Thibault Gayral

David Daney

Experimental calibration of an active space telescope with flexure joints

Keywords: Calibration, Sources of inaccuracy, Flexure joints, Micrometer accuracy, Photogrammetry, Observability, Parallel robots, Space telescope Étalonnage, Sources d'erreur, Joints flexibles, Précision micrométrique, Photogrammétrie, Observabilité, Robots parallèles, Télescope spatial

In order to reach a micrometer accuracy, a suitable model of a space telescope needs to be developed and validated through calibration. Since high velocity is not required in such an application, the dynamic effects can be neglected and only geometric and/or static calibration has to be considered. Moreover, measurements for calibration are performed in a clean room under controlled pressure, temperature and humidity conditions to minimize the influence of the non-geometric errors. Thus, two possible static inaccuracy sources are identified and modelled: one from the deformation of the mobile platform and the other resulting from the behaviour of the flexure joints. Their influences on the final accuracy of the telescope are compared through results of calibration using an accurate measurement system of photogrammetry. Results show that the flexure joints can be modelled by perfect spherical joints due to the small workspace of the telescope. As for the mobile platform deformation, the developed models allow us to explain how the model errors are directly accounted in the parameter identification during the experimental calibration. This results in different sets of identified parameter values which all enable a good positioning accuracy. Those differences are explained and results of calibration allow a proper choice of the most suitable model for the telescope. Considering this model, a positioning accuracy of some micrometers is finally reached.

INTRODUCTION

Robotic calibration aims at improving the robot accuracy through identification of its model parameters. However, the robot model can only approximate the real robot behaviour. Indeed, many sources of inaccuracy influence the functioning of the robot. The classically considered inaccuracy sources are geometric, static, dynamic and/or non-geometric errors due to environmental interactions. Identifying all the inaccuracy sources, if possible, leads to the use of a huge number of specific sensors. For example, static calibration is often performed using force or frequency sensors [START_REF] Dumas | Joint stiffness identification of six-revolute industrial serial robots[END_REF][START_REF] Rognant | A systematic procedure for the elastodynamic modeling and identification of robot manipulators[END_REF]. Because of the cost and difficulty of such a procedure, many studies were performed to analyse the main sources of inaccuracy that need to be considered in the robot modelling [START_REF] Mooring | The effect of kinematic model complexity on manipulator accuracy[END_REF][START_REF] Wang | On the accuracy of a stewart platform -part i. the effect of manufacturing tolerances[END_REF][START_REF] Masory | Kinematic modeling and calibration of a stewart platform[END_REF]. However, these analyses conclude that influence of systematic errors on the robot accuracy mostly depend on the robot application, manufacturing tolerances and environmental conditions, which means that no a priori can be given on the influence of one or another inaccuracy source.

Thus, for calibration of a specific robot, certain assumptions have to be made on the inaccuracy sources that need to be considered. Such assumptions can be derived from the specific environmental conditions or from results of previous work dealing with the same robot architecture or the same applications. Following this, specific measurement systems need to be considered.

The objective of this report is to find the most appropriate model of an active space telescope in order to reach a micrometer accuracy. A space telescope does not manipulate any payload, has a small workspace and does not need high velocity, thus dynamic errors do not need to be considered. Moreover, in order to reduce the influence of the non-geometric errors, measurements for calibration will be performed in a clean room under controlled pressure, temperature and humidity conditions. Thus only geometric and static errors have to be considered for the telescope modelling.

