight in the air. The catenary shape can be determined knowing the cable's length and its deflection. Depending on the available sensors, its shape can be estimated either from the relative position of the two attachment points such as in [START_REF] D'antonio | The catenary robot: Design and control of a cable propelled by two quadrotors[END_REF], [START_REF] D'antonio | Folding knots using a team of aerial robots[END_REF] or from the local cable tangent orientation [START_REF] Sellier | Inertial-Shape Estimation of Underwater Cables for Tethered Robots[END_REF].

However, the extension of the catenary model to an underwater cable is not straightforward in the presence of hydrodynamic effects. In static conditions, submarine cables are only subject to weight and buoyancy. While the quasistatic assumption remains applicable for aerial cables at low speed when air friction can be neglected, the underwater drag involves hydrodynamic forces that move the cable from its vertical plane, even at low speed [START_REF] Drupt | Validity of the catenary model for moving submarine cables with negative buoyancy[END_REF].

The main contribution of the present study is the introduction of a new enhanced catenary model that incorporates full tilting of the tether during surge or sway motion of the 1 COSMER Laboratory, Univ. de Toulon, France 2 Univ. de Toulon, Aix Marseille Univ, CNRS LIS, Marseille, France 3 CNRS I3S Laboratory, Univ. Côte d'Azur, Sophia Antipolis, France underwater robot. The relevance of this geometric model is confirmed by experiments conducted on eight cables with different mechanical properties (Fig. 1).

The paper is organized as follows. Section II presents related work on underwater tethered robots and cable shape estimation. Section III introduces the augmented catenary model. The experimental protocol and data analysis are detailed in Section IV. Section V concludes the study and outlines future work.

II. RELATED WORK

Tethered robots are particularly common in underwater robotics, where cables are the only way to provide real time communication between ROVs and the surface [START_REF] Tortorici | Towards active self-management of umbilical linking rov and usv for safer submarine missions[END_REF]. Multiple underwater robots can also be connected together in order to communicate freely and share resources. In [START_REF] Yu | Armless underwater manipulation using a small deployable agent vehicle connected by a smart cable[END_REF], small vehicles are connected to a bigger autonomous submarine. In addition, underwater robot chains are being investigated in order to limit the mechanical impact of tethers on ROV navigation and control by the addition of multiple mini-ROV on the cable to control its shape [START_REF] Laranjeira | Catenary-based visual servoing for tether shape control between underwater vehicles[END_REF], [START_REF] Sellier | Inertial-Shape Estimation of Underwater Cables for Tethered Robots[END_REF]. It results in an additional underwater tethered robot configuration in which small devices are linked together by short cable portions.

The safe control of tethered robotic systems strongly relies on cable shape knowledge: on the one hand, controlling tethered robots requires knowledge of the cable's state [START_REF] Laranjeira | Catenary-based visual servoing for tether shape control between underwater vehicles[END_REF], on the other hand, cable shape knowledge gives information about the localization of the robot at its end point [START_REF] Mcgarey | Tslam: Tethered simultaneous localization and mapping for mobile robots[END_REF], [START_REF] Viel | ROV localization based on umbilical angle measurement[END_REF]. Some underwater cables are designed as proprioceptive sensors in order to measure their own shape. Opticalfiber cables measure their deformations using reflectometry techniques [START_REF] Duncan | High-accuracy fiber-optic shape sensing[END_REF], [START_REF] Floris | Fiber optic shape sensors: A comprehensive review[END_REF]. Such cables can be used to locate connected underwater vessels [START_REF] Yu | Armless underwater manipulation using a small deployable agent vehicle connected by a smart cable[END_REF]. Another version of optical fiber cable shape sensing consists in adding an external optical fiber sensor coating on a cable [START_REF] Xu | Underwater cable shape detection using shapetape[END_REF]. Such cables are, however, fragile and extremely expensive. The cable shape can also be estimated from the measurements of inertial sensors placed along it [START_REF] Frank | Smart tether system for underwater navigation and cable shape measurement[END_REF]. However, these methods require the use of a specially designed cable. In addition, the accuracy of these methods decreases with the cable's length due to error accumulation, e.g. approx. 1 cm error for a 1 m long optical fiber cable [START_REF] Duncan | High-accuracy fiber-optic shape sensing[END_REF].

Other strategies make use of physical or geometrical cable models. A physical model can be developed based on the cable's hydrodynamics and the forces acting upon it [START_REF] Schjølberg | Finite element cable-model for remotely operated vehicles (rovs) by application of beam theory[END_REF], [START_REF] Meng | Dynamics calculation of complex deep-sea cable system based on hybrid optimization algorithm[END_REF], [START_REF] Soylu | Dynamics and control of tethered underwater-manipulator systems[END_REF], [START_REF] Hong | Dynamics modeling and motion simulation of usv/uuv with linked underwater cable[END_REF]. Although such models are the most comprehensive and designed to closely match the physics, they are computationally intensive and require an extensive understanding of multiple parameters which are difficult to measure under actual circumstances, such as water current or thruster parameters. As a result, simpler models are often preferred. The cable may be constrained to a simplified geometric shape artificially, for instance by introducing weights or sliding floaters to make it piecewise linear [START_REF] Viel | Self-management of the umbilical of a rov for underwater exploration[END_REF], [START_REF]Self-management of ROV umbilical using sliding buoys and stop[END_REF], [START_REF] Viel | ROV localization based on umbilical angle measurement[END_REF]. Some studies employ the catenary model, an hyperbolic curves, for underwater or aerial settings [START_REF] Laranjeira | Catenary-based visual servoing for tether shape control between underwater vehicles[END_REF], [START_REF] Sellier | Inertial-Shape Estimation of Underwater Cables for Tethered Robots[END_REF], whereas parabolic curves may also be utilized in aerial scenarios [START_REF] Smolentsev | Shape visual servoing of a tether cable from parabolic features[END_REF].

However, in the presence of external forces, including ambient currents, hydrodynamic damping during motion, the geometric models that represent quasi-static cable shapes may not be accurate enough. A preliminary study [START_REF] Drupt | Validity of the catenary model for moving submarine cables with negative buoyancy[END_REF] investigated the accuracy of an inclined catenary model for short cables connecting a pair of moving robots, using motion tracking of genuine cables. This paper expands on the findings of the study by presenting an enhanced parameterized model that considers complete cable inclination caused by external forces underwater.

III. MODELING AND ESTIMATION

Let F w = (O w , x w , y w , z w ) be a fixed Cartesian reference frame with the z w -axis vertical and pointing upwards.

A. Standard catenary model

The shape of a homogeneous hanging cable, with fixed length L, only subjected to its weight and buoyancy, is defined by a standard catenary curve (Fig. 2). The catenary lies in a vertical plane

P v = (O v , x v , z v ), associated to the Cartesian frame F v = (O v , x v , y v , z v )
where O v is the lowest point of the curve, x v is horizontal and z v is collinear to z w . Curve points coordinates ( v X, v Z) ∈ P v are given by:

( v X, v Z) ∈ [ v X i , v X f ] × R such that v Z = cosh ( v XC) -1 C (1)
where C ∈ R * + is the catenary parameter, and indexes i and f refer to the curve's endpoints. The left superscript indicates the frame in which a coordinate is given. 2: Catenary curve of length L defined in P v . Parameters are the sag H, the difference of elevation ∆H and the horizontal distance l between attachment points.

O v z v x v ( v X i , v Z i ) ( v X f , v Z f ) H ∆H l Fig.
P v P γ e ( v X i , v Z i ) ( v X f , v Z f ) x v z v O v x γ z γ O γ g a γ γ x w y w z w O γ

B. Augmented catenary model

As soon as the underwater cable is moved fast enough, hydrodynamics forces become non-negligible. The drag force dampens the movement with an amplitude that is proportional to the square of the speed. This effect is amplified by the added mass when significant accelerations are present [START_REF] Fossen | [END_REF], [START_REF] Schjølberg | Finite element cable-model for remotely operated vehicles (rovs) by application of beam theory[END_REF].

