Guillaume Pot
email: guillaume.pot@ensam.eu

Catherine Coutand
email: catherine.coutand@clermont.inra.fr

Evelyne Toussaint
email: evelyne.toussaint@univ-bpclermont.fr

Jean-Benoit Le Cam
email: jean-benoit.lecam@univ-rennes1.fr

Marc Saudreau
email: marc.saudreau@clermont.inra.fr

Jean-Benoît Le

J.-B Le Cam

A

Keywords: Gravitropism, Biomechanical model, Poplar

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Introduction

Trees can maintain or modify the orientation of their trunk or branches in relation to gravity. These gravitropic movements are enabled by the asymmetrical production of reaction wood in plant parts that have achieved their elongation [START_REF] Scurfield | Reaction wood: Its structure and function[END_REF]. In most deciduous species, reaction wood is called tension wood (TW) because this wood is subjected to longitudinal tensile stresses within the living tree. Stresses in trees are mainly due to the wood cell maturation process. During their maturation, wood cells tend to shrink in their longitudinal direction, but these maturation strains (MS) are impeded by the cells created earlier, which are already stiffer, thus producing internal maturation stresses [START_REF] Kubler | Studies on growth stresses in trees. 1. the origin of growth stresses and the stresses in transverse direction[END_REF]. The high internal stresses caused by TW can induce damage during wood exploitation and processing [START_REF] Cassens | Growth stress in hardwood timber. 14th Central Hardwood Forest conference[END_REF].

At the beginning of the righting-up movement of a tilted trunk, deciduous trees create TW with high internal stresses on the upper face of the tilted trunk, and so-called opposite wood (OW) with low internal stresses on the lower face (on the opposite side). In fact, a recent study on the kinematics of the gravitropic movement of artificially tilted poplars [START_REF] Coutand | The gravitropic response of poplar trunks: Key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF] revealed three phases: a latency phase (no curving up of the tilted trunk), a gravitropic phase (curving up) and an autotropic phase (decurving enabling the trunk to become straight and vertical). Within this last phase TW is produced on the opposite side.

Several biomechanical models concerning tree gravitropism can be found in the literature. They were built to simulate the movement of growing trees and/or to compute the internal stresses with a technological aim (because of the splitting or twisting of planks due to stresses). [START_REF] Kubler | Studies on growth stresses in trees. 1. the origin of growth stresses and the stresses in transverse direction[END_REF] was the first author to calculate analytically the internal stresses in the case of a trunk growing symmetrically for an elastic and transversally isotropic material. Then [START_REF] Archer | On the distribution of tree growth stresses. part i: An anisotropic plane strain theory[END_REF] extended these formulae for asymmetrical growth. Later, Fournier et al (1991a,b) developed a semi-analytical model which showed that maturation stresses are much greater than stresses due to the self-weight of the tree. Numerical finite element modelling of tree shape regulation was also developed by [START_REF] Fourcaud | Numerical modelling of shape regulation and growth stresses in trees[END_REF]. Recently, biomechanical models have shown that viscoelasticity can increase the righting-up efficiency of trees [START_REF] Dlouhá | Biomechanical performances of trees in the phase of active reorientation[END_REF][START_REF] Coutand | Twig: A model to simulate the gravitropic response of a tree axis in the frame of elasticity and viscoelasticity, at intra-annual time scale[END_REF]. These biomechanical models consider that the maturation process is instantaneous: when cells are created in the model, they immediately acquire the mechanical properties of a matured wood cell and they produce a given level of MS.

As noticed by [START_REF] Coutand | The gravitropic response of poplar trunks: Key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF], gravitropic righting in young poplar trees occurs over a few weeks; thus the kinetics of maturation (i.e. changes with time in terms of mechanical properties and MS) could have an effect on the gravitropic response. Nevertheless, none of the existing biomechanical models considers a progressive maturation. Moreover, they do not usually use experimentally determined properties as input data, and thus cannot be compared to experimental measurements of gravitropic movement.

In the present paper, a model is built with the aim of simulating the righting-up of tilted trees which create TW on the upper face of the trunk during a single season of growth. Both progressive maturation and the viscoelasticity of green wood are taken into account. The objectives are (i) to compare the proposed model featuring progressive maturation to models with instantaneous maturation and (ii) to compare the model to experimental measurements to estimate MS by inverse identification.

After the present introduction, the second part of this work presents the biomechanical model, with special emphasis on the modelling of progressive maturation. In the third part, the experimental data used as the input to the model and the identification of MS are presented. Then the results are discussed and conclusions are drawn concerning the value of MS obtained and the ability of the model to simulate tree gravitropism.

Biomechanical modelling

General description of the model

The biomechanical model presented in the following is an enhancement of the model of [START_REF] Coutand | Twig: A model to simulate the gravitropic response of a tree axis in the frame of elasticity and viscoelasticity, at intra-annual time scale[END_REF] (TWIG). As in this previous work, it is programmed using MATLAB version 7.5 (2007). The same principles of modelling are used, typically the geometrical definition of the tree and the incremental formulation of the viscoelastic behaviour. They are briefly recalled hereafter.

The TWIG model [START_REF] Coutand | Twig: A model to simulate the gravitropic response of a tree axis in the frame of elasticity and viscoelasticity, at intra-annual time scale[END_REF]) is a semi-analytical model based on Euler-Bernoulli beam theory.

The mechanical properties used are those of wood in the longitudinal direction. According to Fournier et al (1991b), such a model is sufficient to approach the longitudinal internal stresses that are the main cause of tree gravitropic movements.

The tree is modelled by a beam whose diameter evolves discontinuously along its length L in n seg cylindrical segments, themselves numbered i seg (Fig. 1a,b). This model does not account for the longitudinal growth. Indeed, experimental measurements show that longitudinal growth is small during gravitropic reaction [START_REF] Pot | Mechanical characterization of green wood during maturation process and modeling of gravitropic reaction of young poplar[END_REF]. However, it could be modelled using an additional mass at the tip of the beam.

Asymmetrical radial growth is modelled considering the eccentricity of the outer cylinder of each segment relatively to the pith (Fig. 1). The coefficient δ is the distance between the centre of the pith and the centre of the new outer cylinder. The difference in mechanical properties between TW and OW is taken into account thanks to the division of the cross section of the beam into two angular sectors, defined by angle γ (Fig. 1c,d). Model parameters depend on the face considered and are named using the index "up" for parameters relative to the upper face of the tilted tree and "low" for parameters relative to the lower face of the tilted tree. For the sake of simplicity, the "f ace" index is used in the following as a generic term for "up" or "low".

The beam in Fig. 1a is anchored at its base and the external forces are only due to the self-weight of the structure. Branches are not represented, but their effect in terms of weight is considered as a distributed mass which can evolve during the growing season (see section 3.3 for details). Weight induces a normal force N w , a shear force T w and a bending moment M w on the structure. Shear forces are neglected, in keeping with the Euler-Bernoulli assumption. Maturation induces internal normal forces N m and bending moments M m which bend the beam. Finally, the total normal forces and bending moments are noted in a vector {F } as the sum of the normal forces and bending moments due to weight and maturation:

{F } =      N w + N m M w + M m      Oxyz (1)
These normal forces and bending moments produce a longitudinal strain along the axis that is noted ε O and a curvature (χ) of the beam. It is also noted in vector form (named the vector of generalised strains) as follows:

{D} =      ε O χ      Oxyz (2)

Spatio-temporal description

In order to model the maturation process, the variations over time of wood mechanical properties must be taken into account. Since the tree is growing in diameter, there are also spatial variations in the whole structure. This is the reason why it is necessary to consider the spatio-temporal variation of the parameters.

In the model, radial growth is considered as an incremental problem: growth is modelled by the addition at each time step i t of a new radial growth increment i dr = i t (Fig. 1). By this means, every radial growth increment is denoted at every time step using index i dr . This enables the spatial heterogeneity of the beam to be represented.

The duration of growth that is simulated, denoted T , is divided into n growth increments. Thus, the duration of a growth increment is ∆t dr = T n . Thanks to the distinction between spatial and temporal increments, it is possible to define the time at the beginning of the current growth step as:

t(i t ) = i t ∆t dr (3)
and the time of creation of a given growth increment i dr as:

t(i dr ) = i dr ∆t dr (4)
With these two characteristic times, it is possible to define the age of each growth increment, denoted a.

The age of a growth increment is defined as the difference between the time at the end of the current growth step and the time of creation of the increment. Thus, the age of a given growth increment at time step i t is:

a = (i t + 1 -i dr )∆t dr (5)
By this means, when a growth increment is created at a given time step, it immediately has an age of ∆t dr .