The telescope requires the use of flexure joints to be compatible with space environments. In order to model the flexure joints behaviour, it was found that their parasitic shift and generated stress have to be considered [START_REF] Zelenika | Optimized flexural hinge shapes for microsystems and high-precision applications[END_REF]. Parasitic shift occurs when the flexure joint is bent: because of large deflections, the center of rotation of the joint changes during the motion which induces deviations from ideal pivot kinematics. However, such displacements are often small, and so the model parameters often non-identifiable. Concerning the generated stress, a stiffness modelling of the device is required in order to calculate the structure deformation. Nonetheless, small changes in the stiffness parameters can yield to great variations of forces and torques which directly influence the device deformation calculation. Because of the two above drawbacks, flexure joints are often used for micro-manipulation, where high accuracy over only a small workspace is required, as in [START_REF] Zelenika | Optimized flexural hinge shapes for microsystems and high-precision applications[END_REF][START_REF] Yi | Design and experiment of a 3-DOF parallel micromechanism utilizing flexure hinges[END_REF]. However, a proper design of the joints [START_REF] Kang | Analysis and design of parallel mechanisms with flexure joints[END_REF] allows to consider larger workspaces of some millimetres while keeping a good accuracy considering proper models of deformation. Fazenda et al. use neural-network calibration to reach a sub-micrometer accuracy [START_REF] Fazenda | Calibration of the 6 DOF highprecision flexure parallel robot "Sigma 6[END_REF]. Dong et al. consider stiffness matrices of the joints to simulate the control of a wide-range flexure hinge-based parallel manipulator [START_REF] Dong | Conceptional design and kinematics modeling of a wide-range flexure hinge-based parallel manipulator[END_REF]. Cléroux and Gosselin [START_REF] Cléroux | Modeling and identification of non-geometric parameters in semi-flexible parallel robots[END_REF] simulate a calibration process to identify non-geometric parameters in semi-flexible parallel robots.

In this report is proposed three incremental models to consider the flexure joints behaviour of the telescope: a spherical joint model, one issued from equations of the beam theory and a stiffness model. However, it was found that the generated stresses of the flexure joints imply a deformation of the telescope mobile platform, that also needs to be considered in the telescope modelling.

In order to allow the models comparison, the values of the model parameters have to be optimized in order to improve the telescope accuracy. Indeed, because of manufacturing and assembly errors, the model parameter values are not well-known and need to be estimated through Inria additional measurements: this is the goal of calibration. The way those redundant measurements are obtained defines three types of calibration: external [START_REF] Vischer | Kinematic calibration of the parallel delta robot[END_REF][START_REF] Daney | Interval method for calibration of parallel robots : Vision based experiments[END_REF][START_REF] Patel | Calibration of a hexapod machine tool using a redundant leg[END_REF][START_REF] Takeda | A DBB-Based kinematic calibration method for inparallel actuated mechanisms using a fourier series[END_REF][START_REF] Zhuang | Method for kinematic calibration of stewart platforms[END_REF][START_REF] Abtahi | Experimental kinematic calibration of parallel manipulators using a relative position error measurement system[END_REF], constrained [START_REF] Bennett | Autonomous calibration of single-loop closed kinematic chains formed by manipulators with passive endpoint constraints[END_REF][START_REF] Ikits | Kinematic calibration using a plane constraint[END_REF][START_REF] Khalil | Self calibration of stewart-gough parallel robots without extra sensors[END_REF] or self-calibration [START_REF] Wampler | An implicit loop method for kinematic calibration and its application to closed-chain mechanisms[END_REF][START_REF] Besnard | Calibration of parallel robots using two inclinometers[END_REF][START_REF] Daney | Kinematic calibration of the gough platform[END_REF]. Constrained calibration does not ensure the manipulator accuracy over the entire workspace [START_REF] Merlet | Parallel Robots[END_REF] and self-calibration is required only if the device has to be calibrated while working. Thus we choose to perform external calibration for the telescope. Photogrammetry is chosen as a measurement system because of its high accuracy over a wide number of object points, which allows to reach a micrometer final accuracy.

The space telescope is presented in section 2. Two models are proposed for the mobile platform deformation in section 3: a linear and a matrix approximation. As for the flexure joints, their parasitic shift are modelled in section 4 and their generated stress is taken into account in the stiffness model of section 5. Finally, the experimental setup of calibration is explained in section 6 and results are analysed and compared in section 7.

ACTIVE SPACE TELESCOPE

In this section we describe the active space telescope under consideration. First its application for space observation is explained. Then, its parallel kinematic structure and the definition of notations are given in section 2.2. Finally, requirements for the telescope to provide the required image quality are provided and the use of flexure joints is explained.