In the steady state case where all points on the cable move at the same velocity, they are subjected to the same drag forces. As a consequence, the resulting uniform acceleration of the cable is tilted, and its intensity is increased depending on the velocity. The addition of two degrees of freedom in the cable shape model to account for this effect is used here to extend the standard catenary model.

The first degree of freedom is a rotation of angle γ around the catenary axis e, which is defined as the unit vector that connects cable endpoints [START_REF] Sellier | Inertial-Shape Estimation of Underwater Cables for Tethered Robots[END_REF] (Fig. 3). This is to consider the deformations of a tether subjected to sway motions, i.e. the cable ends move out of the vertical plane, namely P v . This rotation applied to F v and P v gives F γ = (O γ , x γ , y γ , z γ ) and P γ , respectively.

The second degree of freedom, a rotation of angle θ in P γ (Fig. 4), considers tether deformations caused by surge motions, i.e. the two cable ends get closer or further away. The approximation of uniform hydrodynamic forces along the cable results in a uniform tilted acceleration a θγ with respect to gravity. Let F θγ = (O θγ , x θγ , y θγ , z θγ ) be the associated Cartesian frame where z θγ is parallel to a θγ and y θγ is parallel to y γ . The two ends of the cable can be expressed in F θγ and ( 1) is used to construct the oriented catenary. This 2-DOF model is named the θγ-augmented model.

Note that zero values for γ and θ yield the standard catenary model. If θ only is set to zero, the model presented in [START_REF] Sellier | Inertial-Shape Estimation of Underwater Cables for Tethered Robots[END_REF] can be obtained, and is referenced as γ-augmented.

O γ z γ x γ O θγ z θγ x θγ ( γ X i , γ Z i ) ( γ X f , γ Z f )

C. Curvilinear discretization and residual

Let us consider n + 1 3D measurement points distributed along the cable's length. Accordingly, the model is discretized with respect to the curvilinear abscissa.

Henceforth, the symbols m, v, γ and θγ are used as indexes referring to the measurements and the estimates of the standard, γ-augmented and θγ-augmented catenary model variations, respectively. Model points are written:

w P * k = ( w X * k , w Y * k , w Z * k ) k ∈ {0, . . . , n}, * ∈ {m, v, γ, θγ}
where indexes k = 0 and k = n are the tether endpoints, also indexed i and f . Model variants accuracies are computed by means of the residuals:

ε * P = 1 n n k=0 ∥ w P m k -w P * k ∥, * ∈ {v, γ, θγ} (2)
D. Model parameters estimation 1) Catenary parameter: Let L be the known tether length, l the horizontal distance and ∆H the difference in elevation between the tether attachment points calculated from the measurements of the initial and final attachment points ♢ P m i and ♢ P m f , with ♢ ∈ {v, γ, θγ} the frame in which the points are expressed (Fig. 2). The catenary parameter C is estimated by finding the root of function f which relates the curve's length and C [1] (using Brent's method [START_REF] Brent | Algorithms for minimization without derivatives[END_REF]):

f (C) = C 2 L 2 -∆H 2 -4 sinh 2 (lC/2)
2) Augmented model parameters: Let ε θγ P (θ, γ) be the accuracy written as a function of θ and γ (2). Angles θ and γ are estimated by minimizing this function by means of the trust region reflective algorithm [START_REF] Branch | A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems[END_REF]:

(θ, γ) = argmin (θ,γ)∈[-π,π] 2 ε θγ P (θ, γ)
The initial estimates θ 0 and γ 0 are respectively set to zero and to the angle between P γ0 and P v , where P γ0 is the closest plane to all measured points.

IV. EXPERIMENTS

A. Cables description

The experiments involve eight 3 m long cables presented in Fig. 5. These cables are meant to reflect the full range of mechanical characteristics found in robot umbilicals and tethers: stiffness from the most flexible to the most rigid; neutral, positive or negative buoyancy; elastic or nonelastic (Table I). Overall, these cables can be classified into two categories based on their capacity to retain plastic deformation: cables 1, 2 and 3 do, and cables 4, 5, 6, 7 and 8 do not. Cables 4, 6 and 7, are expected to give the best results in catenary shape fitting due to their clear positive or negative buoyancy, highlighted in Table I.

B. Experimental setup

The experiment is set up in a 7.2 m long, 4.2 m wide and 3 m deep tank. One end of the cable, namely w P m n , is attached to a ROV (BlueROV), while the other, namely w P m 0 , is fixed and attached to the side of the pool (Fig. 1). The cable's and robot's positions are recorded by a 5camera, 100 Hz Qualisys 1 underwater motion capture system. The robot as well as each cable are equipped with passive reflective markers and the cable's markers are evenly spaced out by 0.2 m (n = 15). The calibration of the motion capture system gives a standard deviation of 2.2 mm across the pool's volume for a known object's length. The points w P m k are measured in the Qualysis world Cartesian frame which is defined by the positioning of a calibration target that may not be perfectly vertical in the real experiment. This introduces a small offset (±0.05 rad) for the steady state angles as it can be seen in Fig. 8.

The initial position of the robot is defined such that the robot is stabilized at 1 m depth and approximately 1 m from the lateral pool wall facing the fixed attachment point (Fig. 6). Surge and sway movements are conducted individually to investigate the cable's response movement within or without its original vertical plane. The robot repeats the same open-loop control in sway or surge with the same step profile one after the other for each cable: 1) stabilisation with auto-depth at 1 m depth for 2 s, 2) abrupt start with constant surge or sway command applied for 2.5 s, 3) reversing of the thrusters for 0.3 s with a doubled velocity of opposite direction, to obtain an abrupt stop, 4) slow down the thruster for half the initial velocity during 0.2 s and 5) deactivation of the auto-depth control and complete stop of the thrusters after 13 s. The robot then drifts slowly.

The starting position of the mini-ROV is manually determined with the help of a visual reference point. It is not always perfectly aligned with the fixed point (see starting circles and trajectories in Fig. 6). To address this problem, closed-loop control could be considered. However, conventional position sensors such as IMUs, DVLs and USBLs are not suitable for an experiment in a compact metal tank, and the use of a visual target limits the range of possible movements. Another option would be to use a traction bench. However, such a setup would not be representative of the dynamics of the mini-ROV in interaction with its tether. Nevertheless, the depth sensor allows for closed-loop depth control and the desired direction of motion was dominant with its open-loop controller.

The experiment is repeated three times for each cable, under six experimental conditions, which makes it possible to test the reactions of more or less taut cables with accelerations of different amplitudes. Each experimental condition is a combination of a direction of motion (sway or surge), a speed (0.3 m s -1 or 0.6 m s -1 ) and a starting point characterized by its horizontal distance from the fixation point (about 1.5 m or 2 m).

C. Results

1) General study: Figure 7 represents the residuals (2) for the three model variants, namely standard Catenary, γaugmented and θγ-augmented, for all sequences and all experimental conditions. There is a general trend showing that the θγ-augmented model has lower residuals than the standard catenary model and the γ-augmented one. The γaugmented variant is also more accurate than the standard catenary model.

Regardless of the experimental condition, cable 6 and 7 show lower residuals for all three variants, as expected regarding their physical properties (see Table I). Indeed, these cables are the closer to the definition of the catenary because they are flexible and negatively buoyant. Sail rope 5 and elastic rope 8 are also negatively buoyant but lighter with regards to their volume so their residual is higher. Indeed, the drag forces damping effects wins over the weight. The floating cable 4 behaves like an inverted catenary. It has a high residual for the standard catenary model (2 nd worst median) but shows a good fit for the θγ-augmented model (3 rd best median).

