Isabelle Dufour
email: isabelle.dufour@ims-bordeaux.fr.

F Josse

S M Heinrich

C Lucat

C Ayela

F Ménil

O Brand

Unconventional uses of microcantilevers as chemical sensors in gas and liquid media (Invited paper)

Keywords: Cantilever, chemical detection, resonant sensor, in-plane mode, viscoelasticity, static mode

The use of microcantilevers as (bio)chemical sensors usually involves the application of a chemically sensitive layer. The coated device operates either in a static bending regime or in a dynamic flexural mode. While some of these coated devices may be operated successfully in both the static and dynamic modes, others may suffer from certain shortcomings depending on the type of coating, the medium of operation and the sensing application. Such shortcomings include lack of selectivity and reversibility of the sensitive coating and a reduced quality factor due to the surrounding medium. In particular, the performance of microcantilevers excited in their standard out-of-plane dynamic mode drastically decreases in viscous liquid media. Moreover, the responses of coated cantilevers operating in the static bending mode are often difficult to interpret. To resolve these performance issues, the following emerging unconventional uses of microcantilevers are reviewed in this paper:

(1) dynamic-mode operation without using a sensitive coating, (2) the use of in-plane vibration modes (both flexural and longitudinal) in liquid media, and (3) incorporation of viscoelastic effects in the coatings in the static mode of operation. The advantages and drawbacks of these atypical uses of microcantilevers for chemical sensing in gas and liquid environments are discussed.

Introduction

In recent years, interest in microcantilever-based chemical and bio-chemical sensing systems has risen due to their projected high sensitivity [START_REF] Sepaniak | Microcantilever Transducers: A new Approach in Sensor Technology[END_REF][START_REF] Goeders | Microcantilevers: Sensing Chemical Interactions via Mechanical Motion[END_REF][START_REF] Waggoner | Micro-and nanomechanical sensors for environmental, chemical, and biological detection[END_REF][START_REF] Lange | Complementary metal oxide semiconductor cantilever arrays on a single chip: Mass-sensitive detection of volatile organic compounds[END_REF][START_REF] Datar | Cantilevers: Nanomechanical tools for diagnostics[END_REF]. The large ratio of surface area to volume makes the microcantilever extremely sensitive to surface processes. For (bio)chemical detection, the microcantilever is usually coated with a (bio)chemically sensitive layer that aims to selectively sorb the analyte or molecule of interest. The sorbed substances can then be detected by monitoring either the resonant frequency shift (dynamic mode) or the quasistatic deflection (static mode). A simple way to explain the basic principle of such sensors is to say that in the case of dynamic mode operation, the change in mass associated with the sorption of analytes into the sensitive coating causes a shift in resonant frequency, which may be correlated to the ambient concentration of the target substance, while for static-mode operation, the sorption of analyte causes a cantilever deflection that is induced by the tendency of the sensitive coating to expand (or contract) upon analyte sorption (modification of strain and stress in the coating). However, the behavior of such cantilevers may actually be much more complicated as illustrated in Fig. 1.

As illustrated in Fig. 1 and as noted in the previous paragraph, the predominant effect in the dynamic mode is the mass modification, while in the static mode the stress and strain modification; however, secondary effects may include static deflection due to sorbed mass and frequency shifts due to analyte-induced coating stress/strain. Another effect which is not necessarily negligible is the effect of the state of the surrounding medium on the coating sorption properties (e.g., fluid temperature on polymer partition coefficient) or on the static or dynamic response of the microcantilever (e.g., "thermal drift" in the output signal due to Young's modulus temperature dependence).

While such cantilever-based sensors generally operate relatively well, they have shortcomings depending on the type of coating, the medium of operation and the application. The main performance issues can be summarized as follows:

 Drawbacks due to the sensitive coating: In addition to being subjected to environmental effects including temperature, humidity and aging effects, most chemically sensitive coatings, often polymers, only show partial selectivity to various analytes. Ideal selectivity is often achieved at the expense of reversibility. Moreover, the limited long-term stability of the viscoelastic coatings and the resulting aging affect the reliability of the sensor.  Drawbacks of the commonly used flexural out-of-plane dynamic mode in liquid media: Dynamically excited microcantilevers are well suited for operation in low-viscosity media such as air. Their usefulness as a sensing platform is limited when operating in viscous liquid media. This is due to the large decrease in the device's quality factor caused by the additional viscous losses in the fluid. While the viscous damping reduces the quality factor the effective displaced fluid mass causes a significant decrease in the resonant frequency. The result is a decrease in the device sensitivity as well as an increase in the system's susceptibility to frequency noise, thus raising the sensor's detection limit. It is noted that the sensitivity of the sensors in the dynamic mode increases with the resonant frequency, whereas the quality factor Q of the resonance mode influences the sensor resolution.

 Drawbacks due to the interpretation of the static-mode responses: In static-mode operation, polymercoated microcantilevers may exhibit either a monotonic transient response or a response exhibiting an overshoot [START_REF] Loui | Chemical vapor discrimination using a compact and low-power array of piezoresistive microcantilevers[END_REF]. As the stresses in the cantilever coated with a viscoelastic material may depend on time, the coated cantilever's response (bending or deflection) will therefore exhibit more complex behaviors. Such complexities in the sensor response make it difficult to easily interpret the measurements, analyze and predict the sensor's response, and perform signal processing for rapid and error-free detection of chemical analytes.

In the present paper, we will review unconventional uses of microcantilevers as chemical sensors to solve some of the performance issues and avoid or minimize the associated drawbacks. To eliminate problems associated with the sensitive coatings, an alternative method to detect and quantify chemical species without the use of a sensitive coating will be presented (section 2). To resolve the drawbacks associated with the commonly used out-of-plane flexural dynamic mode in viscous liquid media, the characteristics of alternative vibrating modes, such as the in-plane (flexural and longitudinal) vibrating modes, will be described (section 3). Finally, an interpretation of the static-mode cantilever response taking into account the viscoelasticity of the sensitive coating will be introduced in section 4.

Notation

The cantilever geometry is shown in Fig. 2a and the actuation and deflection in Fig. 2b.

The geometry of the cantilever is defined by the width b, thickness h, and length L. Coordinate x is measured along the beam length. The actuation force F and the resulting deflection w occur at the angular frequency  . The properties of the cantilever material are the Young's modulus E and the beam mass density  . The fluid (gas or liquid) properties are defined as f  , the fluid mass density, and  , the fluid viscosity.

Uncoated microcantilever-based chemical sensors operating in dynamic mode

Basic equations for dynamic mode operation

When a microcantilever vibrates in a viscous fluid (gas or liquid), the fluid offers resistance to the motion. The force per unit length, fluid F , which is the consequence of all normal and tangential stresses (hydrodynamic pressure and viscous shear) exerted by the fluid on all the surfaces of the cantilever, can be written in the frequency domain (underlined notation) as [START_REF] Stokes | On the effects of the internal friction of fluids on the motion of pendulums[END_REF]:

        2 1 2 fluid F j g ,x w ,x g ,x w ,x          (1)
where x is the longitudinal coordinate,  the radial frequency of the vibration,   w ,x  the microcantilever deflection, and 1 g and 2 g are functions specific to the fluid properties and the microcantilever cross-sectional geometry. They may be interpreted, respectively, as the distributed damping coefficient of the fluid and the distributed effective fluid mass. In general, these quantities may be x-dependent; however, in what follows they will be assumed uniform in x as this assumption has been shown to be sufficiently accurate for modeling microcantilever flexural vibrations in viscous fluids (e.g. [START_REF] Sader | Frequency Response of Cantilever Beams Immersed in Viscous Fluids with Applications to the Atomic Force Microscope[END_REF]).

The fluid effects (viscous and inertial) influence the dynamic response of the beam, in particular, the resonant frequency f r and the quality factor associated with viscous losses, visc Q , as [START_REF] Sader | Frequency Response of Cantilever Beams Immersed in Viscous Fluids with Applications to the Atomic Force Microscope[END_REF][START_REF] Chon | Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids[END_REF]:

0, 2 2 1 1 1 2 1 r v a c f f Q Lg m    (2) 2 0, 1 2 1 / / visc vac Lg m Q f Lg m    (3)
where 0,vac f is the undamped natural frequency of the microcantilever in vacuum, Q is the total quality factor of the cantilever/fluid system (incorporating all losses), m is the microcantilever mass and L is the microcantilever length.

The above equations illustrate the fact that in the general case of dynamic mode operation of a microcantilever, the resonant frequency depends on the fluid properties via the terms 1 g and 2 g . Such dependency can be used, in some cases, to characterize the fluid, thus eliminating the need for a chemically sensitive coating.