This age has a physical meaning, since it corresponds to the age of the wood cells contained in the modelled tree. It enables the modelling of the change in mechanical properties according to the maturation stage of each increment.

Finally, any parameter P of the model depends on spatial and temporal increments, and can be written as P f ace (i seg , i dr , i t ). It varies with the circumferential position ("f ace" index), the longitudinal position (i seg ), the radial position (i dr ), and the time (i t ). The decoupling between spatial and temporal increments enables the parameters of each spatial increment to vary freely with time increments. This decoupling is the main contribution of the present work, since other models usually consider instantaneous maturation for which the parameters of spatial increments do not change with time after their creation.

Incremental formulation of viscoelastic behaviour

As in [START_REF] Coutand | Twig: A model to simulate the gravitropic response of a tree axis in the frame of elasticity and viscoelasticity, at intra-annual time scale[END_REF], a viscoelastic constitutive law of green wood mechanical behaviour and an incremental calculation method are used to model tree gravitropism. The fundamental principle of this modelling is to decompose time into increments of ∆t and calculate Boltzmann's equation over the finite time interval [t, t + ∆t], under the hypothesis of linearity over this interval. This leads to the following equation:

{∆F } = [ K * ] {∆D} + { F hist } (6)
where {∆F } is the variation in normal force and bending moment during time increment ∆t; [K * ] is the fictitious stiffness matrix; {∆D} is the variation in longitudinal strain and curvature during time increment ∆t; and

{ F hist }
is the normal force and bending moment which accounts for the effect of the total load history since the beginning of the loading period.

The viscoelastic constitutive law used is based on a Burgers' rheological model (Fig. 2), in accordance with the results of Pot et al (2013b). By this means, Burgers' rheological parameters appear in mathematical expressions of [K * ] and [START_REF] Pot | Mechanical characterization of green wood during maturation process and modeling of gravitropic reaction of young poplar[END_REF], but the fundamental principle of incremental formulation do not change with respect to [START_REF] Jurkiewiez | Incremental analysis of time-dependent effects in composite structures[END_REF] or [START_REF] Coutand | Twig: A model to simulate the gravitropic response of a tree axis in the frame of elasticity and viscoelasticity, at intra-annual time scale[END_REF].

{ F hist } (
The time increments of the viscoelastic calculation (∆t) are different from time increments that describe the growth of the trunk (denoted ∆t dr ). Indeed, the time increments of the incremental viscoelastic formulation must be short enough to observe the hypothesis of the linearity of the equations [START_REF] Jurkiewiez | Incremental analysis of time-dependent effects in composite structures[END_REF]. The duration of growth increments cannot be short enough to verify linearity, because it would lead to a high number of spatial increment and thus high dimensions for matrix and vectors, which would overload memory. Consequently, growth time increments are subdivided into viscoelastic calculation time increments. Thus, the calculation is divided into two steps:

1. The first calculation is performed at the beginning of each growth step, thus at time t(i t ). The variation in force and moment due to maturation and weight are then calculated and equation 6 is solved.

2. Next, a calculation loop is performed without considering growth and maturation. However tree movements and changes in internal forces can appear because of viscoelasticity and weight. Time t is incremented in the viscoelastic formulation with the short time increment ∆t until the next growth increment.

During these two calculation steps, the viscoelastic formulation (equation 6) is inverted and used for each segment in order to obtain the variation in generalised strains {∆D(i seg , t)}:

{∆D(i seg , t)} = [ K * (i seg , t) ] -1 ( {∆F (i seg , t)} - { F hist (i seg , t) }) (7)
The same formulation is used for steps where growth and maturation appear and steps where only the effect of viscoelasticity is concerned. However, when growth occurs, maturation forces appear in {∆F (i seg , t)} and the mechanical parameters are changed.

2.4 Determination of time step size and computational errors

Influence of time step size on viscoelastic calculation

The influence of time step size on viscoelastic calculation was determined thanks to a preliminary study on the creep of a wood beam. This beam had similar characteristics as those used in the present study. A 3 m long cylindrical homogeneous beam of radius r = 15 mm and density of ρ = 500 kg.m -3 was tilted to α = 35 • from the vertical and its deflection under self-weight was modeled. The analytical expression of the curvature can be easily calculated in the case of a non-growing beam:

χ(x, t) = J(t) M w (x) I(x) (8) with J(t) = 1 E 0 + t E 0 τ ∞ + 1 E 1 ( 1 -e -t/τ 1 )
the compliance function of Burgers' model (with notations of The evolution with time of curvature at stem base obtained with the model is presented in Fig. 3a for different sizes of time step (∆t), and compared to the analytical result. The less the ∆t, the more the accuracy of the model. With the lowest time step size presented here, ∆t = 0.0063 days (9 minutes), the maximum relative error between model and theoretical formula is 0.17%.

Influence of growth step size

The discretization of the viscoelastic calculation is not the only source of computational error: the quantity of growth increments may have an influence on the accuracy of the computation because of the decoupling between growth and mechanical effect (see [START_REF] Guillon | Numerical methods for the biomechanics of growing trees[END_REF] for more details about the numerical methods for the biomechanics of growing trees). However, there are no analytical results for a viscoelastic growing beam, thus no comparison with the model can be completed. A simple way to evaluate the effect of growth increment size is to compare the results obtained for different quantity of growth increments.

The temporal evolution of the curvature at trunk base during radial growth and gravitropic reaction has been compared to computation with 10, 50, 111 and 1 110 growth increments, each of them being divided in 100 viscoelastic increments. Here, the input data of the model are those defined in part 3 for tree A, that is the same input data as those used for discussing the results in part 4. The results presented in Fig. 3b show that curves obtained for 50, 111 and 1 110 growth increments are superimposed. Between the 111 and 1 110 growth increments calculations, the relative difference in curvature is about 0.82%.

This low difference allows us to ignore computational errors concerning the comparison between model results and experimental measurements of curvature. Moreover, in all following calculations, 111 growth increments (∆t dr ) divided in 100 viscoelastic increments (∆t) will be used during the calculation period of 70 days. This quantity of growth increments corresponds to the limit of a classical 32-bit operating system because of the maximum memory size of 2 GB allocated to MATLAB on such systems.

Modelling normal forces and bending moments due to maturation

In equation 7, the variation in normal forces and bending moments {∆F (i seg , t)} is calculated as the sum of the normal forces and bending moments due to weight and maturation (see equation 1).

Normal forces and bending moments due to weight can easily be calculated using knowledge of wood density and radial growth increments. Thus, the variation is calculated as the difference between two time steps.

Normal forces and bending moments due to maturation are calculated by considering that maturation stresses are generated at the time of the creation of growth increments by the full locking of MS. Thanks to the model presented above, these MS can vary with each spatial or growth temporal increment; they are denoted ∆ε m f ace (i seg , i dr , t(i t )). The maturation stresses created because of the impediment of these MS are calculated thanks to the modulus of elasticity (MOE) of Burgers' model, denoted E 0,f ace (i seg , i dr , i t ), as follows:

∆σ m f ace (i seg , i dr , t(i t )) = E 0,f ace (i seg , i dr , t(i t ))∆ε m f ace (i seg , i dr , t(i t )) (9)
Normal forces ∆N m and bending moments ∆M m produced by these maturation stresses are calculated by integration onto the surface of each spatial increment denoted S f ace (i seg , i dr ):

         ∆N m f ace (i seg , i dr , t(i t )) = ∫ S f ace (i seg ,i dr ) ∆σ m f ace (i seg , i dr , t(i t ))dS ∆M m f ace (i seg , i dr , t(i t )) = ∫ S f ace (i seg ,i dr ) ∆σ m f ace (i seg , i dr , t(i t ))zdS (10)
Since MOE and MS are constant within a single spatial increment, normal forces and bending moments due to wood maturation can be written as follows:

       ∆N m f ace (i seg , i dr , t(i t )) = E 0,f ace (i seg , i dr , t(i t ))∆ε m f ace (i seg , i dr , t(i t ))S f ace (i seg , i dr ) ∆M m f ace (i seg , i dr , t(i t )) = E 0,f ace (i seg , i dr , t(i t ))∆ε m f ace (i seg , i dr , t(i t ))A f ace (i seg , i dr ) (11)
where

A f ace (i seg , i dr ) = ∫ S f ace (i seg ,i dr )
zdS is the static moment of a given spatial increment. The total increments of normal forces and bending moments due to wood maturation on a given segment and time increment are computed as the sum of the normal forces and bending moments on each radial increment of the upper and lower face. This gives finally:

             ∆N m (i seg , t(i t )) = i dr =i t ∑ i dr =1 ∆N m sup (i seg , i dr , t(i t )) + ∆N m inf (i seg , i dr , t(i t )) ∆M m (i seg , t(i t )) = i dr =i t ∑ i dr =1 ∆M m sup (i seg , i dr , t(i t )) + ∆M m inf (i seg , i dr , t(i t )) (12)
3 Input data of the model

The model described above is able to consider any spatio-temporal variations in mechanical parameters.

