Hubert Jóźwiak
email: hubert.jozwiak@stud.umk.pl

Franck Thibault

Nikodem Stolarczyk

Piotr Wcis Lo

Ab initio line-shape calculations for the S and O branches of H 2 perturbed by He

Keywords: H 2, hydrogen, He, helium, line-shape parameters, molecular collisions, quantum scattering

We report the results of ab initio investigation of the S and O branches of the hydrogen molecule perturbed by helium atom. The calculations, based on closecoupling scheme, were performed for vibrational bands from v=0 to v=5, and rotational levels up to j=5. We provide the pressure broadening and shifting coefficients and the real and imaginary parts of Dicke contribution to the Hess profile. The results are important for astrophysical studies, for example as a reference data for studying the gas giants' atmospheres.

Hydrogen molecule is the most abundant molecule in the universe. Its importance in astrophysical studies has been thoroughly reviewed in Ref. [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF]. Let us remind that the accurate values of pressure broadening and shifting coefficients are needed for the spectroscopic studies of the gas giants' atmospheres 5 [START_REF] Baines | The abundances of methane and ortho/para hydrogen on Uranus and Neptune: Implications of new laboratory 4-0 H 2 quadrupole line parameters[END_REF][START_REF] Hayden | 395 The H 2 4-0 S(0,1, and 2) quadrupole features in Jupiter[END_REF]. Moreover, the simplicity of the H 2 -He system allows highly accurate ab initio potential energy surface to be calculated and verified experimentally.

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The goal of this paper is to provide the fully ab initio line-shape parameters for H 2 in helium baths. In a recent paper [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF], some of the present authors reported a new potential energy surface for this system and provided line-shape 10 parameters for Raman isotropic and anisotropic Q lines. The latter parameters can be also used for electric quadrupole Q lines. In addition, this PES [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF] was recently successfully tested on the very first purely rotational Stokes lines of D 2 in He [START_REF] Martinez | Testing the ab initio quantum-scattering calculations for the D 2 -He benchmark system with stimulated Raman spectroscopy[END_REF]. Here, we investigated the anisotropic Raman S and O lines of H 2 in He, for vibrational bands from 0-0 up to 0-5, for rotational levels 15 from j=0 to j=5 for S lines, and from j=2 to j=5 for O lines. We report the pressure broadening and shifting coefficients and both real and imaginary part of the Dicke parameter. We have calculated them by averaging the generalized spectroscopic cross sections for 18 temperatures, ranging from 5 to 2000 K. The generalized spectroscopic cross sections were calculated using S-matrix elements 20 provided by the MOLSCAT [START_REF] Hutson | Molscat version[END_REF] code.

Quantum scattering calculations

In this section we report the details of ab initio quantum scattering calculations for the H 2 -He system. The potential energy surface (PES) in the examined system depends on the distance between the hydrogen molecule and the helium 25 atom, R, the intramolecular distance, r HH , and the angle between these two axis, θ. We made use of the PES which was recently reported in Ref. [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF], which is an extension of the Bakr et al. PES [START_REF] Bakr | Highly accurate potential energy surface for the He-H 2 dimer[END_REF]. It has recently been shown [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF] that the PES from Ref. [START_REF] Bakr | Highly accurate potential energy surface for the He-H 2 dimer[END_REF] is more accurate than the previous PESs ( [START_REF] Schaefer | Quantum calculations of rotational and NMR relaxation, depolarized Rayleigh and rotational Raman line shapes[END_REF][START_REF] Muchnick | The HeH 2 energy surface[END_REF][START_REF] Boothroyd | Accurate analytic HeH 2 potential energy surface from a greatly expanded set of ab initio energies[END_REF]). However, it was pointed out, that the PES from Ref. [START_REF] Bakr | Highly accurate potential energy surface for the He-H 2 dimer[END_REF] does not cover large 30 enough vibration amplitude in H 2 and therefore should not be used to study the overtones. Not only does the most recent PES [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF] cover the sufficient range of the interatomic distance, r HH , but it also properly describes the asymptotic behavior of the interaction energy in this system. For more details see [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF].

The PES is expanded over Legendre polynomials, P λ :

V (R, r HH , θ) = λ V λ (R, r HH )P λ (cos(θ)).

(1)

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Since H 2 is a homonuclear molecule, the λ index takes only even values, and due

35

to the small anisotropy of the PES, it is sufficient to retain only the first 4 terms of the expansion. The obtained terms V λ (R, r HH ) are averaged over rovibrational wavefunctions of the H 2 molecule χ j,v (r HH ), which gives the radial coupling terms V λ,v,j,v ,j (R). For transitions between different vibrational states, one can neglect the centrifugal distortion [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF], and write the radial coupling terms 40 as V λ,v,0,v ,0 (R). Thus for rovibrational transitions we consider 2*4 radial coupling terms. While considering purely rotational transitions, one should take into consideration the centrifugal distortion. This phenomenon was firstly studied by Shafer and Gordon [START_REF] Shafer | Quantum scattering theory of rotational relaxation and spectral line shapes in H 2 -He gas mixtures[END_REF], and latter by Dubernet and Tuckey [START_REF] Dubernet | Raman Q and S line broadening and shifting coefficients: some commonly used assumptions revisited[END_REF], who have shown that it has a relatively small effect effect for the broadening but a 45 rather large one for the shift. This is due to the fact, that the line shift is very sensitive to the difference in the elastic scattering in the two different rotational states. Therefore, in this case, one cannot average the terms V λ (R, r HH ) over rovibrational wavefunctions with j set to 0. In order to obtain proper results, we have prepared the radial coupling terms V λ,0,j,0,j (R) with j up to 15 and 14,

50

for ortho and para states, respectively. Thus for purely rotational transitions we have to consider 144 radial coupling terms.

We performed close-coupling calculations for this system using the MOLSCAT [START_REF] Hutson | Molscat version[END_REF] code, obtaining the S-matrix elements. We used them to calculate the generalized spectroscopic cross section [START_REF] Monchick | Diatomic-diatomic molecular collision integrals for pressure broadening and dicke narrowing: a generalization of Hess ´s theory[END_REF][START_REF] Schaefer | Line broadening of HD immersed in He and H 2 430 gas[END_REF]:

σ q λ (v i , j i , v f , j f , E kin ) = π k 2 Ji,J f ,l,l , l, l (-1) l-l -l+ l [J i ] [J f ] × [l] [l ] l l   l l λ 0 0 0     l l λ 0 0 0        j i j i l l j f l j f l q J f J i λ      ×i l-l -l+ l [δ ll δll -v i j i l |S Ji (E Ti )|v i j i l v f j f l |S J f (E T f )|v f j f l * ]. (2)
The rovibrational transition is denoted as v i , j i → v f , j f , where i and f are the indexes of the initial and final states, respectively. J is the total angular

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momentum, l is the relative kinetic momentum, and J = j+l. k is the wavevector 55 associated with the initial kinetic energy. [x] denotes (2x+1), quantities in parentheses are Wigner 3j and the quantity in bracket is a Wigner 12j symbol of the second kind [START_REF] Yutsis | Theory of angular momentum, Israel Program for Scientific Translations[END_REF]. The total energies associated with each of the rovibrational energy levels are denoted as

E Ti = E kin + E viji and E T f = E kin + E v f j f . The
tensor order of the radiation-matter interaction is denoted as q, and for the 60 investigated here anisotropic (or equivalently absorption quadrupole) S and O lines, q=2. Equation ( 2) is a quite complex formula which involves a number of angular momenta and S-matrix elements. The summation over the six angular momenta reduces to a summation over only four of them for the case of λ=0.

Moreover, in this case, the total angular momenta are in a triangular relation 65 with q, and the Wigner 12j symbol can be transformed into a product of two Wigner 6j symbols. Thus, for λ=0, Eq.( 2) reduces to the standard generalized spectroscopic cross section σ q 0 [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF][START_REF] Shafer | Quantum scattering theory of rotational relaxation and spectral line shapes in H 2 -He gas mixtures[END_REF][START_REF] Ben-Reuven | Symmetry considerations in pressure-broadening theory[END_REF][START_REF] Ben-Reuven | Impact broadening of microwave spectra[END_REF], whose real and imaginary parts are referred to as the pressure broadening (PBXS) and pressure shifting (PSXS) cross sections, respectively. For λ=1, Eq.( 2) transforms into the complex Dicke 70 cross section (DXS) σ q 1 [START_REF] Martinez | Testing the ab initio quantum-scattering calculations for the D 2 -He benchmark system with stimulated Raman spectroscopy[END_REF][START_REF] Monchick | Diatomic-diatomic molecular collision integrals for pressure broadening and dicke narrowing: a generalization of Hess ´s theory[END_REF][START_REF] Schaefer | Line broadening of HD immersed in He and H 2 430 gas[END_REF][START_REF] Corey | Dicke narrowing and collisional broadening of spectral lines in dilute molecular gases[END_REF]. Considering the vector nature of the velocity (a rank-1 tensor), the velocity-changing collisions induce additional angular momenta coupling. The summation is still over six angular momenta and there is no simple triangular rule for J i , q and J f . Thus, Dicke cross sections are more averaged quantities than PBXS and PSXS.