For the stiffer cables 1, 2 and 3, the improvement in accuracy for the θγ-augmented variant still exists, albeit to a lesser extent. Indeed, the stiffer the cable, the higher the residuals. These cables have permanent deformations as a result of being coiled. Furthermore they are neutrally buoyant. As a result, the impact of weight is insufficient to generate the catenary shape.

The section below examines the three model variants in greater details, distinguishing between the experiments involving predominant surge and sway motion. Cables 3, is chosen as representative of the stiff cables with neutral buoyancy and cable 6 is chosen among the flexible cables that are negatively buoyant (as defined in IV-A).

2) Detailed analysis of cables 3 and 6 as group representatives: Tables II andIII show the box plots of the model residuals for cable 3 and 6 according to experimental parameters. First, the sway movements are displayed for a distance of 2 m and two different velocities. Then the sway movements are detailed for two different velocities and two starting points. These Tables show that the improvements of accuracy of the θγ-augmented model is consistent across experimental parameters.

For both groups representatives, the residuals of the models are smaller for slower movements and greater distance from the attachment point (i.e. more tension in the cable). The γ-augmented model displays a higher accuracy for lateral sway movements compared to forward surge movements. Conversely, our θγ-augmented is significantly better CONFIDENTIAL. Limited circulation. For review only. for surge movements in comparison of the γ-augmented. The gain in accuracy is even greater for cable 6's group.

3) Detailed analysis for experimental conditions of 0.6 m s -1 speed and 2 m distance for cables 3 and 6 as group representatives: Fig. 8 shows the three model variants residuals, angles θ and γ, as well as the robot's speed. These figures show in detail the different phases of the movement: an initial stable position (hydrostatic equilibrium phase), an abrupt start (dynamic phase), a phase of continuous application of a constant command (steady state phase), an abrupt stop (dynamic phase) and a subsequent hydrostatic phase.

A clear correlation appears between the robot's dynamics and the angle γ for sway motion. The estimated γ angle consistently increases with speed during sway movements, whereas θ remains close to zero (Fig. 8a and8b). As anticipated, surge movements have a noticeable impact on cable 3's estimated angle θ, with γ maintaining a small value (Fig. 8c). For cable 6, both angles increase with velocity, with θ being greater than the one observed for sway movement (Fig. 8d). However, since γ is larger than expected, let us examine the trajectory displayed in Fig. 6 relative to surge motion with 0.6 m s -1 and 2 m. While the primary motion is surge, the trajectory of the robot is not precisely aligned with the fixed point causing the cable to move laterally. Although this effect is present throughout all sequences, it is particularly pronounced in this trajectory which explains the values for γ. This demonstrates that our θγ-augmented model is capable of simultaneously managing significant sway and surge movements. Finally, one can clearly see that cable 6 restores balance towards the vertical plane and the standard model at a faster rate compared to the lighter cable 3. As a result, physics-based modelling of angle dynamics opens up promising perspectives for model-based control synthesis.

V. CONCLUSION

This work introduced an augmented catenary model that accounts for the hydrodynamic effects on the tether when the ROV performs surge and sway motion. It incorporates the models presented in the state of the art, which can be found by zeroing one or both of the angles proposed as augmentation parameters.

The results drawn from the experiments carried out on eight different cables show that the augmented model provides a better estimate of the shape of all cables during dynamic phases, compared with the standard catenary model. The accuracy is best for flexible cables with adequate negative buoyancy. In addition, cables with the lowest residuals relative to the standard catenary model lead to the best improvements in accuracy using the augmented model. Their measured shape features greater planarity and homogenous tilting of their support plane during motion. In light of these results, the augmented catenary model will be used in model-based controllers to estimate cable shape and its lowest position for deployment scenarios of tethered robots.

Fig. 1 :

 1 Fig. 1: Experimental setup. A dark red cable connects the ROV to a fixed point. Optical markers are regularly placed on the cable for underwater motion tracking. The slack light yellow tether is used for communications with the ROV.

Fig. 3 :

 3 Fig. 3: Standard (blue) and γ-augmented (red) catenary for γ = 10 • .

Fig. 4 :

 4 Fig. 4: γ-augmented (red) and θγ-augmented (green) catenary for θ = 45 • .

Fig. 5 :

 5 Fig. 5: Cables ends: (1) coaxial cable, (2) four-pair network cable, (3) two-pair network cable, (4) floating braided rope, (5) sail rope, (6) leaded seamstress rope, (7) plain steel chain, (8) elastic.

Fig. 6 :

 6 Fig. 6: Top view of typical trajectories of cable 6 end. Circles refer to initial positions and crosses refer to final positions.

10 1 10 2

 2 surge, 0.3 m s -1 , 2.0 m surge, 0.6 m s -1 , 2.0 m sway, 0.3 m s -1 , 1.5 m sway, 0.3 m s -1 , 2.0 m sway, 0.6 m s -1 , 1.5 m sway, 0.6 m s -1 , 2.0 m error / mm

  CONFIDENTIAL. Limited circulation. For review only. Manuscript 633 submitted to 2024 IEEE International Conference on Robotics and Automation (ICRA). Received September 15, 2023.

Fig. 8 :

 8 Fig. 8: Model residuals, robot speed and θ, γ angles over time. Speed command of 0.6 m s -1 and 2 m distance.

TABLE I :

 I Cables characteristics

	cable #	1	2	3	4
	diameter / m	0.007	0.005	0.004	0.008
	weight / N	1.079	0.687	0.392	0.294
	buoyancy / N	0.049	0.147	-0.049	-0.491
	wet weight / N	1.079	0.687	0.392	0.981
	plastic deformation	✓	✓	✓	×
	cable #	5	6	7	8
	diameter / m	0.006	0.004	-	0.005
	weight / N	1.079	1.373	3.120	0.589
	buoyancy / N	0.245	1.177	2.708	0.098
	wet weight / N	1.570	1.521	3.120	0.687
	plastic deformation	×	×	×	×

  Manuscript 633 submitted to 2024 IEEE International Conference on Robotics and Automation (ICRA). Received September 15, 2023.

		8
		7
		6
	cable #	4 5
		3
		2
		1
		10 1	10 2
			error / mm
	Fig. 7: Box plots of all 8 cables under all experimental
	parameters (speed, direction, distance) with the median, 1 st
	and 3 rd quartile, the 1 st and 99 th percentile of model residuals
	ε v P (blue), ε γ P (red) and ε θγ P (gray).

TABLE II :

 II Box plots of the two-pair network cable, cable 3 residuals ε v P in blue, ε γ P in red and ε θγ P in gray

	Experimental parameters	Box plot

TABLE III :

 III Box plots of the leaded seamstress rope, cable 6 residuals ε v P in blue, ε γ P in red and ε θγ P in gray

	Experimental parameters	Box plot
	surge, 0.3 m s -1 , 2.0 m
	surge, 0.6 m s -1 , 2.0 m
	sway, 0.3 m s -1 , 1.5 m
	sway, 0.3 m s -1 , 2.0 m
	sway, 0.6 m s -1 , 1.5 m
	sway, 0.6 m s -1 , 2.0 m
	error / mm	10 1	10 2

Manuscript 633 submitted to 2024 IEEE International Conference on Robotics and Automation (ICRA). Received September 15, 2023.

VI. ACKNOWLEDGMENT

This work is funded by the French Research Ministry, the CARTT de l'Université de Toulon.