Input data are needed for this model: spatio-temporal variation in geometry, mass, viscoelastic properties, and MS must be known. Several prior experimental studies [START_REF] Coutand | The gravitropic response of poplar trunks: Key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF]Pot et al, 2013a,b;[START_REF] Pot | Mechanical characterization of green wood during maturation process and modeling of gravitropic reaction of young poplar[END_REF] are used to this end. However, it is not possible to measure MS variations during wood maturation.

Since the variations of the other parameters were known, MS were obtained by inverse identification thanks to the comparison between tree righting calculated with the model and the experimental observations.

The MS temporal variation obtained from this identification is presented in this section, as well as input parameters that were measured experimentally (mechanical properties, cross section geometry and mass distribution). The principal input parameters of the model with progressive maturation are summarized in table 2.

The experimental data are measured on two different sets of two-year-old hybrid poplars (Populus give temporal variations in the parameters of Burgers' model according to the age of the wood cells. In this study, it is assumed that all cells have the same viscoelastic properties once they are matured. This approximation is consistent with the results of Pot et al (2013a). In addition, negligible differences were found by Pot et al (2013b) between the viscoelastic behaviour of OW and that of TW.

Consequently, in the present model, the same values of viscoelastic parameters are used for both faces.

Without any information about longitudinal variations, it is assumed that mechanical properties do not vary with the longitudinal position in the tree. For the sake of simplicity, longitudinal increments i seg and "f ace" index are omitted in expressions of viscoelastic parameters. Finally, viscoelastic parameters depend only on the maturation state, i.e. wood cell age. In the model, the relation between age and radial increments i dr or temporal increments i t is obtained thanks to equation 5.

Temporal variations in each viscoelastic parameter of Burgers' model are approximated by appropriate mathematical functions that smooth the variations of the experimental data (see Fig. 9 in Pot et al (2013b)).

Thus, E 1 is modelled by a linear function: E 1 ( a) = 1.1 + 0.17 a where E 1 is in GPa and a in days. η 1 is obtained by the relation η 1 = E 1 τ 1 with a constant value of τ 1 = 38 hours. 1 η ∞ is modelled by a slightly decreasing exponential function: 1 η ∞ = 6.75 10 -12 exp (-9.4 10 -3 a), where η ∞ is in Pa.days and a in days.

The temporal variation of E 0 was studied in [START_REF] Pot | Mechanical characterization of green wood during maturation process and modeling of gravitropic reaction of young poplar[END_REF]. It was modelled thanks to the addition of an exponential function and a Gaussian function that are schematically presented in Fig. 4. Contrary to other parameters of Burger's model, it was shown that there is a difference between TW and OW: the height of the peak due to the Gaussian function was higher for TW (6500 MPa) than for OW (3800 MPa).

Cross section geometry

The diameter of trees A, B, C and D was regularly measured during the season from the base to the top of the tree as described in [START_REF] Coutand | The gravitropic response of poplar trunks: Key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF]; [START_REF] Pot | Mechanical characterization of green wood during maturation process and modeling of gravitropic reaction of young poplar[END_REF]. The diameter at the trunk base on the day of tilting is presented in table 1. The spatial and temporal variations in diameter for each of these trees are used as input data in the biomechanical model.

The cross section is supposed to be perfectly circular. The geometry is defined thanks to the eccentricity of the growth increments and a TW sector angle (Fig. 1). Eccentricity is defined by a coefficient of eccentricity set to 0.4, which means that the centre of each growth increment is located at a distance from the pith centre of 40% of the growth ring thickness. This value was found in Pot et al (2013b). The angle of the TW sector is set to 180 • thanks to experimental measurements which show that it does not significantly vary with longitudinal position in the tree and that the mean value is 180±20 • [START_REF] Pot | Mechanical characterization of green wood during maturation process and modeling of gravitropic reaction of young poplar[END_REF]. As a first approximation, both of these geometrical parameters are considered constant with space and time.

The pith is considered in the cross section as a cylinder of constant radius of r pith = 2.65 mm, according to experimental measurements all along the trunk [START_REF] Pot | Mechanical characterization of green wood during maturation process and modeling of gravitropic reaction of young poplar[END_REF]. Note that pith is considered as an elastic material with a MOE of 30 MPa, which is one hundred-fold lower than wood, as suggested by [START_REF] Alméras | Biomechanical design and long-term stability of trees: morphological and wood traits involved in the balance between weight increase and the gravitropic reaction[END_REF].

Mass distribution

Mass distribution along the trunk

The mass distribution of the two sets of trees (11 trees) was measured by weighing different parts of the trees. After tree felling, the trunk was divided into logs of 35 cm length. They were weighed when the trees were felled (i.e. in green state). Density was then calculated thanks to diameter measurements under the hypothesis of a perfectly conical shape. The results for trees felled at different dates along the season are shown in Fig. 5a. It appears that green wood density increases from about 600 to 1100 kg.m -3 along the length of the trunk. This result could be due to a higher water content in the upper parts of the trunk.

Indeed, the pith is in greater proportion in the upper part of the trunk since its diameter remains almost constant along the tree. This highly porous material must be full of water and then have a density close to 1000 kg.m -3 , which finally induces a higher density at the top of the tree.

In the following, the longitudinal variation in wood density is modelled thanks to a cubic polynomial equation that fits the experimental data, as follows:

ρ(x) = 17.42x 3 -0.91x 2 -3.25x + 627.50 ( 13
)
where ρ is the density in kg.m -3 and x is the distance from trunk base in m. Since the density is used in the model, the mass of the trunk automatically increases when growth occurs.

Mass distribution of the foliage

The mass of the foliage of the set of seven trees felled at different dates along the season is presented in Fig. 5b. It appears that there is an increase in foliage mass along the season. A linear approximation gives a rate of mass increase of 3.6 10 -3 kg.day -1 . This rate was used to model the temporal variation in foliage mass of each tree.

The mass of the foliage of trees A, B, C and D was measured at the end of the season. The foliage was divided in two parts along the trunk, and the length of these parts was measured. The total mass for each tree is presented in table 1. Note that foliage mass can vary by 30% between trees. In the model, the foliage mass of each tree is used in input by distributing linearly the mass of the two measured parts over their length for each tree. The variation in foliage mass along the season is modelled thanks to the rate of mass increase found above. It is important to take this variation into account, because over a period of 70 days the foliage mass can vary by more than 50%. In terms of mass it represents about 40% of the total tree mass.

Maturation strains

Definition of the temporal variation

To the best of the authors knowledge, no method for measuring MS during wood maturation exists, whereas it is the most influential parameter according to various authors [START_REF] Alméras | Effect of circumferential heterogeneity of wood maturation strain, modulus of elasticity and radial growth on the regulation of stem orientation in trees[END_REF][START_REF] Coutand | Twig: A model to simulate the gravitropic response of a tree axis in the frame of elasticity and viscoelasticity, at intra-annual time scale[END_REF].

Measurements of residual longitudinal maturation strains (RLMS) exist, but they give information on strains in the periphery of the tree only, i.e. cells that are always in the same maturation state. As mentioned by [START_REF] Gril | Tree mechanics and wood mechanics: relating hygrothermal recovery of green wood to the maturation process[END_REF], MS must not appear instantaneously; therefore it is necessary to consider their temporal variation with maturation (wood cell age in our case). Moreover, the level of MS produced by wood cells may vary along the season and with the gravitropic response because of different qualities of TW, as mentioned by [START_REF] Coutand | The gravitropic response of poplar trunks: Key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF]. Thus, it seems necessary to model temporal variations in terms of MS that are due both to maturation and variations along the season.

A Gaussian cumulative distribution function was chosen to model the temporal variation in MS with wood cell age (Fig. 6a). The shape of such a function enables the modelling of a smooth increase in MS during a customizable period (the maturation time), and then an asymptote is reached, which models the half of MS creation when the wood cells are mature. MS variation with wood cell age for a given face is defined by the equation:

ε m f ace ( a, t) = ε m tot,f ace (t) ∫ a 0 exp ( (u -τ a,f ace ) 2 ln (0.01) (δ a,f ace /2) 2 ) du ( 14
)
where τ a,f ace is the characteristic time which defines the middle of the distribution, δ a,f ace is the width of the distribution at 1% of its height, and ε m tot (t) is the final value of MS when the total strain has been reached.

