Alberto Imparato

Pierre Jop

Artyom Petrosyan

Sergio Ciliberto

Probability density functions of work and heat near the stochastic resonance of a colloidal particle

We study experimentally and theoretically the probability density functions of the injected and dissipated energy in a system of a colloidal particle trapped in a double well potential periodically modulated by an external perturbation. The work done by the external force and the dissipated energy are measured close to the stochastic resonance where the injected power is maximum. We show a good agreement between the probability density functions exactly computed from a Langevin dynamics and the measured ones. The probability density function of the work done on the particle satisfies the fluctuation theorem.

I. INTRODUCTION

The study of fluctuations of the injected and dissipated power in a system driven out of equilibrium by an external force is nowadays a widely studied problem which is not yet completely understood. This is a very important and general issue within the context of Fluctuation Theorems (FT) which constitute extremely useful relations for characterizing the probabilities of observing entropy production or consumption in out of equilibrium systems. These relations were first observed in the simulations of a sheared fluid [7] and later proved both for chaotic dynamical systems [8] and for stochastic dynamics [9]. These works lead to different formulations which find powerful applications for measuring free-energy difference in biology (see e.g. [10] for a review). The hypothesis and the extensions of fluctuation theorems [11] have been tested in various experimental systems such as colloidal particles [12,13,27], mechanical oscillators [14], electric circuits [15] and optically driven single two-level systems [16]. The effect of anharmonic potential on the motion of a colloidal particle has been tested by Blickle et al [13,16,17]. In a recent experiment [26] it has been shown that FT holds for a colloidal particle confined in a double well potential and driven out of equilibrium near the stochastic resonance (see next section). The purpose of this article is to compare the probability density function (PDF) for work and heat measured in this experiment with those analytically computed from a non-linear Langevin dynamics. We also discuss the difference between the PDF obtained from a phase average of the driving and those obtained from a fixed phase only. Such a precise comparison between theory and experiment in a double well potential driven out of equilibrium has never been done before. There is only a numerical study, which has explored the distributions of the dissipated heat and of the work in a Langevin dynamics near the stochastic resonance [18,19]. The paper is organized as follow. The properties of the stochastic resonance are recalled in the next section. The experimental set-up is described in section 3. The analytical PDF are derived in section 4. The comparison between the theoretical predictions and the experimental measurements is done in section 5. Finally we conclude in section 6.

II. STOCHASTIC RESONANCE

A colloidal particle, confined in a double well potential, hops between the two wells at a rate r k , named the Kramers' rate, which is determined by the height δU of the energy barrier between the two wells, specifically r k = τ -1 0 exp(-δU kB T ), where τ 0 is a characteristic time, k B the Boltzmann constant and T the heat bath temperature [1]. When the double well potential U is modulated by an external periodic perturbation whose frequency is close to r k the system presents the stochastic resonance phenomenon [2], i.e. the hops of the particle between the two wells synchronize with the external forcing. The stochastic resonance has been widely studied in many different systems and it has been shown to be a bona fide resonance looking at the resident time [2,3], the Fourier transform of the signal for different noise intensity [4]. Numerically, the stochastic resonance has been characterized by computing the injected work done by the external agent as a function of noise and frequency [5,6]. In a recent experiment [26] some of us have studied experimentally the Steady State Fluctuation Theorem (SSFT) in a system composed by a Brownian particle trapped in a double well potential periodically modulated by an external driving force. We have measured the energy injected into the system by the sinusoidal perturbation and we have analyzed the distributions of work and heat fluctuations. We find that although the dynamics of the system is strongly non-linear the SSFT holds for the work integrated on time intervals which are only a few periods of the driving force. In this paper we will compare this measured PDF with those derived analytically from a Langevin dynamics in which the exact potential of the experiment has been used.

III. EXPERIMENTAL SET-UP

The experimental setup is composed by a custom built inverted optical tweezers made of an oil-immersion objective (63×, N.A.=1.3) which focuses a laser beam (wavelength λ = 1064 nm) to the diffraction limit for trapping glass beads (2 µm in diameter). The silica beads are dispersed in bidistilled water in very small concentration. The suspension is introduced in the sample chamber of dimensions 0.25 × 10 × 10 mm 3 , then a single bead is trapped and moved away from others. The position of the bead is tracked using a fast-camera with the resolution of 108 nm/pixels which gives after treatment the position of the bead with an accuracy better than 20 nm. The trajectories of the bead are sampled at 50 Hz. The position of the trap can be easily displaced on the focal plane of the objective by deflecting the laser beam using an acousto-optic deflector (AOD). To construct the double well potential the laser is focused alternatively at two different positions at a rate of 5 kHz. The residence times τ i (with i = 1, 2) of the laser in each of the two positions determine the mean trapping strength felt by the trapped particle. Indeed if τ 1 = τ 2 = 100µs the typical diffusion length of the bead during this period is only 5 nm. As a consequence the bead feels an average double-well potential:

U 0 (x) = ax 4 -bx 2 -dx , (1)
where a, b and d are determined by the laser intensity and by the distance of the two focal points. In our experiment the distance between the two spots is 1.45 µm, which produces a trap whose minima are at x min = ±0.45 µm. The total intensity of the laser is 58 mW on the focal plane which corresponds to an inter-well barrier energy δU 0 = 1.8 k B T .

Starting from the static symmetric double-trap, (τ 1 = τ 2 ) we modulate the depth of the wells at low frequency by modulating the residence times (τ i ) during which the spot remains in each position. We keep the total intensity of the laser constant in order to produce a more stable potential. The modulation of the average intensity is harmonic at frequency f and its amplitude (τ 2 -τ 1 )/(τ 2 + τ 1 ), is 0.7 % of the average intensity in the static symmetric case. Thus the potential felt by the bead has the following profile in the axial direction:

U (x, t) = U 0 (x) + U p (x, t) = U 0 + c x sin(2πf t), (2)
with

ax 4 min = 1.8 k B T , bx 2 min = 3.6 k B T , d|x min | = 0.44 k B T and c|x min | = 0.81 k B T .
The amplitude of the time dependent perturbation is synchronously acquired with the bead trajectory. The parameters given here are average parameters since the coefficients a, b and c ,obtained from fitted steady distributions at given phases, vary with the phase (δa/a ≈ 10%, δb/b ≈ δc/c ≈ 5%).

An example of the measured potential at t = 1 4f and at t = 3 4f is shown on the Fig. 1a). The figure is obtained by measuring the two steady state probability distribution function P (x) of x, (corresponding to U 0 + c and U o -c respectively) and by taking U (x) = -ln [P (x)]. A. The equation of motion.

The x position of the particle can be described by a Langevin equation:

γ ẋ = - ∂U ∂x + ξ, (3)
with γ = 1.61 10 -8 N s m -1 the friction coefficient and ξ the stochastic force. The natural Kramers' rate (c = 0) for the particle is r k = 0.3 Hz at T = 300 K. When c = 0 the particle can experience a stochastic resonance when the forcing frequency is close to the Kramers' rate. An example of the sinusoidal force with the corresponding position are shown on the figure 1b).

B. The work and the heat.

Since the synchronization is not perfect, sometimes the particle receives energy from the perturbation, sometimes the bead moves against the perturbation leading to a negative work on the system.

In the following, all energies are normalized by k B T . From the trajectories, we compute the stochastic W s and the classical W cl works done by the perturbation on the system and the heat Q exchanged with the bath. These three quantities are defined by the following equations as in ref. [20]:

W s (t 0 , t f ) = t0+t f t0 dt ∂U (x, t) ∂t W cl (t 0 , t f ) = - t0+t f t0 dt ẋ ∂U p (x, t) ∂x (4)
Q(t 0 , t f ) = - t0+t f t0 dx ∂U (x, t) ∂x
where in this case t f = n f is a multiple of the forcing period. We use both W s and W cl because they give complementary information on the fluctuations of the energy injected by the external perturbation into the system (see ref. [24] and reference therein for a discussion on this point). For example, as discussed in ref. [26] W s /T is the total entropy production rate in this specific case [22]. The heat and the work, defined in eq.4, are related through the first principle of thermodynamics: Q = -∆U + W s , where ∆U = U (x(t f + t 0 ), t 0 + t f ) -U (x(t 0 ), t 0 ), whereas the two works are related by a boundary term W cl = -∆U p + W s , where ∆U p = U p (x(t f + t 0 ), t f + t 0 ) -U p (x(t 0 ), t 0 ). Since the characteristic time evolution of the perturbation is small compared to the fluctuation of position and due to the harmonic form of the perturbation, the integrals are computed as follows:

W s (t 0 , t f ) = ω c δt t f /δt i=1 x(i) cos(ω(t 0 + t i )) W cl (t 0 , t f ) = -∆U p + W s (5) Q(t 0 , t f ) = -∆U + W s
where δt is the sampling time. We checked that the direct computing of integrals of Q and W cl gives the same results. It is important to stress that t 0 can either take any value (as it has been done in ref. [26]) or be a multiple of 1/f : the fluctuations in the two cases exhibit different PDFs, as we will see in the next sections. To compute the works and heat from experimental data for a given duration t f , we thus divide a single trajectory into different segments starting either with a fixed phase, or with different phases, before averaging the results over the whole trajectory, and then over different runs. In ref. [26] the average work received over one period has been measured for different frequencies (t f = 1 f in eq. 4). Each trajectory is here recorded during 3200 s in different consecutive runs, which corresponds to 160 up to 7500 forcing periods, for the range of frequencies explored. It has been found that the maximum injected energy is around the frequency f ≈ 0.1 Hz, which is comparable with half of the Kramers' rate of the fixed potential r K = 0.3 Hz. This maximum of transferred energy shows that the stochastic resonance for a Brownian particle is a bona fide resonance, as it was previously shown in experiments using resident time distributions [3,21] or directly in simulations [5,6]. It is worth noting that the average values of work in the case of a periodic forcing do not dependent on their definitions: only the boundary terms, which vanish in average with time, are different [26]. As the difference between W cl and W s has been discussed in ref. [26] we will focus here only on W s that, in order to simplify the notation, will be indicated by W .