75

We took advantage of the fact, that the PES does not couple ortho and para states of H 2 . While constructing the basis for the calculations, we dealt with even and odd rotational levels separately. The calculations of the cross sections were performed for 263 kinetic energies, ranging from 0.1 to 9000 cm -1 . The basis included rotational levels up to j=15 and j=14 for ortho and para states 80 of H 2 molecule, respectively. One asymptotically closed level was retained in the calculations, apart from the kinetic energies larger than 7000 cm -1 . The energy levels of the rovibrational states were taken from Komasa et al. [19].

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Line-shape parameters and collisional integrals

In this section we use generalized spectroscopic cross section to calculate the 85 spectral line-shape parameters: pressure broadening and shifting coefficients, Γ 0 and ∆ 0 , as well as the real and imaginary part of the Dicke coefficient (which is also referred to as the frequency of optical velocity-changing collisions), ν r opt and ν i opt , respectively. The first step is to calculate ab initio collision integrals ω s,s λ :

ω s,s λ (q) = v dxx (s+s +2)/2 e -x σ q λ (v i j i v f j f ; E kin = xk b T ) , (3)
where v = 8k b T /πµ is the mean relative speed of the particles. The spectroscopic line-shape parameters are related to the collision integrals in the following way [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF]:

γ 0 = Re n b ω 00 0 (q) κ, (4)
δ 0 = -Im n b ω 00 0 (q) κ, (5)
νr opt = Re n b m b 2 3 ω 11 1 (q) -ω 00 0 (q) m a + m b κ, (6)
νi opt = -Im n b m b 2 3 ω 11 1 (q) -ω 00 0 (q) m a + m b κ, (7)
where m a and m b are the masses of the active molecule and perturber atom respectively, and n b is the number density of the bath. Those parameters are rescaled with κ coefficient, which stands for the definition of units, such that these are in the units of cm -1 /atm, with

κ = 1 2πc × T 0 p 0 T , (8)
where T 0 = 273.15 K and p 0 = 1 atm= 101.325 kPa.

90

The final line-shape parameters can be obtained in the following way: Γ 0 = γ 0 p, ∆ 0 = δ 0 p, ν r opt = νr opt p, ν i opt = νi opt p, where p is the helium pressure.

Results

In this section we report the results of our calculations of the pressure broadening, shifting cross sections along with the Dicke cross sections, as well as the 95 obtained line-shape parameters. Our calculations produced a large quantity of data, and we only present some representative examples here. The full set of results can be found in the supplementary materials [START_REF] Jóźwiak | Supplementary material 445 for PB-, PS-and Dicke coefficients[END_REF].

Generalized spectroscopic cross-sections

The goal of the following discussion is to give a brief look into the rotational on the rotational number j (Figs. 3,4). The rest of the data can be found in the supplementary materials for this paper [START_REF] Jóźwiak | Supplementary material 445 for PB-, PS-and Dicke coefficients[END_REF]. ) consider the purely rotational S 0 (1) and O 0 (3) lines. As expected, the pressure broadening cross sections are identical, while the pressure shifting cross sections have opposite sign. This phenomenon was observed for all investigated purely 115 rotational lines during this investigation, and was previously noticed in earlier studies [START_REF] Martinez | Testing the ab initio quantum-scattering calculations for the D 2 -He benchmark system with stimulated Raman spectroscopy[END_REF][START_REF] Ben-Reuven | Symmetry considerations in pressure-broadening theory[END_REF]. The values of the PBXS and PSXS are small, comparing to the other vibrational bands, due to the fact, that there is no vibrational phase shift. Moreover, the dephasing between j=1 and j=3 may be small. For kinetic energies in range from 0.1 cm -1 to 10 -20 cm -1 , values of the PBXS decrease, ) sample the long range attractive part of the potential, and as the kinetic energy increases, the repulsive part of the potential plays a more important role (recall that the overall PES well depth is about 10 cm -1 ). Thus, it is possible that the 125 shift, which result from the difference of scattering phase shifts (see Eq. (A.4) in

S1(0) S1(1) S1(2) S1(3) S1(4) S1 (5)
S1(0) S1(1) S1(2) S1(3) S1(4) S1 (5)
Ref. [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF]), may change its sign while passing from the very low kinetic energies to higher ones. Note also, that analogously to vibrational transitions (see below)

the difference V 0,0,ji,0,ji (R) -V 0,j f ,0,j f (R) changes its sign while passing from the long to the short range part of the potential. As the kinetic energy increases,

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the role of the inelastic collisions increases and the values of the PBXS increase (see the inelastic cross sections in the Fig. 1). Ultimately, the purely rotational phase shift should tend to 0.

A similar discussion has already been reported around Figs. 13 and 14 of

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Ref. [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF] for the pressure broadening and shifting cross sections of the S 0 (1) line.

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In addition, the role of the isotropic components of the PES was discussed for both cross sections.

Let us now consider rovibrational transitions. In these cases, at least at low kinetic energies, a large contribution to the pressure broadening and shifting cross sections comes from the vibrational dephasing, which arises from the dif-140 ference in the isotropic components of the PES in v = 0 and in v (see Fig. 4 of Ref. [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF])

: ∆V 0 (R) = V 0,0,0 (R) -V 0,v,v (R)
. This difference increases for 0-v transitions with v. Since this difference varies with R, its contribution to the pressure broadening and sifting cross sections varies with the kinetic energy.

Moreover, as the kinetic energy increases, the contribution from the inelastic 145 cross section increases. The inelastic cross sections increase with v because the spacing between the rotational energy levels in a given vibrational state decreases with v. One should also realize, that the inelastic contribution to the PBXS, presented in Fig. 1 is the same for S 0 (j) and O 0 (j + 2) lines, while for rovibrational S v (j) and O v (j + 2) lines this contribution is different. This is due 150 to the slight differences between the radial terms of the potential in v = 0 and in v (as v increases, so does the interaction energy) and a strong variation of the energy spacing between the rotational levels with v. For example the difference between the rotational levels E v=0 (j = 3) -E v=0 (j = 1) is about 587 cm -1 , while the difference between E v=5 (j = 3) -E v=5 (j = 1) is about 444 cm -1 .

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These considerations explain why the absolute values of the pressure broadening and shifting cross sections increase with v. As the energy increases (above about 600 cm -1 ) the inelastic cross sections increase abruptly, and the magnitude of the PSXS decreases. Finally, a sign change of the PSXS is observed around 20 cm -1 , that may come from a sign change of the difference between 160 the isotropic parts of the PES in v = 0 and in v, ∆V 0 (R). Such analysis was performed for isotropic and anisotropic Q v (2) lines in Figs. 5 and7 of Ref. [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF],

in terms of the vibrational dephasing, full dephasing and inelastic contributions. Re(DXS) are equal for the S 0 (1) and O 0 (3) lines, while their imaginary parts have opposite sign [START_REF] Martinez | Testing the ab initio quantum-scattering calculations for the D 2 -He benchmark system with stimulated Raman spectroscopy[END_REF]. Similar to the Q v (2) lines, discussed in Ref. [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF] (see Fig. 1 See Table V of Ref. [START_REF] Forsman | H 2 -He vibrational line-shape parameters: Measurement and semiclassical calculation[END_REF], which provides thermally averaged values of the cross sections at various temperatures for Q lines of the fundamental band and Fig. 13 of Ref. [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF], which provides values for isotropic and anisotropic Raman Q lines at room temperature.

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A N U S C R I P T considered. The dephasing contribution to the PBXS (Eq. (A4) of [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF]) must be considered too. If we assume that it does not depend on j (which is the case for a purely vibrational dephasing, coming from the differences of the isotropic parts 195 of the PES in v=0 and in v=1), then its contribution would be the same for S 1 (j) lines (or O 1 (j) lines). This is roughly observed for kinetic energies larger than about 1000 cm -1 . At the lowest kinetic energies (except below 1 cm -1 )

the main contribution to the PBXS comes from the dephasing contribution.

Since it slightly depends on j [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF][START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF], this is mainly a vibrational contribution.

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Concerning the PSXS, the values for the O and S lines, all together, do not differ very much for a very wide range of kinetic energies 2 (between about 1 and 1000 cm -1 ) because they mainly originate from the vibrational phase shift. In the highest energy domain, above 1000 cm -1 , we observe a splitting of the S 1 (j)

and O 1 (j) shifts, due to the competition between the inelastic transitions and The behavior of the cross sections for kinetic energies below 1 cm -1 is difficult to interpret. At first sight, these calculations are expected to be more simple. In fact, this is not the case because the wavelength of the scattered 215 wavefunction strongly varies from infinity to the small R distance. Moreover, the results are very sensitive to the input parameters of the propagator. However, it can be seen from Figs. 1 and2, where the probability density function of the Maxwell-Boltzmann distribution is presented, that the results below 1 cm -1

do not influence the final line-shape parameters for temperatures larger than 220 2 See Table II of Ref. [START_REF] Forsman | H 2 -He vibrational line-shape parameters: Measurement and semiclassical calculation[END_REF] for Q 1 (j) lines and Fig. 14 of Ref. [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF], which provide thermally averaged values of the cross sections at room temperature.