Gas detection in binary mixture

Using the previous equations, analytical expressions for the relative frequency shift due to small changes in both mass density and viscosity of the fluid can be obtained. In the case of different binary mixtures ( 2He / N , 2 2

CO / N and 2 2 H / N ), it has been shown that the predominant effect for the change in the microcantilever resonant frequency is due to the gas mass density variation. The sensitivity to mass density can be approximated by [START_REF] Tétin | Modeling and performance of uncoated microcantilever-based chemical sensors[END_REF]

1 8 r r f df a b f d h      (4)
From Eq. ( 4) one may conclude that the relative frequency shift due to the mass density variation of the gas increases with the ratio of the microcantilever width to the microcantilever thickness. Examples for the detection of hydrogen in nitrogen and carbon dioxide in nitrogen using silicon cantilevers with electromagnetic actuation and piezoresistive measurements are presented in Fig. 3. The silicon cantilever has a length of 2 mm, a width of 400 µm and a thickness of 15 µm. The resonance frequency of the uncoated sensor is approximately 5 kHz.

As predicted by Eq. 4, due to the fact that hydrogen is lighter than nitrogen, the resonant frequency increases with the hydrogen concentration, whereas it decreases in the case of carbon dioxide detection because this gas is heaver than nitrogen.

In order to estimate the limit of detection with this cantilever in the case of hydrogen detection in nitrogen, measurements with smaller hydrogen concentrations have been conducted (Fig. 4). It can be seen that, without any signal processing, the noise in the resonant frequency measurement is approximately 2 mHz and that the detection of 200 ppm of hydrogen is possible.

Compared to classical resonant microcantilever-based chemical sensors with sensitive coatings, the uncoated microsensors exhibit shorter response times because there is no analyte sorption into the coating affecting the sensor response. Moreover, the absence of the sensitive coating leads to a more reliable and reversible behavior because there is no significant absorption and desorption phenomenon. The above results indicate that uncoated microsensors may serve as viable devices for the detection of specific concentrations of one gas in a binary mixture. The sensitivity and resolution of such sensors will be larger for those cases in which the difference between the mass densities of the two gases is higher. The major drawback of such sensors is that there is no intrinsic selectivity; thus, these sensors can only be used for specific applications such as when monitoring environments in which it is known that only one gas concentration can vary while the relative concentrations of the other gases remain fixed.

In-plane vibration modes in liquid media

From Eq. 2, it can be seen that the decrease of the resonant frequency when the microcantilever is immersed in a fluid is essentially due to the mass effect 2 g ; this effect can be minimized if the term 2 g is small in liquid. On the other hand, the strong decrease in the quality factor in liquid is due to the stronger influence of the viscous term 1 g compared to the influence of the displaced fluid mass (term due to 2 g ) in Eq. 3. In order to limit the decrease of both the resonant frequency and the quality factor upon immersion in liquid media, alternative vibration modes that essentially shear the surrounding fluid rather than exerting normal stress on it have been studied and tested. In particular, two in-plane vibration modes (lateral flexural mode and elongation mode) (Fig. 5) have been theoretically studied [START_REF] Cox | Resonant Microcantilevers Vibrating Laterally in Viscous Liquid Media[END_REF][START_REF] Castille | Longitudinal vibration mode of piezoelectric thick-film cantilever-based sensors in liquid media[END_REF] in order to better understand the "intuitive" advantages offered by the in-plane mode of microcantilever-based sensors over their out-of-plane counterparts, as well as to quantify these advantages. Example analysis and measurements of the resonant frequency and quality factor associated with these two unconventional modes are presented in what follows.

In-plane (lateral) flexural mode

Exciting the microcantilever in the in-plane flexural vibration mode (or lateral vibration) rather than the outof-plane flexural mode would reduce the amount of fluid resistance (combined effects of fluid-related inertial and viscous forces), which could potentially improve the sensitivity and limit of detection of microcantileverbased chemical sensors. The characteristics of the in-plane flexural mode can be obtained by evaluating the hydrodynamic forces ( 1 g and 2 g ) acting on a laterally vibrating microcantilever.

A first modeling approach consists of considering that the displacement of the fluid is the same as in the case of an infinite vibrating plane. Then the hydrodynamic force acts only on the two larger surfaces of the cantilever and the expressions of 1 g and 2 g are [START_REF] Heinrich | An Analytical Model for In-Plane Flexural Vibrations of Thin Cantilever-Based Sensors in Viscous Fluids: Applications to Chemical Sensing in Liquids[END_REF][START_REF] Heinrich | An Analytical Model of a Thermally Excited Microcantilever Vibrating Laterally in a Viscous Fluid[END_REF]:

1 2 g g   and 2 2 f g b    (5)
Modeling the fluid resistance using Eqs. ( 5) corresponds to the limiting case of high Reynolds's number (high frequency), i.e., Re→∞, and thin microcantilevers vibrating in the lateral flexural mode, i.e., cross-sectional aspect ratio h/b→0. This type of fluid resistance is described as "Stokes resistance" and leads to simple analytical expressions for the quality factor due to viscous fluid losses and the relative frequency change due to immersion in a viscous fluid [START_REF] Heinrich | An Analytical Model for In-Plane Flexural Vibrations of Thin Cantilever-Based Sensors in Viscous Fluids: Applications to Chemical Sensing in Liquids[END_REF][START_REF] Heinrich | An Analytical Model of a Thermally Excited Microcantilever Vibrating Laterally in a Viscous Fluid[END_REF].

A more complete modeling approach consists of taking into account both the shear forces acting on the two larger surfaces and the pressure forces acting on the two other surfaces (Fig. 6). Each force is composed of two terms: one proportional to velocity and one proportional to acceleration. Then it is also theoretically possible to extract the two terms 1 g and 2 g . But due to edge effects, as shown in Fig. 6, each force is not uniformly distributed. Recently, finite-element results combined with semi-analytical models [START_REF] Cox | Resonant Microcantilevers Vibrating Laterally in Viscous Liquid Media[END_REF][START_REF] Brumley | Oscillation of cylinders of rectangular cross section immersed in fluid[END_REF] have appeared in the literature to provide more accuracy results for the hydrodynamic force. These studies show that 1 g and 2 g are functions of both Reynolds number (dimensionless frequency involving fluid properties) and cantilever aspect ratio (h/b). Obtaining the hydrodynamic forces allows for the resonant frequency and Q-factor to be investigated as functions of both beam geometry and medium properties [START_REF] Cox | Resonant Microcantilevers Vibrating Laterally in Viscous Liquid Media[END_REF]. Trends in these characteristics can be used to optimize device geometry and maximize the frequency stability in sensing applications.

Several cantilevers with widths of 45, 60, 75 and 90 µm, lengths of 200, 400, 600, 800, 1000 µm and silicon thicknesses of 8 and 12 µm, corresponding to in-plane resonant frequencies up to about 3 MHz, have been characterized in air and water. The modeling of the fluid interaction terms, 1 g and 2 g , according to Eq. 5 for the lateral mode result in the resonant frequency being proportional to and the quality factor (in water) being proportional to h b L [START_REF] Heinrich | An Analytical Model for In-Plane Flexural Vibrations of Thin Cantilever-Based Sensors in Viscous Fluids: Applications to Chemical Sensing in Liquids[END_REF][START_REF] Heinrich | An Analytical Model of a Thermally Excited Microcantilever Vibrating Laterally in a Viscous Fluid[END_REF]. The results of a simple, closed-form analytical expression for Q gives excellent quantitative agreement for the quality factor in water for aspect ratios up to h/b=0.2 (see, e.g., Fig. 7a) and a reasonable upper-bound estimate for thicker beams [START_REF] Heinrich | An Analytical Model for In-Plane Flexural Vibrations of Thin Cantilever-Based Sensors in Viscous Fluids: Applications to Chemical Sensing in Liquids[END_REF][START_REF] Heinrich | An Analytical Model of a Thermally Excited Microcantilever Vibrating Laterally in a Viscous Fluid[END_REF]. In addition, the analytical formula for resonant frequency in fluid [START_REF] Heinrich | An Analytical Model for In-Plane Flexural Vibrations of Thin Cantilever-Based Sensors in Viscous Fluids: Applications to Chemical Sensing in Liquids[END_REF] matches the trend exhibited by measurements made on the different cantilever geometries when operated in water (Fig. 7b). While the simple theory predicts qualitatively certain aspects of the behavior of the relative frequency drop from air to water (see, e.g., the slope in Fig. 7c), the quantitative results for the relative change in frequency significantly underestimate the data due to the neglected pressure effect on the smaller faces (i.e., the beam thickness effect). Therefore, the analytical result for the fluid-induced relative frequency change [START_REF] Heinrich | An Analytical Model for In-Plane Flexural Vibrations of Thin Cantilever-Based Sensors in Viscous Fluids: Applications to Chemical Sensing in Liquids[END_REF] should only be used as a general guide for the firstorder dependency of relative frequency drop on beam geometry. (An extension of the analytical formula for % frequency change that accounts for the pressure effect has been achieved recently and will be submitted for publication in the near future.) Other interesting features of the experimental data include (1) for the first lateral flexural mode the microcantilevers exhibit quality factors in air up to 4300 and up to 67 in water (which usually doesn't exceed 10-20 in water for out-of-plane modes); (2) the resonant frequencies of the in-plane mode are only lowered by 5 % to 10 % in water compared to the values in air, whereas this reduction is approximately 50 % for the out-of-plane modes. These two characteristics of the lateral mode operation are promising for future detection in liquid media.