IV. EQUATIONS FOR THE W AND Q PDFS

In this section we discuss the equation governing the time evolution of the work and heat PDFs. For a stochastic process described by eq. ( 3) the Fokker-Planck equation reads

∂ t p(x, t) = Γ ∂ ∂x [U ′ (x, t)p] + k B T Γ ∂ 2 p ∂x 2 , (6)
where p(x, t) is the PDF associated to the coordinate x, and Γ = 1/γ. Here and in the following, the prime denotes derivative with respect to x.

Let us consider the joint probability distribution function of the position and of the stochastic work φ(x, W, t): in ref. [29] it has been shown that the time evolution of such a function is governed by the partial differential equation

∂ t φ(x, W, t) = Γ ∂ ∂x [U ′ (x, t)φ] + k B T Γ ∂ 2 φ ∂x 2 - ∂U ∂t ∂φ ∂W , (7)
with the starting condition φ(x, W, t 0 ) = p(x, t 0 )δ(W ). The unconstrained probability distribution of the work is given by

Φ(W, t) = dxφ(x, W, t). (8)
By introducing the Fourier transform ψ(x, λ, t) = dW e -λW φ(x, W, t),

eq. ( 7) becomes

∂ t ψ(x, λ, t) = Γ ∂ ∂x [U ′ (x, t)ψ] + k B T Γ ∂ 2 ψ ∂x 2 -λ ∂U ∂t ψ, (10)
with the starting condition ψ(x, λ, t 0 ) = p(x, t 0 ). Note that, by using the Fourier transform definition eq. ( 9), the wavenumber λ associated to W is a purely imaginary number, λ = i|λ|.

Let us now consider the joint probability distribution ϕ(x, Q, t) of the position x and the heat Q exchanged by the brownian particle whose motion is described by eq. ( 3). The Fokker-Planck-like equation, governing the time evolution of such a function, reads [27,28]

∂ t ϕ(x, Q, t) = ∂ x (ΓU ′ ϕ) -∂ Q (ΓU ′2 ϕ) -∂ x [(ΓU ′ k B T )∂ Q ϕ] -∂ Q [(ΓU ′ k B T )∂ x ϕ] +Γk B T ∂ 2 ϕ ∂x 2 + Γk B T U ′2 ∂ 2 ϕ ∂Q 2 , (11)
with the starting condition ϕ(x, Q, t 0 ) = p(x, t 0 )δ(Q). Note that we use here the opposite sign convention for Q, eq. ( 5), with respect to that adopted in ref. [27].

The unconstrained probability distribution functions of the heat reads

Φ(Q, t) = dx ϕ(x, Q, t). (12)
Let χ(x, λ, t) be the Fourier transform of ϕ(x, Q, t), as given by

χ(x, λ, t) = dQ e -λQ ϕ(x, Q, t), (13)
eq. ( 11) becomes

∂ t χ(x, λ, t) = Γk B T ∂ 2 χ ∂x 2 + ∂ x (ΓU ′ χ) -λΓU ′2 χ -λ∂ x [(ΓU ′ k B T )χ] -λ(ΓU ′ k B T )∂ x χ + λ 2 Γk B T U ′2 χ, (14)
with the starting condition χ(x, λ, t 0 ) = p(x, t 0 ). In ref. [27], it has been shown that by introducing the function g(x, λ, t) defined by

χ(x, λ, t) = g(x, λ, t) exp - β -2λ 2 U (x, t) , (15)
equation ( 14) simplifies, as one gets rid of the first order derivatives with respect to x, and it becomes

∂ t g = Γk B T ∂ 2 g ∂x 2 -Γβ U ′2 4 g + Γ 2 U ′′ g + β -2λ 2 g ∂ t U, (16)
with the starting condition as given by g(x, λ, t 0 ) = p(x, t 0 ) exp[(β -2λ)U (x, t 0 )/2]. We now show that a similar simplification can be obtained for eq. ( 10): let h(x, λ, t) be defined as

ψ(x, λ, t) = h(x, λ, t) exp[-βU (x, t)/2], (17)
substituting h(x, λ, t) into eq. ( 10), we obtain the following equation for h(x, λ, t):

∂ t h = Γk B T ∂ 2 h ∂x 2 -Γβ U ′2 4 h + Γ 2 U ′′ h + β -2λ 2 h ∂ t U. ( 18
)
The last equation is identical to ( 16), however its starting condition reads h(x, λ, t 0 ) = p(x, t 0 ) exp[βU (x, t 0 )/2], which is different from the starting condition of eq. ( 16).

As λ is an imaginary number, one can split eq. ( 16) and eq. ( 18), in a set of two equations for the real and imaginary part. For example, eq. ( 18) becomes

∂ t h R = Γk B T ∂ 2 h R ∂x 2 -Γβ U ′2 4 h R + Γ 2 U ′′ h R + β 2 h R + |λ|h I ∂ t U, (19)
∂ t h I = Γk B T ∂ 2 h I ∂x 2 -Γβ U ′2 4 h I + Γ 2 U ′′ h I + β 2 h I -|λ|h R ∂ t U, (20)
where h R and h I are the real and the imaginary part of h, respectively. Equations ( 19) and ( 20), and the analogous equations for g R and g I , can be solved numerically for any value of |λ|, and for any choice of the initial condition p(x, t 0 ) using, e.g., MATHEMATICA [30]. The functions ψ(x, λ, t) and χ(x, λ, t), can be thus obtained by using equations ( 17) and (15). The target functions φ(x, W, t) and ϕ(x, Q, t) can then be obtained by taking the Fourier inverse transform, i.e. by inverting eqs. ( 9) and ( 13), respectively. Also in this case the computation can be performed numerically.

V. COMPARISON WITH EXPERIMENTS

In this section, we compare the results for the work and heat PDFs, as obtained by solving the differential equations introduced in the previous section, with the experimental outcomes. We take the external driving frequency to be equal to f = 0.25 Hz, to have a good statistic, by allowing the observation of the system over a sufficient number of periods. Such a value is close to the natural Kramers' rate, which ensures that the system is in the stochastic resonance regime.

We first consider the distribution of the work over a single period of the external potential (2). The unconstrained probability distribution of the work Φ(W, t), is obtained as follows. In order to solve eq. ( 10), we solve numerically eq. ( 6) up to the time t 0 = 50/f ≫ 1/f , so as to ensure that the solution p(x, t 0 ) of eq. ( 6) represents the steady state distribution of the position of the particle. Such a solution is used as a starting condition for eq. ( 10), which is solved numerically up to time t 1 = t 0 + 1/f , i.e. along a single period, and for different values of λ, so as to obtain ψ(x, λ, t 1 ). The resulting unconstrained Fourier transform of the work PDF, defined as Ψ(λ, t 1 ) = dx ψ(x, λ, t 1 ), is plotted in fig. 2 . As discussed in the previous section, by inverting eq. ( 9) and exploiting eq. ( 8), we finally obtain Φ(W, t 1 ). In fig. 3, the experimental histogram of the work done on the particle along a single period 1/f is plotted; in the same figure the unconstrained probability distribution of the work Φ(W, t 1 ) is also plotted. We find a good agreement between the experimental distribution of the work and the expected PDF. It is worth noting first that we start sampling W only few periods 1/f after the beginning of each experiment and second that the experimental results are averaged over different values of t 0 in order to improve the statistics (either with a fixed phase or over different phases).

We now check that the work PDF satisfies the SSFT, which we expect to be true for any time t 1 which is an integer multiple of the period 1/f as discussed in [17,23]. Given the periodicity of the external driving, the stochastic work here considered corresponds to the total entropy production, as defined by Seifert in [17,23]. In figure 4, we plot the symmetry function for the work PDF, defined as together with the corresponding work PDF, for two values of the final time t 1 . The fluctuation theorem states that the simmetry function has to satisfy the equality S(W, t 1 ) = W , for any value of W . Inspection of the figure indicates that the fluctuation theorem is satisfied even at short times t 1 , the small deviations for the largest value of W being due to numerical and experimental difficulties in the sampling of the tails of the work PDF. More details on SSFT have been already discussed in ref. [26]. We now consider the work distribution averaged over several phases. In order to obtain the probability distribution of the work, we solve eq. ( 6) up to the time t 0 (k) = (50 + k/N )/f , where N = 20 and k = 0, . . . N -1, i.e. we consider N initial conditions with a phase difference 1/(N f ). For each value of k we solve eq. ( 10), up to time t 1 (k) = t 0 (k) + 1/f , so as to obtain ψ(x, λ, t 1 (k)|p(x, t 0 (k)), i.e. the Fourier transform of the work PDF with the starting condition p(x, t 0 (k)), and by antitransforming, we obtain φ(x, W, t 1 (k)|p(x, t 0 (k)). Finally the average work PDF is obtained as Φ(W ) = 1/N k dxφ(x, λ, t 1 (k)|p(x, t 0 (k)). In fig. 5, this distribution is compared with the experimental outcomes. We find a good agreement between the expected PDF and the histogram of the work.