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A N U S C R I P T 10 K. )

• • • • • • ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ○ ○ ○ ○ ○ ○ • S v (0) ■ S v (1) ◆ S v (2) ▲ S v (3) ▼ S v (4) ○ S v (5)
• • • • • • ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ • O v (2) ■ O v (3) ◆ O v (4) ▲ O v (5)

Line-shape parameters

In this subsection we present the dependence of the line-shape parameters on the final vibrational number v of the transition. We chose to present in graphical form the results at the temperature of 296 K. The coefficients at 225 different temperatures can be found in the supplementary materials [START_REF] Jóźwiak | Supplementary material 445 for PB-, PS-and Dicke coefficients[END_REF]. The pressure broadening coefficients, shown in the Fig. 5, increase with v for both S ) and O lines. However, in case of O lines, the difference between the rotational numbers of the initial states, j, is more significant. We have no clear explanation for that. However, we note here, that the inelastic contribution to the pressure 230 broadening cross sections for S(j) and O(j+2) lines are quite different. This could cause the difference in the dependence of γ 0 on j for the S and O lines.

A C C E P T E D M A N U S C R I P T • • • • • • ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ○ ○ ○ ○ ○ ○ • S v (0) ■ S v (1) ◆ S v (2) ▲ S v (3) ▼ S v (4) ○ S v (5)
• • • • • • ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ • O v (2) ■ O v (3) ◆ O v (4) ▲ O v (5)
The pressure shift coefficients follow similar trend, increasing with v in case of both lines, as predicted. lines, as the vibrational quantum number v increases, the dependence on the rotational quantum number is quite significant. For the imaginary part, the dependence on the vibrational quantum number is very similar for both S and O lines, with a quite small j dependence for a given vibrational band.

The line-shape parameters were calculated at 18 temperatures between 5 and 240 2000 K. Figure 7 presents the temperature dependence of all the four reported coefficients of the S v (1) line in various vibrational bands. The behavior of the pressure broadening and shift coefficients is similar to the one observed for Q lines [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF]. The strong dependence on T and v can also be observed for the real and imaginary parts of the Dicke parameter, with a slight decrease of the absolute 245 values of νi opt for higher vibrational bands in temperatures above 500 K.

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Table 1: Half width at half maximum (HWHM) for the S 0 (j) lines in 10 -3 cm -1 amg -1 .

Besides the typical Lorentzian contribution, γ 0 , we present the effective half width at half maximum, γ † , taking into account all the line-shape effects (both the velocity-changing collisions and speed-dependent effects). The inhomogeneous component of the broadening due to the speed-dependence of the line shift (see Eq. ( 9)) is denoted as γ δ . The values in parenthesis indicate the 1σ standard uncertainty. b Experimental values from Ref. [START_REF] Michaut | Investigations of pure rotational transitions of H 2 self-perturbed and perturbed by He. I. Measurement, modeling, and quantum calculations[END_REF].

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Table 2: Pressure shift coefficients of the S 0 (j) lines in 10 -3 cm -1 amg -1 . Besides the typical Lorentzian contribution, δ 0 , we present the effective shift of the spectral line, δ † , taking into account all the line-shape effects (both the velocity-changing collisions and speed-dependent effects). The values in parenthesis indicate the 1σ standard uncertainty. b Experimental values from Ref. [START_REF] Michaut | Investigations of pure rotational transitions of H 2 self-perturbed and perturbed by He. I. Measurement, modeling, and quantum calculations[END_REF].

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Comparison with the experimental data

We compared the values of the pressure broadening and shift coefficients presented here with the available experimental data for S 0 (j = 0 -4) lines, reported by Hermans et al. [START_REF] Hermans | Measurements on the influence of binary collisions on the depolarized Rayleigh and rotational Raman lines for H 2 -noble gas systems at low temperatures[END_REF] and Michaut et al. [START_REF] Michaut | Investigations of pure rotational transitions of H 2 self-perturbed and perturbed by He. I. Measurement, modeling, and quantum calculations[END_REF]. The data are gathered 250 in the Tables 1 and2. Due to the fact that some of the experimental data were reported for temperatures outside of our grid, we performed additional calculations for them.

Overall, the agreement between our calculations and the experimental results is quite well. For the pressure broadening coefficient, a slight disagreement be-

255

tween theory and experiment can be observed in two cases: for temperatures below 50 K and for temperatures above 600 K. The disagreement in the lowtemperature range was already reported in Ref. [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF], for the S 0 (1) line and we still have no explanation for that. In the high-temperature case, the discrepancy is reduced if the contributions from the inhomogeneous broadening due to 260 the speed dependence of the collisional shift and narrowing due to the speed dependence of the collisional broadening [START_REF] Wcis Lo | Strong competition between velocity-changing and phase-or state-changing collisions in H 2 spec-460 tra perturbed by Ar[END_REF] are taken into account (see Tables 1, 3 and 5 of Ref. [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF]). The values of the line broadening and shift corrected for these line-shape effects, γ † and δ † , are shown in Tables 1 and2. The detailed discussion on how to calculate the speed dependence of collisional broadening 265 and shift from the generalized spectroscopic cross section is given in Ref. [START_REF] Wcis Lo | Accurate deuterium spectroscopy for fundamen-465 tal studies[END_REF],

see Eq. ( 23) therein. Apart from the temperatures below 50 K, the largest disagreement between γ † and the experimental results is 1.8σ combined standard uncertainty.

In case of the pressure shift coefficient, for low temperatures the uncertainty 270 of the experimental data is quite large, and our results lie within the error bars.

For larger temperatures we observe a slight disagreement with the experimental data, that was also noticed for the S 0 (1) line in Ref. [START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF]. The shift calculated after taking into account the speed-dependent effects has reduced this disagreement, and the largest discrepancy is 2.3σ.

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In Ref. [START_REF] Michaut | Investigations of pure rotational transitions of H 2 self-perturbed and perturbed by He. I. Measurement, modeling, and quantum calculations[END_REF] the authors state that the inhomogeneous broadening could be neglected for the S 0 lines in the He-perturbed H 2 . The results, presented

A C C E P T E D M A N U S C R I P T
in Table 1 contradict this statement. In order to more explicitly present the influence of the inhomogeneous broadening and the speed dependence of the lineshape parameters, let us recall that if the speed dependence is approximated by the simple quadratic function and the velocity-changing collisions by the hardcollisions model, then the above mentioned inhomogeneous broadening is [START_REF] Wcis Lo | The implementation of non-voigt line profiles in the hitran database: H2 case study[END_REF]:

γ δ = 3 2 δ 2 2 νr opt , (9)
where δ 2 is a parameter quantifying the magnitude of the speed dependence of the collisional shift within the quadratic approximation. Equation ( 9) is true for νr opt δ 2 , which is well obeyed for molecular hydrogen. Additionally, if this condition is satisfied, the shape of the inhomogeneous component is Lorentzian, and can be simply added to γ 0 . We present the values of γ δ in Table 1. If there

280
would not be any narrowing due to the speed dependence of the pressure broadening coefficient then the difference between γ † and γ 0 would exactly match γ δ .

For some lines this difference is almost as large as γ δ (for instance for the S 0 (4) line at 995 K), which means that the speed dependence of the collisional shift is considerably larger than the speed dependence of the collisional broadening 285 (which is not explicitly presented here). For other cases, the speed dependence of the broadening dominates, effectively resulting in narrowing of the spectral line, and thus γ † < γ 0 .

Accuracy of the calculations

The close-coupling scheme is the most accurate method of obtaining the 290 line-shape parameters. However, there are several factors that could influence the accuracy of the results presented in this paper. The most important one is the accuracy of the potential energy surface used in the calculations. We made use of the most accurate PES available for the H 2 -He system [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF], which is an extension of the previous one [START_REF] Bakr | Highly accurate potential energy surface for the He-H 2 dimer[END_REF]. Very conservative estimations of the

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for the S 0 (1) and S 1 (1) lines to check how such slight change of the isotropic radial terms V λ=0,v,j,v ,j (R) (multiplication by 1.01) could influence the final line-shape parameters. For the purely rotational line, for temperatures above 300 50 K, the relative differences between the parameters obtained with modified and correct radial terms were below 0.4%. For the remaining temperatures (5, 10, 25 and 50 K), the differences were 2%. For the S 1 (1) line, the relative differences were below 1% for all temperatures. However, as stated previously, the estimates of the uncertainty of the PES were quite conservative.