Aymeric Hénard
email: aymeric.henard@univ-brest.fr

Etienne Peillard
email: etienne.peillard@imt-atlantique.fr

Jérémy Rivière
email: jeremy.riviere@univ-brest.fr

Sébastien Kubicki
email: sebastien.kubicki@enib.fr

Gilles Coppin
email: gilles.coppin@imt-atlantique.fr

Human Perception of Swarm Fragmentation

Keywords: Human-centered computing → User studies, Graphical user interfaces, • Computing methodologies → Multi-agent systems, Perception, Self-organization Human-Robot Interaction, Human perception, Robotic swarm, Swarm Intelligence

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Robot swarms are autonomous and self-organised systems that possess properties that make them useful for carrying out complex tasks beyond the abilities of a single robot. However, in scenarios where multiple swarm behaviours are required depending on the situation [START_REF] Coppin | Autonomy spectrum and performance perception issues in swarm supervisory control[END_REF] (e.g. reach an area using a flocking behaviour, then spread out using an expansion behaviour and finally cover it dynamically using a swarming behaviour), or where specific real-time adjustments are required (e.g. densify the swarm to pass through a hole), human intervention/assistance may sometimes be necessary. Human beings can adopt different roles when interacting with a robot swarm [START_REF] Scholtz | Theory and evaluation of human robot interactions[END_REF]: supervisor (high-level control, actions on the medium/long-term), operator (low-level control, actions on the short-term), mechanic (physical control, solves physical problems), peer (part of the swarm, interacts passively and actively while performing its tasks) and bystander (only an observer of the swarm functioning). Our goal is to offer to an operator means of control to influence a self-organised swarm efficiently, preventing undesirable effects that negatively affect the swarm. One of these undesirable effects is the swarm fragmentation [START_REF] Olfati-Saber | Flocking for multi-agent dynamic systems: algorithms and theory[END_REF][START_REF] Punzo | Characteristics of swarms on the edge of fragmentation[END_REF], which corresponds to a loss of communication and coordination among robots in the swarm. Fragmentation can be observed when the swarm loses one or more robots or splits into several groups. Sometimes, causing fragmentation is necessary as it allows the group to be divided to perform concurrently different tasks. However, when the fragmentation is not intended, it is harmful to the swarm and can, in the worst case, stop the swarm's self-organisation as a whole: the loss of all interactions between robots prevents the emergence of collective behaviours such as flocking or area coverage [START_REF] Hénard | A unifying method-based classification of robot swarm spatial self-organisation behaviours[END_REF]. Fragmentation can occur for various reasons such as the apparition of technical problems, changes in the environment, coordination problems between the robots, wrong initial parameter settings or unfitting control by the operator [START_REF] Kolling | Human Interaction With Robot Swarms: A Survey[END_REF]. When an operator interacts with the swarm's robots, (s)he interferes with their self-organising behaviour, potentially causing an unwitting fragmentation, for example, if (s)he reacts at the wrong time or frequency [START_REF] Nagavalli | Neglect benevolence in human control of robotic swarms[END_REF]. If the swarm is to be controlled by an operator who can cause or prevent fragmentation, it is first necessary to check whether and under what conditions humans can perceive (i.e. detect or anticipate) swarm fragmentation. In order to test several fragmentation situations, the use of multiple behaviours is necessary. It will allow us to check if different behaviours may impact the human perception of fragmentation differently. Moreover, in order to know what information humans may use to perceive fragmentation, we define two metrics characterising two aspects of the swarm: the separation speed and distance of the fragmented groups. It will allow us to see if these aspects are important to perceive fragmentation as a human being. We first review the literature related to fragmentation and human perception of swarms in section 2, before detailing our experimental protocol in section 3 and presenting our results in section 4. Finally, we provide an interpretation of the results and discuss limitations and perspectives in section 5, and we conclude in section 6.

RELATED WORKS

As fragmentation can be harmful to the integrity of the swarm, developing solutions to avoid it is necessary. The first solution may be to reinforce communications within the swarm, by limiting the individual autonomy of the robots or by establishing strict relationships between them, preventing the robots from wandering off and losing track of the rest of the group [START_REF] Rosalie | Coverage optimization with connectivity preservation for uav swarms applying chaotic dynamics[END_REF][START_REF] Herbert G Tanner | Stable flocking of mobile agents, Part I: Fixed topology[END_REF][START_REF] Herbert G Tanner | Stable flocking of mobile agents part II: dynamic topology[END_REF]. Rather than constraining the self-organisation, an alternative is to ask an operator to monitor the swarm and interact if necessary. Previous works on Human Swarm Interaction (HSI) show that it is only necessary to interact with the swarm when needed and at the right time. In [START_REF] Coppin | Autonomy spectrum and performance perception issues in swarm supervisory control[END_REF], the authors show that human intervention reduced the efficiency of the swarm, seemingly due to a massive intervention from the operator. Moreover, the concept of Neglect Benevolence [START_REF] Ma | On the timing of operator commands for the navigation of a robot swarm[END_REF][START_REF] Nagavalli | Bounds of Neglect Benevolence in Input Timing for Human Interaction with Robotic Swarms[END_REF][START_REF] Nagavalli | Neglect benevolence in human control of robotic swarms[END_REF][START_REF] Walker | Neglect benevolence in human control of swarms in the presence of latency[END_REF] implies that it is sometimes necessary to wait for the right moment before exercising control in order to get the swarm to perform a task as efficiently as possible. This also implies waiting for the right time to avoid harmful scenarios for the swarm. However, the operator must understand the swarm's dynamics and its impact on the swarm behaviour to figure out what the right time is.

More generally, we can assume that to avoid swarm fragmentation, the operator must first be able to anticipate it, or at least to perceive it, before applying the right control to prevent it or to correct it. Regarding human perception capabilities of swarms, previous studies show that a human can discriminate between flocking, rendezvous and dispersion behaviours [START_REF] Walker | Characterizing human perception of emergent swarm behaviors[END_REF], as well as swarming [START_REF] Harvey | Assessing human judgment of computationally generated swarming behavior[END_REF]. In addition, the concept of Legibility [START_REF] Capelli | Communication through motion: Legibility of multi-robot systems[END_REF][START_REF] Capelli | Understanding multi-robot systems: on the concept of legibility[END_REF] shows that the trajectory of a swarm is more predictable, without the need to add additional information, when the swarm is dense, rigid and adopts a direct trajectory. Moreover, the study from [START_REF] Seiffert | Motion perception of biological swarms[END_REF] shows that a human is also able to identify and differentiate in a very efficient way between a self-organised pattern and a pattern simulated by random dots, but struggles to discriminate it when the dots form a solid rigid structure. The authors suggest that humans can identify the overall movement of the swarm, but not its nature. For example, humans do not seem to be able to determine whether the swarm's movement is self-organised or simulated (i.e. agents do not interact and follow a predefined trajectory, mimicking the desired self-organised behaviour). In [START_REF] Kaduk | Effects of Human-Swarm Interaction on Subjective Time Perception: Swarm Size and Speed[END_REF], the authors show that an increase in the number of robots and their speed influences how the time is perceived by the participants, which appears to be faster. More broadly, previous works on human perception show that a human can make visual discrimination among moving points, for example by identifying a point moving in a fixed direction while all the other points are moving randomly [START_REF] Scott Nj Watamaniuk | Detecting a trajectory embedded in random-direction motion noise[END_REF]. In cognitive sciences, the Gestalt theory [START_REF] Wagemans | A century of Gestalt psychology in visual perception: I. Perceptual grouping and figure-ground organization[END_REF] suggests that people have a natural tendency to perceive objects in their environment in terms of global shapes or coherent sets rather than isolated parts. In particular, the notion of Perceptual Grouping describes the ability of humans to "perceive some elements of the visual field as "going together" more strongly than others". This notion encompasses several object properties such as Common Fate, Synchrony or Proximity. Common Fate [START_REF] Stürzel | Perceptual limits of common fate[END_REF][START_REF] William R Uttal | Motion and shape in common fate[END_REF][START_REF] Wagemans | A century of Gestalt psychology in visual perception: I. Perceptual grouping and figure-ground organization[END_REF] refers to the ability of a human to perceive among points going in random directions, moving points that are close and heading towards the same point or direction as if they had a common destiny. This identification is influenced by some factors that can be found in [START_REF] William R Uttal | Motion and shape in common fate[END_REF], such as the proximity of the points, the similarity of their movement or their number. Synchrony [START_REF] Wagemans | A century of Gestalt psychology in visual perception: I. Perceptual grouping and figure-ground organization[END_REF] refers to the tendency to group elements that change simultaneously, even if they are not following the same direction. Proximity [START_REF] Ali | The Effect of Gestalt Laws of Perceptual Organization on the Comprehension of Three-Variable Bar and Line Graphs[END_REF] refers to the tendency to group together elements near to each other, than others further away. In another HSI study [START_REF] St-Onge | Collective expression: how robotic swarms convey information with group motion[END_REF], the authors used Gestalt theory principles to study how humans perceive the cohesion of a swarm and showed that this perception of cohesion is linked to the proximity of the robots, the synchronisation of their movements and their tendency to follow a leader. As fragmentation is a loss of swarm cohesion, these results give us a good indication of the factors that can prevent humans from perceiving fragmentation. However, to our knowledge, nothing in the literature has shown that humans can perceive swarm fragmentation. As fragmentation causes the appearance of several groups that do not communicate, humans may be capable of perceiving them using agents' position (e.g. proximity) and movement (e.g. common fate). However, it is still unclear whether this subjective grouping represents the actual state of the swarm, or whether it misleads the human about the real nature of the interactions within the swarm. Humans could, for example, misperceive fragmentation in swarm behaviours exhibiting agents with long-distance sensing capacities and erratic movement. On the contrary, they might misperceive a cohesive swarm in behaviours exhibiting agents with short-distance sensing capacities and homogeneous movement and position.