S(W, t 1 ) = log [Φ(W, t 1 )/Φ(-W, t 1 )] , (21)
Similarly to what has been done for the work, we now consider the PDF of the heat Φ(Q), both with a single value of the initial phase, and averaged over several initial conditions. Also in this case, we use the function p(x, t 0 ), solution of eq. ( 6), as a starting condition for the numerical integration of eq. ( 14), then by antitransforming the function χ(x, λ, t 1 ), we obtain ϕ(x, Q, t 1 ) and finally Φ(Q, t 1 ), as discussed in the previous section. The results for the case of a single value of the initial phase is plotted in fig. 6, where we show a nice agreement with the experimental data. In order to obtain the heat PDF averaged over several initial phases, we solve numerically eq. ( 14) in the time intervals [t 0 (k), t 1 (k)], where t 0 (k), and t 1 (k) have been defined above. Then by antitransforming the function χ(x, λ, t 1 (k)), we obtain ϕ(x, Q, t 1 |p(x, t 0 (k)), and finally the unconstrained PDF of Q defined as Φ(Q) = 1/N k dxϕ(x, Q, t 1 |p(x, t 0 (k)). In fig. 7 we compare this expected PDF with the experimental outcomes, the agreement between the numerical PDF and the experimental data is good also in this case.

VI. CONCLUSIONS

In conclusion, we have experimentally and theoretically investigated the power injected in a bistable colloidal system by an external oscillating force.

We have compared the PDFs of the stochastic work and of the heat measured in the experiment with those obtained theoretically. They exhibit large tails toward negative values and their shape are non-gaussian close to the resonance when averaging over a single period.

In refs. [27], it has been shown that if the external potential is quadratic, an analytical gaussian solution can be obtained for equation (7) at any value of t and for eq. (11) in the long time limit. Here we have shown that the same equations can be solved numerically, also at finite time and for a more complex potentials, and they lead predictions on the system energy exchange that can be verified experimentally. We have shown that the Fourier transform of the heat and work PDFs obey the same differential equations ( 16) and (18), and the only difference lies in the initial conditions. This ensures Φ(W ) and Φ(Q) to be different at short time, while one has Φ(W ) = Φ(Q) in the long time regime, as one would expect, since U = 0 for long times.

Thus, our results suggest that the set of equations ( 7) and ( 11) represents an useful and general tool to investigate the energy balance of microscopic systems in arbitrary potentials. Finally, we find that the work PDF satisfy the fluctuation theorem, even at short times.

FIG. 1 :

 1 FIG.1: a) The perturbed potential at t = 1 4f and half a forcing period later. b) Example of trajectory of the glass bead and the corresponding perturbation at f = 0.1 Hz.

FIG. 2 :FIG. 3 :

 23 FIG.2: Unconstrained Fourier transform of the work PDF as a function of λ, as obtained by numerical solution of eq. (10), with t0 = 50/f and t1 = t0 + 1/f . Full line ΨR(λ, t1), dashed line ΨI(λ, t1).

FIG. 4 :FIG. 5 :

 45 FIG.4: Right panel: Symmetry function for the work PDF S(W, t1), eq. (21), for two different values of t1. The symbols corresponds to the experimental data, the lines to the numerical solution of eqs. (7)-(10) with t0 = 50/f . Red boxes and red dashed-dotted line: symmetry function obtained with t1 = t0 + 2/f . Blue diamonds and blue dashed line: symmetry function obtained with t1 = t0 + 4/f . The full line corresponds to the expected behaviour S(W, t1) = W . Left panel: PDF of the work for t1 = t0 + 2/f (red dashed-dotted line), and t1 = t0 + 4/f (blue dashed line).

FIG. 6 :FIG. 7 :

 67 FIG.6: Diamonds: experimental probability distribution of the heat exchanged by the particle, along a single period, and with a single value of the initial phase. Full line, Φ(Q, t1) as obtained by numerical solution of eqs. (11)-(14), with t0 = 50/f and t1 = t0 + 1/f .

Acknowledgments

AI is grateful to L. Peliti for long and interesting discussions.


Nicolas Plihon
email: nicolas.plihon@ens-lyon.fr

Gauthier Legrand

Francis Pagaud

Arsène Chemin

Jérémy Ferrand

Nicolas Taberlet

The physics of Magnus gliders

Magnus gliders are spinning toys displaying spectacular looped trajectories when launched at large velocity. These trajectories originate from the large amplitude of the Magnus force due to translational velocities of a few meters per second combined with backspin of a few hundred radians per seconds. In this article we analyse the trajectories of Magnus gliders built from paper cups, easily reproducible in the laboratory. We highlight an analogy between the trajectory of the glider and the trajectory of charged particles in crossed electric and magnetic fields.The influence of the initial velocity and the initial backspin on the trajectories is analyzed using high speed imaging. The features of these trajectories are captured by a simple model of the evolution of the Magnus and drag forces as a function of the spin of the gliders. The experimental data and the modeling show that the type of trajectories -for instance the occurence of loops -depends mostly on the value and orientation of the initial translational velocity, regardless of the value of the backspin, while the maximum height of the apex depends on both the initial translational velocity and initial backspin.

I. INTRODUCTION

Spinning objects moving rapidly in air often display complex trajectories due to the non-linear evolution of the aerodynamic forces they experience. In addition to the drag (present regardless of the spin rate), the rotation induces a force that is perpendicular to the velocity, known as the Magnus effect. The first explanation of the Magnus effect is often attributed to German physicist Heinrich Gustav Magnus [START_REF] Magnus | On the Deflection of a Projectile, Poggendorf's[END_REF], although a number of scientists (including Sir Isaac Newton) described the phenomenon far earlier [START_REF] Johnson | The magnus effect -early investigations and a question of priority[END_REF].

The combined effect of spin and drag on translating objects is extraordinarily important in sports [START_REF] Clanet | Sports ballistics[END_REF][START_REF] Frohlich | Resource letter ps-2: Physics of sports[END_REF], and has attracted much attention in the context of soccer [START_REF] Goff | Creating drag and lift curves from soccer trajectories[END_REF][START_REF] Goff | Trajectory analysis of a soccer ball[END_REF], baseball [START_REF] Nathan | The effect of spin on the flight of a baseball[END_REF], golf [START_REF] Thomson | The dynamics of a golf ball[END_REF][START_REF] Penner | The physics of golf[END_REF], tennis [START_REF] Štěpánek | The aerodynamics of tennis balls-the topspin lob[END_REF], but also badminton and the dynamics of shuttlecocks [START_REF] Peastrel | Terminal velocity of a shuttlecock in vertical fall[END_REF][START_REF] Cohen | The physics of badminton[END_REF] or other games involving spinning objects without spherical symmetry, exemplified by frisbees or boomerangs [START_REF] Lorenz | Spinning Flight[END_REF]. Indeed, the trajectory of lifted balls are sometimes puzzling for sport players, such as free kicks in soccer [START_REF] Dupeux | Football curves[END_REF], paradoxical pop-ups in baseball [START_REF] Mcbeath | Paradoxical pop-ups: Why are they difficult to catch?[END_REF] or zigzagging paths at very low spin [START_REF] Nathan | Analysis of knuckleball trajectories[END_REF]. In this context, the dimensionless number controlling the amplitude of the Magnus force is referred to as the spin number S p [START_REF] Bush | The aerodynamics of the beautiful game[END_REF].

In this article, we analyze and model the trajectories of a spinning toy, known as the Magnus glider (or Magnus flyer), widely disseminated thanks to a popular video by Bruce Yeany [START_REF] Yeany | Physics of toys-cup flyers[END_REF] (see also supplementary video). A Magnus glider typically consists of two light cups with their bottoms glued together, launched (using a rubber band) at a high velocity (typically 10 m/s) with high backspin (typically 100 rev/s). Due to its initial backspin, the flyer experiences a Magnus force that remains perpendicular to the velocity, sometimes resulting in looped trajecto- ries. It should be emphasized that, contrary to conventional gliders, the Magnus flyer only experiences lift due to its initial rotational kinetic energy which eventually dissipates -Magnus glider is therefore a misnomer, but the term is retained here due to its popular use in spite of its inadequacy. Further technical details of the Magnus glider itself and launching procedure are given in section II and in the supplementary video. Fig. 1 shows chronophotography of the successive positions of the Magnus glider (every 75 ms), initially launched at a velocity of 15 m/s and a spin rate of 660 rad/s: the backspin generates a Magnus lift force strong enough to lead to this spectacular looped trajectory. Magnus gliders are thus a unique tool to investigate the various complex trajec-tories observed for sports-balls at the laboratory scale, with two major advantages: the first one being that trajectories remain two dimensional, since a Magnus glider is much closer to a cylinder than to a ball; this is a strong asset for imaging trajectories. In this paper, we will only work in the vertical 2D-plane and any lateral motion is ignored. This is in general a very good assumption since experimentally all flights remains planar, but any asymmetry in the flyer itself may lead to lateral instability. The second advantage is that complex trajectories may be easily imaged since the typical spatial scale is of the order of a few meters (as compared to tens of meters or more for usual sports-balls trajectories) and launching Magnus gliders does not require advanced technical skills (as compared, for instance, to the long practice required to pitch baseballs).

The importance of the Magnus effect is not restricted to sports-balls trajectories, and several important studies were led in the context of aeronautical or marine engineering applications such as Flettner boats [START_REF] Flettner | The Flettner Rotorship[END_REF] or airplanes [START_REF] Seifert | A review of the magnus effect in aeronautics[END_REF]. Here, we introduce a simple educational tool allowing for the investigation of the Magnus force and its features, accessible at the laboratory scale.

The article is organized as follows. The details of the experimental protocol and image analysis are provided in section II. In section III, the trajectories of several experimental launches are computed using a model of the aerodynamic forces using data compiled from previous aerodynamical studies. Other features of the trajectories, such as oscillations and the spin-down of the glider due to drag, are then analyzed. Finally, the conditions for obtaining looped trajectories similar to the one shown in Fig. 1 as a function of the various control parameters are discussed in section IV. Section V concludes the article.