305

Other factors that could influence the accuracy of the calculations come from the numerical methods of solving the close-coupling equations. Let us remind that we used MOLSCAT code for that purpose [START_REF] Hutson | Molscat version[END_REF]. Firstly, there are several parameters of the used propagator that could slightly modify the output of the program. We made use of the hybrid modified log-derivative Airy propagator or Appendix C of Ref. [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF]). However, increasing this parameter to 200 a 0 has 320 rather small influence on the output cross sections in that case, and the lineshape parameters for the S 0 (1) and S 1 (1) lines obtained this way differ from the ones reported in this work only by less than 0.02%. Thus, we can safely state that the error coming from this parameter is negligible.

The start point of the propagation, controlled by RMIN, should be placed of the highest kinetic energy in the asymptotic region [START_REF] Hutson | Molscat version[END_REF]. Through the DR variable the user can declare the initial step size explicitly. A wrong step could strongly modify the results, especially for kinetic energies below 1 cm -1 . We note here that both pressure shifting cross section and imaginary part of the 340 Dicke cross section are very sensitive to the changes of these parameters. We performed tests using different step sizes for several lines from 0-5 band and we could determine the error associated with this factor: 0.5% for the pressure broadening coefficient and the real part of the Dicke parameter and 1% for the pressure shift coefficient and the imaginary part of the Dicke parameter.

345

The number of closed levels used in the calculations could also contribute to the uncertainty of the results. For all investigated lines we kept one closed level, but we made additional calculations with two closed levels to check how this could influence the final line-shape parameters. The relative differences between these results where about 0.5%. Moreover, we checked how truncating 350 the basis in the calculations for highest kinetic energies (above 8000 cm -1 ) could modify the results. We found that such truncation could influence the line-shape parameters only for T = 2000 K. This error is 0.5% large.

Taking into consideration the most significant factors: the accuracy of the PES, the error coming from the step size of the propagator and from the number distribution at this temperature is still significant for kinetic energies about 360 9000 cm -1 , and calculations should be performed for even larger kinetic energies. Therefore, we estimate that the uncertainty for T = 2000 K is 3% large.

For temperatures below 50 K, the uncertainties of the line-shape parameters are 3.5% large. All the error bars reported in this work are 1σ standard uncertainties. 365

Conclusion

We provided a set of line-shape parameters for S and O branches of H 2 in He for 18 temperatures between 5 and 2000 K. We investigated 60 transitions in total, for vibrational bands ranging from 0-0 up to 0-5 and for rotational levels from j=0 to j=5 for S lines, and from j=2 to j=5 for O lines. We used the 370 most accurate available PES for this system [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF] and performed close-coupling calculations using the MOLSCAT code [START_REF] Hutson | Molscat version[END_REF]. The generalized spectroscopic cross sections were obtained from the S-matrix elements, for kinetic energies ranging from 0.1 up to 9000 cm -1 . The final line-shape parameters reported here are the standard pressure broadening and shifting coefficients and the real and imagi-375 nary parts of Dicke parameters. All these quantities, along with the generalized cross sections, can be found in the supplementary materials [START_REF] Jóźwiak | Supplementary material 445 for PB-, PS-and Dicke coefficients[END_REF]. The results of this investigation provide a reference data for astrophysical and planetary studies. The present work completes the previously reported calculations on the Q-branches [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF] and together with Ref. [START_REF] Thibault | Rovibrational line-shape parameters for H 2 in He and new H 2 -He potential energy surface[END_REF] provide data that will be used 380 in our forthcoming article [START_REF] Wcis Lo | First 475 comprehensive dataset of beyond-voigt line-shape parameters from ab initio quantum scattering calculations for the HITRAN database[END_REF] to generate the first comprehensive dataset of line-shape parameters for the HITRAN database [START_REF] Gordon | The HITRAN2016 molecular spectroscopic database[END_REF].

Acknowledgement

The research is supported by the National Science Centre, Poland, Project

•

  Close-coupling calculations of pressure broadening and shifting coefficients • Close-coupling calculations of Dicke parameters • Data for anisotropic Raman S (j=0-5) and O (j=2-5) lines for purely rotational transitions, fundamental band and first four overtones 1. Introduction

Figure 1 :

 1 Figure 1: Comparison of pressure broadening and shifting cross sections between Sv(1) and Ov(3) lines in various vibrational bands at different kinetic energies (solid lines). The index v denotes the final vibrational quantum number. The inelastic contributions to the PBXS are presented in the upper panels (dashed lines). The Maxwell-Boltzmann distributions at 10, 296 and 1000 K are represented as dashed gray lines in the lower panels.

100

  and vibrational dependences of the cross sections, as well as the similarities

Figure 2 :

 2 Figure 2: Comparison of real and imaginary parts of Dicke cross sections between Sv(1) and Ov(3) lines in various vibrational bands at different kinetic energies. The index v denotes the final vibrational quantum number.

Figure 1

 1 Figure1shows the pressure broadening and shifting cross sections as a function of the kinetic energy for various vibrational bands for the S v (1) and O v (3) lines, as well as the contributions to the broadening from the inelastic cross 110

Figure 3 :

 3 Figure 3: Comparison of pressure broadening and shifting cross sections between S and O lines in the fundamental band at various kinetic energies. The inelastic contributions to the PBXS for the S 1 (0-3) and O 1 (2-5) lines are presented in the upper panels (dashed lines). The Maxwell-Boltzmann distributions at 10, 296 and 1000 K are represented as dashed gray lines in the lower right panel.

120

  before growing up. At the minimum, a change of the sign of the PSXS is observed. A tentative explanation is the following: the very low kinetic energies

Figure 4 :

 4 Figure 4: Comparison of real and imaginary part of Dicke cross sections between S and O lines in the fundamental band at various kinetic energies.

Figure 2

 2 Figure 2 shows an analogous comparison but for the real and imaginary parts of Dicke cross sections (DXS). It can be seen, that the values of the 165

9

  Fig. 3 presents the comparison of various S and O lines in the fundamental band in order to show the rotational dependence of the cross sections with j.Generally speaking, the PBXS for low j are sensitive to both the long and short 180

205

  the elastic ones (see Fig.11of Ref.[START_REF] Thibault | A test of H 2 -He potential energy surfaces[END_REF]).

Figure 4

 4 Figure 4 shows the comparison between Dicke cross sections for different lines of the fundamental band. The real part of these XSs are almost identical for the whole examined kinetic energy range, and there is no clear j dependence. Similar situation occurs while discussing the imaginary part of these cross sections. The210

Figure 5 :

 5 Figure 5: Comparison of pressure broadening, γ 0 , and pressure shift, δ 0 , coefficients at 296 K as a function of the final vibrational state for the 0→ v bands.

Figure 6 :

 6 Figure 6: Comparison of real and imaginary parts of Dicke parameter, νopt, at 296 K as a function of the final vibrational state for the 0→ v bands.

Figure 6 A

 6 provides comparison for the real and imaginary parts of the Dicke parameter. The real part clearly depends on v, and for O 235

Figure 7 :

 7 Figure 7: Pressure broadening, γ 0 , and pressure shift, δ 0 , coefficients, real and imaginary parts of the Dicke parameter, νopt, for the Sv(1) lines as a function of temperature.

a

  Experimental values from Ref.[START_REF] Hermans | Measurements on the influence of binary collisions on the depolarized Rayleigh and rotational Raman lines for H 2 -noble gas systems at low temperatures[END_REF].

310of

  Manolopoulos and Alexander (setting INTFLG=8 in the MOLSCAT code), which uses the diabatic modified log-derivative method for intermolecular distances R between RMIN and RMID, and the Airy propagator between RMID and RMAX (when the propagator reaches R = RMAX, the calculations are terminated), where RMIN, RMID and RMAX are parameters that are set man-315 ually. In all calculations the values of these three parameters where set to 1 a 0 , 20 a 0 and 100 a 0 , respectively. The change of the RMAX parameter could potentially influence the values of the cross sections for small kinetic energies, as the long range interactions are mostly important in that region (see Sec. 4.1

325

  deep in the potential wall, and the value of 1 a 0 satisfies this condition. Additional tests were performed for several kinetic energies for the S 1 (1) line, to determine if increasing the range of the propagation (by setting RMIN = 0.5 a 0 )A C C E P T E D M A N U S C R I P Tcould influence the outcoming results. No significant difference was observed for any of the final line-shape parameters. Thus we believe that there is no signif-330 icant error coming from this parameter. Note that MOLSCAT allows RMIN and RMAX not to be fixed but to be determined internally for each new total angular momentum. No significant changes were obtained.Another factor influencing the results comes from the step size of the chosen propagator, which, in the MOLSCAT code, can be controlled by variables STEP 335 or DR. STEPS is the number of steps per half-wavelength for the open channel

355

  of closed levels, we can conservatively estimate the accuracy of the line-shape parameters presented here. The uncertainties for temperatures above 50 K and below 2000 K are 2.5% large. We stress here, that the accuracy of the calculations deteriorates at 2000 K, due to the fact, that the Maxwell-Boltzmann A C C E P T E D M A N U S C R I P T