EXPERIMENT ON SWARM FRAGMENTATION PERCEPTION

This study aims to investigate the ability of humans to perceive swarm fragmentation. More precisely, two research questions are addressed here: 1) How swarm fragmentation is perceived depending on different swarm behaviours? 2) Do humans use the distance between agents and the speed at which they separate to assess whether there is fragmentation? To answer these questions, we create a binary discrimination task in which participants have to determine whether the presented simulated swarms displayed fragmentation. Videos are used to display the swarm, showing expansion, densification, flocking and swarming behaviour.

Swarm simulation

Studies from [START_REF] Walker | Characterizing human perception of emergent swarm behaviors[END_REF] and [START_REF] Harvey | Assessing human judgment of computationally generated swarming behavior[END_REF] show that humans can discriminate between densification, expansion, flocking and swarming behaviours in a swarm. In this study, we chose to use the same behaviours, for which we detail our implementation below, to obtain diverse fragmentation situations and to see if distinct behaviours have an impact on the human perception of swarm fragmentation. The four behaviours are defined as follows:

• In densification behaviour, swarm agents move closer to each other, causing an increase in swarm density. • Expansion is the opposite of densification. Swarm agents move away from each other, causing a decrease in swarm density. This behaviour is useful for achieving area coverage. • Flocking is the coordinated movement (i.e. similar speed and direction) of an aggregate of autonomous agents. • Swarming is the opposite of flocking. Agents move without any coordination, going in all directions and at different speeds. However, they stay aggregated.

To obtain all these behaviours, we implemented the Boids model [START_REF] Reynolds | Flocks, Herds and Schools: A Distributed Behavioral Model[END_REF], on the Unity game engine. Our implementation uses a motion system based on virtual forces, which does not take into account the mass of the simulated robots. Acceleration ( ì 𝐴), speed ( ì 𝑆) and position ( ì 𝑃) are represented by two-dimensional vectors, as the agents are mobile robots that navigate on a plane.

ì 𝑆 (𝑡) = ì 𝑆 (𝑡 -1) + ì 𝐴(𝑡) × Δ𝑡 ì 𝑃 (𝑡) = ì 𝑃 (𝑡 -1) + ì 𝑆 (𝑡) × Δ𝑡
Each agent has a field of view and a blind spot. Neighbours in the agent's field of view are perceived only if they are not in the agent's blind spot (see Figure 2). We call N the set of the agent's neighbours and ì 𝑃 𝑎 the position of the neighbour a. In this controller, the acceleration of an agent at time t is obtained by adding the forces resulting from six predefined rules:

(1) Cohesion rule: brings the agent closer to its neighbours

ì 𝐶𝑜ℎ𝑒𝑠𝑖𝑜𝑛(𝑡) = ( 𝑎∈𝑁 ì 𝑃 𝑎 (𝑡) 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑢𝑟𝑠𝐶𝑜𝑢𝑛𝑡 -ì 𝑃 (𝑡)) × 𝑐𝐼
(2) Separation rule: keeps the agent away from its neighbours

ì 𝑆𝑒𝑝𝑎𝑟𝑎𝑡𝑖𝑜𝑛(𝑡) = 𝑎∈𝑁 ì 𝑃 (𝑡 ) -ì 𝑃 𝑎 (𝑡 ) ∥ ( ì 𝑃 (𝑡 ) -ì 𝑃 𝑎 (𝑡 ) ) ∥ 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑢𝑟𝑠𝐶𝑜𝑢𝑛𝑡 × 𝑠𝐼 (3)
𝐴 = {𝑎 ∈ 𝑁 , ∥ ì 𝑃 -ì 𝑃 𝑎 ∥ < 𝑠𝑑 } ì 𝐴𝑣𝑜𝑖𝑑 (𝑡) = ∑ 𝑎∈𝐴 ì 𝑃 (𝑡) -ì 𝑃 𝑎 (𝑡) ∥( ì 𝑃 (𝑡) -ì 𝑃 𝑎 (𝑡))∥ × 𝑜𝐼
The rules' intensity can be adjusted at any time, as well as the size of the field of view and of the blind spot angle. This allows to seek a specific swarm state and then update the agents' parameters to obtain the desired swarm behaviour. Table 1 details the bounds of each parameter and the values used to obtain a flocking behaviour.

To achieve densification or expansion, it is necessary to create an imbalance between the intensities of the cohesion (𝑐𝐼 ) and separation (𝑠𝐼 ) rules: in favour of 𝑐𝐼 for densification, and 𝑠𝐼 for expansion. To obtain the flocking behaviour, increasing the intensity of the alignment rule (𝑎𝐼 ) is necessary. It is also important that 𝑠𝐼 is not too strong compared to 𝑐𝐼 , preventing the agents from staying together. The same goes for swarming, where the agents must remain in a group. However, it is necessary to decrease 𝑎𝐼 to a minimum and increase the intensity of the random movement rule (𝑟𝐼 ), allowing agents to move more freely, without aligning themselves with their neighbours. Finally, agents must be aggregated (i.e. form a single swarm) in the initial state to obtain each of these collective behaviours. The swarm used in this study is composed of 40 agents, whose behaviour comes from the rules we previously presented.

The agents move within an area bounded by impassable walls and with no obstacles. This area measures 7x7 units and is white to contrast with the colour of the agents, which are represented by black cylinders with a diameter of 0.08 units. The observation is made from a distance of 7.0 units with a birds-eye view aligned normally to the plane 1 . Using a bird's-eye perspective eliminates some of the factors that can influence human perception, such as perspective or occlusion. This view from above is often used in previous works on perception (e.g. [START_REF] Roundtree | Analysis of human-swarm visualizations[END_REF][START_REF] Walker | Characterizing human perception of emergent swarm behaviors[END_REF]). In this swarm simulation, when an agent does not perceive any neighbour and is not perceived by any other agent, it becomes isolated and fragmentation appears in the swarm. An isolated agent is not the only possible source of fragmentation. Sometimes the swarm divides into two or more groups, and all agents have neighbours belonging to their respective groups. Replacing agents with nodes and their neighbours with adjacent nodes makes it possible to define a graph that connects the agents. Consequently, if only a disconnected graph with two or more components can be obtained, it implies that the swarm is fragmented, because there is no coordination or communication possible between the two groups. Figure 1 shows an example of fragmentation for each swarm behaviour.