II. EXPERIMENTAL TRAJECTORIES OF MAGNUS GLIDERS

A sketch of the Magnus glider used in our study is provided in Fig. 2(a). It is made from two paper cups of height L/2 = 9 cm, bottom radius 2.75 cm, and top radius 4 cm, glued together at their bottoms. The glider has an average radius R = 3.37 cm and a total length L = 18.8 cm. In order to ease the detection of trajectories and angular velocities from high speed video-imaging, the Magnus glider is painted black and paper end-caps with two large white disks are glued to its sides (see Fig. 1 and the supplementary video for further details, as well as the appendix A). The total mass of the Magnus glider is m = 36.8 g. The initial spin and propulsion are provided using a 1-m-long rubber band wrapped around the center of the Magnus glider (the bottom of each cup), attached to a fixed point and stretched to launch the assembly, as sketched in Fig. 2(b) (see supplementary video). The rubber band is wrapped around the glider with constant tension provided by a weight attached at the bottom the band (see the supplementary video and appendix A for details). The Magnus glider is then held at a distance l from the fixed point, providing an initial elastic tension to the remaining part of the rubber band which will create forward thrust. As the rubber band contracts and unwinds from the glider, the glider gradually acquires back-spin and linear velocity, as sketched in Fig. 2(c)-(f).

For each run, the reference time t = 0 is defined as the time at which the rubber band is completely unwound and has no contact with the glider. At time t = 0, the glider has an initial velocity V 0 and initial angular velocity ω 0 . The initial velocity V 0 has an amplitude V 0 , and makes an angle α 0 with the horizontal (Fig. 2(e)). The values of V 0 and ω 0 depend on three parameters: (1) the number of turns of the rubber band around the Magnus glider, (2) the mass of the weight attached to the bottom of the band, (3) the tension of the remaining part of the rubber band. Note that during the launch phase (steps (c)-(f) in Fig. 2), a (Magnus) lift force is exerted on the Magnus glider. Its origin is that a flow is driven by the rotation of the glider: outside of the boundary layer, and in the center of mass reference frame, the flow velocity at the top of the glider is larger than the flow velocity at the bottom of the glider due to the addition of the translational speed to the flow driven by rotation. Due to the difference in flow velocity, a pressure gradient exerts a vertical force directed upwards. As a consequence, the initial launch angle is not directly set by the angle at which the rubber band is stretched since the glider experiences a lift force during the launch procedure (t < 0). The values of V 0 , α 0 and ω 0 are therefore computed a posteriori from image processing of the videos.

The trajectories are recorded using a Phantom v2511 high speed camera, at a rate of 4000 fps, with a resolution of 1280 by 800 pixels. We stress here that the trajectories of the Magnus gliders could be imaged with cameras available on mid-range smartphones as discussed in appendix B. Each image is processed to extract the locations of the centers r 1 (t) and r 2 (t) of the two white disks on the end-plate [21]. For each image, the location of the center r(t) of the Magnus glider is determined as the average (r 1 (t) + r 2 (t))/2, while the angular position θ(t) of the Magnus glider is determined as the angle between r 2 (t)-r 1 (t) and the horizontal (see supplementary videos for details). The location of the Magnus glider at time t is noted r(t) = x(t)x + z(t)ẑ, where x is the horizontal direction and ẑ is the upward vertical direction. The instantaneous velocity V(t) and the instantaneous angular velocity ω(t) are computed from r(t) and θ(t).

For the regimes reported in this article V 0 is of the order of 10 m/s, while the typical angular velocity ω 0 is of the order of 500 rad/s. Introducing ν as the kinematic viscosity of air, the features of the flow created by the Magnus glider, and its trajectory, depend upon two dimensionless numbers, namely the Reynolds numbers Re defined as

Re = 2RV ν
and the Spin number S p defined as

S p = ωR V .
The Reynolds number at time t 0 is in excess of 5 × 10 4 , which means that the flow is fully turbulent. For the experiments reported in this study, the Spin number is always larger than 2. From previous studies in wind tunnels [START_REF] Swanson | The Magnus Effect: A Summary of Investigations to Date[END_REF][START_REF] Badalamenti | On the application of rotating cylinders to micro air vehicles[END_REF], we know that the values of the drag and lift forces depend on the geometrical parameters and on the values of S p and Re.

Figure 3 shows the three types of trajectories of the Magnus glider obtained for increasing values of the initial velocity V 0 and angular velocity ω 0 from top to bottom (see caption). The trajectory in the top panel (lowest speed and spin rate) has the shape of an arch and will be referred to as an arch, that in the central panel as a cusp and that in lower panel as a loop.

The time evolution of the angular velocity ω of the Magnus glider is reported in the second column of Fig. 3. The red dots indicate ω = dθ(t)/dt, computed using a Savitsky-Golay of first order on 81 points (which corresponds to a low-pass filter with 50 Hz cut-off frequency) and the black dashed line is an empirical fit of the form ω 0 exp (-t/τ 0 ). Although the decrease in rotation speed is very clear, the fractional decrease is small over the duration of a typical experiment as τ 0 is of the order of a few seconds. In the following, the change in angular velocity is taken into account but a simpler assumption of constant rotation speed yields very satifactory results as well (see next section for a thorough discussion). The rightmost column displays the velocity V (t) (solid red lines), computed using a Savitsky-Golay of first order on 401 points (which corresponds to a low-pass filter with a 10 Hz cut-off frequency) as well as the time evolution of the Spin number S p (dotted black line), using the functional fit ω 0 exp(-t/τ 0 ) for the angular velocity. The velocity initially decreases due to drag and since the Magnus glider flies upward, part of its initial kinetic energy is converted into potential energy. Clearly, a strong energy dissipation is caused by the drag forces: the velocity at the end of the trajectories, at z = 0 (i.e. at time t 1.3 s) typically only amounts to one third of the the initial velocity, also at z = 0. We note that the Spin number is always greater than 2, and its initial value quickly increases since the velocity decreases. We also note that, for the cusp regime, the velocity vanishes at the apex, and as a consequence the Spin number diverges. A first observation is that regimes displaying a cusp or a loop require high initial linear and angular velocities; the exact conditions will be elucidated in section IV.

III. MODELING THE AERODYNAMIC FORCES ACTING ON MAGNUS GLIDERS A. Predicting trajectories from drag and lift forces

In this section, we discuss the features of the aerodynamic forces required to adequately model the trajectory of the Magnus glider. The equations of motion applied to the center of mass of the Magnus glider r(t) are:

ma = mg + F L + F D , (1)
where a is the Magnus glider acceleration, F L is the lift force due to the Magnus effect and F D the drag force. Before discussing realistic models, it is instructive to address the simplest model of an ideal fluid of density ρ, for which the flow features around the Magnus glider may be described using potential flow theory [START_REF] Tritton | Physical Fluid Dynamics[END_REF]. In the (non-inertial) center of mass reference frame, the flow is expressed as the addition of a vortex flow surrounding the glider with an uniform incoming flow (due to the glider translation). Since there is no viscosity, there is no drag exerted on the Magnus glider, and, restricting the computation to a two-dimensional case, the lift force due to Magnus effect reads F L = ρV(t) × Γ, where Γ is the circulation of the flow along the cylinder, defined as Γ = 2πω 0 R 2 L. Therefore, in the framework of potential flow theory, the Magnus force is proportional to the product of the translational velocity with the rotational velocity. In the presence of forces perpendicular to the velocity, trajectories obtained from integration of the equations of motion are known as trochoids (a generic family of curves including the cycloid). These trajectories are well known in the context of plasma physics for the guiding center dynamics of single charged particles in the presence of a constant magnetic field B 0 [START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF][START_REF] Bellan | Fundamentals of Plasma Physics[END_REF]. Indeed, in the presence of a constant electric field E 0 perpendicular to B 0 , the equations of motion for a particule of mass m q and charge q reads m q a = qE 0 + qV × B 0 , which is analogous to that of the Magnus glider in the framework of an ideal fluid (the electric force qE 0 being analogous to gravitational force mg and the Lorentz force qV×B 0 being analogous to the Magnus force ρV(t)×Γ). The charged particle gyrates at the cyclotron frequency around a guiding center drifting perpendicularly to the electric field and the magnetic field -an effect known as the electric drift in plasma physics. The radius of gyration is known as the Larmor radius, and is by the amplitude of the velocity perpendicular to the magnetic magnetic. Applying this analogy to the context of Magnus gliders leads to a drift perpendicular to the gravitational field and spin axis. This simple model of inviscid flow has the pedagogical virtue of being analytically solvable, with trajectories symmetric around the apex. As already pointed out, due to the existence of drag, the experimental trajectories reported here are not symmetric around the apex and a further degree of refinement is required to model the dynamics of the Magnus glider.