No. 2015 /

 2015 19/D/ST2/02195. The research is part of the program of the National 385 Laboratory FAMO in Toruń, Poland. A C C E P T E D M A N U S C R I P T for H 2 (HD)He mixtures, Physica A 129 (1985) 469. doi:10.1016/ 0378-4371(85)90181-5.

accuracy of the used PES indicate the uncertainties of the interaction energies, for instance, near the minimum are not larger than 1%. We repeated calculations

A C C E P T E D M A N U S C R I P T


Adrien Malaisé
email: adrien.malaise@inria.fr

Pauline Maurice

Francis Colas

Serena Ivaldi

Clélie Amiot

Activity Recognition for Ergonomics Assessment of Industrial Tasks with Automatic Feature Selection

Keywords: Recognition, Human Factors, Ergonomics, Feature Selection

HAL is

I. INTRODUCTION

W ORK-related musculoskeletal disorders (WMSDs) are a major health issue, affecting almost 50 % of industrial workers in developed countries, and representing an important cost for companies and society as a whole [START_REF] Schneider | Osh in figures: occupational safety and health in the transport sector-an overview[END_REF]. In order to reduce the prevalence of WMSDs, the ergonomics of the workplace should be evaluated and improved. Additionally, workers should be made aware of which movements and postures to avoid. Vignais et al. [START_REF] Vignais | Innovative system for real-time ergonomic feedback in industrial manufacturing[END_REF] indeed showed that realtime feedback about the ergonomics of a posture could help decrease the risk of developing WMSDs. Providing real-time ergonomic feedback to workers is one of the objectives of the European project AnDy [START_REF] Ivaldi | Anticipatory models of human movements and dynamics: the roadmap of the andy project[END_REF]. Within the AnDy project, online ergonomic assessment will also be used to control a collaborative robot in a way that guides the worker towards more ergonomic movements.

Currently, ergonomics is evaluated with standard pen-andpaper worksheets filled by experts observing the workers doing their job [START_REF] Li | Current techniques for assessing physical exposure to work-related musculoskeletal risks, with emphasis on posture-based methods[END_REF]. Digital human modeling software provide automatic filling of these ergonomic worksheets [START_REF] Bossomaier | Scientific approaches for the industrial workstations ergonomic design: A review[END_REF], but only based on a digital simulation of the activity to evaluate. These software are intended for workstation design: they do not work Fig. 1. Human activity recognition module. Data from both wearable sensors form the set of all available features, among which the feature selection module enables to select the best performing ones with either a filter-based method or a wrapper-based method. Parameters of the HMM models are learned based on the selected features and using an annotated training dataset (supervised learning). The feature selection and training processes are both performed offline. Recognition can be done either offline or online. in real-time, and identification of actions performed is done manually by the user. A possible solution to automatize the filling of ergonomic worksheets is to use activity recognition algorithms to infer the actions and postures that are considered in these worksheets. In a previous paper [START_REF] Malaisé | Activity recognition with multiple wearable sensors for industrial applications[END_REF], we proposed a method based on wearable sensors and Hidden Markov Models (HMM) to automatically recognize different activities during an industrial pick-and-place task. The activities were, however, high-level descriptions of the actions specific to the task considered and were not relevant for ergonomics evaluation. Furthermore, the features (i.e., task and movement descriptors, such as joint angles, center of mass position, etc.) used as input of the recognition model were manually selected among all the available data, which requires expert knowledge of the relevant movement descriptors for the tasks. To obtain robust and generic activity recognition models, one cannot rely on a small set of hand-picked features, because there is no guarantee that these features are the optimal set that maximizes the performance of the learned recognition models. On the other hand, using all the possible features is not a viable solution, since it requires to use a large number of wearable and environmental sensors, whereas in industrial applications the number of sensors should be optimized.

Motivated by these problems, in this paper we present two contributions to address the limitations of our previous work. First, we propose a generic taxonomy (i.e., a set of activities to recognize) that is compatible with the postures and actions evaluated in standard ergonomic assessment worksheets. Two ways of using the proposed taxonomy with activity recognition models based on supervised learning of an HMM are compared in order to identify the one that achieves the best recognition performance. Second, we propose to identify the best performing subsets of features used as input for our recognition algorithm. To be used in industrial environment, the number of required sensors for our activity recognition module must be minimized. Therefore, the most informative features regarding our proposed taxonomy need to be identified to select the optimal set of sensors. We compare different feature selection and dimensionality reduction methods in order to improve the robustness of the resulting list of relevant features.

We apply our proposed method to the recognition of activities relevant for ergonomics evaluation of industry tasks, that we recorded in a lab environment with 13 participants. The outcome of our study is the list of features to be used to classify the activities according to our taxonomy. Both the dataset and our software are also publicly available.

The paper is organized as follows: Section II reviews the related work. Section III presents the proposed taxonomy and the activity recognition module depicted in Fig. 1, and describes the creation of the dataset used to train and test the recognition module. The two ways of using the proposed taxonomy and the subsets of relevant features extracted from feature selection are compared and discussed in Section IV. Conclusions are presented in Section V.

II. RELATED WORK

This paper proposes to use an activity recognition algorithm to identify postures and actions considered in standard ergonomic assessment methods, in order to provide a basis for an automatic assessment of ergonomics. This section reviews how ergonomics is currently assessed in industry, and how such assessments could be automatized. Work related to recognition of human activities and feature selection is also presented.

A. Assessment of ergonomics in industry

Standard ergonomic assessment methods used in industry rely on pen-and-paper worksheets filled by an expert [START_REF] Roman-Liu | Comparison of concepts in easy-to-use methods for msd risk assessment[END_REF], e.g., RULA (Rapid Upper Limb Assessment) [START_REF] Mcatamney | Rula: a survey method for the investigation of work-related upper limb disorders[END_REF], REBA (Rapid Entire Body Assessment) [START_REF] Hignett | Rapid entire body assessment[END_REF], OWAS (Owako Working Posture Analysis System) [START_REF] Corlett | A technique for assessing postural discomfort[END_REF], OCRA (Occupational Repetitive Actions) [START_REF] Occhipinti | Ocra: a concise index for the assessment of exposure to repetitive movements of the upper limbs[END_REF] or EAWS (Ergonomic Assessment Worksheet) [START_REF] Schaub | The european assembly worksheet[END_REF]. These ergonomic worksheets consists of grids or checklists that evaluate the main biomechanical factors of MSDs: posture, external force or manipulated load, and task repetitiveness. Posture is evaluated mainly based on the elevation of the arms and flexion of the torso.

Currently, no tools are used in industry to provide workers with real-time ergonomic information on their postures and gestures. Yet, Vignais et al. [START_REF] Vignais | Innovative system for real-time ergonomic feedback in industrial manufacturing[END_REF] showed that real-time ergonomic feedback could help reduce the factors of risk for WMSDs. Kim et al. [START_REF] Kim | Ergotac: A tactile feedback interface for improving human ergonomics in workplaces[END_REF] recently developed a wearable device with vibrotactile elements to warn workers about joint overloading. Their method focuses however only on load handling. Yet, awkward postures increase the risk of developing WMSDs even in the absence of a load. In order to assess the postural risk, motion capture systems can be used to track workers' whole-body motion in real-time and simultaneously fill in standard ergonomic assessment worksheets. Busch et al. [START_REF] Busch | Postural optimization for an ergonomic human-robot interaction[END_REF] used optical marker-based motion capture to fill in the REBA ergonomic assessment worksheet automatically and online. They however noted that marker-based motion capture systems are ill-adapted to industrial settings due to occlusion issues. Fully wearable sensors such as inertial measurement units (IMUs) can be a solution to the occlusion problem.

B. Automatic recognition of human activities

Aside from the motion tracking technique mentioned in the previous section, most methods proposed in the literature to automatically fill in ergonomic assessment worksheets rely on direct measurement of joints angles and segment positions [START_REF] Busch | Postural optimization for an ergonomic human-robot interaction[END_REF] [START_REF] Plantard | Validation of an ergonomic assessment method using kinect data in real workplace conditions[END_REF]. Such methods enable to identify postures, but they cannot identify gestures or actions (i.e., time-series of specific postures) which are present in some ergonomic assessment worksheets (e.g., walking in EAWS worksheet). Moreover, action-related information can be needed to fill in the ergonomic worksheet (e.g., is the person carrying a load, does the task involves vibrations...). Activity recognition models based on machine learning can be used to address the action recognition issue. Classification algorithms have been widely used to recognize human daily activities, such as walking, sitting or lying [START_REF] Dubois | Human activities recognition with RGB-Depth camera using HMM[END_REF]. Among all the existing techniques, Hidden Markov Models (HMM) have the advantage of efficiently modeling time-series data such as whole-body human motions [START_REF] Yamato | Recognizing human action in time-sequential images using hidden markov model[END_REF]. In a previous paper, we successfully used HMM models along with an inertial motion capture system to identify the phases of a pick-and-place activity [START_REF] Malaisé | Activity recognition with multiple wearable sensors for industrial applications[END_REF].