Telescope functioning

The telescope under consideration will be used for space observation. It is composed of two mirrors and as shown in figure 1, the image of the object is reflected in the two mirrors before being view by the camera. Thus the quality of the image greatly depends on the positioning of the secondary mirror (M2) with respect to the primary one (M1). In order reach a micrometer positioning accuracy, a parallel kinematic structure is considered to link M1 and M2, as described in section 2.2.

Kinematic structure

The telescope structure is based on an active wrist platform shown on figure 2, first designed by Merlet [START_REF]Articulated device, for use in particular in robotics[END_REF][START_REF] Merlet | Nouvelle architecture pour un manipulateur parallele a six degres de liberte[END_REF]. The device thus looks like a 6-PUS parallel robot with all prismatic actuators linked to the base and in a vertical position, where P, U and S stand for prismatic, universal and spherical joints respectively.

Frames F o = (x o , y o , z o ) and F E = (x E , y E , z E ) are attached to the base and the mobile platform respectively. Those frames are linked together through a position vector p and a rotation matrix R which describe the position and orientation of the mobile platform with respect to the base. The base and mobile platforms are linked together through 6 legs of fixed length l i ≈ 0.5 m, with i = 1...6. Each leg i is attached at one side to the mobile platform at point b i , whose coordinates are given in the mobile frame F E , and to point c i at the other side. Coordinates of point c i are always given in the base frame F o and follow the relation c i = a i + ρ i u i where ρ i relates the displacement of actuator i along direction u i and a i stands for the attachment point of the i th actuator on the base platform. To each leg i is attached a frame F i = (x i , y i , z i ), where z i is the principal direction of the leg and x i is such as z o belongs to the plane (c i , x i , z i ).

Inria

The initial pose X 0 of the device is defined as the position p 0 and orientation R 0 of the mobile platform where all the actuators are at their initial position ρ 0 i . In this pose, the orientation of frame

F i with respect to F o is R 0 i = [x 0 i y 0 i z 0 i ].
The design geometry is given in figure 3, with design parameters α 0 , r 0 , α E and r E .

Telescope requirements

The space telescope does not manipulate any payload and only needs a small workspace. The device workspace W and desired accuracy are given in table 1. Workspace requirements of the telescope do not consider θ z since motions around z E do not influence its functioning. For the telescope to fit the workspace requirements, the motor ranges are set to ∆ρ i = ±1.3 mm with an accuracy of ±1 µm. Considering a perfect 6-PUS parallel robot with dimensions of figure 3 and noting the above motor specifications, a pose accuracy of, at worst, 6 µm and 10 -6 degree was calculated over the entire workspace W .

x y z θ x θ y (mm) (mm) (mm) ( • ) ( • ) Workspace 2 2 0.

Flexure joints

In order to avoid backlash, friction, hysteresis effects, to improve the device repeatability and for the telescope to be compatible with the aerospace environment [START_REF] Henein | Flexure pivot for aerospace mechanisms[END_REF], circular-fillet hinge flexure joints [START_REF] Chen | Elliptical-arc-fillet flexure hinges: Toward a generalized model for commonly used flexure hinges[END_REF] are considered in place of U and S joints, as shown in orange on figure 2. Even if flexure joints are of interest for the manipulator design, two major drawbacks penalize the device accuracy: generated stress and parasitic shift [START_REF] Zelenika | Optimized flexural hinge shapes for microsystems and high-precision applications[END_REF] that need to be taken into account in the telescope modelling.

In this report, different models of kinematic and static behaviour of the flexure joints are proposed and compared in order to enhance the manipulator accuracy after calibration. However, an other source of inaccuracy was observed using photogrammetry data. Indeed, the mobile platform of the telescope lose its shape during the telescope functioning and this deformation also needs to be considered in the telescope modelling, as explained in section 3.

Inria

MOBILE PLATFORM DEFORMATION MODELLING

Photogrammetry was performed on the entire telescope structure as shown in figure 4. For 40 poses, the position of each object point attached to the telescope parts is measured. However, each position is measured in the frame of measurement with noise. Thus, a robust method for pose determination from corresponding data points is required to obtain the rigid body displacement with the best possible accuracy. The data analysis highlights a deformation of the mobile platform. However, it is not possible to change the design of the mobile platform since it directly influences the image quality of the telescope. Consequently, those deformations have to be taken into account: a linear and a matrix model are proposed below.