Metrics

To characterise the human perception of swarm fragmentation, we defined two metrics inspired by the way humans group individualities according to the Gestalt theory. The first metric defines the distance separating fragmented groups. The second metric defines the separation speed of two fragmented groups. If several fragments appear, the two metrics qualify the higher score they can measure between two fragments.

Distance visibility score.

Based on the notion of Proximity from the Gestalt theory, humans seem to group individualities that are close to each other. Fragmented groups often move away from 1 see the video provided as supplementary material each other, increasing the distance between them. However, the distance among agents of a single group remains unchanged. Consequently, a difference may appear between inter-group distance and intra-group distance. So, we defined a metric measuring this difference to verify whether it affects the human perception of swarm fragmentation. For this purpose, we measured the intergroup and intra-group distances when a fragmentation occurred.

To measure the inter-group distance, clusters are identified, based on the perception of the agents, in a fragmented swarm. Once the different clusters are obtained, we seek the distance between two close clusters calculated as follows:

(1) We look for the smallest distance between two clusters among all the clusters. (2) The two closest clusters are merged to form a larger one.

(3) The operation is repeated so that only two clusters remain. [START_REF] Capelli | Understanding multi-robot systems: on the concept of legibility[END_REF] The smallest distance between the two remaining clusters is defined as the inter-group distance. To measure the intra-group distance, we identify the cluster containing the most agents and average the distance of the K Nearest Neighbours (KNN) of each agent (K=3). Finally, we divide the intergroup distance by the intra-group distance to obtain the distance value visibility score.

Separation speed.

Based on the notion of Common fate from the Gestalt theory, humans seem to group together elements that are heading toward the same point. Consequently, when two distinct directions are perceived, two distinct groups may be identified. The more the directions are different and the speed of the agents is high, the faster the groups will separate from each other. So, we defined a metric to verify whether separation speed affects the perception of swarm fragmentation. As humans prefer to see coherent sets rather than isolated parts, this metric measures the separation speed of fragmented groups based on the separation speed of their centre of mass at the time of the fragmentation. To do this, we identified the group leaving the aggregate and measured the separation speed of its centre of mass compared to the centre of mass of the rest of the swarm. In the case where multiple groups were separating, we took the group that was moving away the fastest.

Experimental protocol

Since the dynamics of swarms are very sensitive to the initial state and random disturbances, it is difficult to replay a simulation identically from one subject to another. Consequently, to ensure that all participants view the same content, we selected and recorded videos of the swarm simulation at 60 frames per second. There are two types of videos, those containing a swarm fragmentation, whose presence has been verified using a function based on the agents' perceptions throughout the video, and those that do not. Fragmentation never appears at the same time across different videos. Moreover, the videos are classified into four categories according to the displayed swarm behaviour: expansion, densification, flocking and swarming (see section 3.1). All videos begin with a fully aggregated swarm, meaning that there is no fragmentation at the initial state. Furthermore, once the swarm agents' parameters are adjusted to exhibit the desired swarm behaviour, they don't change during the recording of the video. Among these parameters, we find the intensity of the rules, as well as the perception radius ([0.6;1] meter), the blind angle ([0;180] degrees), and the speed of the agents ([0.1;0.3] m/s), which varies between videos. Varying the perception radius and the blind angle prevents participants from retaining a reference distance as a fragmentation distance. This also allows us to approach a real context where robots have different perception ranges and angles depending on their sensors [START_REF] Chen | A survey of robot swarms' relative localization method[END_REF]. Finally, the videos end once the swarm appears to be stabilised, thus preventing the swarm's dynamics from causing to the participants an impression of imminent fragmentation. Additionally, when a fragmentation occurs, the group does not reform, the fragmentation is present until the end of the video.

First task.

The first task of the participants is to watch videos presented in random order and determine if the video they are watching contains fragmentation or not. There are 48 videos with varying lengths between 6s and 50s, for a total duration of 19 minutes. A progress bar is constantly displayed on the screen indicating the percentage of time left before the end of the video. Each of the categories presented in section 3.3 is represented by 12 videos, 6 of which contain fragmentation and 6 of which do not. In the videos containing a fragmented swarm, the fragmentation occurs on average 11.6 seconds after the beginning of the video (with a minimum of 2 seconds), and there are on average 12.3 seconds left until the end of the video (with a minimum of 3.15 seconds). This gives participants a reasonable amount of time to react and answer. To avoid differences in learning, all participants watch the videos until the end. There is a pause between each video, which ends with a participant's input to launch the next video. While viewing a video, the user can press one of the two buttons to answer the question "Is there a fragmentation in this video?". Participants are aware that we choose to record a negative answer if they don't give any answer. We make this choice because (1) they are informed beforehand that they should give the correct answer as quickly as possible, as soon as they think that either a fragmentation has occurred (or will occur) or no fragmentation has occurred (or will not occur), and (2) they always have access to the remaining time in the video. In addition, participants are warned that their answer is final. In this task, the participants' answers are collected and marked as correct or incorrect, according to their alignment with the result of the numerical calculation verifying the presence of fragmentation. Response times of participants are also recorded to assess how effective they are at anticipating fragmentation.

Second task.

The second task of the participants aims to collect qualitative data on the participants' perception of the swarm's dynamics and to identify important factors that contribute to the human perception of fragmentation in a swarm. This task involves watching 8 additional videos, two of each category. They differ from the other task in that they not only show the behaviours and fragmentation previously used but also additional fragmentation situations, with properties from other categories. In this way, we can identify which properties seem to be important in their perception and how they interact. Among these videos, two have particular characteristics to check if participants are sensitive to them. One contains a swarm that perpetually splits into several groups and reassembles into one aggregate. The other contains an already fragmented swarm. This task takes the form of an interview, recorded with the prior agreement of the participants. The participants' speech and the content viewed are recorded, giving us a context for their comments. This recording is deleted after analysis, as agreed with them. During the interview, participants express themselves freely and can watch the videos as many times as necessary. However, in some situations, the interviewer may trigger comments by asking them open and non-oriented questions: "Can you describe what you see?" if they don't express themselves spontaneously, or "Can you detail your reasoning?" if their answer is too brief. After watching and commenting on the 8 videos, the participants were asked three final oral questions: "What strategy do you use to identify fragmentation?", "What makes it easy to see whether or not there is going to be fragmentation?" and "What makes it difficult to see whether or not there is going to be fragmentation?". At the end of the task, we asked the participants to write down the elements that indicate whether or not that there is or will be fragmentation. The interviews and written comments are analysed after all participants have taken part in the experiment. The interviews are viewed, and the terms used by the participants are collected manually and classified in a table according to whether they indicate the presence or absence of fragmentation. In addition, the answers to the last 3 questions are also collected, indicating the participants' strategy, and what they found easy and hard. The obtained results are discussed in section 5.

Preparations.

A total of 49 participants took part in the experiment, of which 60% were male and 40% were female. The participants ranged in age from 20 to 80 (median of 27). We chose participants without prior knowledge of robot swarms. We assumed that studying their perception provides us with a fundamental basis which can be generalised. All participants had a correct or corrected vision. Collected data were anonymised. The study was conducted in a quiet room exclusively dedicated to the study for its duration. The ethics committee of the authors' institution has approved this experimental protocol.

RESULTS

Two factors were tested in this experiment. First, we checked the impact of the swarm behaviour on human perception. Non-parametric tests were used, as the experiment uses a within-subject pattern and as a Shapiro test has confirmed that data do not follow normal distribution. The Friedman test followed by a Nemeyi post-hoc were used to check differences in correct answers and anticipation among categories. Secondly, we checked the relationship between the proposed metrics (see sections 3.2.1 and 3.2.2) and the rate of correct answers, thanks to a logistic regression approach.