For the microcantilevers designed and fabricated for operation in the lateral flexural vibration mode, thermal excitation and piezoresistive detection, both based on the use of diffused resistors, have been chosen as actuating and sensing mechanisms, respectively. The design of the actuation resistors and integrated Wheatstone bridge allow a more efficient excitation of the in-plane flexural mode than the more classical out-of-plane flexural mode [START_REF] Beardslee | Thermal Excitation and Piezoresistive Detection of Cantilever In-Plane Resonance Modes for Sensing Applications[END_REF]. It is noted that, for example, at a given driving voltage, the free-end vibration amplitude, measured using a Polytec MSA-500 laser vibrometer, is 118 nm at the first transverse flexural resonant frequency whereas it is 750 nm for the first lateral flexural resonant frequency.

For dynamically driven microcantilever chemical sensors, the limit of detection is usually defined as the analyte mass or concentration corresponding to a frequency shift equal to three times the frequency noise of the system measurement, (or is defined as three times the ratio of the device/system frequency noise to the chemical sensitivity). It is difficult to directly measure and compare the LOD because of the system noise for various transducers. However, when operating in an oscillator configuration, normalizing the chemical (mass) sensitivity, the limit of detection is proportional to 3 f r /( S m Q) [START_REF] Fadel | Signal to noise ratio of resonant microcantilever type chemical sensors as a function of resonant frequency and quality factor[END_REF]. Using this definition with the values of the operating frequencies, together with the values of the quality factor and the mass sensitivities, one can assess the LOD for each operational mode. For example, when vibrating the microcantilever in the lateral direction compared to the transverse direction, the quality factor (Q) is larger due to the decreased viscous damping and the ratio of the resonant frequency to the chemical sensitivity (f r /S m ) is smaller due to the decreased effective mass of the system. Furthermore, due to the increased resonant frequency and decreased effective mass of the beam (compared with transverse mode operation), the estimated mass sensitivity of a laterally excited microcantilever is found to be much larger. For example, the mass sensitivity of a 200x60x5.8 m 3 cantilever is found to be roughly two orders of magnitude larger in water (4.65 Hz/pg vs. 0.0456 Hz/pg) (and even better in more viscous solutions) when laterally excited compared to transversely excited. In an oscillator configuration, the reduced frequency noise (due to the increased quality factor) and improved mass sensitivity are expected to yield much lower limits of detection. These improvements in device characteristics indicate that microcantilevers excited laterally are more suited for operating in media of high viscosities compared to microcantilevers excited transversely, and thus will result in a lower minimum detectable mass. Based on measurements made in liquid media with the cantilevers presented in this section with various analytes, estimated LODs on the order of 100ppb were determined, which represents roughly an order of magnitude improvement in LOD compared to values reported for out-of-plane cantilevers in water [START_REF] Beardslee | Liquid-phase chemical sensing using lateral mode resonant cantilevers[END_REF].

In-plane longitudinal mode

For the same reason as noted in the previous section, the longitudinal (axial) mode may be of potential interest for detection in liquid media. To assess the characteristics of this mode, self-actuated resonantmicrocantilevers based on a thick-film piezoelectric layer associated with two electrodes have been processed and tested. Screen-printed microcantilevers comprising Au/PZT/Au layers are partially released from an alumina substrate (Fig. 8) using a sacrificial layer process [START_REF] Lucat | New sacrificial layer based screen-printing process for free-standing thick-films applied to MEMS[END_REF].

Using an impedance analyzer (HP4194A), frequency spectra have been measured for three different cantilever geometries (piezoelectric layers of 10x2x0.075 mm 3 , 8x2x0.075 mm 3 and 6x1x0.075 mm 3 coated with 5-10 µm thick symmetrical electrodes) in air and in various other fluids with viscosities ranging from 1.55 to 300 cP. Even though the quality factor in air is relatively low (probably due to internal mechanical losses and piezoelectric losses), ranging from 130 to 300, the quality factors in liquid are significantly higher compared to those of classical out-of-plane modes; the measured total quality factors for the three cantilever geometries range from 70 to 107, 41 to 51, and 20 to 22 for the 1.55, 20, and 300 cP fluids, respectively. The decrease of the resonant frequency of the first longitudinal mode from air to dodecane (1.55 cP) is only 1 % and from air to 300 cP silicone oil is a mere 2.8 %.

As was the case with lateral-mode silicon cantilevers having an integrated scheme for actuation and measurement, these thick-film piezoelectric cantilevers actuated in the longitudinal mode and utilizing impedance measurement are promising in terms of sensitivity and limit of detection for (bio)chemical detection in liquid media.

Static-mode cantilever response: Influence of the viscoelasticity of polymer coatings

A static-mode, polymer-coated microcantilever undergoes a quasi-static deformation when analyte sorption causes either a change in the surface free energy (adsorption) or coating expansion (absorption). In either case, it is necessary to have an accurate model of the beam's deflection in order to design devices for high sensitivity and to analyze or predict the sensor's response. Unlike the elastic case, the stresses in the cantilever coated with a viscoelastic material will depend on time even if the absorbed analyte concentration remains constant. The coated cantilever's transient response will therefore exhibit more complex behaviors [20]. The transient response in the static mode can be of various shapes when polymeric sensitive coatings are used, which include monotonic response and response with an overshoot. An analytical solution for predicting deflection and stress in a cantilever with viscoelastic coating when the coating is subjected to an exponential eigenstrain history (i.e., sorption-induced swelling strain) has been developed (Fig. 9) which provides an explanation for the different shapes observed in the transient responses [START_REF] Heinrich | An Analytical Model for Transient Deformation of Viscoelastically Coated Beams: Applications to Static-Mode Microcantilever Chemical Sensors[END_REF].

As shown in Fig. 9, the model considers that the viscoelastic material is characterized by its biaxial relaxation modulus which is defined by three parameters: the initial biaxial modulus 0 M , the steady-state biaxial modulus M  and the relaxation constant time R  . The time constant of the swelling strain is denoted by S  and corresponds to the diffusion time of the analyte into the coating. Using this notation, the analytical expression of the cantilever deflection allows one to obtain an "overshoot criterion" for determining if an overshoot will occur in the quasi-static beam response. Overshoot is predicted (Fig. 10) to occur if

0 1 R S M M     (6)
The validation of this analytical model and its limitations has been made using finite element modeling [22].

Based on the results of the analytical model, an optimization of the choice of the mechanical viscoelastic properties of the polymer coating (instantaneous and asymptotic biaxial moduli and corresponding relaxation time) and of the coating thickness (determining the time constant of the chemical sorption into the coating and then of the eigenstrain) can be envisaged in order to have larger transient deflection and an overshoot whose characteristics could possibly be used to perform more rapid chemical detection, possibly eliminating the need to wait for the steady state to be achieved. The reduction in detection time could be especially large in the case of thick coatings and/or polymeric coatings that exhibit slow creep/relaxation.

Conclusion

The limitations of classical microcantilever-based microsensors are essentially due to the low selectivity and reliability of the sensitive coating, the limitation of the dynamic mode in liquid media, and the difficulty in interpreting complex static-mode responses. In the present paper, some emerging alternative uses of microcantilevers as chemical sensors have been presented. The feasibility of these unconventional uses has been confirmed by physical analytical modeling, in addition to experimental observations and/or numerical validation. These unconventional uses have some advantages (shorter response time, better reliability when sensitive coatings are not used, higher quality factor and resonant frequency in liquid for in-plane resonant modes relative to the classical out-of-plane mode, and higher signal and shorter response time utilizing the overshoot phenomenon in the static mode). However, these advantages are accompanied by some drawbacks, including lack of selectivity when sensitive coatings are not employed and the difficulty in determining the viscoelastic properties of polymer coatings for static-mode applications. The preliminary results are promising, but further measurements and modeling must be performed in order to fully realize the benefits that these unconventional uses may offer for some applications.