Photogrammetry data analysis

In order to extract the position and orientation of each rigid body of the telescope structure from photogrammetry object points, two methods are compared in this section. The first one considers the geometrical shape of the rigid body. The second one is based on the algorithm proposed by Haralick et al. [START_REF] Haralick | Pose estimation from corresponding point data[END_REF]. To allow the comparison and without loss of generality, let consider a cylinder on which is fixed a variable number of object points.

The first algorithm finds the position and direction of the straight line that minimizes the sum of square distances to the object points. The orientation of this line defines the z i axis. The projection of the barycentre of the object points on this line gives the position of the rigid body and the direction of this projection the x i axis.

Contrary to the first algorithm, the Haralick method gives only the displacements of a scatter of points between two poses, with noise considerations on each pose. The reader is referred to [START_REF] Haralick | Pose estimation from corresponding point data[END_REF] for the algorithm details. Please note that Haralick et al. considers only Gaussian noise of variance σ on each object points. However, simulations with uniformly distributed noise over the range [-2σ; 2σ] give the same results, for both algorithms.

To process simulations, first a perfect scatter of object points is generated in an initial position P 0 . All the object points belong to a cylinder of length 50 mm and radius 1 mm. Simulations are performed for different number of object points, from 6 to 20 as it can be seen on the abscissa of figure 5. Each of those scatters is then translated to a desired pose P and measurement noise is added to each object points. The measurement noise follows a normal distribution of variance σ = 5 µm which corresponds to the photogrammetry uncertainties. Finally, the two methods are applied on those scatters in order to calculate the cylinder displacement. For each number of object points, the process is repeated 20 times. The mean of the errors and their variance are represented on figure 5 as an error bar. Figure 5 shows that the Haralick method is more robust to measurement noise and presents the advantage to have no a priori knowledge of the body geometry. Thus, this method will be applied on each scatter of object points of the platform in order to obtain its position and orientation in the measurement frame.

With Haralick method, the data analysis shows a deformation of the mobile platform of the telescope. Indeed, deformations of about ten micrometers from a rigid model are observed on almost every measurement configurations.

Figure 6 illustrates a deformation of the ring on which the legs are attached at points b i . The deformed ring looks like a buckled wheel for almost all measurement configurations. Such a deformation is due to the flexure joints that generate stresses. The measured deformations at points b i are higher than the 5 µm given by finite element modelling simulations, and unfortunately above the desired accuracy.

Linear model

Figure 7 shows the plot of the measured displacements ∆b z i of points b i as functions of the actuator displacements ∆ρ i . Thanks to the small displacements ∆b z i (±60 µm), a linear relation between ∆b z i and ∆ρ i is supposed, considering the coefficient κ i : The initial value of κ i is identified to 0.011 for i = 1...6.

∆b z i = κ i ∆ρ i (1)

Matrix model

The mobile platform deformation is due to the stress generated by the flexure joints. Those stresses depend on the static equilibrium of the mobile platform which involves both forces acting on the platform at points b i . Moreover, those forces should mostly depend on the actuator displacements ∆ρ i that induce deformations of the flexure joints. So displacements of point b i must depend on all the actuator displacements such as:

∆b z = K∆ρ (2)
with ∆b z = [∆b z 1 ...∆b z 6 ] T , ∆ρ = [∆ρ 1 ...∆ρ 6 ] T and K a coefficient matrix. In order to identify the terms of the coefficient matrix K, data from figure 7 is considered. For each leg i, the i th row k i of matrix K is used to calculate the displacements ∆b z i such as:

∆b z i = k i ∆ρ ( 3
)
The objective is to obtain the best estimate of ∆b z i using equation (3) considering the measurements of figure 7. In order to minimize the distances between the measurements and the matrix estimate, a least-square optimization is performed on each leg of the telescope. The identified matrix is given in equation 4.