Impact of swarm behaviour on correct answers

Firstly, we verified if the swarm behaviour has an impact on the rate of correct responses from participants.Regarding the mean score for each category, densification (91%), flocking (89%) and swarming (87%) have similar results, but the expansion category scored lower (67%). Then, for each video category, the results are differentiated according to whether or not there is fragmentation, forming eight categories. Figure 3 shows the obtained results. The Friedman test performed on the eight categories shows statistically significant • Fragmentation from the expansion category was found to be significantly harder to identify than either densification (p<0.001), flocking (p=0.003) or swarming (p=0.001). • Fragmentation from the densification category was found to be significantly easier to identify than either expansion (p<0.001), flocking (p=0.011) or swarming (p=0.023). • Videos containing fragmentation from the expansion category were found to be significantly harder to identify than videos containing no fragmentation within the same category (p<0.001). Similar effects were found for flocking (p<0.001) and swarming (p=0.032). Only the densification category was found to have no significant difference between the scores for fragmented and non-fragmented swarms (p=0.804).

Focusing only on the videos showing a fragmented swarm, we looked further at the perception of fragmentation without taking into account the anticipated responses. Figure 4 shows the obtained results. Regarding the mean score for each category, densification (91%), flocking (87%) and swarming (79%) have similar results, but the expansion category scored lower (37%). The Friedman test shows statistically significant differences in correct answers (Q=63.44, p<0.001). The Nemenyi post-hoc test shows that fragmentation from the expansion category was found to be significantly harder to perceive than either densification (p<0.001), flocking (p<0.001) or swarming (p<0.001).

Impact of swarm behaviour on anticipation

Secondly, we verified if the swarm behaviour has an impact on the participants' anticipation. The recorded response times are used to compute the anticipated responses and the percentage of video remaining at the time of the responses for all participants. Other analyses using directly the response times were not possible, because the videos are not equivalent (see section 3.3). First, we looked at the anticipation rate of responses on videos containing fragmentation (cf. Figure 5.1). Regarding the mean anticipation rate for each category, expansion (29%), densification (28%) and flocking (30%) have similar results, but the swarming category scored lower (14%). The Friedman test shows statistically significant differences in fragmentation anticipation (Q=37.82, p<0.001). The Nemenyi post-hoc test shows that fragmentation from the swarming category was found to be significantly less anticipated than either expansion (p<0.001), densification (p=0.001) or flocking (p<0.001).

Then, we looked at the percentage of video remaining specifically on the videos that contain non-fragmented swarms. Regarding the mean for each category, expansion (26%), densification (29%) and flocking (33%) have similar results, but the swarming category scored lower (19%). The Friedman test shows statistically significant differences in percentage of video remaining (Q=44.23, p<0.001). Figure 5.2 shows the obtained results. The Nemenyi post-hoc test shows that answers from the swarming category were found to be less anticipated than either expansion (p=0.043), densification (p<0.001) or flocking (p<0.001).

Finally, we looked at the wrong answers given due to anticipation from videos containing fragmented swarms (cf. Figure 5.3). As the number of wrong answers is different for each category, the Friedman test cannot be used. Instead, we used the non-parametric alternative of the One-way ANOVA, the Kruskal-Wallis test. Regarding the mean anticipation rate for each category, expansion (18%), densification (12%) and swarming (17%) have similar results, but the

Effect of distance visibility score

To verify if the distance separating fragmented groups may influence the human perception of fragmentation, we did a logistic regression of the rate of correct answers as a function of the distance visibility score metric defined in section 3.2.1. To spread the data correctly, a log of 10 is applied to the distance score.

As shown in Figure 6.1, characterising the videos by the best score they achieve significantly impacts the rate of correct answers (p<0.001, 𝛼=3.4662, SE=0.263). Moreover, we also tested the distance score at the time of response. It significantly impacts the rate of correct responses (p<0.001, 𝛼=3.5850, SE=0.328).

Effect of separation speed

To verify if the separation speed of fragmented groups may influence the human perception of fragmentation, we performed a logistic regression of the rate of correct answers as a function of the separation speed metric (see section 3.2.2). By observing the results, we noticed that this metric does not allow us to characterise the videos showing densification behaviours, because the mass centres of the different groups do not move. The densification category is therefore removed from the analysis for this section.

As shown in Figure 6.2, characterising the videos by the highest separation speed they achieve significantly impacts the rate of correct responses (p<0.001, 𝛼=8.6938, SE=0.843). Moreover, we also tested the speed of separation at the time of response. It significantly impacts the rate of correct responses (p<0.001, 𝛼=10.3325, SE=1.075).