C. Feature selection for activity recognition

Motion capture systems provide a large set of data that can be used as input of an activity recognition algorithm, e.g., position, velocity and/or acceleration of all the joints and body segments. These features are however not all relevant: depending on the type of activities that are targeted and the classifier that is used, some features are useless or simply redundant. Relevant features can be selected without requiring any a priori knowledge by using feature selection algorithms. Feature selection algorithms allow to automatically select, from a high dimensional set of features, the subset that maximizes the recognition performance of a classifier, hence reducing the dimensionality of the input data. Two main types of methods exist for automatic feature selection: wrapperbased methods and filter-based methods [START_REF] Guyon | An introduction to variable and feature selection[END_REF].

1) Wrapper-based methods: With wrapper-based methods, the selection of relevant features is based on the recognition performance of a classifier and is thus classifier-dependent [START_REF] Kohavi | Wrappers for feature subset selection[END_REF]. The general idea is overcome the combinatorial explosion by iteratively building a few best-performing candidates of feature subsets. More precisely, let N be the total number of features. First, the classifier is independently trained with each feature (i.e., one model for each feature), and the recognition performance of each model is evaluated with standard metrics such as accuracy or F1-score. The M features that correspond to the best-performing models are retained for the next iteration, where M is a number chosen by the user. At the next iteration, a model is trained for each combination of one of the M retained features with any of the N -1 other features. The M subsets of 2 features offering the best performance are then retained for the next iteration. The process is repeated for subsets of 3 features, and so on until the recognition performance plateaus. This iterative selection process has the advantage of taking into account relations between features, without requiring to test each possible subsets within the initial set of features. In the original method M = 1, i.e., only the best subset of features is retained at each iteration. However, Mandery et al. [START_REF] Mandery | Dimensionality reduction for whole-body human motion recognition[END_REF] proposed to keep several subsets at each iteration to reduce the risk of eliminating a subset offering a high performance when later combined with one or more other features. They used the wrapper-based method with kinematic features of human motion, but each of their features was a high-dimensional vector. For instance, they considered all the joint angles of the human body as one single feature, hence not evaluating the relevance of each joint individually.

2) Filter-based methods: The selection of relevant features with filter-based methods is not classifier-dependent, therefore no time-consuming training of classifier is required. Each feature is individually assigned a score in relation with the observations. Interaction between features are not considered [START_REF] Li | Feature selection: A data perspective[END_REF]. A score commonly used to rank the features in the Fisher score [START_REF] Duda | Pattern classification[END_REF]: it depends on how this feature allows to regroup observations corresponding to the same activity (low intraclass variance), and differentiate observations that correspond to different activities (high interclass variance). Once each feature is assigned a score, the selection process consists in taking the highest-ranked features up to a given number.

III. METHOD

This section first presents the taxonomy and the recognition algorithm used to identify ergonomically critical postures and actions. The data collection campaign that was conducted to generate a dataset for training our models is then described.

A. Taxonomy

We base our work on the EAWS ergonomic assessment worksheet, as it is widely used in the industry. The taxonomy we propose, i.e., the set of activities that need to be recognized by our activity recognition algorithm, is therefore inspired by the items evaluated in EAWS. The ergonomic assessment of EAWS is largely based on the evaluation of the posture. Nevertheless, posture is not the only risk factor for WMSDs, and other factors evaluated in EAWS require contextual information about the action being performed (e.g., does the task involve vibrations, does the person carry a load). Our taxonomy is therefore composed of several levels to identify both postures and goal-oriented actions (Table I).

The postural assessment in EAWS identifies 4 main postures: standing, kneeling/crouching, sitting, lying. Each main posture is then divided into subcategories corresponding to arms and torso configuration: upright, bent, strongly bent, shoulder level work, and overhead work (see Fig. 2 for a sample of the EAWS postural grid). The labels in the postural level of our taxonomy (level DEPOS) mainly reproduce the EAWS postural classification, e.g,. standing bent forward, walking upright (Table I). EAWS does, however, not distinguish between kneeling and crouching. These two postures may generate the same overall level of WMSDs risk, but is not necessarily located in the same joints or body-parts. In order to provide workers with joint-specific ergonomic feedback, our taxonomy distinguishes between kneeling and crouching.

The postural information contained in the DEPOS level of the taxonomy can be retrieved either directly, or by combining the identification of the main posture (standing, kneeling...) with the identification of the torso and arms configuration (bent, overhead work...). Separating the identification of the main posture from the identification of the torso and arms configuration allows to use a different recognition model for each of these two categories, which might improve the recognition performance. We therefore define 2 additional levels in the taxonomy: GEPOS and DET. GEPOS contains the labels corresponding to the main posture, while DET contains the labels corresponding to the torso and arms configuration. The two ways of inferring the full postural information is compared in this work to select the best-performing method (i.e., using one single model with DEPOS labels or combining two models, one using GEPOS labels and one using DET labels).

The fourth and last level of the taxonomy (level CUACT) contains labels representing goal-oriented actions, e.g., carrying a load, screwing. Unlike the postural levels of the taxonomy, the list of labels in the CUACT level is not exhaustive. The labels described in Table I are sufficient to describe the actions performed in our experimental dataset (see Section III-C). New labels can however be added if other actions relevant for ergonomic assessment need to be identified.

B. Activity recognition with feature selection

In order to automatically infer the postures and actions proposed in our taxonomy, we use classifiers for human activity recognition which are described hereinafter. The software implementing our methods is available on Github (http://github. com/inria-larsen/activity-recognition-prediction-wearable).

1) Classifier: We build on our previous method [START_REF] Malaisé | Activity recognition with multiple wearable sensors for industrial applications[END_REF], where activity models were represented by HMM with manually selected features, trained in a supervised way using action labels. Here, the activity models are also based on HMM, but with a set of features that is optimized by the feature selection algorithm. The models are trained in a supervised way to recognize the activities from the different levels of the taxonomy I). S = {s 1 , s 2 , ..., s N } represents the set of possible states. The model is trained based on several time series sequences, each represented by a series of discrete states Q = {q 0 , q 1 , ..., q t , ...q T }, obtained by manual annotations (see Section III-C4), and a series of T observations X = {x 1 , ..., x t , ..., x T } corresponding to features data extracted from the motion capture suit or the glove. Section III-C2 presents the sequences of activity included in the dataset. For each instant t, the goal is to infer the activity, such as q t = s i .

To train and evaluate a model, the dataset is split randomly into a training set (70% of the sequences) and a testing set (30% of the sequences). Two parameters of the HMM are learned based on the data from the sequences of the training set: the modeled observations and the transition matrix. The The "norm" features (marked with a *) correspond to the norm of the vector containing the velocity or acceleration of all joints or segments.

observations are modeled based on a Gaussian probability distribution, with a full covariance matrix used. A cross validation with ten iterations is computed in order to recognition performance for each model with the mean and standard deviation of the F1-score.

2) Data processing: To reduce noise, a sliding window filter is applied to the recorded motion capture and glove data. The observation vector contains the mean of the data across each time window. Banos et al. [START_REF] Banos | Window size impact in human activity recognition[END_REF] showed that precise recognition could be obtained with short time windows (250 ms -500 ms). A 250 ms window is used, with an overlap of 50% (125 ms) between each window, as it reduces the risk of missing data and can improve the performance [START_REF] Bao | Activity recognition from userannotated acceleration data[END_REF]. Therefore, there is an observation every 125 ms.

3) Feature selection: The features represent the observation used to train the model. Table II presents the different features evaluated in our work, and Section III-C3 describes how they are obtained from our measurement devices. Both feature selection methods presented in Section II-C have advantages and drawbacks in terms of recognition performance and computation time. To optimize the choice of the relevant features to use as input of the classifier, three methods are used to extract relevant features from all possible features: a wrapper-based method, a filter-based method and a standard dimensionality reduction based on Principal Component Analysis (PCA). The resulting subsets of features are evaluated in terms of recognition performance.

As presented in Section II-C, the wrapper-based method is an iterative algorithm. The method used is inspired by the work of Mandery et al. [START_REF] Mandery | Dimensionality reduction for whole-body human motion recognition[END_REF]. The HMM models presented above are used as classifier to evaluate the subsets of features, and the metric performance used is the F1-score. At each iteration k, the 10 best subsets of features of dimension k are kept to be tested at iteration k + 1. The filter-based method used is based on the Fisher score with regard to II-C2. For each subset of features obtained with the three different methods, a crossvalidation of three iterations is performed with the dataset split randomly at each iteration with 70% of the sequences used to train the model, and 30% to test the model.

C. Experimental protocol

The activity recognition algorithm presented in section III-B is based on supervised learning. Thus, a labeled dataset must be provided for the training stage. We conducted a campaign of in-lab data collection where participants performed various manual tasks. Whole-body kinematics and hand contact forces were recorded. The data were then manually annotated to generate the training dataset. The dataset is publicly available on Zenodo (http://zenodo.org/record/1471975). The experiment was approved by Inria's ethical committee COERLE.