ε m tot (t) can vary with time in order to model variations in MS along the season. Its variations are arbitrarily defined with the same type of Gaussian distribution function (Fig. 6b):

ε m tot,f ace (t) = C f ace ∫ t 0 exp ( (u -τ t,f ace ) 2 ln (0, 01) (δ t,f ace /2) 2 ) du ( 15
)
where τ t,f ace is the characteristic time which define the middle of the distribution, δ t,f ace is the width of the distribution at 1% of its height, and C f ace is a constant which is adjusted in order to obtain the desired maximum value of MS (which is noted ε m tot,f ace (t → +∞)). With this second function, wood cells created at the beginning of the gravitropic reaction produce lower MS than wood cells created later in the season, which simulates a variation in TW quality at the beginning of the gravitropic reaction. This function determines the regulation of MS along the season regardless of the righting of the tree, thus it must be identified from experimental measurements.

Inverse identification of MS temporal variation

Because of the lack of experimental methods to determine the MS temporal variations, the parameters defined above were identified by fitting the modelled temporal variations of the curvature at the trunk base of tree A to experimental data. To identify model parameters, some assumptions were made, because the numerous parameters induce an indeterminate system. Firstly, the same temporal parameters were used for both upper and lower faces, which means that maturation was presumed to occur at the same rate in both TW and OW. Secondly, the temporal parameters relative to MS variations with wood cell age were fixed according to experimental data:

δ a,f ace , which is the duration of maturation in terms of MS, is determined thanks to an experimental study (see [START_REF] Pot | Mechanical characterization of green wood during maturation process and modeling of gravitropic reaction of young poplar[END_REF]) which showed that the MOE increases while maturating up to 13 to 25 days after cell creation. In the present study, it is assumed that the average of this duration corresponds to the duration of the maturation in terms of MS, thus: δ a,f ace = 19 days.

τ a,f ace , which is the time when MS reach 50% of their final value, can be reasonably bounded between 5 and 12 days. Indeed, higher value than 12 days would lead to very low deformations for wood cells aged of 8 days (cf. τ a,f ace = 15 days in Fig. 6a), which is not consistent with the fact that RLMS measurements provide significant deformations in the first millimeter of the outer part of growing trees (see [START_REF] Coutand | Mechanosensing is involved in the regulation of autostress levels in tension wood[END_REF]). Lower value than 5 days would signify that maturation is almost instantaneous, this particular case is discussed in part 4.

Thirdly, the maximum MS of the lower face, denoted ε m tot,inf (t → +∞) was set to 0.005. By doing so, it represents a moderate proportion (between 11% and 36%) of the maximum MS of the upper face that is found below, which seems probable.

By considering these assumptions, the maximum MS of the upper face (ε m tot,sup (t → +∞)) and the temporal parameters relative to its variation along the season (τ t,f ace and δ t,f ace ) were identified by minimizing the squared differences between experimental and modelled curvature at trunk base of tree A up to 32 days after tilting. The optimization was done thanks to a constrained interior-point algorithm. Lower and upper boundaries of the optimization were chosen to keep a physical meaning of the parameters, that is : 1 < τ t,f ace < 30, 2 < δ t,f ace < 70 and 0.005 < ε m tot,sup (t → +∞) < 0.1. Different initial values of the parameters were tested close to these bounds, without significant change in the optimized result.

Three different values of τ a,f ace were tested in input (5, 8 and 12 days, cf. Fig. 6a). The results are presented in table 3. The identification gives an interval of [0.014;0.044] for ε m tot,sup (t → +∞), which is discussed in section 4.4. The three results are comparable in term of fitting of the temporal evolution of curvature. In the following discussions, the parameters obtained for τ a,f ace = 8 days are chosen. These parameters are listed in table 2. The corresponding MS temporal variation along the season is presented in Fig. 6b. Thanks to these parameters, the temporal variation in tree A curvature at the trunk base was correctly simulated up to 32 days after tilting (Fig. 7a). The MS parameters obtained for this tree were then used to simulate the gravitropic reaction of trees B, C and D (see section 4.3).

Input data for the elastic model and the model with instantaneous maturation

The parameters used for the viscoelastic model with progressive maturation are described in the above sections. To the best knowledge of the authors, it is the first time that such a model has been proposed; it is useful, therefore, to compare it to models with more usual hypothesis. To do this, the same model was used but the input parameters were modified.

Firstly, the effect of viscoelasticity was studied in comparison with an elastic model with progressive maturation. The same MOE and MS parameters as viscoelastic with progressive maturation calculation were used. The only difference was the viscoelastic behaviour, which was not taken into account.

Secondly, the proposed model was compared to a classical elastic model with instantaneous maturation. MOE were set instantaneously to their mean value according to experimental bending tests, that is E 0,inf (i seg , i dr , t) = 3 GPa for the lower face and E 0,sup (i seg , i dr , t) = 5.1 GPa for the upper face. This induces better righting-up efficiency; thus, it was necessary to use lower MS to obtain the same curvature. This is why, by maintaining the same value of MS of 0.005 on the lower face, MS of 0.0052 on the upper face enabled us to obtain the same curvature value as with experimental measurements 28 days after tilting.

Results and discussion

Comparison of curvature

Results of the model in terms of curvature at the trunk base are presented in Fig. 7a. They are compared to experimental data (solid black line). Despite the better righting-up efficiency regarding the level of MS, an elastic model with instantaneous maturation (dash-dotted green line) cannot correctly simulate the temporal curvature variation. Indeed, the lag period during which there is no tree curvature cannot be simulated. For this instantaneous model, the curvature increases quickly at the beginning of growth, and then the rate of curvature decreases progressively because of the increase in diameter of the trunk. Indeed, this increase in diameter induces an increase in flexural rigidity, which counterbalances the effect of MS.

Conversely, the viscoelastic model with progressive maturation (solid red line) can simulate the slow decrease in curvature during the first days after tilting. This decrease is explained by the shape of the temporal variation in MS maximum value, increase in tree weight, and viscoelastic creep. The elastic calculation (dashed blue line) shows that the decrease in curvature is lower than that obtained with the viscoelastic model with progressive maturation in this zone. The quick increase in curvature between 18 and 32 days after tilting is also steeper with the viscoelastic model with progressive maturation than for the elastic model with progressive maturation. This shows that viscoelasticity has a significant effect on tree gravitropism, which was already highlighted by the models of [START_REF] Dlouhá | Biomechanical performances of trees in the phase of active reorientation[END_REF] and [START_REF] Coutand | Twig: A model to simulate the gravitropic response of a tree axis in the frame of elasticity and viscoelasticity, at intra-annual time scale[END_REF]. However, this is the first time to the authors knowledge that this result has been confirmed by using experimental data as an input to the model.

From 32 days after tilting to 70 days after tilting, the rate of curvature in the model remains almost constant, while experimental results show a decrease in this rate. The model's behaviour is consistent with the temporal variation in total MS used, which reaches its maximum value about 32 days after tilting (Fig. 6b) and then remains constant. To model the decrease in curvature rate that is observed experimentally, it would be necessary to change the temporal variation of MS along the season, with a decrease 32 days after tilting. This result shows that there is a decrease in the intensity of MS, which can be interpreted as the regulation of the level of MS by the tree in order to avoid becoming too curved and exceeding the vertical limit. This latter explanation is consistent with the results of [START_REF] Coutand | The gravitropic response of poplar trunks: Key roles of prestressed wood regulation and the relative kinetics of cambial growth versus wood maturation[END_REF], which show that there exists an autotropic phase when the trunk is decurved to finally become straight and vertical.

Comparison of stress profiles

In Fig. 7b, the stress profiles of tree A obtained with the model for the same three assumptions discussed above are presented (no experimental data are available). The stress profiles are compared to the only temporal position when the curvature at the trunk base is almost the same for each model, that is 28 days after tilting. Consequently, these 3 different stress profiles give rise to the same trunk curvature.

All three models present a Kubler-like profile [START_REF] Kubler | Studies on growth stresses in trees. 1. the origin of growth stresses and the stresses in transverse direction[END_REF] in the central part, with a radius of between 2.65 mm and 17.6 mm from the centre of the pith. This is due to the initial calculation of the model, for which the tree is considered to grow symmetrically with OW mechanical properties up to the radius measured when tree was tilted, that is 17.6 mm for tree A. Stresses in the pith are not presented, since its MOE is very low.