List of Figure Captions

2 bL

 2 (where b is the cantilever width and L the cantilever length), the relative resonant frequency shift from air to water being proportional to   L h b

Fig. 1 .

 1 Fig. 1. Principle of microcantilever-based sensors for chemical detection: dynamic and static modes.

Fig. 2 .

 2 Fig. 2. a) cantilever geometry, b) Example of actuation force and deflection for an out-of-plane flexural mode.

Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .

 34567 Fig. 3. Example of (a) detection of hydrogen in nitrogen and (b) carbon dioxide in nitrogen with an uncoated 2000 x 400 x 15 µm 3 silicon cantilever

Fig. 8 .Fig. 9 .

 89 Fig.8. Example of screen printed piezoelectric cantilever designed for axial-mode operation[START_REF] Castille | Longitudinal vibration mode of piezoelectric thick-film cantilever-based sensors in liquid media[END_REF]

Fig. 10 .FiguresFig. 1 .Fig. 2 .

 1012 Fig. 10. Cantilever deflection obtained with the analytical model

  Fig. 10. Can

Acknowledgements

The authors would like to thank the many students who have worked on this subject in the three different labs: Luke Beardslee, Christophe Castille, Russell Cox, Vincent Cuisset, Rabin Maharjan, Sébastien Tétin and Michael Wenzel.

This work has been partially supported by the French National Agency (ANR) in the frame of its program in Nanosciences and Nanotechnologies (MicRheo project n°ANR-08-NANO-004); by the CPER Pôle 4N Nanosciences en Aquitaine (GP 206-action 216/1) with the contribution of the Conseil Régional d'Aquitaine, the FEDER and the Ministry of Education and Research; and by the Conseil Régional Aquitaine (project MicRhéo-Aquitaine 2009-1102001); by PACT and Andra; by the Fulbright Commission/Région Aquitaine (Research Scholar Award, France/Aquitaine Regional Program #6281) and by the National Science Foundation Grant No. ECCS-0824017.

Biographies of Authors

Isabelle Dufour graduated from Ecole Normale Supérieure de Cachan in 1990 and received the Ph.D. and H.D.R. degrees in engineering science from the University of Paris-Sud, Orsay, France, in 1993 and 2000, respectively. She was a CNRS research fellow from 1994 to 2007, first in Cachan working on the modelling of electrostatic actuators (micromotors, micropumps) and then after 2000 in Bordeaux working on microcantileverbased chemical sensors. She is currently Professor of electrical engineering at the University of Bordeaux and her research interests are in the areas of microcantilever-based sensors for chemical detection, rheological measurements and materials characterisation. His main research interests were in thick-film materials for hybrid components and chemical sensors. Now, his research efforts are on the fabrication and characterization of free-standing structures for Microsystems applications, using the association of screen-printing technology and sacrificial layer method specifically developed at IMS. Microcantilevers or self-supporting devices for gas sensing, force sensors, energy harvesting, etc. are studied.

Fabien

Cédric Ayela was born in


Marie-Lou Barnaud
email: marie-loubarnaud@wanadoo.fr

Nicolas Morgado
email: nicolas.morgado@upmf-grenoble.fr

Richard Palluel-Germain

Julien Diard
email: julien.diard@upmf-grenoble.fr

Anne Spalanzani
email: anne.spalanzani@inria.fr

Proxemics models for human-aware navigation in robotics: Grounding interaction and personal space models in experimental data from psychology*

In order to navigate in a social environment, a robot must be aware of social spaces, which include proximity and interaction-based constraints. Previous models of interaction and personal spaces have been inspired by studies in social psychology but not systematically grounded and validated with respect to experimental data. We propose to implement personal and interaction space models in order to replicate a classical psychology experiment. Our robotic simulations can thus be compared with experimental data from humans. Thanks to this comparison, we first show the validity of our models, examine the necessity of the interaction and personal spaces and discuss their geometric shape. Our experiments suggest that humanlike robotic behavior can be obtained by using only correctly calibrated personal spaces (i.e., without explicit representation of interaction spaces and therefore, without the need to detect interactions between humans in the environment).

I. INTRODUCTION

Human navigation, however mundane its context and purpose, is a complex activity, subject to a large number of external constraints. Avoiding bumping into two people discussing together, and avoiding disturbing these people by interrupting their interaction, are two examples of such constraints. The first is an instance of a safety constraint, useful to prevent situations potentially harmful to the navigating agent; the second is an instance of a social constraint, useful to prevent situations potentially annoying to others. Trying and avoiding being a nuisance, both to self and to others, would therefore seem to be a basic requirement for humanlike navigation.

In the context of human-aware robotic navigation, autonomous robots should respect these two types of constraints. Early robotics research have mostly focused on safety constraints [START_REF] Borenstein | The vector field histogram-fast obstacle avoidance for mobile robots[END_REF], [START_REF] Fox | The dynamic window approach to collision avoidance[END_REF], [START_REF] Khatib | Real-time obstacle avoidance for manipulators and mobile robots[END_REF]. Recent developments have investigated incorporating social constraints in navigation algorithms and space representations [START_REF] Kirby | Companion: A constraintoptimizing method for person acceptable navigation[END_REF], [START_REF] Rios-Martinez | Understanding human interaction for probabilistic autonomous navigation using Risk-RRT approach[END_REF], [START_REF] Sisbot | A human aware mobile robot motion planner[END_REF]. In particular, robotics research has integrated various models of personal spaces (PS) and interaction spaces (IS).

For instance, PS has been modeled by asymmetrical combinations of 2D Normal distributions, centered on the navigating agent's position [START_REF] Kirby | Social robot navigation[END_REF], [START_REF] Rios-Martinez | Understanding human interaction for probabilistic autonomous navigation using Risk-RRT approach[END_REF], or by considering several regions around humans in the robotic environment, assigning each region to accompanying and following tasks [START_REF] Pandey | A framework towards a socially aware mobile robot motion in human-centered dynamic environment[END_REF]. PS models have also been expanded, integrating explicit attraction towards interaction or, on the contrary, avoidance of interaction, in order to differentially weigh four 2D Normal distributions and obtain different PS shapes [START_REF] Svenstrup | Pose estimation and adaptive robot behaviour for human-robot interaction[END_REF]. Visibility has also been considered, for instance so that robots would avoid zones that humans do not see (e.g., zones behind a corner, behind other people) [START_REF] Sisbot | A human aware mobile robot motion planner[END_REF]. Interaction space has also been modeled, for instance with Normal probability distributions, either one-dimensional or two-dimensional, according to the geometrical configuration of the interacting group [START_REF] Rios-Martinez | Understanding human interaction for probabilistic autonomous navigation using Risk-RRT approach[END_REF].

Outside of robotics, experiments on animal navigation and social conventions have led to a large number of theories about personal space representation, with applications from livestock herding [START_REF] Hediger | Studies of the psychology and behavior of captive animals in zoos and circuses[END_REF] to understanding the psychology of interaction in humans.

Hall [START_REF] Hall | The hidden dimension[END_REF], for instance, proposed the concept of proxemics to describe observations and theories of the use of space in humans. Assuming that social relationships are usually reflected in physical distances (between people), Hall defined four proxemic distances: the intimate, personal, social and public distances, which are represented by four concentric boundaries. PS conceptually matches Hall's model of personal distance.

The boundary of PS is influenced by contextual parameters (e.g., culture) or internal parameters (e.g., speed) making its direct observations difficult [START_REF] Hayduk | Personal space: An evaluative and orienting overview[END_REF], [START_REF] Sommer | From personal space to cyberspace[END_REF]. Moreover, several authors proposed that personal space would not be concentric but would be elongated to the front [START_REF] Hayduk | The shape of personal space: An experimental investigation[END_REF], [START_REF] Helbing | Social force model for pedestrian dynamics[END_REF]. PS is the most studied space, but is not the only one. Indeed, more recently, psychology studies have focused on other spaces, like IS [START_REF] Kendon | Spacing and orientation in co-present interaction[END_REF], either for static interactions (e.g., discussing at the water cooler) or interactions during movement (e.g., walking in group), or the affordance space [START_REF] Lindner | Towards a formalization of social spaces for socially aware robots[END_REF].

So far, and to the best of our knowledge, it appears that human-aware robotics uses models of social constraints that are somewhat inspired from known representations of social spaces in humans, but not systematically grounded in experimental data from social and cognitive psychology. This is a possible shortcoming, as it could limit the ability of robots to mimic navigation trajectories and behaviors that appear socially acceptable and natural to surrounding humans. Developing formal models of human-like social navigation could also, in turn, help social psychology identify mathematical properties of models that would be useful for understanding human behavior.