DISCUSSION

This section aims to discuss the results presented in section 4. It is structured as follows: first, the results of each behaviour will be discussed, followed by the results of the two metrics and the qualitative results, to finally conclude. Swarm behaviours. Firstly, participants perceive fragmentation with greater difficulty in expansion category videos. Some videos in this category score 22%, which is significantly lower than the 50% that would be expected if participants were responding randomly. These results suggest that the information presented in these videos gives the impression that the swarm is not fragmented. When a swarm is expanding, it tends to spread homogeneously and regularly, both in terms of movement and position. Even when agents become isolated, they continue their trajectory with inertia, which remains homogeneous with the others, and stop at the same time because agents are homogeneous and have the same maximum speed. These results are consistent with the Gestalt theory presented in section 2, especially with the concepts of Synchrony and Proximity. To summarise, swarms that expand homogeneously and fragment provide conflicting information about their true state. This could be problematic because expansion allows for area coverage, a useful behaviour in practical applications [START_REF] Brambilla | Swarm robotics: a review from the swarm engineering perspective[END_REF]. Furthermore, we observe that participants identify fragmentation more easily in videos containing fragmented swarms from the densification category. Indeed, the movement of agents grouping towards the swarm's centre of mass facilitates its identification, notably thanks to the notion of Common Fate. In the case where fragmentation occurs, multiple centres of gravity are perceptible, making the identification of multiple groups simpler as 67% of subjects reported in the qualitative feedback. Furthermore, due to the nature of densification behaviour, the distance between agents in the swarm decreases, without changing the distance at which communication between two agents breaks down. As a result, the difference between the two is as remarkable as the gap widens. Similarly, the videos in the flocking category were found to have significantly good results as shown in Figure 3. Taking only answers given when the fragmentation already occurs, results are even better than the densification category as shown in Figure 4. This difference can be explained by the results presented in Figure 5. Indeed, half of the wrong answers came from bad anticipations (i.e. participants wrongly answered that no fragmentation would occur). These results are consistent with the literature [START_REF] Ali | The Effect of Gestalt Laws of Perceptual Organization on the Comprehension of Three-Variable Bar and Line Graphs[END_REF][START_REF] Wagemans | A century of Gestalt psychology in visual perception: I. Perceptual grouping and figure-ground organization[END_REF]: flocking behaviour exhibits several Perceptual Grouping properties such as Common fate, Synchrony and Proximity, and shows little or no evidence of any risk of fragmentation that might make participants doubt. In a real context, this is not a problem because the operators can reevaluate their choices continuously. However, these results show that participants do not seem to perceive the fragility of the swarm. Otherwise, it was observed that videos in the swarming category make participants more cautious than for other categories. Indeed, they are less likely to anticipate their response, whether it is anticipating the fragmentation or responding before the end of the video (see Figure 5). This can be explained by the fact that swarms within this category have erratic behaviour, making it difficult to perceive divergent agents. In addition, when fragmentation occurs, it is often unexpected according to the participants. The qualitative feedback shows that they found it difficult to answer the videos from the swarming category and tend to say "You never know what will happen". Metrics. When considering the introduced metrics, the difference between inter-group and intra-group distances seems to influence the probability that a participant responds correctly (see Figure 6.1). As participants do not have access to the agents' perception distance, they must use other cues to determine the fragmentation. Since the swarm is composed of agents with homogeneous behaviour, the appearance of a remarkable difference in distance is a useful cue to identify the appearance of a second group. This also justifies why it is difficult to perceive fragmentation in the expansion category, as fragmented agents tend to stay at a homogeneous distance from each other, there is no remarkable distance. The choice of this metric is also reinforced by qualitative results. Indeed, 57% of participants mentioned comparing the distance between groups to the inter-group distance to estimate if they perceived fragmentation. Similarly, the separation speed of the centres of mass appears to influence the probability that a participant correctly identifies the video (see Figure 6.2). However, as the densification behaviour does not involve the movement of its centres of mass, humans probably rely on other clues for this behaviour. Using these two metrics, we have shown that humans seem to use separation distance and speed to infer the presence or absence of swarm fragmentation. Determining the existence of other clues will be the subject of future works. Qualitative results. Of the 19 explicit strategies obtained from the participants, 26% involved looking at the swarm as a whole, 57% keeping an eye on the outside, and 42% spotting anomalies. Some participants seemed to be more sensitive to anomalies arising from the distance between agents, and others to the movements of the agents. Moreover, of the 29 responses obtained, 58% of the participants reported finding it easier to evaluate videos displaying dense swarms moving in a common direction, specifically videos in the flocking and densification categories. They also found it easier to assess the fragmentation when the inter-group distance is very marked (17%), opposite directions are visible (41%) and the group separates quickly (37%). Conversely, of the 31 responses obtained, 80% reported finding it difficult to evaluate videos with expanding swarms, as the agents have a homogeneous behaviour and positioning and there is no clue to assess a fragmentation distance. In addition, they found it difficult to evaluate videos containing swarms in a swarming behaviour (32%) because fragmentation is deemed unpredictable (19%). In both cases, the task seemed more difficult when the swarm density was low (29%). Finally, when several contradictory pieces of information are visible, the participants seemed to choose the non-fragmentation answer. This qualitative result may explain the different obtained results within the same category between fragmented and no fragmented videos for expansion, swarming and flocking (section 4.1). In the expansion category, agents show a dispersion behaviour, but homogeneous and synchronised. In the swarming category, agents show erratic behaviour, but they keep being attracted to a centre of gravity. In the flocking category, agents move together in a coordinated way but sometimes show few instabilities inside the swarm. Only the densification category show different results, probably because its fragmentation is neat and visible almost from the start as multiple centres of gravity appear in term of movement and density. Synthesis. Our results strengthen results from [START_REF] St-Onge | Collective expression: how robotic swarms convey information with group motion[END_REF] and the assumption from [START_REF] Seiffert | Motion perception of biological swarms[END_REF] suggesting that "motion perception of swarms is supported by a global motion system that is not specialized for the biological nature of the individual agents and their interaction, but instead, capitalize on the motion redundancy from multiple individuals to code the overall pattern. ". As long as agents are moving in the same way, they are seen as part of the same group even if they no longer interact and only apply the inertia of their past interaction, as shown by the results from the expansion category. However, even if the four behaviours came from the same self-organisation method, the redundancy of the agent movement is neither evidence of the cohesion of the swarm nor proof that the agents interact. As such, this redundancy is not enough to inform humans about the stability of the aggregated swarm. Consequently, we suggest that humans need supplementary information to anticipate fragmentation and understand its causes in order to prevent it. Limitation and future work. The experimentation in this document has been conducted under perfect conditions and does not take into account the issues that an operator may encounter in real conditions. Changes in the operator's point of view can modify the perception of distances and complicate the perception of certain fragmentation. Additionally, in a real environment, there could be variations in brightness, colours, and elements, as well as occlusion that could hinder the operator's perception of the robots. In future experiments, the impact of several parameters may need to be studied. In particular, the number of robots, their speed, the size of their blind spot or their perception range could affect the way an operator perceives fragmentation. In addition, the influence of the operator's background would also need to be evaluated in future work. Moreover, it would be interesting to study more specifically the anticipation abilities of fragmentation in humans. Additionally, reproducing this experiment with a swarm of drones would allow us to see the impact of a 3D-navigating swarm on fragmentation perception. Moreover, testing a greater variety of behaviours would help to refine the obtained results. Finally, the use of eye tracking could help us to know what the human is looking at in the swarm.

CONCLUSION

In this document, we studied the human perception of fragmentation by presenting 49 participants with a discrimination task among videos containing swarms that fragment or not. The results show that humans perceive correctly swarm fragmentation (around 90% of the time) for flocking, densification and swarming. Yet, they have difficulties perceiving fragmentation when the swarm adopts an expansion behaviour, as it does not present any cue that would allow the estimation of a loss of communication among agents. In addition, the results show that it is difficult for humans to correctly anticipate fragmentation in all categories of behaviour. Moreover, humans take more time to respond and struggle to anticipate fragmentation when the swarm adopts a swarming behaviour, as the erratic movement of the agents prevents the anticipation of the appearance of fragmentation. We also have shown that the inter/intra-group distance and the separation speed metrics influence the chances of correctly identifying swarm fragmentation, suggesting that humans use among other cues distance separating groups to perceive fragmentation, as well as the separation speed of groups centre of mass. These results highlight the situations where it is most difficult for an operator to perceive and anticipate fragmentation. We have also proposed metrics to quantify and qualify these situations. These results provide a better understanding of the internal logic of robot swarm operators and will guide future research to provide relevant information for robot swarm control.

Figure 2 :

 2 Figure 2: Perception capability of an agent. Only agents in the perception area are perceived. The letter d represents the field of view size and 𝛼 the blind spot angle.

( 4 )

 4 Alignment rule: aligns the agent's speed and direction with its neighbours ì 𝐴𝑙𝑖𝑔𝑛𝑚𝑒𝑛𝑡 (𝑡) = 𝑎∈𝑁 ì 𝑆 𝑎 (𝑡) 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑢𝑟𝑠𝐶𝑜𝑢𝑛𝑡 × 𝑎𝐼 Random movement rule: allows the agent to move on its own, randomly (Here, X and Y are two different numbers randomly generated between -0.5 and 0.5, which change each time the rule is called). ì 𝑅𝑛𝑑𝑀𝑜𝑣𝑒𝑚𝑒𝑛𝑡 (𝑡) = ì 𝑉 (𝑋 ; 𝑌 ) × 𝑟𝑚𝐼 (5) Friction rule: reduces agent speed over time ì 𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛 (𝑡) = -ì 𝑆 (𝑡) × 𝑓 𝐼 (6) Overlapping avoidance rule: prevents agents from overlapping

Figure 3 :

 3 Figure 3: Correct answer rate according to the category of videos and whether they are fragmented or not

Figure 4 :

 4 Figure 4: Rate of correct answers given after fragmentation occurrence in videos containing a fragmentation, according to video category

Figure 6 :

 6 Figure 6: (1) Probability of correct answer as a function of the best distance visibility score achieved by the video (2) Probability of correct answer as a function of the best separation speed achieved by the video

Table 1 :

 1 Example of values used to obtain a flocking behaviour

	Variable Value Bounds	Description
	𝑐𝐼	2	[0;∞[	cohesion intensity
	𝑎𝐼	4	[0;∞[	alignment intensity
	𝑠𝐼	1	[0;∞[	separation intensity
	𝑟𝑚𝐼	0	[0;∞[	random movement intensity
	𝑓 𝐼	0.1	[0;1]	friction intensity
	𝑜𝐼	20	[0;∞[	overlapping avoidance intensity
	𝑠𝑑	0.09	[0;∞[	safety distance (in meters)
	-	1	[0;∞[	perception radius (in meters)
	-	0	[0;360]	blind spot (in degrees)

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ACKNOWLEDGMENTS

This work is part of the ANR ARTUISIS 2 project, which received funding under the reference ANR-21-CE33-0006