1) Participants: 13 healthy adults took part in the data collection (9 males and 4 females, 11 right handed and 2 left handed, age: 25.7 ± 5.0 yrs, body height: 175.4 ± 7.9 cm, body mass: 72.3 ± 14.4 kg). Participants were naive to the purpose of the experiment and had no or limited experience of industrial work. All participants gave written informed consent before starting the experiment.

2) Task description: Our activity recognition software specifically targets tasks and postures that are common in industrial environments, following the taxonomy proposed in section III-A. Therefore, participants were asked to perform a series of manual activities mimicking industrial activities and involving postures of the EAWS ergonomic worksheet. Six activities were defined and are depicted in Fig. 3: • Screw high: Take a screw and a bolt on a 75 cm-high table, walk to the shelf, screw at a height of 175 cm. • Screw middle: Take a screw and a bolt on a 75 cm-high table, walk to the shelf, screw at a height of 115 cm. • Screw low: Take a screw and a bolt on a 75 cm-high table, walk to the shelf, screw at a height of 25 cm (6 participants) or 60 cm (7 participants). • Untie knot: Untie a knot placed on a 45 cm-high table.

• Carry 10 kg: Take a 10 kg load on a 55 cm-high table, walk to the shelf, put the load on a 20 cm-high shelf. • Carry 5 kg: Take a 5 kg load on a 55 cm-high table, walk to the shelf, put the load on a 110 cm-high shelf. One trial consisted in performing all 6 activities successively in a specified order. The order in which the activities had to be performed was selected randomly among a list of 6 predefined orders. Each participant performed the activities in 3 different orders, with 5 consecutive trials for each order, resulting in a total of 15 trials. Each trial lasted approximately 90 s. A break of about 2 min was allowed between each trial to limit fatigue. Two spatial organizations of the items involved in the activities were used to avoid any direct association of one activity with one position in space (set-up A: 6 participants; set-up B: 7 participants). Participants were given no instructions regarding the way to perform each activity. The positions of the different items (holes and tables heights, position of the knot on the table) were, however, chosen to encourage participants to adopt some specific EAWS postures (e.g., crouching, working with hand above head level...).

3) Instrumentation: Two kinds of wearable sensors were used to record whole-body motion and contact forces (Fig. 1). Participants were equipped with an Xsens MVN Link inertial motion tracking suit consisting of 17 IMUs to record whole-body kinematics (Xsens Technologies, Enschede, The Netherlands). Kinematic data were recorded at 240 Hz with the Xsens MVN software. The MVN software outputs the orientation and linear acceleration of each IMU. In addition, the MVN software uses a scalable kinematic body model composed of 23 segments linked together by 22 3-DoF joints to compute the position, velocity and acceleration of the origin of each segment and of the center of mass of the model. Participants were also equipped with an instrumented e-glove on their right-hand (Emphasis Telematics, Athens, Greece). The e-glove embeds 3 flexion sensors -on the thumb, index, and middle finger -and 4 pressure sensorson the palm and on the fingertip of the thumb, index and middle finger. Data of the e-glove were recorded at 50 Hz with a proprietary software. The Xsens and e-glove data were timestamped to enable their synchronization. All Xsens and e-glove outputs were integrated in the features used in the activity recognition algorithm (Table II). In addition to these local features concerning individual joint, we used "norm" features that correspond to the norm of the vector containing the velocity or acceleration of all joints or segments. These features are inspired from Mandery et al. [START_REF] Mandery | Dimensionality reduction for whole-body human motion recognition[END_REF] who identified them as relevant for their activity recognition. Note that no calibration of the e-glove signal was performed: only the raw sensors data were used. In addition to the wearable sensors, participants were videotaped with 2 cameras. Data of the cameras were not used in the activity recognition algorithm. They only served to help annotators resolve ambiguities when labeling the training dataset.

4) Annotations: In order to generate a training dataset, the collected data must be manually labeled using the labels defined in Table I. Labeling by human annotators is intrinsically subjective, especially during transitions between two successive postures or tasks. Therefore three annotators independently labeled the data. The ground truth is extracted from these annotations. For each time window of 250 ms (see Section III-B2) there are six related frames and annotations per annotators. Thus, 18 labels are associated to a window of observations. The ground truth is defined by the label which had the majority among the 18 labels. An absence of majority can happen during transition or when an annotator adds an intermediate state (e.g., a bent before a strongly bent). To resolve this conflict, the segments without majority are cut with the first half labeled as the label of the previous state, and the second half labeled as the label of the next state. Annotations were performed with the Anvil software (http://www.anvil-software.org/) on the videos of the Xsens avatar replaying participants' motions (video framerate: 25 fps). Each frame of the video was given 4 different labels: one for each level of the taxonomy defined in Section III-A. The annotators give labels corresponding to the level: GEPOS, DEPOS and CUACT. The DET is directly extracted from the DEPOS labels, by keeping only the labels about the additional postural information (bent, overhead work...).

IV. RESULTS First, the performance of the subsets of features identified with the 3 feature selection methods are compared. Then these taxonomy-specific subsets are compared with generic sets of features. A third part compares the use of a direct model and a combined model to represent the taxonomy. The last part presents the results of generalization between participants. III then presents the dimension and recognition performance of the best subsets of features identified with each of the 3 selection methods. In both cases, we limited the size of the subsets to 15 features, since we aim to use low-dimensional sets of features to increase computational efficiency and reduce the number of sensors needed. For all 4 levels of the taxonomy, the wrapper-based method identifies better performing subsets than both the filter-based method and the PCA. The recognition scores are nevertheless close with all 3 methods, except for the CUACT level where the wrapper-based method set largely outperforms the 2 other sets of features. Importantly when aiming to reduce the number of sensors, the wrapper-based method identifies well-performing sets of features even in small dimension: the recognition performance starts to plateau around 3 features (Fig. 4). It should also be noted that though subsets of features identified with the simple PCA method give good recognition performances in low dimension, they cannot be used to reduce the number of sensors. Outputs of a PCA are linear combinations of the input data. Therefore all initial features (hence all sensors) are needed to compute the features selected with the PCA.

A. Comparison of the feature selection methods

Fig. 4 shows that adding new features to subsets selected with the filter-based method sometimes degrades the recognition performance (e.g., for DEPOS and DET). This is likely due to the underlying Gaussian assumption in the observation model of the HMM: if the distribution of the data related to one feature strongly differs from a Gaussian distribution, inclusion of this feature in the model might deteriorate the performance. The wrapper-based method is less sensitive to the violation of the Gaussian assumption in features, because features are then selected directly based on the recognition performance of the model. Hence the wrapper-based method is more robust, but its feature selection process is computationally much more expensive than the filter-based and PCA methods. This last concern nevertheless fades if optimal sets of features associated with the proposed taxonomy are identified once and for all, and the selection process does not need to be run again.

B. Comparison of subsets of features

In order to check whether identifying sets of features specific to the proposed taxonomy is needed, we compared our best-performing subsets of features (with the wrapperbased method) with subsets proposed by Mandery et al. [START_REF] Mandery | Dimensionality reduction for whole-body human motion recognition[END_REF] for human activity recognition, and with high-dimensional sets of features (all features, 3D position of each segment, and angle of each joint). Since the 2 subsets proposed in Mandery et al. [START_REF] Mandery | Dimensionality reduction for whole-body human motion recognition[END_REF] are of dimension 4 and 8, we used our best-performing subsets of similar dimension to make First, using high-dimensional sets of features does not result in better performance compared to using reduced sets of selected features, while it is likely computationally less efficient and may require more sensors. Hence smaller sets of features are advantageous. Second, among the reduced sets of features, our best-performing subsets greatly outperform the generic subsets proposed by Mandery et al. [START_REF] Mandery | Dimensionality reduction for whole-body human motion recognition[END_REF] for all 4 levels of the taxonomy. Hence, using a set of features dedicated to our taxonomy is recommended in order to optimize the performance of the recognition module. It should nevertheless be noted that the scores presented here are obtained with subsets of features optimized separately for each level of the taxonomy. If all levels are to be used simultaneously, and if a goal is to reduce the number of sensors required, a subset of features common to all 4 levels may be preferable. Table V lists the subsets of features that allows to reach 2% of the maximum recognition rate for each taxonomy level (See Table III). The GEPOS and DET subsets both contain only features that can be obtained with a reduced number of sensors. GEPOS features correspond to lower-body movements, while DET features are related to back and arm movements, in accordance with the labels of each of these 2 taxonomy levels (Table I). Conversely, several features in the DEPOS and CUACT subsets require all IMUs sensors for their computation (center of mass and "norm" features). These "aggregated" features were included in the initial set because Mandery et al. [START_REF] Mandery | Dimensionality reduction for whole-body human motion recognition[END_REF] identified them as useful for human motion recognition which is confirmed by our results. Nevertheless, when the number of sensors is a concern, a new set of features should be identified with excluding all "aggregated" features from the initial set of available features.