A more detailed model incorporates realistic drag and lift forces. For non-spinning objects, the drag force is usually expressed as

F D = 1 2 C d (Re)ρV 2 S
where S is the cross sectional area of the Magnus glider (here, S = 2RL) and C d (Re) the drag coefficient, which depend on Re [START_REF] Tritton | Physical Fluid Dynamics[END_REF]. Here, and following the notations widely used in the literature, the drag and lift forces are expressed as [START_REF] Clanet | Sports ballistics[END_REF][START_REF] Bush | The aerodynamics of the beautiful game[END_REF][START_REF] Swanson | The Magnus Effect: A Summary of Investigations to Date[END_REF][START_REF] Badalamenti | On the application of rotating cylinders to micro air vehicles[END_REF]:

F D = C d (Re, S p ) ρSV 2 2 ṽ F L = C l (Re, S p ) ρSV 2 2 ω × ṽ,
where ṽ is the velocity unit vector and ω the angular velocity unit vector and where the drag and lift coefficients C d and C l depend on the Reynolds and Spin numbers. These coefficients have been investigated extensively for cylinders of various aspect ratio, since the seminal work of Magnus on a brass cylinder, followed by successive investigations by Lafay [START_REF] Lafay | Sur l'inversion du phénomène de magnus, comptes-rendus[END_REF][START_REF] Lafay | Contribution Expérimentale à l'Aérodynamique du Cylindre[END_REF] and by the group led by Prandtl in Göttingen [START_REF] Prandtl | Application of the magnus effect to the wind propulsion of ships[END_REF]. The experimental and numerical investigations of the evolution of drag and lift coefficients of spinning cylinders have been compiled by Swanson [START_REF] Swanson | The Magnus Effect: A Summary of Investigations to Date[END_REF], and more recently by Baladamenti [START_REF] Badalamenti | On the application of rotating cylinders to micro air vehicles[END_REF].

Most of these studies were led in the context of technical applications such as Flettner boats [START_REF] Flettner | The Flettner Rotorship[END_REF] or other aeronautical ships or airplanes [START_REF] Seifert | A review of the magnus effect in aeronautics[END_REF] and provide a useful database for the modeling of the aerodynamical forces applied to the Magnus glider. In the next paragraph, we extrapolate the values of the drag and lift coefficients presented in previous studies to the geometry of our Magnus glider.

For non-spinning objects, and in the turbulent regime (i.e. Reynolds numbers in excess of 10 3 ), it is well-known that the drag coefficient is Re-independent [START_REF] Tritton | Physical Fluid Dynamics[END_REF]. A similar turbulent regime was also observed for spinning objects [START_REF] Swanson | The Magnus Effect: A Summary of Investigations to Date[END_REF][START_REF] Badalamenti | On the application of rotating cylinders to micro air vehicles[END_REF]; we will thus consider drag and lift coefficients to be Re-independent (below the drag crisis occurring around Re ∼ 4 × 10 5 for cylinders, an order of magnitude larger than the Reynolds numbers reached in this study). The values of the drag and lift coefficients for cylinders strongly depend on various control parameters [START_REF] Badalamenti | On the application of rotating cylinders to micro air vehicles[END_REF] such as the the cylinder aspect ratio, the roughness of the surface, the presence of endcaps, etc. Among the measurements of the drag and lift coefficients in wind- tunnels for spinning cylinders of various aspect ratios, the closest configuration to our case was studied for an aspect ratio 1.7 [START_REF] Busemann | Messungen an Rotierenden Zylindern[END_REF] in the Göttingen wind tunnel, nearly 90 years ago. We extracted the experimental data from the Göttingen study, shown in Fig. 4 as red crosses. Note that, in the framework of the potential flow theory discussed above, the lift coefficient reads C pf l = 2πS p and its evolution is reported as a blue dash-dotted line in Fig. 4. Note that C l ∝ S p means that the Magnus force scales as the product of the translational velocity with the rotational velocity. The first important observation is that the lift coefficient from the Göttingen windtunnel experiment is at one order of than the one computed from potential flow theory. The second observation is that both C l and d strongly depend on the Spin number S p . of the experimental data for S p > 1 (i.e. in the range of Spin numbers reached here) are shown as dashed black lines in Fig. 4, respectively as C AR=1.7 d = -0.35 + 0.88S 0.55 p and C AR=1.7 l = -0.31 + 1.3S 0.34 p . We stress that the Göttingen experimental points are somewhat scattered around the fits. The compilation provided by Badalamenti [START_REF] Badalamenti | On the application of rotating cylinders to micro air vehicles[END_REF] shows that both C l and C d depend on the aspect ratio of the cylinder. We thus choose to model the aerodynamic forces in Eq. 1 using a drag and a lift coefficients proportional to the Göttingen data shown in Fig. 4 The values of β l and β d are obtained from the best fits of nine experimental trajectories, reported in Fig. 5 (red solid lines), for various values of the initial kinematic parameters of the Magnus glider. Equation 1 is integrated numerically using as initial conditions the values of ω 0 , V 0 and α 0 obtained experimentally for the nine trajectories [START_REF]Using the ode45 Matlab function[END_REF]. A further approximation is to consider that IG. 5. Nine experimental trajectories (solid red line) and best fit (dashed black line) using the model of Eq. 1, with the drag and lift coefficients shown in Fig. 4. Initial conditions ω0, V0 and α0 at (x, z) = (0, 0), computed from the experimental data, are specified for each trajectory.

the angular velocity is kept constant during the flight of he Magnus glider -this assumption remains valid as long as the flight duration is lower than τ 0 and since the drag and lift coefficients only weakly depend on the Spin number for values larger than 2. The numerically-integrated trajectories are displayed in Fig. 5 (dotted black lines), and correspond to β l = 1.03 and β d = 0.9. The evolution of the drag and lift coefficients used for the numerical integration are shown in Fig. 4 (solid black line) as a function of the Spin number, and are in agreement with the wind-tunnel measurements. The trajectories computed from numerical integration of the model and displayed in Fig. 5 show a very good agreement with the experimental trajectories. As a partial conclusion, we showed here that the modeling of the experimental trajectories is very accurate when using the lift and drag coefficients obtained from wind tunnel measurements [START_REF] Busemann | Messungen an Rotierenden Zylindern[END_REF] for cylinders, in configurations close to our Magnus glider and assuming a constant angular velocity. A possible extension to this work would be a precise experimental characterization of the aerodynamic of joined truncated cones. We stress here that the remarkable agreement between the experiments and the model is observed although the Magnus glider is a very light projectile whose trajectory may be strongly influenced by external perturbations (weak draft or air convection).

Finally, we note that a further refinement of the model could be incorporated from an added mass effect, since the mass of air m air in the volume of the Magnus glider is 4% of the mass of the Magnus glider. This refinement would lead to consider the following equation of motions: (m + m air )a = mg + F L + F D . This refinement only marginally affects the results and does not modifies the message carried here: incorporating the added mass effect leads to β am l = 1.12 and β am d = 1.

B. Fast oscillations of the glider

A closer observation of the trajectories (or the velocities) in Fig. 3 shows strong oscillations around the time-averaged trajectory. The power spectral density of the horizontal component x(t) of a typical trajectory is displayed in Fig. 6, showing a well-defined peaks at f p = 21 Hz for the fast oscillations of the glider. The origin of these oscillations is now discussed. The inset of Fig. 6 shows the evolution of the peak frequency f p , expressed as a Strouhal number 2f p R/V 0 as a function of the Spin number at time t = 0 for the nine trajectories reported in Fig. 5. While we cannot rule out that these oscillations are excited by perturbations during the launching phase of the Magnus glider, a probable origin lies in the wake detachment from the Magnus glider at early times. Indeed, the evolution of 2f p R/V 0 is compatible with the evolution of the Strouhal number as a function of the Spin number, where the Strouhal number is defined as 2f R/V 0 with f the vortex-shedding frequency. The compilation by Badalamenti [START_REF] Badalamenti | On the application of rotating cylinders to micro air vehicles[END_REF] indeed shows that the Strouhal number lies between 0.1 and 0.25 and increases with the Spin number: all the features observed here are thus compatible with oscillations excited from vortex-shedding at early times of the trajectory.

C. Friction decreases angular velocity

Let us now discuss the time evolution of the angular velocity of the Magnus glider, as reported in the central panel of Fig. 3. From top to bottom, the characteristic times τ 0 are respectively 7.1 s, 6.1 s, and 4.4 s. The decay of the angular velocity is due to the friction within the boundary layer of the spinning Magnus glider. In wind-tunnel experiments, the amplitude of the dissipation is usually quantified from torque measurements. The torque T required to spin a cylinder at angular velocity ω in the presence of a cross-flow V is expressed as T = C T ρV 2 R 2 L with a torque coefficient C T and assuming a turbulent friction force, scaling as ρV 2 , applied uniformly on the surface of the cylinder. The compilation of the estimates of C T from previous studies [START_REF] Badalamenti | On the application of rotating cylinders to micro air vehicles[END_REF] shows that C T scales linearly with S p as C T ∼ c S p , with c between 0.02 and 0.06 (this indeed strongly depends on the roughness of the surface). Assuming the moment of inertia of the Magnus glider being that of a cylindrical shell of mass m and radius R, Newton's law of rotation reads

mR 2 dθ 2 dt 2 = -C T (S p )ρV 2 R 2 L ∼ -c ωρV R 3 L,
and since ω = dθ/dt, dω dt ∼ - c ωρV RL m .
The angular velocity thus decays exponentially with a characteristic time m c ρ V RL . Assuming c = 0.05 (in the range from the literature), the estimates for the decay times are respectively 8.1, 5.8 and 4.5 s for V 0 equal 8.4, 11.9 and 15.1 m/s respectively, which is in excellent agreement with the experimental observations.

IV. WHEN DOES THE MAGNUS GLIDER

LOOP BACK ?