In this paper, we present a study of several models of social space representations, grounding their comparison in experimental data from the social psychology literature. More precisely, we consider a classical experiment by Efran and Cheyne [START_REF] Efran | Shared space: The co-operative control of spatial areas by two interacting individuals[END_REF], where human participants had to walk down a corridor, choosing whether they would pass between two confederates involved in an interaction, or avoid them and pass behind. We then define mathematical models of personal and interaction spaces. They provide navigation costs to a cost-based trajectory planning algorithm.

Our experiment addresses three questions. Can the experiment and experimental observations be replicated using our simulated robot? We demonstrate it does, in some portions of the parameter space we explored, and it does not, elsewhere. To replicate experimental data, is it necessary to involve an interaction space representation, or are personal spaces sufficient? We demonstrate that personal spaces are sufficient, and that interaction space models may be superfluous. Finally, what geometrical shapes can personal spaces have in order to explain experimental data? We demonstrate that personal spaces need to have an asymmetrical shape, elongated to the front.

To substantiate these claims, the rest of this paper is structured as follows. We first describe Efran and Cheyne's experimental protocol and data collection, then we introduce the mathematical models for personal and interaction spaces. Our experiment, and the simulator it is implemented in, are then described, followed by the presentation and analysis of our experimental data.

II. EFRAN AND CHEYNE'S EXPERIMENTS

We summarize here the main elements and findings of the classical experiment by Efran and Cheyne [START_REF] Efran | Shared space: The co-operative control of spatial areas by two interacting individuals[END_REF].

Two confederates, involved in a mock discussion, stood in narrow corridors. The experiment took place in three different corridors, of widths 2.28 m, 2.49 m and 2.36 m. Confederate A was back to one wall, confederate B was facing him (or her), some distance d away. From trial to trial, distance d varied, from 0.61 m to 1.37 m. We considered 13, 15 and 15 different distances for the three corridors, respectively, for a total of 43 possible different spatial configurations. The space "outside" of the interacting pair depended on the corridor: for instance, it was only 0.21 m at its smallest (in the first corridor).

In this setting, participants were people who simply happened to walk down this corridor. They could do so either by walking outside of the interacting pair, or between confederates (see Fig. 1, top). When participants were gone from the corridor, confederates took note of the passage chosen. However, some inclusion criteria had to be met (e.g., no-one else was in the corridor during the observation). Participants were unaware that they were observed for the purpose of the experiment. There were respectively 132, 901 and 223 participants, in the three corridors. Observations were the probabilities, for each distance d, that a participant would pass between confederates: experimental data are shown in Fig. 1 (bottom). Observations indicate, of course, that a larger distance d between confederates increased the probability that participants would pass between them, and decreased the probability of passing behind confederate B. These experimental data quantify precisely how this probability varied as a function of distance d between confederates. These are the data that we aim to mathematically replicate, in our experiment.

III. PERSONAL SPACE AND INTERACTION SPACE MODELS

A. Personal space model

We model PS in the form of a real-valued cost. Its value is 0 for positions where nothing hinders the navigating agent; on the other hand, its value is 1 for positions where something maximally disturbs or hinders it (i.e., unwanted contact or collision). Between these extremes, the cost varies gradually as a function of distance and angle to the navigating agent. Following Kirby [START_REF] Kirby | Social robot navigation[END_REF], this mathematically takes the form of two Normal probability distributions on the x, y plane, joined together seamlessly on the coronal plane of the navigating agent (i.e., the plane separating what is in front to what is to the rear of the navigating agent). These two 2D Normal distributions have independent front variance σ h and rear variance σ r , but, by construction, they have the same side variance σ s . Their covariance matrices are Σ h for the front 2D Normal and Σ r for the rear 2D Normal, with:

Σ h = σ h 0 0 σ s , Σ r = σ r 0 0 σ s . (1)
Therefore, the personal space cost at position (x, y) relative to the navigating agent, and angle α = 0 is given by:

P S(x, y) ∝ N (x, y ; [x 0 , y 0 ], Σ h ) if x 0 ≥ 0 N (x, y ; [x 0 , y 0 ], Σ r ) otherwise . (2)
The general formulation, for any angle α, can be found elsewhere [START_REF] Kirby | Social robot navigation[END_REF]: it basically amounts to rotating the front/rear axis by angle α. An example of a personal space cost function is shown in Fig. 2.

B. Interaction space model

In this paper, we explore two mathematical definitions for the interaction space (IS) model. We only consider onedimensional functions, because the geometry of Efran and Cheyne's experiment only provides information about passage through the "critical line", that is to say the axis crossing the corridor at the position of the interacting confederates.

The first mathematical model was already used in previous works [START_REF] Rios-Martinez | Understanding human interaction for probabilistic autonomous navigation using Risk-RRT approach[END_REF], and consists in a Normal probability distribution, defined by:

IS N (x) ∝ N (x ; center, dist/s int ) , (3)
with center the middle position between the interacting people, dist the distance between them, and s int a free parameter. The proportionality coefficient is defined so that the maximum value taken by this function is 1 to represent maximum hindrance.

The second mathematical model is simply a constant value between interacting people, defined by:

IS c (x) = h = min(p int /dist, 1) , (4)
with h the "height" of the constant, given by the ratio between p int , a free strictly positive parameter, and dist the distance between the interacting people, provided it does not exceed a maximum value of 1 (for representing maximum hindrance, as previously). This model represents a constant valued hindrance between interacting people, whatever the position that would be chosen by the navigating agent to pass between them.

The two mathematical definitions of interaction space we investigate are illustrated in Fig. 3.

IV. EXPERIMENT

A. Simulation of Efran and Cheyne's experiment

We have replicated Efran and Cheyne's experiment in a robotic simulator. To do so, we have created three different environments, that are corridors of the same dimensions as described by Efran and Cheyne. Static, simulated confederates A and B, and a moving simulated participant (i.e., the simulated robot) were placed in these environments. Each was implemented as a rectangle of lengths 0.3 m from "front" to "back" and 0.4 m long from left to right.

Simulated confederates were placed as in the original experiment, with the distance d between them a controlled parameter. The simulated robot was placed, at the beginning of each simulation, at the entrance of the corridor, and was provided a navigation goal at the other end. To solve the navigation task, it would therefore have to traverse the corridor, either by passing between or behind confederates, as human participants had to.

B. Trajectory generation

We used the RiskRRT (Risk Rapid-exploring Random Tree) trajectory generation algorithm [START_REF] Fulgenzi | Risk based motion planning and navigation in uncertain dynamic environment[END_REF], which is a variant of the RRT (Rapid-exploring Random Tree) algorithm [START_REF] Lavalle | Rapidly-exploring random trees a new tool for path planning[END_REF] adapted for dynamic environment, using partial planning. The RiskRRT trajectory generation algorithm deploys a tree of nodes with associated costs, taking into account possible collisions with static and dynamic obstacles, along with social costs, and selects the branch of lower maximal cost that also progresses toward the specified goal. A sample navigation scenario, illustrating the RiskRRT algorithm, is shown in Fig. 4.

C. Collision costs and social costs

In RiskRRT, costs associated with nodes represent risk, either of collision, or of social hindrance. This cost for a position (x, y) is therefore defined as a function of the probability that position (x, y) is collision free, PWithoutColl(x, y), and the probability that it is "hindrancefree", PWithoutHind(x, y), with:

cost(x, y) = 1 -(PWithoutColl(x, y) • PWithoutHind(x, y)) . (5)
In the simulation of Efran and Cheyne's experiment, the probability of collision involves the probability of collision with walls in the occupancy map given to the robot, pCollW(x, y), or the confederates pCollC A (x, y) and pCollC B (x, y):

PWithoutColl(x, y) = (1 -pCollW(x, y)) • (1 -pCollC A (x, y)) • (1 -pCollC B (x, y)) . ( 6
)
The probability of social hindrance involves several components: the probability that the robot at position (x, y) would invade personal spaces of confederates A and B pHindC A (x, y) and pHindC B (x, y), the probability of interrupting their interaction pInt(x, y), and the probability that outside elements would invade the personal space of the navigating agent pNav(x, y):

PWithoutHind(x, y) = (1 -pHindC A (x, y)) • (1 -pHindC B (x, y)) (7) • (1 -pInt(x, y)) • (1 -pNav(x, y)) .
The first two terms refer to the PS models centered on each confederate, using the asymmetric 2D Normal model of PS given by Eq. ( 2). pInt(x, y) refers either to the Normal model of IS of Eq. ( 3), or to the constant model of IS of Eq. ( 4), or can be replaced by a constant zero value for testing experimental scenarios where the interaction space model is disabled. Finally, pNav(x, y) is computed by attaching a personal space model to the navigating agent, and finding the object, in the navigating agent's neighborhood, of maximal hindrance (i.e., the navigating agent avoids being close to static or dynamic obstacles as they would invade its personal space). Again, because of the simple geometry of the navigation scenario in Efran and Cheyne's experiment, costs only need to be computed on the "critical line" passing through confederates and crossing the corridor (see Fig. 5, top; the critical line is, because of the geometry of Efran and Cheyne's experiment, parallel with body and gaze orientations of confederates, and normal to the walls). In other words, we only used, from the RiskRRT algorithm, the cost computation for these positions of interest, ignoring the whole trajectory generation before or after the critical line, etc. We computed costs along this critical line, every 0.02 m, excepting positions occupied by confederates.