It is worth noting that compressive stresses close to the pith reach very high negative values for the elastic model with instantaneous maturation (less than -80 MPa), whereas taking maturation into account leads to maximum compressive stresses of -40 MPa and adding viscoelasticity leads to maximum compressive stresses of -20 MPa. The results of the elastic model with instantaneous maturation are consistent with theoretical model of [START_REF] Kubler | Studies on growth stresses in trees. 1. the origin of growth stresses and the stresses in transverse direction[END_REF], for which stresses tend towards infinity close to the pith. However, this result is not physically admissible, contrary to the stress profile obtained thanks to the proposed model, for which both progressive maturation and viscoelasticity induce lower compressive stresses.

A discontinuity in stresses is observed when the tree is tilted, corresponding to a radius of 17.6 mm in Fig. 7b. This is due to the inclination of the tree, which induces a brutal change in stresses. When the tree is tilted, the upper face is tensed and the lower face is compressed. Then new growth increments begin their maturation on a structure for which internal stresses have changed instantaneously, bringing about the observed discontinuity.

In the part of the tree that grows when the tree was tilted (r > 17.6 mm), the elastic model with instantaneous maturation presents a quasi-linear shape for the stress profile, while models with progressive maturation show a more complex profile. For the model with instantaneous maturation, stress is maximum at the periphery of the tree i.e. newly formed cells. This is not realistic, because cells that have just been created are not mature and have a very low MOE, thus they cannot exhibit such a stress level. Conversely, models with progressive maturation show zero stress for new cells, and it increases progressively as the cells become older, i.e. the distance from the periphery increases. This is due to the progressive appearance of both MS and MOE variations, which are defined in Fig. 4 and 6a. Moreover, stresses in the zone of 17.6 to 19 mm from the pith on the upper face of the tree are lower for models with progressive maturation than for models with instantaneous maturation. This is due to the progressive increase in total MS after tilting (Fig. 6b), which enables the lag period to fit the curvature. The initial deflection just after tilting is well described for each tree all along its length (dashed-dotted blue lines in Fig. 8). This shows that MOE measurements by bending tests are appropriate to modelling tree mechanical behaviour, and that the hypothesis of a constant MOE all along the trunk is admissible before gravitropic reaction.

The modelled shape of tree A 28 days after tilting is compared with experimental measurements (dashed green lines in Fig. 8a). The modelled curvature fits the experimental data not only at the trunk base as shown in Fig. 7a, but also all along the trunk. This validates the hypothesis of identical maturation in terms of MS and MOE all along the trunk up to 28 days after tilting.

For trees B, C and D, experimental and modelled shapes 28 days after tilting are in good agreement.

The same observation can be made 7 days after tilting, when the 4 trees present, experimentally and by modelling, a similar sagging. The same observations can be made for intermediate times. Thus, by using exactly the same mechanical properties and MS temporal variation, the model can simulate the gravitropic movement of different trees up to 28 days after tilting. However, from 28 to 32 days after tilting, the model and experimental results start to diverge, and finally the modelled shape 70 days after tilting is very different from the experimental shape, as shown for each tree of Fig. 7. This divergence appears while the temporal variation of total MS defined here remains constant over time (Fig. 5b). Thus, this behaviour shows that MS must start to decrease about 30 days after tilting, possibly because of an autotropic reaction that prevents the tree to overshoot the vertical. Indeed, in a recent paper [START_REF] Coutand | Mechanosensing is involved in the regulation of autostress levels in tension wood[END_REF], it is shown that RLMS of the upper side start to decrease about 35 days after tilting.

Discussion on MS modelling

By using the same mechanical parameters and MS in input, the viscoelastic model with progressive maturation presents satisfying results in terms of shape modelling for different trees up to 32 days after tilting.

The maximum MS values of the upper face, which are obtained by inverse identification, are between 0.014 and 0.044 (with maximum MS of the lower face set to 0.005). These level of strain are far greater than the RLMS measurements of TW performed on the same tree clones (less than 0.003 in [START_REF] Coutand | Mechanosensing is involved in the regulation of autostress levels in tension wood[END_REF]).

However, such strains are necessary to obtain the high curvature increase between 18 and 32 days after tilting.

This discrepancy could be explained by the fact that RLMS are not a good indicator of the MS that exist in the tree. Indeed, RLMS measurements were performed on several wood cells located in the periphery of the trunk (see [START_REF] Yang | Growth stress, its measurement and effects[END_REF] for a review). In the part of the tree where these measurements were performed, there are cells in different maturation states, and thus the RLMS are not representative of the MS of wood cells that are fully matured. Moreover, RLMS measurements are elastic measures:

the deformation due to stress release is measured instantaneously just after the grooves are made. Thus, viscoelastic deformations do not have time to occur during the measurement, whereas the viscoelastic behaviour of wood is significant, as it was shown in a previous work (Pot et al, 2013b). Indeed, it was shown in this latter study that longitudinal slats of green TW that are harvested from poplar trees and maintained in water curve over a period of several weeks because of internal maturation stresses and the viscoelastic behaviour of green wood. This curvature reach a value of 0.019 mm -1 in average for slats that are in the periphery of the tree. Assuming that maturation stresses are fully released when this autonomous curvature is reached, the radius of curvature, i.e. wood deformations, are linked to the differential of MS between the two sides of the slat. With a curvature of 0.019 mm -1 and a thickness of 1.3 mm as in Pot et al (2013b), a value of 0.025 for this differential of MS between the two sides of the slat is obtained. Since these strains are only due to the release of internal stresses, this shows that wood cells can reach strains of the same order of magnitude as those found by inverse identification thanks to the present model. Consequently, it can be concluded that RLMS cannot be assimilated to the MS that really occur during wood cell maturation. It would be useful to establish a relationship between RLMS and MS. Finally, the assumption of the model concerning MS seems acceptable in order to model quantitatively the first steps of gravitropic reaction.

Conclusion

A biomechanical model of tree gravitropism is proposed in this work. The main contribution is the modelling of the progressive maturation of wood while the tree is growing. This is done by taking into account spatio-temporal changes in the mechanical properties. The model uses experimental data of temporal and spatial variations in geometry and the mechanical properties of young poplar trees. To the best of the authors' knowledge, this is the first time that such comparisons between biomechanical model and experimental measurements of gravitropism have been performed.

MS were identified thanks to the model in order to fit the temporal variation in the curvature of a given tree during its gravitropic reaction. The testing of these MS on the gravitropic reaction of four different trees shows that the hypotheses made concerning temporal variations in MS are relevant. It appears that MS in TW must be higher than those obtained with usual estimations of MS, which are based on RLMS measurements. As a result, the exploitation of the present model shows that RLMS measurements underestimate the MS that are really experienced by wood cells. The main suspected cause is the viscoelastic behaviour of green wood, which is not taken into account by RLMS measurements.

The proposed model provides significant improvements in the modelling of tree gravitropism. Firstly, the modelling of temporal variations in MS over the course of the season enables the simulation of the temporal variation in tree curvature in the latency phase and at the beginning of the gravitropic phase. Secondly, the internal stress profile in the trunk is more realistic than stress profiles obtained with elastic models or models with instantaneous maturation. Further work is currently being carried out to simulate the end of the gravitropic phase (and possibly the autotropic phase) by taking into account the regulation of MS.