K = 10 -3         16 0 - 8
        (4)
The elements of K are of the same order of magnitude as κ i . Analysing matrix K, a strong dependency between legs 1-6, 2-3, and 4-5 are observed. For example, displacements ∆ρ 1 and ∆ρ 6 greatly influence ∆b z 1 and ∆b z 6 . This may be due to the mobile platform geometry. Indeed, as shown in figure 4, there is no strengthening for the mobile platform ring between the attachment points of legs 1 and 6. Thus displacements of point b 1 can have a significant impact on ∆b z 6 . Considering the matrix model, the maximal committed error on the displacement estimation of ∆b z i is 20 µm, and the average error is 5 µm which is close to the measurement noise.

With both the linear and matrix models, the mobile platform deformation can be taken into account in the telescope modelling setting b * i = b i + ∆b z i z o . This allows to consider the mobile platform as a rigid body while modelling the telescope. In order to take into account the behaviour of the flexure joints, kinematic models and a stiffness model are proposed in sections 4 and 5 respectively.

Inria

KINEMATIC MODELS

In this section, only the kinematic behaviour of the telescope is considered. The impact of internal forces over the entire structure is neglected and the manipulator parts are treated as rigid bodies thanks to the previous models of the mobile platform deformation. For the two first models, flexure joints are supposed to behave as perfect spherical joints.

6-PUS model

Considering leg orientations z i of a 6-PUS, geometrical closed-loop equations can be written for each leg i:

p + Rb * i = a i + ρ i u i + l i z i (5)
The device inputs can then be issued from equation ( 5):

ρ i = (p + Rb * i -a i -l i z i ) T u i (6)
However, equation ( 6) considers the leg orientation z i which need information on the motor input ρ i to be calculated. Even if it can be done using an iterative algorithm, motor inputs can also be derived from equation [START_REF] Yi | Design and experiment of a 3-DOF parallel micromechanism utilizing flexure hinges[END_REF].

l i = p + Rb * i -a i -ρ i u i -→ ρ i = h T i u i -(h T i u i ) 2 -( h i 2 -l 2 i ) with h i = p + Rb * i -a i (7)

Jacobian-based model

Differentiating equation ( 5) with respect to time yields to the inverse of the kinematic Jacobian matrix J(X) such as ρ = J -1 (X) Ẋ where ρ is the vector of actuators velocities, Ẋ is the twist of the mobile platform and X = [p T , θv T ] T with v the rotation axis vector and θ the rotation angle of the rotation matrix R [24]:

J -1 (X) =           z T 1 u T 1 z 1 (Rb * 1 ) T × z T 1 u T 1 z 1 . . . . . . z T i u T i z i (Rb * i ) T × z T i u T i z i . . . . . .           6×6 (8)
For small displacements, the typical assumption δρ = J -1 (X)δX can be made. Moreover, because of the small workspace of the telescope, the Jacobian matrix does not vary too much and thus can be approximated by the Jacobian matrix of the initial pose of the device J 0 = J(X 0 ). Thus the kinematic behaviour of the telescope can be approximated with equation ( 9) [START_REF] Zhang | The constantjacobian method for kinematics of a three-DOF planar micro-motion stage[END_REF].

ρ = ρ 0 + J -1 0 (X -X 0 ) (9)

Beam theory-based model

In order to consider the parasitic shift of the flexure joints, their displacements are estimated using equations from the beam theory. Flexure joints are assumed to be equivalent to circular beams, as shown on figure 8. As usual in beam theory, the principal deflection is assumed to be generated by the bending moment M f z . Considering the Euler-Bernoulli hypothesis, the static beam equations can be derived, with E, I Gz and γ being the Young modulus, the second moment of area and the curvature of the flexure joints respectively:

   M f z = EI Gz γ ∆y(x) = M f z EI Gz x 2 2 -→ ∆y(x) = γ x 2 2 (10)
Then, the length l(x) of the deformed beam can be calculated using equation [START_REF] Cléroux | Modeling and identification of non-geometric parameters in semi-flexible parallel robots[END_REF]:

l(x) = x 0 1 + ∆y ′ (x) 2 dx -→ ∆x(x) = x 0 1 + (γx) 2 dx -x (11)
Considering the workspace of the telescope, a flexure joint of length l b cannot be bent at an angle more than δ = 1 • from its home position. For such small deflections, the orientation δ can be approximated by δ = γ l b . Thus the beam displacements ∆y and ∆x can be calculated using equation [START_REF] Vischer | Kinematic calibration of the parallel delta robot[END_REF]:

       ∆y = ∆y(l b ) = l b 2 δ ∆x = ∆x(l b ) = l b δ √ 1 + δ 2 + sinh -1 (δ) 2δ -1 (12)
Writing and analysing the Taylor series expansion of equation ( 12) leads to:

∆x - 3 4 l b 2 + ∆y 2 = 3 4 l b 2 + o(δ 4 ) (13)
which is the equation of a circle with center ( 3 4 l b , 0) and radius 3 4 l b . Both the ideal kinematic pivot of center ( 12 l b , 0) and radius 1 2 l b , the beam theory-based model kinematic and its identification as circle of equation ( 13) are plotted on figure 9

for δ ∈ [0; 1 • ].
Figure 9 shows that the kinematic displacement of the flexure joints in their planes of deformation is perfectly identified by the circle of equation 13. However, their orientations are equal to that of an ideal pivot for the same ∆y.

In order to consider the displacements and orientations of the flexure joints, a kinematic model is proposed in figure 10, where l bi is the length of the flexure joints of leg i and β i a design parameter whose initial value is fixed to 1 4 . Displacements are constrained on a sphere of radius (1β i )l bi while orientations are the same as an ideal spherical joint at a i + ρ i u i , with z 0 i the initial orientation of the flexure joints of leg i. Considering the kinematic model of figure 10, the closed-loop equation for each leg i becomes:

p + Rb * i -a i -ρ i u i = (l i -l bi + ∆l Ai + ∆l Bi )z i (14)
with ∆l Ai and ∆l Bi the displacements of the flexure joints of leg i attached to the base and to the mobile platform respectively. Writing the shortest kinematic closed loop equation involving ∆l Ai leads to:

(1

-β i )l bi = ( 1 2 -β i )l bi z 0 i + ∆l Ai z i -→ (1 -β i ) 2 l 2 bi = ( 1 2 -β i ) 2 l 2 bi + 2( 1 2 -β i )l bi ∆l Ai z T i z 0 i + ∆l 2 Ai -→ ∆l 2 Ai + (1 -2β i )l bi z T i z 0 i ∆l Ai + (β i -3 4 )l 2 bi = 0 (15)
The quadratic equation ( 15) has only one positive solution for ∆l Ai :

∆l Ai = l bi 2 (2β i -1)z T i z 0 i + 3 -4β i + (1 -2β i )z T i z 0 i 2 (16)
Almost the same equation can be found considering ∆l Bi :

∆l Bi = l bi 2 (2β i -1)z T i Rz 0 i + 3 -4β i + (1 -2β i )z T i Rz 0 i 2 (17)
Considering equations ( 14), ( 16), [START_REF] Abtahi | Experimental kinematic calibration of parallel manipulators using a relative position error measurement system[END_REF] and noting that z i is a unit vector yields to a system of algebraic equations in the unknowns ρ i , z i , ∆l Ai and ∆l Bi :

               p + Rb * i -a i -ρ i u i = (l i -l bi + ∆l Ai + ∆l Bi )z i 2∆l Ai l bi + (1 -2β 1 )z T i z 0 i 2 = 3 -4β i + (1 -2β i )z T i z 0 i 2 2∆l Bi l bi + (1 -2β 1 )z T i Rz 0 i 2 = 3 -4β i + (1 -2β i )z T i Rz 0 i 2 z i 2 = 1 (18)
Equations ( 14) are linear in z i , which give one linear equation in the ρ i that can be solved. The two remaining equations involving ∆l Ai and ∆l Bi are of global degree 8 and 7 respectively. Using interval analyses, the uniqueness of the solution in the neighbourhood of the nominal values of ∆l Ai and ∆l Bi has been proved. This allows us to use iterations of Newton-Raphson for rapid calculation of the inverse kinematics of the beam theory-based model.