With these costs, we computed the probability of passing between or outside the confederates. Two methods have been explored, either by selecting the side of lowest average cost ("average" method), or by selecting the side containing the lowest cost overall ("minimum" method). A hypothetical situation illustrating the difference is shown in Fig. 5 (bottom). On this example, the "average" method would select passing outside of confederates (smaller average cost on the left side), whereas the "minimum" method would select passing between confederates (smaller minimum value on the right side). Experimental results showed that the "minimum" method yields similar observations (i.e., candidate models are ranked in the same order), with systematically worse model fit (i.e., there are larger discrepancies between data and model predictions) than the "average" method. Therefore, for the remainder of this paper, and because of lack of space, we only report below experiments with the "average" method.

D. Experimental conditions, parameter spaces, and model fit measure

To answer our questions about the necessity of modeling interaction spaces, we have explored several model combinations, resulting in five experimental conditions:

1) in the "PS Only" condition, personal spaces of the navigating agent and confederates were used in computations, but not interaction spaces (i.e., pInt = 0 in Eq. ( 8) above); 2) in the "PS + Constant IS" condition, personal spaces were used, along with an interaction space of constant value between confederates (i.e., pInt = IS c ); 3) in the "PS + Normal IS" condition, personal spaces were used, along with an interaction space following a Normal probability distribution between confederates (i.e., pInt = IS N ); 4) in the "Constant IS Only" condition, personal spaces were disabled, leaving only an interaction space of constant value between confederates; 5) finally, in the "Normal IS Only" condition, personal spaces were disabled, leaving only an interaction space of Normal shape between confederates. For each condition, the possible parameters dimensions were sampled at discrete values: this allows a grid search for evaluating models. Ranges and precision steps are given Table I.

Enumerating all model combinations and parameter space evaluation points yielded 8, 091 candidate models that we experimentally evaluated. Each of these candidate models was simulated in all of the 43 spatial configurations of our simulation of Efran and Cheyne's experiment (recall there were three corridors, with 13 + 15 + 15 possible distances d between confederates). Overall, we performed 8, 091 • 43 = 347, 913 simulations in our experiment (and again the same number for the "minimum" method, that we discounted previously).

Each of these simulations provided the probability to pass between the confederates, Sim P, for each experimental condition, parameter values, and spatial configurations at hand. We therefore measured the model fit of each simulation, by comparing our simulated data with the experimental data of Efran and Cheyne, Obs P, using the root of the squared error:

Model fit = (Sim P -Obs P) 2 , (8)
(i.e., we measured the RMSE for a single point). Model fit is good when the measured error is small.

V. EXPERIMENTAL RESULTS

The model fit measure we presented was aggregated in various ways during data analysis. For instance, we found out that analyzing results for each of the three simulated corridors, or by averaging over corridors, had minimal impact on observations. Therefore, in the remainder of the paper, we only discuss this global analysis of data (but see Table II).

A. Can we replicate Efran and Cheyne's data?

The first data analysis we report aimed at verifying whether the proposed models and their combinations could account for the experimental data of Efran and Cheyne.

To do so, we studied, for each condition, the 200 parameter values with best and worse model fits found in the parameter space. Fig. 6 shows the results for conditions "PS Only", "PS + Constant IS" and "PS + Normal IS". We observe that, in these conditions, the best parameter values allow the simulated models to closely match experimental data, contrary to the worst parameter values, where the navigating agent passes much more frequently or less frequently between confederates.

We also studied the geometry of model fit across the parameter space, to verify whether better solutions could possibly exist outside of the chosen parameter space. We focus here on model fit in the "PS Only" condition, for all parameter values. Recall that parameter space is threedimensional in this case, with three independent variance coefficients σ h , σ r and σ s : we plot model fit against ratios σ r /σ h and σ s /σ h (Fig. 7, top) and ratios σ r /σ s and σ h /σ s (Fig. 7, bottom).

We observe flat valleys in model fit landscapes, where it is very close to its minimal value of 0.084. It is doubtful that better solutions exist outside of the parameter space we explored; examining data by Efran and Cheyne, we can hypothesize that we reached a minimal, residual value for model fit, that is due to experimental noise and small data sample (e.g., see the odd drop in passage probability around distance d = 1.15 m, for corridor 1, in Fig. 1).

Overall, these results show that the chosen parameter spaces contain good solutions allowing to replicate Efran and Cheyne's data, but not trivially so, as portions of parameter spaces yield simulations that do not correspond to human behavior.

Experimental results and analyses are similar for other conditions. In particular, concerning the "Constant IS Only" and "Normal IS Only" conditions, we observe an overall worse fit, everywhere in the parameter space, compared to the "PS Only" condition (Fig. 8). These two conditions are therefore eliminated from further analysis.

B. Are interaction spaces necessary?

Previous analyses have left three conditions in competition: "PS Only", "PS + Constant IS" and "PS + Normal IS". Analysis of the best parameter values shows similar best model fits for each of the three conditions (see Table II). We have investigated this similarity and found that, for parameter values in models with IS, there was an equivalent parameter value for the personal space in the "PS Only" condition. Intuitively, this equivalent model simply replaces the IS between confederates by an increase in the front variance of the PS. We compared for instance the best parameters in the "PS Only" and the "PS + Normal IS", in the overall analysis (all three corridors combined). Removing a Normal IS with s int = 8.0 modifies the front variance of the PS from σ h = 0.6 to σ h = 0.9. The situation is similar when comparing the "PS Only" and "PS + Constant IS" conditions, but with less marked increases in σ h . This means that models with explicit IS are equivalent to a model without IS, but with different parameters. This is both a good and a bad news. The bad news is that, in terms of scientific analysis, this leads to a negative conclusion: this shows that the data of Efran and Cheyne cannot discriminate between models with or without IS. From these data, it is impossible to know whether humans use an IS representation, or whether they only rely on a PS representation. It could also be the case that humans only use plastic PS representations, that are modified according to context, as some psychological experiments have already suggested [START_REF] Hayduk | The shape of personal space: An experimental investigation[END_REF], [START_REF] Sommer | From personal space to cyberspace[END_REF].

On the other hand, the good news is that using only a PS of fixed parameters yields very good results, with respect to the replication of Efran and Cheyne's experiment: notice that valleys of near optimal parameter configurations are wide (Fig. 7), indicating a robust model. We pursue the previous example: assume removing the IS from the optimal "PS + Normal IS", but not increasing σ h . It can be verified that not increasing σ h would only marginally affect model fit (compare, in Fig. 7 (top), the point at σ s /σ h = 1.5/0.6 = 2.5 and σ r /σ h = 0.1/0.6 = 0.16 with the point at σ s /σ h = 1.5/0.9 = 1.6 and σ r /σ h = 0.1/0.9 = 0.11: both lie in the near-optimal valley).

Taking only PS models into account, ignoring IS altogether, also would yield simpler robotic implementation. First, it reduces parameter space by one-dimension; as we showed that it would not decrease model performance, a parsimony argument supports this simplification. This observation also leads to an intriguing possibility for humanaware navigation in robotics. This suggests that detecting interactions in human populated environments, which is possibly a costly and difficult step, might be superfluous. At least for interactions when people are separated by small distances, a representation of their PS already ensures avoiding interrupting their interactions. Whether this extends to interactions on a larger spatial scale, or involving more that two people, is an open question.

C. What is the geometry of personal space?

Examining the model fit landscapes of the "PS Only" condition (Fig. 7) suggests that only some PS shapes yield adequate replication of Efran and Cheyne's data. Examples in valleys of good solutions all have similar geometries, with an asymmetrical shape, elongated to the front and short to the rear. This result is not compatible with Hall's original proposal [START_REF] Hall | The hidden dimension[END_REF] of PS as concentric circles (a circular PS can be seen in Fig. 7 as a mediocre solution). In contrast, it strongly supports other proposals which have suggested asymmetrical shapes, similar to the ones we obtained in our experiment [START_REF] Hayduk | The shape of personal space: An experimental investigation[END_REF].