Fig. 2 )

 2 Fig. 2), I the second moment of inertia, and M w (x) the bending moment along the position x in the stem due to self weight. Burgers model parameters are mean value find in Pot et al (2013b): E 0 = 3 GPa, E 1 = 3.5 GPa, τ 1 = 38 hours, and τ ∞ = 200 days.

  deltoides x Populus nigra, cv I4551) that were tilted by about 35 • from the vertical at the beginning of the growing season. A first set of seven trees was used in[START_REF] Pot | Mechanical characterization of green wood during maturation process and modeling of gravitropic reaction of young poplar[END_REF] andPot et al (2013b) to measure mechanical properties and mass distribution during 63 days. A second set of four trees was used specifically for the present work for cross section and shape measurement in order to perform a comparison between the experimental and modelled righting of trees during gravitropic reaction. They were felled at the end of the season. They are named A, B, C and D in the following.3.1 Mechanical propertiesPot et al (2013b) performed creep tests on wood samples harvested in the basal part of the trunk. They

4. 3

 3 Fig.8presents both the modelled and experimental shapes of the 4 trees named A, B, C, D at different times after tilting. As explained before, the diameter and foliage mass of each tree were used as an input for

Fig. 1 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8

 1345678 Fig. 1 Schematic representation of the modelling of the trunk in the longitudinal-radial plane (a, b) and radial-tangential plane (c, d) at two consecutive growth steps: it = 1 (a, c) and it = 2 (b, d)

Table 1

 1 Diameter at the trunk base at the time of tilting and foliage mass 70 days after tilting for the 4 different trees studied Tree Diameter at trunk base at Foliage mass 70 days the time of tilting (mm) after tilting (kg)

	A	35.2	0.541
	B	31.4	0.414
	C	32.1	0.470
	D	30.9	0.473

Table 2

 2 Input parameters used for trees A, B, C and D for the viscoelastic model with progressive maturation

	Parameters	Value and unit
	Material properties (section 3.1)
	Burgers' parameter τ 1	38 hours
	Burgers' parameter E 1

Table 3

 3 Results of the identification of MS parameters by fitting the temporal evolution of curvature at trunk base of tree C, for different values of τ a,f ace

	Fixed parameter		Optimised parameters
	τ a,f ace (days)	τ t,f ace (days) δ t,f ace (days) ε m tot,sup (t → +∞)
	5	21	66	0.044
	8	14	59	0.027
	12	1.6	47	0.014

This work was supported by a grant from the Auvergne Regional Council and the European Regional Development Fund.

Conflict of interest:

The authors declare that they have no conflict of interest.


Luca Petrillo

Student Member, IEEE Theodoros Mavridis

Member, IEEE Julien Sarrazin

Aziz Benlarbi-Delaï

Philippe De Doncker

Statistical On-Body Measurement Results at 60 GHz

Keywords: GHz propagation, BAN, channel model

This paper studies the path loss and shadowing between two body mounted devices at 60 GHz. The temporal fading is experimentally investigated and the Doppler spectrum is characterized and modeled. Measurements have been conducted in an anechoic chamber for both horizontal and vertical polarizations.

I. INTRODUCTION

Future Wireless Body Area Networks (WBAN) will allow a variety of applications ranging from health care to entertainment or defense. Researches in the past years have been mainly concentrated on communicating systems working around 2.4 GHz [START_REF] Timmons | Analysis of the performance of ieee 802.15. 4 for medical sensor body area networking[END_REF] and in the Ultra Wideband (UWB) spectrum [START_REF] Fort | An ultra-wideband body area propagation channel model-from statistics to implementation[END_REF] because of the wide availability of the devices. Nowadays, the emergence of 60 GHz communicating systems [START_REF] Rappaport | State of the art in 60-ghz integrated circuits and systems for wireless communications[END_REF] encourages the study and development of new 60 GHz Body Area Networks (BAN) [START_REF] Alipour | Propagation characteristics for a 60 ghz wireless body area network (wban)[END_REF]. Specific on-body 60 GHz channels have been investigated [START_REF] Nechayev | Effect of body motion on propagation path gain at 60 ghz[END_REF], [START_REF] Nechayev | Use of motion capture for path gain modelling of millimetre-wave on-body communication links[END_REF] and some preliminary results for path gain on the body have been presented [START_REF] Wu | Antenna design and channel measurements for on-body communications at 60 ghz[END_REF]. Static propagation models for off-body [START_REF] Mavridis | Theoretical and experimental investigation of a 60 GHz off-body propagation model[END_REF] and on-body [START_REF] Petrillo | Analytic creeping wave model and measurements for 60 GHz body area networks[END_REF] 60 GHz communications have already been studied and a skin-equivalent phantom at 60 GHz has been proposed to validate plane-propagation models [START_REF] Chahat | Skin-equivalent phantom for on-body antenna measurements at 60 ghz[END_REF]. This paper proposes an experimental study to characterize path loss and shadowing between two body mounted devices and the fading due to breathing at 60 GHz. Due to the high attenuation at this frequency, high fading phenomena are expected.

II. MEASUREMENT CAMPAIGN

The experimental campaign has been conducted on a person, with the characteristics presented in table I, in an anechoic chamber to avoid any environment influence. The person under test (PUT) is a male wearing a cotton t-shirt. An Agilent E8361C Vector Network Analyser (VNA) and U-Band open wave-guide antennas have been used. We ensured that the open waveguides were oriented towards each other while collecting measurements, in order to prevent mis-alignment. Effects of small mis-alignments, which cannot be avoided, is palliated by the relative large beamwidth of open waveguides used as antennas in the azimuthal direction. To reduce the losses in the cables and reduce their lengths, the VNA has been placed inside the anechoic chamber and delivers a transmitted power of 0 dBm. A power amplifier with a gain of 30 dB at 60 GHz L. Petrillo, T. Mavridis and P. De Doncker are with OPERA Dpt. -Wireless Communications Group, Université Libre de Bruxelles (ULB), B-1050 Brussels, Belgium e-mail: ( lpetrill@ulb.ac.be ; tmavridi@ulb.ac.be ; pdedonck@ulb.ac.be) T. Mavridis, J. Sarrazin and A. Benlarbi-Delaï are with L2E Dept., Sorbonne Universités, UPMC Univ Paris 06, UR2, L2E, F-75005, Paris, France e-mail: (julien.sarrazin@upmc.fr; aziz.benlarbi delai@umpc.fr) has been used at transmission side to raise link budget. The VNA has been used on continuous wave mode.

Measurements have been performed for both vertical (normal to the body surface) and horizontal (longitudinal to the body surface) polarizations, and for five particular distances between antennas: 5, 10, 15, 20 and 25 cm. For each distance, we took different realizations of the measurements: vertical and horizontal links, with respect to the human body, were investigated, so that for each distance and for each polarization we took forty measurements. The positions have been chosen in order to have equal numbers of measurement on the chest and on the abdomen, as well as vertical and horizontal links. So, for example, for vertical polarization and for 5 cm between the antennas, we measured 10 vertical and 10 horizontal links on the chest, 10 vertical and 10 horizontal links on the abdomen. For 20 and 25 cm distances, vertical measurements correspond to 20 chest-abdomen links. For the distance 25 cm, horizontal measurements have not been taken, since the torso of the PUT was not sufficiently large to obtain such distance. A scheme is shown in Fig. 1, where stars linked by a dashed line indicate a measure for a particular distance, place on the PUT and orientation of the link. We have schematically indicated chest and abdomen by an horizontal line representing the diaphragm. Each measurement has been conducted during 10 seconds which is equivalent to 150 temporal acquisitions. The antennas were located at 2 cm from the body surface for both transmitter (TX) and receiver (RX) antenna. This choice permits to compare the influence of antenna body separation by comparison with [START_REF] Chahat | On-body propagation at 60 ghz[END_REF]. The measurement parameters are summarized in table II. Horizontal links Vertical links

III. ON-BODY PATH LOSS

A. Measurement Results and Path Loss Model

In the following, the experimental results are averaged over time and each measurement is presented as the mean value over the 150 time acquisitions.

Vertical polarization:

The path loss for vertical and horizontal links for vertical polarization are shown in Fig. 2.

The path loss L dB can be easily modeled with distance d as a logarithmic path loss :

L dB (d) = L 0 dB + 10n log 10 (d/d 0 ) (1)
where d 0 is a reference distance, L 0 dB is the path loss at d 0 and n is the path loss exponent. Shadowing can be modeled by a random variable following normal distribution, with zero mean and standard deviation σ L . We propose for both vertical and horizontal links two values of σ L , depending on the distance range. Model parameters are summarized in Table III.

TABLE III: Path loss parameters for vertical and horizontal links for vertical polarization

Vertical links

Horizontal links

d 0 =5 cm d 0 =5 cm L 0 =29.3 dB L 0 =30.87 dB n=2.62 n=3.1 d ≤20 cm d ≤10 cm σ L =2.36 dB σ L =2.13 dB d =25 cm d ≥15 cm σ L =5.85 dB σ L =7.68 dB Path Loss Measurement d [cm] Path Loss Measurement Path Loss [dB] d [cm]
Horizontal links Vertical links It can be deduced that the electromagnetic waves attenuate faster for horizontal links. The reason is that for horizontal links, the surface of propagation is more curved than for vertical ones because of the human body shape, and therefore diffraction is more severe. Moreover, measurements are more scattered for horizontal links, which is also a consequence of diffraction. For vertical links, it can be considered that the main propagation mechanism is the sum of direct and reflected waves off the flat surface of the body between the antennas. Constructive and destructive interference between these two contributions can be significant. With respect to measurement results presented in this section, path-loss as function of distance seems not to be useful due to high dispersion of measurement points. For both vertical and horizontal links, high standard deviation values arise because line-of-sight between the open waveguides can be obstructed by torso curvature (for horizontal links) or by stomach sphericity (for vertical links).