C. Comparison of models associated with the taxonomy

In the remaining of the paper, we only use the subsets presenting in Table V. We compare two ways of inferring the full postural information: using one model to recognize directly the full posture (DEPOS) or combining two models, each one focusing on different aspects of the posture (COPOS = GEPOS + DET). The two methods are evaluated by computing the F1score of each sequence obtained either with the direct model or with the combined model.

Using only a single model to infer directly DEPOS gives a better recognition performance: the F1-score is 92.56 with

Fig. 2 .

 2 Fig. 2. Sample of the EAWS worksheet postural grid. Postures are first classified into main categories, such as standing or kneeling. Each main posture is then divided into subcategories corresponding to arms and torso configuration (e.g., bent forward, elbow at shoulder level).

Fig. 3 .

 3 Fig. 3. The 6 tasks performed by participants, with associated videos and corresponding posture and action following the taxonomy proposed in Section III-A.

Fig. 4

 4 Fig. 4 displays the evolution of the recognition performance (F1-score) with the number of features included in the recognition model, for all 3 feature selection methods (wrapperbased, filter-based and PCA) and for each of the 4 taxonomy levels. TableIIIthen presents the dimension and recognition performance of the best subsets of features identified with each of the 3 selection methods. In both cases, we limited the size of the subsets to 15 features, since we aim to use low-dimensional sets of features to increase computational efficiency and reduce the number of sensors needed. For all 4 levels of the taxonomy, the wrapper-based method identifies better performing subsets than both the filter-based method and the PCA. The recognition scores are nevertheless close with all 3 methods, except for the CUACT level where the wrapper-based method set largely outperforms the 2 other sets of features. Importantly when aiming to reduce the number of sensors, the wrapper-based method identifies well-performing sets of features even in small dimension: the recognition performance starts to plateau around 3 features (Fig.4). It should also be noted that though subsets of features identified with the simple PCA method give good recognition performances in low dimension, they cannot be used to reduce the number of sensors. Outputs of a PCA are linear combinations of the input data. Therefore all initial features (hence all sensors) are needed to compute the features selected with the PCA. Fig.4shows that adding new features to subsets selected with the filter-based method sometimes degrades the recognition performance (e.g., for DEPOS and DET). This is likely

Fig. 4 .

 4 Fig.[START_REF] Li | Current techniques for assessing physical exposure to work-related musculoskeletal risks, with emphasis on posture-based methods[END_REF]. Evolution of F1-score with the number of features used for the recognition. Full lines and shaded areas represent respectively the mean of the F1-Score and the standard deviation. The wrapper-based method allows to reach the maximum of performance with a low number of features. With the filter-based method, addition of certain features can decrease the performance such as for the DET and DEPOS levels.

TABLE I DESCRIPTION

 I OF THE TAXONOMY.

	State		Label Description
	GEPOS			Main posture
	Standing		St	Ends when feet start moving
	Walking		Wa	Starts when one foot start moving, ends
				when both feet are still
	Kneeling		Kn	At least one knee on the floor
	Crouching		Cr	No knee on the floor
	Sitting		Si	Buttock on a chair or support
	Lying		Ly	Torso on the floor or horizontal surface
	DET			Torso and arms configuration
	Upright		U	Torso straight
	Bent forward	BF	Torso flexion angle between 20°and 60°S
	trongly bent	BS	Torso flexion angle greater than 60°S
	forward
	houlder	level	OS	Elbow(s) at or above shoulder level with
	work			hand(s) at or below head level
	Overhead work	OH	Hand(s) above head level
	DEPOS			Full postural information: combination of
				GEPOS and DET
			St U	Standing upright
			St BF Standing bent forward
			St BS Standing strongly bent forward
			St OS Standing shoulder level work
			St OH Standing overhead work
			Wa U Walking upright
			...	...
	CUACT			Goal-oriented action
	Reaching		Re	Moving an arm towards a target, no object
				in hand
	Picking		Pi	Picking-up an object, starts when touching
				the object, ends when arm stops moving
				with respect to the body
	Placing		Pl	Placing an object, similar to Re but with an
				object in hand
	Release		Rl	Bringing arm back after manipulation task
	Carrying		Ca	Carrying an object. Starts at the end of Pi,
				ends at the beginning of Pl
	Fine		Fm	Dexterous manipulation of an object
	manipulation
	Screwing		Sc	A special case of Fm: rotational screwing
				movement of the hand
	Idle		Id	Not doing anything with hands
	with the hmmlearn library (http://hmmlearn.readthedocs.io/)
	in Python. The models are defined by N states representing the
	activities, such as standing (St) or walking (Wa) for a model
	using the GEPOS level of the taxonomy (all activities are
	presented in Table

  DEPOS model, and 89.68 with the combined COPOS model. The DEPOS model relies on a lower number of features (5) than the combined model (9 features since no features are shared between the GEPOS and DET subsets), which may render the DEPOS model computationally more efficient. However, as mentioned in Section IV-B the features in the DEPOS subsets require all input data, hence all sensors. The DEPOS model is not more efficient than the COPOS regarding the number of sensors required. The combined model can also be useful to infer full postures that are not in the dataset, if the corresponding labels of the GEPOS and DET levels are included in the dataset. For example, if the real state is kneeling bent forward and does not appear in the dataset, the combined model could infer the complete posture if both kneeling and bent forward states are known individually.

				TABLE IV
	PERFORMANCE OF DIFFERENT SUBSETS OF FEATURES
				GEPOS	DET	DEPOS	CUACT
	Features		Dim. F1	F1	F1	F1
	Our subsets
	Best subset dimen-	4	95.75	92.01 91.84	75.65
	sion 4*
	Best subset dimen-	8	96.19	94.30 92.72	81.00
	sion 8*
	Mandery et al. [20]
	Center of mass veloc-	4	88.61	32.10 30.63	23.75
	ity, Segments velocity
	norm
	Center of mass veloc-	8	48.59	82.35 67.22	55.12
	ity, Segments velocity
	norm, Velocity norm
	of hands and feet
	High-dimensional
	All available features	779	93.95	93.16 89.30	81.39
	All segment positions 69	93.34	95.62 91.15	71.56
	All joint angles	66	91.37	93.74 90.12	73.51
				TABLE V
	DETAILS OF THE FINAL SELECTED SUBSETS OF FEATURES.
	Taxonomy Dim. F1-score Subsets of features
	GEPOS	3	94.53	Pelvis position (z)
				Pelvis velocity (z)
				Left lower leg acceleration (x)
	DET	6	94.00	Angular acceleration pelvis (y)
				L5 vertebra orientation (q1, q3)
				Right shoulder orientation (q2)
				Left forearm position (z)
				Right hand position (x)
	DEPOS	5	92.61	Center of mass position (z)
				Segments velocity norm
				Right shoulder orientation (q1)
				Right ankle joint angle (z)
				Right upper leg position (z)
	CUACT	11	83.36	Segments velocity norm
				IMU acceleration norm
				Angular acceleration head (z)
				Right elbow joint angle (z)
				Pelvis orientation (q3)
				Left hand position (x)
				Left hand orientation (q3)
				Head velocity (z)
				Neck velocity (z)
				Right forearm velocity (z)
				Right hand velocity (z)

the

This paper was recommended for publication by Editor Allison M. Okamura upon evaluation of the Associate Editor and Reviewers' comments. This work was supported by the European Unions Horizon 2020 Research and Innovation Programme under Grant Agreement No. 731540 (project AnDy).

D. Generalization

The recognition models should be able to recognize activities of users that are not in the training set. We therefore test the generalization capability of the recognition models with a leave-one-out cross validation. For each of the 13 participants, a model is trained on the data of the 12 other participants, and the recognition performance is tested with the data of the participant excluded from the training set (the best performing sets of features identified with the wrapper-based method are used). To be consistent with the previous section, only 70% of the sequences of the 12 participants in the training set are used. The F1-score is computed for each of the 13 validations, and the global evaluation of the generalization corresponds to the average score of the 13 F1-scores. The global F1-score is 94.00 for the GEPOS model (2.62% of max. F1-score), 91.55 for the DET model (3.88% of max. F1-score), 90.96 for the DEPOS model (3.04% of max. F1-score), and 81.91 for the CUACT model (3.12% of max. F1-score). The proposed subsets of features for each taxonomy level show overall a good generalization performance.

V. CONCLUSION

In this paper, we addressed the problem of automatic assessment of ergonomics. We proposed a taxonomy of postures and actions inspired by the EAWS ergonomics assessment worksheet used in industry. We showed that using activity recognition models based on HMM with automatic feature selection, postures can be successfully recognized with only a small number of features. In addition, recognition is better when using subsets of features identified specifically for our taxonomy than when using generic features. Therefore, we provide the list of relevant subsets of features associated with our taxonomy. Recognition of actions did not exhibit similar performance, and might require a more complex model such as one based on Neural Network. In the future, the activity recognition module will be used to provide on-line ergonomic feedback to warn workers of non-ergonomic movements.