VI. CONCLUSION

In this paper, we have presented an experiment comparing robotic simulations of PS and IS with experimental data from psychological literature, in a corridor crossing navigation scenario involving avoiding people in interaction. We have proposed several models of IS, and a model of PS, to investigate their combinations extensively.

Our results indicate that the proposed models, with correct parameter values, can adequately replicate the experimental data of Efran and Cheyne. This could benefit future robotic applications, by providing PS parameters that were demonstrated to yield navigation strategies very close to human behavior (i.e., grounding Kirby's asymmetrical Normal model [START_REF] Kirby | Social robot navigation[END_REF] in data from experimental psychology). We have also demonstrated that PS was sufficient in the navigation scenario we explored, and that omitting an explicit IS would only marginally decrease model fit. In a robotic application context, this suggests that detecting interactions might by superfluous, and only representing PS would be enough to obtain human-like group avoidance.

Finally, the analysis of PS geometry strongly suggests that human PS would be asymmetrical and elongated to the front, supporting some non-mathematical models of PS in psychological literature. However, because of the simplicity of Efran and Cheyne's experiment, some geometrical properties could not be investigated (e.g., the critical line "hides" the 2D shape of IS).

The generalizability of our results thus appears to be an interesting topic for future research. A large number of variations on Efran and Cheyne's initial experiments are easily imagined; we describe a few. For instance, if one of the confederates is not a person but a robot, would a different personal space be needed around the robotic confederate? Would it have a personal space, at all? If there is a single person in a corridor reading a poster on the wall, would a person-object interaction space be needed to correctly describe human navigation behavior? If there is a TV-set in the corridor, but no-one currently watching it, would an "enaction" space be needed? Some of these variants would ask questions already treated experimentally in the psychological literature. For instance, in some of our previous experiments, we have studied the influence of affective variables like closeness or friendship between confederates and the navigating agent on his or her navigation behavior [START_REF] Morgado | Close to me? the influence of affective closeness on space perception[END_REF], [START_REF] Morgado | Does friendship influence space perception? with particular reference to the curse of the suspicious participants[END_REF]. The method we have described here could be applied to this data to refine and enrich our model of social navigation.

A more technical question concerns the generalizability of our work with respect to the choice of RiskRRT as a robotic simulator. In our simulations, it appears that computing navigation costs along the critical line was the only component of RiskRRT that was used. This suggests a certain robustness of our approach, in the sense that it is probably independent of the specificities of RiskRRT, and should extend to other cost-based social navigation method.

At the core of the experiment we presented here, lies the method we used for grounding robotic social navigation in human data. This method appears general enough to be extended to other and more complex scenarios, hopefully yielding testable predictions and helping to close the loop between robotics and experimental psychology.

Fig. 1 .

 1 Fig. 1. Summary of Efran and Cheyne's experiment [18]. Top: Schema of the experimental setup. Bottom: Experimental data is the probability of passing between confederates, as a function of the distance d between them.

Fig. 2 .

 2 Fig. 2. Personal space model of parameters σ h = 1.0, σr = 0.3, σs = 0.5, visualized as a real-valued function. Inset: The same personal space, visualized as a contour plot.

Fig. 3 .

 3 Fig. 3. Interaction space models. Top: Interaction space as a Normal probability distribution. Bottom: Interaction space as a constant function.

Fig. 4 .

 4 Fig. 4. Example of a navigation scenario and the possible trajectories computed by the RiskRRT algorithm. Near the simulated confederates, potential collisions and social costs increase the calculated costs of some branches of the tree. Larger costs are represented by circles of larger diameter. In this example, the simulated participant chooses to avoid passing between confederates.

Fig. 5 .

 5 Fig. 5. Top: Illustration of the critical line, the only portion of interest when deciding whether to pass between or outside confederates (top view of the experimental situation). Bottom: Height of the critical line represents passing probabilities (perspective drawing of the experimental situation).

Fig. 6 .

 6 Fig. 6. Comparison of simulated passage probabilities and experimental observations by Efran and Cheyne, in corridor 2 (data are similar for other corridors). a) The 200 best parameter values in the "PS Only" condition replicate experimental data very well, contrary to the 200 worst (b). Observations are similar in the "PS + Constant IS" (c,d) and "PS + Normal IS" (e,f) conditions.

Fig. 7 .

 7 Fig. 7. Model fit in the "PS Only" condition, as a function of ratios σr/σ h and σs/σ h (Top) and as a function of ratios σr/σs and σ h /σs (Bottom). Insets show the corresponding PS shapes for example positions in these spaces.

Fig. 8 .

 8 Fig. 8. Model fit as a function of the single parameter for the "Constant IS Only" (a) condition and the "Normal IS Only" (b) condition. In both cases, an optimal parameter value lies inside the explored parameter space.

TABLE II SUMMARY

 II OF THE BEST PARAMETERS AND MODEL FITS FOR EACH EXPERIMENTAL CONDITION.

*This work was supported by the Pôle Grenoble Cognition.


Keywords: Microcantilever, Polymers Viscoelasticity, Quality factor, Integrated Mechanical Characterization

The increasing interest in polymer materials creates the need for accurate tools to characterize their mechanical properties. Due to energy dissipation in polymers during deformation, these materials exhibit viscoelastic behavior. Accurate determination of these viscoelastic properties and, more specifically, viscous losses, remains challenging and mainly unknown for thin polymer films. In the present work, a straightforward method to determine the loss modulus of organic materials using resonating microcantilevers has been developed. The extracted results for polyisobutylene (PIB) show the variation of viscous losses over a large range of frequencies (7-350kHz).

Progress in polymer chemistry allows the synthesis of materials with specific properties depending on the application. Indeed, increasing interest in polymers as sensitive coatings for bio-chemical sensing in both gas and liquid phases is found in the literature, and more specifically when associated with microsystems [START_REF] Lochon | An alternative solution to improve sensitivity of resonant microcantilever chemical sensors: comparison between using high-order modes and reducing dimensions[END_REF], [START_REF] Hossenlopp | Characterization of epoxy resin (SU-8) film using thickness-shear mode (TSM) resonator under various conditions[END_REF]. Properties such as selectivity, binding capacity and reversibility are often ensured by organic materials for these applications. Another example concerns polymers as structural layers for MEMS applications [START_REF] Johansson | Polymeric cantilever-based biosensors with integrated readout[END_REF], [START_REF] Greve | Thermoplastic microcantilevers fabricated by nanoimprint lithography[END_REF]. In this case, low stiffness is the main required characteristic. For the optimal use of polymers, characterization of their intrinsic mechanical properties is of particular importance. A direct consequence of incomplete characterization is an inaccurate correlation between theory and experiment on the behavior of microstructures. One major cause of this miscorrelation is the neglected energy dissipation that occurs during deformation due to the material's Manuscript received December 17, 2010. This work was supported in part by the CPER Pôle 4N Nanosciences en Aquitaine GP-206-action 216/1 and the National Science Foundation under Award ECCS-0824017. C. Ayela and I. Dufour are with the Université de Bordeaux, Laboratoire de l'Intégration du Matériau au Système, 33405 Talence, France (e-mail: cedric.ayela@ims-bordeaux.fr, isabelle.dufour@imsbordeaux.fr). S. M. Heinrich is with the Department of Civil and Environmental Engineering, Marquette University, Milwaukee, WI 53201-1881, USA (email: stephen.heinrich@marquette.edu). F. Josse is with the Department of Electrical and Computer Engineering, Marquette University, Milwaukee, WI 53201-1881, USA (e-mail: fabien.josse@marquette.edu). viscoelasticity. To account for this, polymer materials may be characterized mechanically by using a complex Young's modulus, where the real part corresponds to elastic stiffness and the imaginary part to energy dissipation. Moreover, each part is frequency-dependent, making their determination challenging. As a consequence, the database of mechanically characterized viscoelastic materials is restricted to conventional ones, such as polyisobutylene (PIB) or poly(methylmethacrylate) (PMMA) [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF], whose properties are usually determined via experimentation on bulk materials. In this context, a simple and reliable method to determine the viscoelastic properties of polymeric thin films over a large frequency range is of major interest.

In the present work, polymer-coated silicon microcantilevers vibrating in different resonance modes are used for the determination of viscoelastic properties of polyisobutylene thin films. More specifically, values of the imaginary part of the complex modulus were obtained, as these values are often ignored due to lack of data, despite the fact that they are needed in many important applications to accurately quantify the energy dissipation that occurs in such materials.