In the previous section, we introduced a model for the drag and lift forces that accurately describe the aerodynamic forces applied to the Magnus glider. In this section, we solve numerically the equations of motion to identify which of the three types of trajectories are observed as a function of the initial kinematic parameters V 0 , α 0 and ω 0 . Indeed, due to our experimental protocol using a single rubber band to launch and spin the Magnus glider, there is a strong correlation between the amplitude of the initial velocity V 0 and the initial angular velocity ω 0 . More specifically, for three values of ω 0 (respectively to 360, 500 and 660 rad/s), Eq. 1 is solved numerically for various values of V 0 and α 0 , using the values of the drag and lift coefficients introduced in the previous section. The angular velocity was assumed to remain constant. The results are reported in Fig. 7, where the steps are 0.1 • for α 0 and 0.1 m.s -1 for V 0 . The white lines correspond to trajectories where the horizontal velocity vanishes, i.e. at the apex of the cusps: this line is thus a separatrix between arches, on the left, and loops, on the right. The line in the middle and bottom panels corresponds to the as a solid line in the top panel. As a first conclusion, the velocity has a weak effect on type of tory. A second rather counter-intuitive conclusion can be drawn from the inspection of the colorbar of Fig. 7 which represents the altitude of the apex of the trajectory (in meters): higher altitudes are reached at lower angular velocities. These results are easily explained from the evolution of the aerodynamic coefficients with the Spin number as shown in Fig. 4 for Spin numbers larger than 2. Indeed, the drag coefficient increases faster than the lift coefficient with the Spin number. Thus, a higher angular velocity leads to a higher drag, and thus lower altitude, while it does not strongly modifies the lift force. The white crosses labeled 1 to 9 in Fig. 7 correspond to the values of V 0 and α 0 for the trajectories reported Fig. 5 (1 referring to the top-left trajectory and 9 to the bottom right trajectory). Trajectories 1 to 4 indeed correspond to arches. Trajectories 5 and 6 lie very close to the white line, i.e. correspond to cusps, while trajectories 7 to 9 are looped trajectories. As a partial conclusion here, we demonstrated that the type of trajectory only weakly de-pends on ω 0 and mainly depends on the amplitude and orientation of the initial velocity V 0 , while the height of the apex strongly depends on ω 0 .

V. CONCLUSION

In this article we described an experimental investigation of the dynamics of Magnus gliders made from light paper cups. Such gliders are a simple, inexpensive and easy to image in a classroom laboratory allowing to measure the spectacular effect of the Magnus lift on the trajectories of spinning objects. The analysis of high-speed movies revealed three types of trajectories: arches, cusps and loops. These trajectories were accurately modeled using simple analytical descriptions for the drag and lift forces, based on previous aerodynamical studies in wind tunnels. The model predicts that the type of trajectory mainly depends on the amplitude and orientation of the launching velocity V 0 . The altitude of the apex strongly depends on the spin for a given launching velocity, and, counter-intuitively, decreases with increasing spin. These features are readily understood from the evolution of the drag and lift forces with the Spin number.

The investigation of the dynamics of Magnus gliders in the vicinity of the drag crisis, i.e. for Reynolds numbers around 4 10 5 , where the drag force suddenly decreases would undoubtedly be a natural follow-up of our study. The influence of the reverse Magnus effect [START_REF] Bush | The aerodynamics of the beautiful game[END_REF], linked to the difference in the boundary layers on the upstream and downstream sides could also lead to unexpected spectacular trajectories. We addressed the aerodynamic origin of small amplitudes fast oscillations on the trajectories of a symmetric Magnus glider, but similar oscillations could also be triggered by asymmetric gliders i.e., those for which the center of mass is off the axis of rotation. from domestic rubber elastic bands. The domestic rubber bands were 1 cm wide, 1 mm thick and 10 cm long and several of them (between 5 and 10 depending on the experiment) were connected using a loop to loop knots. The resulting band was wrapped around the center of the Magnus glider (i.e. taking advantage of the reinforced rim of the bottoms of the cups) with constant tension provided by a weight (between 1 kg and 3 g, depending on the experiment) attached at the bottom the band, and hanging 1 or 2 m below the Magnus glider. The free end of the rubber was then firmly held while the other end was stretched at a length l between 50 and 80 cm (i.e. roughly twice the length of the un-stretched remaining part of the band).

Several type of cups were tested, motivated by the variety of Magnus gliders reported in the video by [START_REF] Yeany | Physics of toys-cup flyers[END_REF]. Styrofoam extremely light, but also extremely brittle, which restrict the ranges of tension and stretching of the without breaking the Magnus gliders. plastic cups also too fragile to sustain the wrapping the rubber band. We also tested small paper cups and paper towel rolls, but their trajectories were very unstable (i.e. the trajectories did not remain 2D).

The reinforced bottom rims of paper cups were found to be an asset for the mechanical strength of the assembly.

FIG. 1 .

 1 FIG. 1. Chrono-photography showing the positions of the Magnus glider every 75 ms, after launching from the bottom left corner. The trajectory (dashed red line) displays a loop.

FIG. 2 .

 2 FIG. 2. (a) Sketch of the Magnus glider showing the forces acting during free flight. (b)-(f) Time sequence showing the launching protocol using a rubber band. The black point represents a fixed point.

FIG. 3 .

 3 FIG. 3. (left) Trajectory, (center) time evolution of the angular velocity ω, and (right) time evolution of the amplitude of the velocity V (t) (solid red line) and spin number Sp(t) (dotted black line) for three launches of the Magnus glider. The measured initial conditions are (top) ω0 = 362 rad/s, V0 = 8.4 m/s and α0 = 18 • , (middle) ω0 = 492 rad/s, V0 = 11.9 m/s and α0 = 21 • and (bottom) ω0 = 623 rad/s, V0 = 15 m/s and α0 = 29 • .

4 FIG. 4 .

 44 FIG. 4. (Top) Lift and (bottom) drag coefficients as a function of the Spin number Sp from the Göttingen wind tunnel (red crosses) and fitted curves for a 1.7 aspect ratio. The blue dash-dotted line corresponds to the potential flow theory.

  , i.e equal to β d C AR=1.7 d , and β l C AR=1.7 l , with β l,d ∼ 1, since the aspect ratio of the Magnus glider L/(2R) is equal to 2.67.

3 FIG. 6 .

 36 FIG. 6. Power spectral density (PSD) of the horizontal component of position a function f for a typical trajectory, showing a clear peak at fp. Inset: evolution of the frequency fp, expressed as a number, as function if the initial Spin number.

4 FIG. 7 .

 47 FIG. 7. Phase diagram of behavioral regimes, color-coding of the altitude of the apex of the trajectory (in meters) in the parameter space (V0, α0) for ω0 equal to (top) 360 rad/s, (middle) 500 rad/s and (bottom) 650 rad/s. The white lines display the location of the cusps. See text for details.

ACKNOWLEDGMENTS

The authors acknowledge support from the University of Lyon, the Université Claude Bernard Lyon 1, and the École Normale Supérieure de Lyon. The authors are thankful the committees of the French Physicists' Tournament and of the International Physicists' Tournament. The authors are grateful to L. Delance, C. Desgrange, R. Faure, L. Guislain, L. Morlet-Decarnin, H. Mussillon, A. Walbecq and P. Wang for fruitful discussions.

The Magnus gliders used for this study were made out of paper cups, of height L/2 = 9 cm, bottom radius 2.75 cm, and top radius 4 cm, glued together at their bottoms. They were launched using long rubber bands assembled Appendix B: High speed imaging requirements High speed imaging of the trajectories require a resolution larger than 1200 by 700 pixels. The capture of the rotational dynamics (around 100 rev/s) requires a framerate larger than a few hundreds of frames per seconds. In our manuscript we indeed use an ultra high speed camera, but at a moderate frame-rate of 4000 fps. We stress here that many of the cameras available in undergrad labs would allow to correctly image the experiments. Using standard mid-range smartphones which routinely image up to 240 fps in 2021 would be sufficient to image the trajectories (with a far better resolution than what we report in this study). The rotation rate for all of the regimes reported in our manuscript could be imaged using high-end smartphones which image up to 1920 fps in 2021 with a 1280x720 pixels resolution.


Ilya Brodoline

Stéphane Viollet

Julien Serres

Julien Serres An

An energy efficient 3D printed leg for bio-inspired hexapod robots

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

Sized from Cataglyphis Fortis ant

To provide a reliable leg structure able to be used by biologists as a model for hexapod insect-type robots. The joints angles limits and lengths were measured from ants and scaled 1:20.

New 3D printable leg

Mostly composed of 3D printed parts, this new leg has a reduced price of manufacturing. In association with some aluminium parts, the total mass equals to 300 grams.

Materials : SLA resin, PLA, Aluminium.

Energy saving mechanics

Screw-nut mechanism is irreversible. Therefore, there is no energy consumption when a joint is not used.

Mechanics biological ASPECTS

Bio-inspired leg

Perspectives

Force sensors Next step of the project is to implement force sensor at strategic location, most representatives the campaniform sensilla groups.

Tibial receptors (11% leg receptors) Trochanteral receptors (74% leg receptors)

Bio-inspired hexapod robot

New robot leg should be tested on a hexapod robot, to verify its accuracy and e ciency compared to classical servomotor based robot. [1], [START_REF] Wehner | [END_REF] Total mass 2kg 2cm 20cm [START_REF] Dupeyroux | Science Robotics[END_REF] [1], [START_REF] Sangbae | Context Servomotors are the mostly used actuators for small electric legged robots[END_REF] An energy efficient 3D printed leg for bioinspired hexapod robots Ilya Brodoline, Stéphane Viollet, Julien R. Serres Aix-Marseille University, CNRS, ISM, Marseille, France Corresponding author: ilya.brodoline@univ-amu.fr

  In opposition to compact servomotors, this actuator has a large structure in favor of heat dissipation.

  Lorem ipsumDistribution of campaniform sensilla on an insect leg[2]


Bruno Guillaume
email: bruno.guillaume@inria.fr

Guy Perrier
email: guy.perrier@loria.fr

Graph Rewriting for Enhanced Universal Dependencies

This paper describes a system proposed for the IWPT 2021 Shared Task on Parsing into Enhanced Universal Dependencies (EUD). We propose a Graph Rewriting based system for computing Enhanced Universal Dependencies, given the Basic Universal Dependencies (UD).