Horizontal polarization:

The path loss for vertical and horizontal links for horizontal polarization is shown in Fig. 3.

Again, the path loss model is given by equation ( 1) and path loss coefficients are given in Table IV. As compared to vertical polarization, horizontal polarization measurements show greater standard deviation, as is commonly found in lower frequency BAN channel measurements [START_REF] Ghannoum | Characterization of the uwb on-body propagation channel[END_REF]. This can be explained by the reflected wave influence on the total received field, which, for horizontal polarization, is more pronounced than for vertical polarization [START_REF] Chahat | On-body propagation at 60 ghz[END_REF]. Norton's formulations [START_REF] King | Cylindrical antennas and arrays[END_REF] predict this trend for Hertzian dipoles placed several wavelengths above a conductive surface. It is shown that the combination between direct and reflected waves on the body surface gives a more oscillating pattern for horizontal polarization with respect to spatial parameters. For vertical links, the standard deviation linearly increase at the rate of 0.3 dB/cm, from a first value equal to 2.61 dB found at 5 cm. For horizontal links the trend is even worse due to the adverse effects of body curvature. Regardless of polarization, measurement results suggest that path-loss for on-body channel links on the front side of the torso at 60 GHz is not useful to describe radio channel because of too high variability of measurement points around the fitting. This is due to the small wavelength at 60 GHz that significantly affects the channel spatial variation due to different spots of transmitter and receiver on the torso. We are of the opinion that point-to-point scenario analysis should be used instead.

IV. TEMPORAL FADING

To study temporal fading, different distributions (Normal, Lognormal, Gamma, Nakagami, Rayleigth, Rice, Weibull and Uniform) were investigated. For each model, estimation results were evaluated through Mean Square Error (MSE). Results are plotted in Fig. 4. Lognormal distribution is then adopted, which permits to represent temporal fading in dB by a normal distribution. The standard deviation (STD) is evaluated for each polarization, link and distance. It is increasing with the distance, as it has been observed at lower frequency [START_REF] Liu | Dynamic channel modeling at 2.4 ghz for on-body area networks[END_REF]. Therefore, we characterize it by:

σ F = µ σF + σ σF N (0, 1) (2)
µ σF = σ F0 + n σF (d -d 0 ) (3)
Where N (0, 1) is a normal process of zero mean and variance equal to 1. Values of n σF , σ F0 , σ σF for d 0 = 5 cm are reported in Table V.

Due to the small wavelength, on-body channel at 60 GHz on the front side of the torso is subject to dynamical body scattering because of breathing, which cause the thorax and the abdomen to inflate and deflate. Fig. 5 shows two examples of measurements in the temporal domain, normalized with respect to their mean values S 21 . The first one (solid line) corresponds to a distance between antennas equal to 15 cm, vertical polarization and vertical link on the chest. This is almost a static channel, where temporal fading follows a normal distribution in logarithmic scale. On the other hand, the dashed line clearly reveals a periodic movement that affects the channel between the antennas : it corresponds to the same 15 cm distance between antennas for the same polarization and for an horizontal link on the chest. The breathing affects the channel, since the chest naturally expands and compress, varying the distance between antennas. A similar effect is experimentally found for vertical links greater than 20 cm (chestabdomen), caused by stomach's compressing and deflating. For links affected by respiration, the temporal fading is not normally distributed, as it has already been reported in [START_REF] Petrillo | Analytic creeping wave model and measurements for 60 GHz body area networks[END_REF] for abdomen/chest links, but its probability density function is almost uniformly distributed.

We are of the opinion that the reason has to be found in the breathing action that affects the channel and consequently Doppler Spectrum needs to be investigated to characterize this phenomenon.

A. Breathing Doppler Characterization

To characterize the Doppler spectrum we took a new set of measurements by increasing the acquisition length, which has been set to 30 sec. Measurements have been taken for both vertical and horizontal polarizations with the same setup as previous measures. We have investigated an horizontal link on the chest and a vertical chest-stomach link. The Doppler spectrum is estimated with the Welch periodogram method [START_REF] Welch | The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms[END_REF] by normalizing the channel gains to 0 dB. The average Doppler Power Spectral Density (PSD) of a typical body link affected by breathing is shown in Fig. 6: the zero Doppler frequency is found because S 21 oscillates around a mean value, whereas the peaks are at fundamental frequency, which is nearly 0.18 Hz, and its harmonics. The breathing cycle then lasts approximately 5 sec, which it is considered as normal at rest [START_REF] Lindh | Delmar's comprehensive medical assisting: administrative and clinical competencies[END_REF]. Following [START_REF] Liu | Statistical characterization and modeling of doppler spectrum in dynamic on-body channels[END_REF], we propose a model for respiration Doppler PSD:

PSD(f D ) = 1 1 + sf 2 D ∞ m=1 x 2 m 1 1 + 4 m a + s 2 (f D ± mf 0 ) 4
(4) where f 0 is the breathing frequency and s, x and a are independent parameters of the model whose values are estimated by minimal square error (MSE) estimator. The parameters estimation is shown in Table VI for different torso links.

V. CONCLUSION

In this paper, we presented the results of a measurement campaign conducted at 60 GHz on the front of the torso of a human subject for both vertical and horizontal polarizations. A path loss has been derived for horizontal and vertical links, while large deviations around the models have been observed regardless of polarization. Temporal fading is then discussed and investigated: a normal fit in dB scale is adopted and a model for STD is derived. Doppler spectrum is evaluated and an analytical model is introduced. Estimated parameters are reported.

Fig. 1 :

 1 Fig. 1: Measurement scheme. Stars correspond to measurement points on the front of torso. We depicted horizontal and vertical links of different lengths, on the chest, on the abdomen and one vertical chest-abdomen link. The horizontal line represents diaphragm and separate chest from abdomen.

Fig. 2 :

 2 Fig. 2: Measurement results for vertical polarization, vertical and horizontal links. The dots stand for time averaged measurement and the solid line for the path loss.

Fig. 3 :

 3 Fig.3: Measurement results for horizontal polarization, vertical and horizontal links. The dots stand for time averaged measurement and the solid line for the path loss.

Fig. 4 :Fig. 5 :

 45 Fig.4: Average MSE of common distribution fits. The above row presents the vertical polarization and the below row presents the horizontal polarization. The left column is the vertical links while the right column is the horizontal links.

Fig. 6 :

 6 Fig. 6: Normalized Doppler PSD and model of body link affected by breathing influence (vertical polarization, chest).

TABLE I :

 I Physical quantities of the body

	Quantity	Value
	Gender	Male
	Height	172 cm
	Weight	60 Kg
	Torso Perimeter	86 cm

TABLE II :

 II Parameters of the measurement campaign

	Symbol	Value
	f	60 GHz
	Measurement time	10 s
	Acquisitions per measurement	150
	Body-antenna distance	2 cm
	Reference distances	5, 10, 15, 20 and 25 cm
	Path Loss Measurement	Path Loss Measurement
	Path Loss [dB]
	d [cm]	d [cm]

TABLE IV :

 IV Path loss parameters for vertical and horizontal links for horizontal polarization

	Vertical links	Horizontal links
	d 0 =5 cm	d 0 =5 cm
	L 0 =29.98 dB	L 0 =32.14 dB
	n=3.10	n=3.29
	d ≥5 cm	d ≤10 cm
	σ L =2.61+0.3(d-5) dB	σ L =3.77 dB
		d ≥15 cm
		σ L =8.27 dB

TABLE V :

 V Normal fading parameters

	Channel	nσ F	σ F 0	σ sigma F
	V. Pol. V. links	0.028	0.53	0.67
	V. Pol. H. links	0.073	0.43	0.98
	H. Pol. V links	0.053	0.66	1.15
	H. Pol. H links	0.086	0.35	0.9

TABLE VI :

 VI Doppler PSD estimation

	Measurement	f 0 [Hz]	s [Hz -2 ]	x[W ]	a
	V. Pol. chest	0.18	85980	339	287
	H. Pol. chest	0.2	77150	281.9	75.8
	V. Pol. chest-abdomen	0.18	81450	933.5	673.7
	H. Pol. chest-abdomen	0.25	63840	168.2	11.9

ACKNOWLEDGMENT

This work was performed within the Labex SMART supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02. The authors would like to thank Prof. B. Huyart and Mr A. Khy from Telecom ParisTech for their helpful advice and willingness.