With this aim in mind, an analytical model has been developed for calculating the resonant frequency of the out-ofplane flexural mode and the quality factor due to viscoelastic losses in the coating on an elastic cantilever [START_REF] Dufour | Effect of Coating Viscoelasticity on Quality Factor and Limit of Detection of Microcantilever Chemical Sensors[END_REF]. In the present paper we utilize a straightforward extension of the work in [START_REF] Dufour | Effect of Coating Viscoelasticity on Quality Factor and Limit of Detection of Microcantilever Chemical Sensors[END_REF] to calculate the resonant frequency and quality factor of the n th out-of-plane flexural mode. The following notation is used: L, b, h 1 , E 1 and ρ 1 are the length, width, thickness, elastic Young's modulus and density of the microcantilever's base material, respectively, and h 2 , E 2 '+jE'' 2 and ρ 2 are the thickness, complex Young's modulus and density of the viscoelastic coating that is assumed to cover one of the two larger faces of the cantilever.

In the case of a silicon cantilever with polymeric coating, because of the large difference in the values of Young's moduli, the following approximate equations have been used to calculate the resonant frequency and quality factor of the n th mode [START_REF] Dufour | Effect of Coating Viscoelasticity on Quality Factor and Limit of Detection of Microcantilever Chemical Sensors[END_REF]: Note that the "n" subscript appears on the coating moduli because they are dependent on frequency, i.e., on the particular node of vibration. In the case of Eq. ( 2), the frequency-dependence of the coating loss modulus within the 3-dB bandwidth of each resonant peak has been neglected. Due to the fact that the ratio ρ 2 /ρ 1 is much larger than E' 2 /E 1 for polymeric coatings on silicon (values for E' 2 are between 4.7 and 148MPa in the frequency range that is considered [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF]), the resonant frequency formula (1) may be simplified and expressed as a function of the resonant frequency of the uncoated cantilever, f n,0 :

  2 23 22 1 2 2 3 11 1 1 2 1 22 11 13 2 12 2 1 ' ,n n n E hh h h h EE h f L h h              (1)
0 22 11 1 n, n f f h h     (3)
with

2 11 0 2 1 12 2 n n, h E f L     (4)
The total quality factor Q n depends on the quality factor of the cantilever without the coating Q n,0 (viscous losses in the surrounding medium, internal losses, support losses, etc.) and the quality factor associated with the viscoelastic losses in the coating, Q n_ve as follows:

  2 3 22 2 1 2 0 0 0 3 0 01 1 3 1 1 " ,n n, n_ve n, nn , n, n, n_ve n_ve Eh h h h QQ Q QQ Q QQ Eh Q               (5)
According to (3), the coating stiffness has a negligible effect on frequency compared to the coating mass. When this case applies (e.g., polymers on silicon), the shift in resonant frequency due to the coating permits the determination of coating thickness when the coating density is known. Rearranging (5), the measurement of the quality factor with and without the coating permits the calculation of the loss modulus of the coating:

20 2 32 10 1 1 31 ,n n, n n, n EQ Q h , E QQ h ()       (6)
To experimentally determine the viscoelastic properties of thin polymer films over a large frequency range, a set of three silicon microcantilevers was used. Cantilevers 1, 2, and 3 were characterized by lengths, widths and thicknesses of 2000x200x27µm 3 , 3000x200x21µm 3 and 3000x200x57µm 3 , respectively. Integrated actuation of the structures was achieved by electromagnetic actuation, while their frequency response was measured using a laser Doppler optical vibrometer (Polytec MSA-500). The fabrication process and actuation principle are detailed in [START_REF] Fadel | Chemical sensing: millimeter size resonant microcantilever performance[END_REF].

First, uncoated cantilevers were characterized to obtain the resonant frequency and quality factor of the first four transverse flexural modes, defining the overall frequency range. With the three geometries, a frequency range of 7kHz to 354kHz was spanned. Dynamic characterization of uncoated cantilevers allowed the determination of the cantilever thickness from the resonant frequency measurement f n,0 and (4). Quality factor measurement resulted in the determination of Q n,0 , which is necessary to use [START_REF] Dufour | Effect of Coating Viscoelasticity on Quality Factor and Limit of Detection of Microcantilever Chemical Sensors[END_REF] for the loss modulus estimation.

Polyisobutylene (PIB) (Sigma-Aldrich) was next deposited on the silicon cantilevers using a spray-coating technique. The polyisobutylene was dissolved in low vapor pressure chloroform permitting layer-by-layer coating. As a result, successive PIB layers were deposited onto the cantilevers with a 130-170nm per-layer thickness for a 2-sec spraying time. A two-minute solvent evaporation step was performed after each deposition, ensuring that no residual solvent traces remained in the PIB layer. Following deposition, resonant frequency f n and quality factor Q n for each mode were measured for the three cantilevers when 10 PIB layers were applied, and the sequence was repeated for up to 80 layers. Figure 1 shows the evolution of resonant frequency and total quality factor of the third flexural mode on cantilever 1, after successive deposition of PIB layers. For the quality factor, a strong decrease is observed for the first 30 layers coated, while a lower influence occurs when a quality factor close to 100 is measured, a result that is consistent with that predicted by [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF]. Experimental data for the four out-of-plane flexural modes of the three cantilevers have been collected. Using the measured resonant frequency f n and (3), values of h 2 corresponding to the successively deposited PIB layers were obtained, while the measured quality factor Q n permitted the calculation of the loss modulus of PIB at various frequencies via [START_REF] Dufour | Effect of Coating Viscoelasticity on Quality Factor and Limit of Detection of Microcantilever Chemical Sensors[END_REF]. An averaged value of the frequency-dependent loss modulus is obtained for different coating thicknesses (with β<0.4) and plotted in Fig. 2, where error-bars represent the values obtained for these different thicknesses. From this figure, the frequency dependence of E" 2 is exhibited over a large frequency range, from 7kHz to 354kHz. A comparable behavior was obtained for the three cantilevers, allowing fitting of the experimental data using a power-law curve. Values of E" 2 for PIB ranged from 6.5MPa to 219MPa, showing the large range of values achievable with the method. These values of E" 2 are low compared to the Young's modulus of silicon E 1 , which validates the assumption implicit in ( 2) and (3). Also shown in Fig. 2 are data presented in Ferry's book for bulk PIB where the values of E'' are obtained from Ferry's complex shear modulus values by assuming a constant value of Poisson's ratio of 0.49 [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF]. Compared with the values obtained in the present work, there is a similar frequencydependent trend, while the magnitudes of the E" 2 are comparable. The differences in the results are attributable to several sources, including the following: (a) Losses associated with Si-PIB interfacial phenomena, which are expected to become more important for smaller coating thicknesses, are reflected in the E2'' measurements of the proposed method; (b) Viscous losses in the surrounding fluid are assumed to be unaffected by the coating thickness; and (c) Differences in bulk properties and thin-coating properties of polymers are regularly observed due to the arrangement of polymer chains at the microscale differing from those at the macroscale [START_REF] Luengo | Thin Film Rheology and Tribology of Confined Polymer Melts: Contrasts with Bulk Properties[END_REF], [START_REF] Tawa | Out-of-Plane Photoreorientation of Azo Dyes in Polymer Thin Films Studied by Surface Plasmon Resonance Spectroscopy[END_REF]. This difference also depends on the method used for deposition.

To summarize, a new simple method has been proposed for the determination of the loss modulus in soft, viscoelastic materials such as polymers. We believe that such a method can enhance the knowledge base on thin polymer films. Moreover, the method can provide a low-cost means of obtaining properties on a wide variety of materials and may be useful in quantifying the effects of both frequency and thickness on the loss modulus of thin layers of viscoelastic materials.

Fig. 1 .

 1 Fig. 1. Evolution of both the resonant frequency and total quality factor after successive PIB layers depositions (third flexural mode of cantilever 1).

Fig. 2 .

 2 Fig. 2. Frequency-dependent loss modulus of PIB (log-scale).

  =1.875, λ 2 =4.694, λ 3 =7.855, λ 4 =10.996.

	Preprint -IEEE/ASME Journal of Microelectromechanical Systems, vol. 20, 2011, pp. 788-790. (doi:	2
	10.1109/JMEMS.2011.2159102).
	with λ 1	n_ve Q	Resonant Microcantilevers for the Determination of the Loss Modulus of Thin   3 11 2 3 22 2 1 2 3 " ,n Eh Eh h h h  (2)   
			Polymer Films
			Cédric Ayela, Stephen M. Heinrich, Fabien Josse, and Isabelle Dufour

E''= (5.10 6 )f 0,624 E''= (2.10 6 )f 0,762 1,0E+06