Introduction

The IWPT 2021 Shared Task on Parsing into Enhanced Universal Dependencies [START_REF] Bouma | From raw text to enhanced universal dependencies[END_REF] is a second edition of an equivalent shared task in 2020 [START_REF] Bouma | Overview of the IWPT 2020 shared task on parsing into enhanced Universal Dependencies[END_REF]. The goal of the shared task is to produce EUD (Schuster and Manning, 2016), with several new annotation layers expressed on top of UD annotations (Nivre et al., 2020).

In the previous shared task, there were two kinds of approaches: producing EUD annotation from raw text with machine learning methods or producing EUD from UD with a rule-based approach (with or without some learning to optimize rule usage). Like (Heinecke, 2020) or (Dehouck et al., 2020), our proposal corresponds to the second approach: we used an existing tool for producing UD annotations and work only on the conversion from UD to EUD. Unlike other rule-based approaches, we used GREW, a generic Graph Rewriting tool [START_REF] Bonfante | Application of Graph Rewriting to Natural Language Processing[END_REF], in order to describe the rules for enhancement.

Another specificity of our work is that we primarily design our rules by following the guidelines. Even if, in a secondary step and in the context of the shared task, we adapt the system to the corpora which diverge from the guidelines (section 2.6), we can easily provide a system closer to the guidelines, adpatable to languages specificities.

Our system achieved 81.58 ELAS score on the task, starting from UDPIPE annotation with an LAS of 85.77. In the paper, we present the system, analyse the results and make some in-depth analysis on French and English of discrepancies between GOLD EU D data and the output of our system starting GOLD U D annotations.

2 Description of the system 2.1 Parsing to UD For this shared-task, we used the UDPIPE2 (Straka et al., 2016) through the online service1 to produce the UD annotation of the data. The models used for each language are trained in UD version 2.6 on the following corpora: Arabic-PADT, Bulgarian-BTB, Czech-PDT, English-EWT, Estonian-EDT, Finnish-TDT, French-GSD, Italian-ISDT, Lithuanian-ALKSNIS, Dutch-Alpino, Polish-PDB, Slovak-SNK, Swedish-Talbanken, Tamil-TTB, Ukrainian-IU2 .

GREW

The transformation UD to EUD is described with the graph rewriting tool GREW3 . Each rule is defined by a pattern and a set of commands describing how to modify the graph. A dedicated strategy mechanism allows for controlling rules applications (in which order subsets of rules must be applied and how they must be iterated). A global transformation system (rules and strategies) is called a Graph Rewriting System (GRS).

Representation of EUD annotations

We use here the convention already adopted in the Deep-Sequoia project (Candito et al., 2014), in which dependencies are drawn in black when the relation exists both in UD layer and EUD layer, in red when they are present in UD layer only and in blue (and below) when they are present in EUD layer only (see Figure 1). This prevents from having two relations when both layers are identical and make figures easier to read.

2.4

From UD to EUD Our goal was to design a GRS, following the EUD guidelines, to convert the UD annotations into EUD annotations, but we observed that the conversion system has to be adapted to each language and to some particular annotation choices.

As the rules of a GRS are organized in strategies, this adaptation is done by activating or not the applications of subsets of rules in the definition of the main strategy of a generic GRS. The rules are freely available4 .

The six types of enhancement

The EUD guidelines identify 6 types of enhancements, and for each type, we have designed a subset of rules to achieve it. We briefly describe the main features of each subset.

Empty nodes for elided predicates Predicate elision is detected thanks to the presence of orphan dependencies. They are mainly found in coordinations and sometimes with parataxis relations. Figure 1 shows the UD annotation of a sentence with ellipsis5 .

First, a null node N is created and a dependency N -obj→ trees is introduced from N to the head of the second conjunct trees. The label obj is determined by a parallelism with the dependency sold -obj→ strawberries6 . The dependency trees -cc→ and is also raised to N.

In a second stage, all orphan dependencies from the head of the second conjunct are transformed into dependencies from the null node. The labels of the new dependencies are determined from the context. In our example, the dependency trees -orphan→ winter is transformed into the dependency N -obl→ winter, because N is a verb, winter a noun with a case dependent.

Propagation of incoming dependencies to conjuncts The propagation of incoming dependencies to the conjuncts of a coordination is systematical. The only difficulty concerns modifiers: if a word H is modified by a coordination, the label of the dependency from H to the head of the first conjunct may need to be changed depending of the POS of the second conjunct. In Figure 2, the dependency come -obl→ parents has to be propagated to the head of the other conjunct, the adverb separately. Because of this POS, the label has to be changed and the dependency becomes come -advmod→ separately.

Many gold corpora of the task do not take into account differences of POS between the conjunct heads and propagate the incoming dependencies without changing their labels.

Propagation of outgoing dependencies from conjuncts The main problem for this enhancement comes from the ambiguity of the UD annotation schema. It is not possible to distinguish a left dependency on a coordination from a left dependency on the first conjunct of the coordination because both are attached to the head of the first conjunct.

But, it is necessary to remove the ambiguity in order to know if one should propagate a left dependency. This is more or less easy depending on the type of the dependency. In Figure 3, the nsubj and cop dependencies on the noun acteur must be propagated on the head of the second conjunct protagoniste. It is easy to design a specific rule for each type of dependency to perform the propagation, what we have done. But, if the dependency is an advmod dependency, there is no general criterion for removing the ambiguity. The dependency on souvent should be propagated, but not the dependencies on n' and pas. This depends on the modifier adverb but also on the context. This is a point where our rule-based approach marks its limits compared to learning approaches. Of course, the answer depends strongly on the language and we will see how to take into account the specificity of each language in subsection 2.5.

For right dependencies, there is no ambiguity, because a right dependent on the first conjunct that follow all conjuncts is necessarily a dependent on the coordination.

Additional subject relations for control and raising constructions Raising and control verbs take an infinitive as a xcomp dependent and the en- hanced subject of the infinitive is either the subject, the direct object or the indirect object of the main verb, if that argument exists. The choice between the three possibilities depends on the lexical information about the raising or control verbs. In our approach, this means that we need lexicons per language. Most often, we do not have such lexicons and choose the subject of the main verb as the subject of the infinitive, as this is the most likely.

Coreference in relative clause constructions

The relative clause enhancement adds ref dependencies from the antecedent to the relative pronoun and all dependencies targeting the relative pronoun are moved to the antecedent.

In Figure 4, the sentence contains two relative pronouns, dont and qui. Let us focus on qui. First, rules make an upward path from the relative pronoun by following the dependencies until finding an acl:relcl dependency. In the example, this requires crossing a dependency nsubj and then conj. As soon as the dependency acl:relcl is reached, it is possible to add the ref dependency, because its source is known, it is the source of the dependency acl:relcl and the antecedent of the relative pronoun, and the target of the dependency as well, the relative pronoun, which is kept in memory. In a second stage, all dependencies targeting the relative pronoun are moved to the antecedent. In the example, the dependency nécessite -nsubj→ qui is transformed into the dependency nécessite -nsubj→ entreprise.

Modifier labels that contain the preposition or other case-marking information With regard to this enhancement, we strictly follow the guidelines: if a case or mark dependent on a modifier is a multiword expression, we add the form of the expression to the dependency representing the modification; if it is a single word, we add the lemma of the word.

The guidelines don't cover the case of several case or mark dependencies having the same source. In Figure 5, the conjunction because and the preposition in both depend on the same governor zone (dependencies in orange). We decide to add the outermost dependent lemma to the modifier dependency feel -advcl→ zone because it is related to this dependency, whereas in has no relationship to it.

If the two candidate dependents are consecutive, we consider them as a single multiword and add the concatenation of their lemmas to the modifier dependency.

Rule ordering

The six types of enhancements are not totally independent, some of them interact, so the order of application between the six corresponding subsets of rules is not neutral. Figure 1 shows an interaction between the subset implementing ellipsis processing and the subset implementing the propagation of outgoing dependencies of a coordination. If we apply the latter first, it is not possible to propagate the dependency sold -nsubj→ they to the second conjunct of the coordination because its verb is elided. We must apply first the subset creating a null node representing this verb. The subset of rules adding case or mark information to modifier dependencies must be applied at the end because it does not create any possibility to apply another subset later.

The order between the application of the five other subsets is relatively free. Figure 4 shows an example of interaction between the subset relating to relative clauses and the subset implementing the propagation of incoming dependencies of a coordination. We applied the former subset first and to apply the second subset, we only need to propagate the dependency entreprise -acl:relcl→ reparlera to the second conjunct of the coordination. But there is no problem to reverse the order between the two subsets.

Even if the order chosen between the 6 subsets aims at minimizing the number of rule applications, we cannot avoid applying some subsets a second time. Figure 6 shows an example of repetition in the subset application. To add the dependency build -nsubj→ I, we need first to apply the subset related to control verbs. We obtain the dependency sell -nsubj→ I. Then we apply the subset related to the propagation of outgoing dependencies of a coordination. We obtain the dependency use -nsubj→ I. Finally, we apply the subset relative to control verbs a second time and we obtain the last dependency build -nsubj→ I.

All rules presented in this subsection constitute the generic GRS used to convert the UD annotation into the EUD for the 17 languages.

Adaptation to the specificities of languages

Rule packages are added to the generic GRS to express specificities of language groups. In order to be applied, they are inserted in the strategy at carefully chosen positions in the generic strategy.

In this way, strategies can be designed adapted to particular languages, by activating or not these new packages. Now, let us examine which types of rules can be added to express specificities of certain languages.

Null Subject Languages Arabic, Bulgarian, Czech, Estonian, Finnish, Italian, Polish, Russian, Slovak, Tamil and Ukrainian are null subject languages. Their grammar permits verbs to lack an explicit subject. This can be a problem for the propagation of subjects of coordinated verbs.