Inria

STIFFNESS MODEL

The parasitic shift of the flexure joints is considered thanks to equations of the beam theory. However, the pose of the telescope mobile platform also depends on its static equilibrium. The static equilibrium of the platform is written in equation [START_REF] Ikits | Kinematic calibration using a plane constraint[END_REF], PE being the external load acting at the center of the mobile platform, Pi b ′ the load generated by the flexure joints acting on the mobile platform and M i a 6 × 6 matrix taking into account the momentum transport:

PE + 6 i=1 M i Pi b ′ = 0 (19)
Stresses generated by the flexure joints are functions of their deformations. Thus, the stiffness matrices of the flexure joints need to be derived for their deformation influence on the static equations of the mobile platform to be considered.

Local stiffness matrix of a flexure joint

Flexure joints of the telescope can be assimilated as perfect cylinders of small ratio diameter/length as shown on figure 11. Hence, stiffness matrices can be calculated from the finite element analysis theory as it is done by Dong et al. in [START_REF] Dong | Conceptional design and kinematics modeling of a wide-range flexure hinge-based parallel manipulator[END_REF]. The stiffness matrix K loc links together the nodal loads [P T i , P T j ] T and displacements [u T i , u T j ] T of the beam through equation ( 20):

P i P j = K loc u i u j (20)
with

P i = [F i x , F i y , F i z , M i x , M i y , M i z ] T and u i = [u i x , u i y , u i z , ϕ i x , ϕ i y , ϕ i z ] T , (ϕ i z , ϕ i y , ϕ i x )
being the ZYX-Euler angles (same for j).

Considering a beam element of length l, of sectional area S, of Young's and shear modulus E and G, and of rotational and polar moment of inertia I and J, the stiffness matrix can be expressed through equation [START_REF] Wampler | An implicit loop method for kinematic calibration and its application to closed-chain mechanisms[END_REF], with k 1 = EI, k 2 = ES and k 3 = GJ.

K loc = K 11 K 12 K 21 K 22 12×12 (21)
with

K 11 = 1 l         k 2 0 0 0 0 0 0 12k1 l 2 0 0 0 6k1 l 0 0 12k1 l 2 0 -6k1 l 0 0 0 0 k 3 0 0 0 0 -6k1 l 0 4k 1 0 0 6k1 l 0 0 0 4k 1         6×6 K 12 = 1 l         -k 2 0 0 0 0 0 0 -12k1 l 2 0 0 0 6k1 l 0 0 -12k1 l 2 0 -6k1 l 0 0 0 0 -k 3 0 0 0 0 -6k1 l 0 2k 1 0 0 6k1 l 0 0 0 2k 1         6×6 K 21 = 1 l         -k 2 0 0 0 0 0 0 -12k1 l 2 0 0 0 -6k1 l 0 0 -12k1 l 2 0 6k1 l 0 0 0 0 -k 3 0 0 0 0 6k1 l 0 2k 1 0 0 -6k1 l 0 0 0 2k 1         6×6 K 22 = 1 l         k 2 0 0 0 0 0 0 12k1 l 2 0 0 0 -6k1 l 0 0 12k1 l 2 0 6k1 l 0 0 0 0 k 3 0 0 0 0 6k1 l 0 4k 1 0 0 -6k1 l 0 0 0 4k 1         6×6

Inverse stiffness model

For ease of calculation, each leg l i of the device is considered as a rigid beam of stiffness matrix K l depending of the form and material property of the legs, and each flexure joint is identified as a stiffness matrix K b . The stiffness model of one chain of the device is presented in figure 12.