Pavel Arapov
email: arapov@i3s.unice.fr

Michel Buffa
email: buffa@i3s.unice.fr

Amel Ben Othmane
email: amel.ben-othmane@inria.fr

A Wiki Way Of Programming For The Web Of Data

Keywords: K.4.3 [Organizational Impacts]: Computer-supported collaborative work Algorithms, Performance, Design, Experimentation, Languages Semantic Web, Web2.0, Wikis, Semantic Wikis, Knowledge Management, Web Applications

WikiNEXT is a wiki engine that enables users to write rapidly applications directly from the browser, in particular applications that can exploit the web of data. WikiNEXT relies on semantic web formalisms and technologies (RDF/RDFa lite) to describe wiki page content and embedded metadata, and to manipulate them (for example, using the SPARQL language). WikiNEXT is a mix between a web-based IDE (Integrated Development Environment) and a semantic wiki. It embeds several editors (a WYSIWYG editor, and an HTML/JavaScript editor + a JavaScript library manager) for coding in the browser, provides an API for exploiting semantic metadata, and uses a graph based data store and an object oriented database for persistence on the server side. It has been specially designed for writing online programming tutorials (i.e. an HTML5 tutorial, a semantic web tutorial on how to consume linked data, etc.), or more generally for developing web applications that can be mixed with more classical wiki documents (in fact all WikiNEXT pages are web applications). The tool is online 1 , open source 2 ; screencasts are available on YouTube (look for "WikiNEXT").

INTRODUCTION

The evolution of the Internet from static pages to web applications changed the paradigm of web programming. Furthermore, technologies and trends changed at a very fast pace these two last years, with the rise of HTML5 and the appearance of powerful JavaScript frameworks, making web-based IDEs (Integrated Development Environments) possible, such as jsbin.com, jsfiddle.com or c9.io. These online applications make possible to write and test JavaScript code directly into the browser. Their online status makes easy to share your code or clone someone else's code. With WikiNEXT, we tried to mix the writing of web application proposed by web based IDEs, with features proposed by semantic wikis: formal representations of data in a graph based knowledge base, reasoning, requesting the web of data, annotation of documents, etc. We will show in this paper the dynamic creation of data models for the knowledge base and the manipulation of these data by web applications written directly using the wiki online editors.

STATE OF THE ART

Web-Based IDEs: JavaScript based source code editors like Code Mirror3 , ACE Cloud94 or ternjs 5 propose features such as syntax highlighting, auto-completion, and have been used by many online IDEs , well known by web developers, such as jsbin.com, jsfiddle.net or tinkerbin.com. These IDEs enable real time editing of HTML/CSS/JavaScript code in the browser, instant preview when files are updated, etc. Most of these tools, however, are sandboxes for testing code online or for writing small examples.

Semantic Wikis:

The first wave of semantic wikis started in 2005, with wiki engines such as Semantic Media Wiki [START_REF] Krötzsch | Semantic mediawiki[END_REF], IkeWiki [START_REF] Schaffert | IkeWiki: A semantic wiki for collaborative knowledge management[END_REF], OntoWiki [START_REF] Auer | OntoWiki-A tool for social, semantic collaboration[END_REF] or SweetWiki [START_REF] Buffa | SweetWiki: A semantic wiki[END_REF]. They all proposed to add semantic metadata to documents, and used internally some formal language to store them, such as RDF (Resource Description Framework) or OWL (Web Ontology Language). These metadata can be exploited to add new functionalities to the wiki: augmented navigation, visualization of metadata, search and retrieval or reasoning. The SPARQL language6 is often used internally for querying RDF data. Today, semantic wikis are still active and there are new solutions, such as the Halo7 extension of Semantic Media Wiki, that proposes forms, auto-completion, a WYSIWYG editor; or more complex extensions such as MoKi [START_REF] Rospocher | Lindstaedt: MoKi: the Modelling wiKi[END_REF] (focuses on enterprise modeling) and OWiki [START_REF] Di Iorio | Ontology-driven generation of wiki content and interfaces[END_REF] (ontology-driven generation of wiki content), also based on Semantic Media Wiki. The KiWi project (Knowledge in Wiki [START_REF] Schaffert | Kiwi -knowledge in a wiki[END_REF]), successor of IkeWiki, or SweetDeki [START_REF] Buffa | ISICIL: Semantics and Social Networks for Business Intelligence[END_REF] the wiki that succeeded to SweetWiki [START_REF] Buffa | SweetWiki: A semantic wiki[END_REF], and has been integrated in the ISICIL ANR project [START_REF] Buffa | ISICIL: Semantics and Social Networks for Business Intelligence[END_REF] are more like Content Management Systems with semantic enhancements. In the next section we present WikiNEXT, which is a mix between a semantic wiki and a Web-based IDE.

WikiNEXT

Enterprise wikis, such as MindTouch, Confluence, Sharepoint, powerful solutions for knowledge management, can be considered more as content management systems (CMS) that provide different modules: wiki, blogs, social network, etc. These modules are very often developed as an extension or as a plugin and often require server side development, as well as defining a data model.

WikiNEXT makes this possible using only client side development, directly in the browser.

A WikiNEXT page has two parts and two editors consensually: HTML and JavaScript. The HTML part holds the view part and the metadata. We think that it fits well the MVC pattern: where Model -metadata, View -HTML and Controller -JavaScript code of a wiki page. See Figure 1.

Figure 1 MVC pattern of a wiki page

Each page is represented semantically as a named graph, whose name is based on its URI (Uniform Resource Identifiers), see Figure 2. We call this graph the Local Knowledge Base. The full stack of wiki pages give us the Global Knowledge Base, the union of all the named graphs.

Figure 2 Semantic Graph of a wiki page

WikiNEXT pages are associated with a set of metadata that describes their basic characteristics: Title, Article, Author, Contributor, Date, etc. But pages are also containers: they can hold metadata created manually or added automatically by an application. WikiNEXT allows defining the data model as a template in a wiki page. This page could then be reused by wiki applications. The template is based on HTML syntax with RDFa/RDFa lite8 annotations and uses the Mustache9 logic-less template engine. WikiNEXT uses schema.org10 vocabularies to describe the default structure of articles, users' details, etc.

Implementation

WikiNEXT has been written from scratch in JavaScript and relies on the Node.js HTTP server. Persistence uses two databases: a MongoDB database for storing JavaScript objects and RDFstorejs for storing semantic metadata and SPARQL requesting. Node.js11 (Node) is an I/O environment built on top of Google Chrome's JavaScript runtime -essentially, a server-side implementation of a JavaScript interpreter. MongoDB is a No SQL document-oriented database. RDFstore-js triplets are also persisted in MongoDB12 , then, RDF triplets are both accessible through RDFstore-js using SPARQL, but also directly through MongoDB for programmatic use. With these technological choices, we tried to minimize the number of languages and the number of data formats used in WikiNEXT: JavaScript can be used in the whole process of development both for programming and for persistence, both on the client side and on the server side.

Evaluation and conclusion

WikiNEXT proposes a new approach to programming the semantic web in JavaScript, by mixing a web-based IDE and a semantic wiki. It's currently used at the University of Nice. We wrote some online tutorials, using WikiNEXT itself, and set up a testing protocol13 . We choose a group of people with a diverse level of web programming knowledge, and asked them to complete various tutorials available in WikiNEXT, that go from writing very simple examples to developing more complex applications. We measured the time spent for completing each tutorial, observed behaviors of our testers and took note of all the problems encountered. Finally we conducted an interview, and asked them to compare this experience with classical web development techniques for performing the same tasks (e.g. using Java/Eclipse). Results14 showed that the system is easy to use, saves time and enables easy data reuse between applications, in particular writing HTML/CSS/JavaScript code, using the WikiNEXT API for creating pages, talking to external or internal data sources was very simple.

http://wikinext.gexsoft.com

https://github.com/pavel-arapov/wikinext

http://codemirror.net/

http://ace.c9.io/

http://ternjs.net/

http://www.w3.org/TR/sparql11-query/

http://www.projecthalo.com/

http://www.w3.org/TR/xhtml-rdfa-primer/, http://www.w3.org/TR/rdfa-lite/

http://mustache.github.io/

http://schema.org/, http://schema-rdfs.org have been created by the main search engine vendors and propose several RDF/S schemas.

http://nodejs.org

http://www.mongodb.org

http://wikinext.gexsoft.com/wiki/52304bb57a61be9c29000010

http://wikinext.gexsoft.com/wiki/52304d287a61be9c29000011