Consider the Polish sentence "Moje gospodarstwo daje mi zabezpieczenie, mam gdzie wrócić (My farm gives me security, I have a place to come back to)". The general rules of subject propagation will propagate the subject gospodarstwo from the verb daje to the coordinated verb mam, which is incorrect because mam is at first person and does not require any explicit subject. In order to avoid the propagation, a specific rule marks all first and second person finite verbs, so that they cannot receive a subject dependency. For the third person, there is an ambiguity. In the Polish sentence "Chłopiec wstaje, otwiera drzwi", there are two correct translations in English: "The boy gets up, opens the door" and "The boy gets up, he opens the door", because one can propagate the subject boy or not. We chose to propagate the subject, which means that in this case, there is no difference between null subject languages and others.

Case addition to the dependency labels for modifiers For case-based languages, the labels of the dependencies targeting modifiers are augmented with their cases. The rule package implementing this enhancement is trivial.

Left dependents of a coordination

We designed rules to propagate the left dependents of a coordination by dependency types. As we said before, a left dependent on the first conjunct of a coordination is ambiguous: it can depend on the whole coordination or only on the first conjunct. In order to determine, for a given language if a given type of dependency must be propagated or not, we tested the two alternatives on the dev corpus of the shared task and keep the alternative yielding the highest score.

This method has important limits because it depends on the annotation of the gold corpus. Moreover it is very coarse; for a given dependency type, not all dependencies have the same behaviour: some must be propagated, others not. It would be necessary to refine the conversion rules but for that, we need linguistic knowledge about the concerned language.

Raising and control verbs The default rule we use is to consider that the subject of the raised or controlled verb is the subject of the main verb but this is not always true. A language-specific lexicon should indicate for each of these verbs which argument of the main verb is the subject of the raised or controlled verb. From the training and development corpora available for the task, we have created lexicons for a five languages: Dutch, English, French, Italian and Polish.

Adaptation to annotation choices

Some annotators of the gold corpora do not strictly follow the guidelines. We have adapted our GRS to their choices on some very specific points.

Dependency label extension specific to one language In Dutch, enhancements for relative clauses distinguish antecedents of relative pronouns that play the deep role of subject and direct object in the relative clause with the extensions relsubj and relobj. We have designed specific rules to add this extension.

Enhancements partially taken into account

We have taken into account the fact that enhancements are only partially achieved for Arabic, Bulgarian, Estonian, French, Russian and Tamil.

Coordinating conjunction raising For the Arabic, Dutch, English, Italian and Swedish treebanks, the names of coordinating conjunctions are added to the corresponding conj dependencies, in the same way as for prepositions and subordinating conjunction. We have taken this into account even though it is not indicated in the guidelines.

Propagation of root dependencies According to the guidelines, root incoming dependencies of a coordination should be propagated like all incoming dependencies, but some treebanks do not and we take this into account.

The French and the Polish treebanks, the latter partially, not only add subjects for raising and control verbs, as mentioned in the guidelines, but add deep subjects for modifier infinitive and participial clauses. Since this goes beyond the guidelines, we do not consider these enhancements.

Table 1 summarizes the three kinds of adaptation to the different languages.

Results

Data we submit to the task, called GREW(UDPIPE), is the output of the application of a language spe- cific GRS to the output of UDPIPE. Figure 7 shows for each language, the LAS of UDPIPE and the ELAS of GREW(UDPIPE). We can observe that the final result highly depends on the UD annotation quality produced by UDPIPE.

In addition to the results provided by the orgnanisers, we make complementary analysis, focusing on the UD to EUD transformation. In Table 2, we report the ELAS score obtained by our system applied on the gold UD annotation as input. We also recall the same measure reported last year by two others systems (Dehouck et al., 2020;Heinecke, 2020). The score of our system is in most of the case between the two other proposal, closed to Dehouck's system except for Estonian and Swedish.

Analysis of discrepancies with the gold annotation In French and English

In order to better understand the behaviour of our GRS, we made some manual inspection of the difference between the annotation GREW(GOLD U D ) and GOLD EU D . We focused on the two languages for which we had the grammatical skills to analyze the discrepancies: French and English.

Discrepancies in French

For the French corpus, we observed 589 discrepancies7 of the computed annotation with the gold annotation, and we manually analyzed the first 100. Table 3 details this analysis.

In order to explain mislabeling in the propagation of incoming dependencies of a coordination, let us return to the example in Figure 2. In propagating incoming dependencies of a coordination that is a modifier, we cannot automatically propagate the label related to the first conjunct to the other conjuncts, because we have to take into account the POS of the heads of these conjuncts. This was not taken into account for the two errors mentioned in the table.

In the table, we have also distinguished errors related to subject or object attributives from errors related to raising and control verbs, because of their particular property: the attributives may have other POS than verb. For exemple, consider the sentence ils laissent les troupes de la KFOR en paix (they leave the KFOR troops in peace). The noun peace has an enhanced subject, which is troops. The gold annotation ignores this type of subject.

44 discrepancies come from the fact that the gold annotation implements enhancements that are not considered by the guidelines. Columns ¬EUD gives a detailed analysis of these enhancements with the number by type. Let us give an example to explain the last type of these non-standard enhancements. Consider the phrase l'occasion également pour J.-P. Bruneau de présenter ses voeux (the occasion also for J.-P. Bruneau to present his wishes). The gold annotation indicates that J.-P. Bruneau is the enhanced subject of present, which is not considered by the guidelines.

Discrepancies in English

For the English test corpus, we observed 815 discrepancies of the computed annotation with the gold annotation, and we also manually inspected the first 100. Table 4 gives a detailed analysis of these errors with their number by type.

Conclusion

We have observed that many conversion problem arise with the CASEDEPREL layer. This layer is of course highly dependent of the language (because lexical information is used in relation definitions). This prevent the new relation to be universal and we believe that this is counterproductive in the objective of a universal description among a large set of languages.

In this paper, we have proposed a rule-based system for computing EUD annotation from UD. Our raw results are far behind the best systems of the task. This can be explained by the fact that we are dependent of the basic UD annotation provided by another tool. Moreover, the manual inspection we have made shows that, at least on English and French, the GOLD test data used in the task are not error-free and contains several annotations that are not described in the guidelines. We can suspect that this is in favour of the learning based approaches which are designed to adapt to the annotated data, completely ignoring the guidelines.

Despite its weakness, we believe that our system have several benefits:

• It has highlighted some places where the guidelines require precisions, like the presence of several case or mark on the same head;

• It can be used for improving the existing EUD data in the project by identiying annotation error in the current EUD annotations; using a different approach, we can guess that the errors reported will be complementary to the ones that can spotted with other methods;

• Thanks to the modular aspect of the GRS with rules packages adpated to language specificities, it is usable as a starting point for adding EUD annotation layer on languages where there is no such data and where learning based methods cannot be used.

Figure 1 :Figure 2 :

 12 Figure 1: They sold strawberries in summer [,] and [Christmas] trees in [the] winter

Figure 3 Figure 4 :Figure 5 :

 345 Figure 3: souvent [, l'] Occident n' est pas seulement [l'] acteur [,] mais [aussi le] protagoniste [des violations des droits de l' homme] (often[, the] West is not only [the] actor[,] but [also the] protagonist [of human rights violations])

Figure 6 :

 6 Figure 6: [So] I decided to sell [eHow] and use [the proceeds] to build wikiHow.

Table 1 :

 1 The three kinds of adaptation of the system. Top: adaptation to languages; middle: adding lexical information; bottom: adaptation to specific annotations observed in dev data.

	Language Dehouck Heinecke Our
	Shared task	2020	2020	2021
	Arabic	98.8	95.2	98.5
	Bulgarian	98.6	97.8	97.6
	Czech	97.9	94.7	97.6
	Dutch	98.9	94.4	97.6
	English	99.5	98.0	99.0
	Estonian	99.2	92.6	93.9
	Finnish	97.3	94.4	96.9
	French	98.9	96.4	99.0
	Italian	99.5	98.4	98.8
	Latvian	95.7	91.0	92.1
	Lithuanian	98.8	94.6	98.2
	Polish	94.9	91.1	95.2
	Russian	98.6	95.4	98.2
	Slovak	98.8	95.4	98.1
	Swedish	98.8	96.1	94.7
	Tamil	99.3	97.0	98.3
	Ukrainian	95.8	94.6	95.9
	Average	98.2	95.1	97.0
	Table 2: Evalutation of the rule-based systems on
	Gold UD data: Dehouck (Dehouck et al., 2020), Hei-
	necke (Heinecke, 2020) and our system

Table 3 :

 3 Manual inspection of the 100 first discrepancies in French between GOLD EU D and GREW(GOLD U D ). Columns are: errors in EUD gold annotation, errors in UD gold annotation, errors produced by our GRS, EUD gold annotations non described in guidelines.

	EUD UD GRS ¬EUD

Proceedings of the 17th International Conference on Parsing Technologies (IWPT 2021), pages 175-183Bangkok, Thailand (online), August 6, 2021. ©2021 Association for Computational Linguistics

https://lindat.mff.cuni.cz/services/ udpipe/

The only modification done was on the Czech output where 7 obvious errors on the lemmatisation makes the conversion producing non valid data.

3 https://grew.fr/

https://gitlab.inria.fr/grew/udtoeud

All examples are extracted form the corpora used in the task. Due to lack of space, expressions in square brackets, which are not essential to our purpose, are skipped in the annotation.

Even if the subject is also a candidate, we always favour the object in this case.

Number of differences computed by the diff Unix tool: differences in consecutive lines are merged as one difference.

ar bg cs nl en et fi fr it lv lt pl ru sk sv ta uk null subject