Note that contrary to the kinematic models of section 4 where points b i and c i relates to the center of the spherical joints while a i is attached to the base, here points a ′ i , b ′ i , c ′ i and d ′ i all stands for nodes of the flexure joints. Thus we have the following relations:

       a ′ i = (a i + ρ i u i ) -1 2 l bi z 0 i b ′ i = b * i + 1 2 l bi z 0 i c ′ i = c i + 1 2 l bi z 0 i d ′ i = b * i -1 2 l bi z 0 i ( 22
)
Considering figure 12, the total local stiffness matrix of each chain i follows equation ( 23) and can be calculated (suffix i is omitted for clarity reasons):

    P a ′ P c ′ P d ′ P b ′     = K     u a ′ u c ′ u d ′ u b ′     (23) Inria
K =     K d 11 K d 12 K d 21 K d 22 + K l 11 K l 12 K l 21 K l 22 + K d 11 K d 12 K d 21 K d 22     24×24 (24)
Then equations ( 23) and ( 24) can be written by regrouping the known and the unknown parameters [START_REF] Chevalier | Mécanique des systèmes et des milieux déformables[END_REF]:

P kno = [P T c ′ , P T d ′ ] T , P unk = [P T a ′ , P T b ′ ] T , u kno = [u T a ′ , u T b ′ ] T and u unk = [u T c ′ , u T d ′ ] T . P kno P unk = K u K f K T f K c u unk u kno (25)
with

K u = K d 22 + K l 11 K l 12 K l 21 K l 22 + K d 11 , K f = K d 21 K d 12 and K c = K d 11 K d 22 ( 26
)
Since P c ′ and P d ′ are the external loads applied on nodes c ′ and d ′ , we have P c ′ = P d ′ = 0 6×1 . Equation ( 25) thus becomes:

0 12×1 = K u u unk + K f u kno P unk = K T f u unk + K c u kno -→ P unk = K c -K T f K -1 u K f u kno (27)
However, equation ( 27) is still in the local coordinate system of the chain under consideration. In order to obtain the stiffness equations in the base frame F o , the initial orientation matrix R 0 i of the i th leg needs to be considered. Thus the global loads Pi unk and displacements ũi kno can be calculated for each leg i through equation ( 28), with

T i0 = diag(R 0 i , R 0 i , R 0 i , R 0 i )
where diag() stands for the block diagonal matrix:

Pi unk = Ki ũi kno = T i0 K c -K T f K -1 u K f T -1 i0 ũi kno ( 28
) Pi b ′ = Ki 21 ũi a ′ + Ki 22 ũi b ′ (29)

RR n°8096

Analysing equation ( 29), we see that Pi b ′ is a function of only the unknown ∆ρ i in ũi a ′ since the displacements ũi b ′ can easily be calculated for a given pose X using ∆b

′ i = p + Rb ′ i -p 0 + R 0 b ′ i .
The static equilibrium condition of the mobile platform can thus be written as a system of 6 equations for the 6 unknowns ∆ρ i considering equation [START_REF] Ikits | Kinematic calibration using a plane constraint[END_REF].

The inverse kinematic of the stiffness model can thus be calculated using an optimization function that aims to minimize equation [START_REF] Ikits | Kinematic calibration using a plane constraint[END_REF] through changes of the variables ∆ρ i . Note that contrary to Dong et al. in [START_REF] Dong | Conceptional design and kinematics modeling of a wide-range flexure hinge-based parallel manipulator[END_REF], the global stiffness matrices are not re-evaluated during the optimization process. Even if those matrices depend on the position and orientation of the flexible beams, taking into account those changes in the optimization loop does not increase the final accuracy in our case. Indeed, the small changes of legs orientations over the workspace imply variations of ∆ρ i of less magnitude than the measurement noise.

Inria

EXPERIMENTAL SETUP

Different models of the device are proposed in sections 4 and 5. For each model, different objective functions can be chosen, for sometimes a different number of identifiable parameters. In this section, first an overview of the objective functions that will be used for experimental calibration is given. The set of geometrical parameters of each model is also clearly identified. Then the choice of the photogrammetry process is explained. Finally, in order to increase the robustness of calibration with respect to measurement noise, the measurement configurations are chosen in the telescope workspace through an optimization algorithm.

Overview

Calibration aims at identifying m model parameters per leg by minimizing an objective function F considering information obtained from N P measurement configurations.

A common way to perform implicit calibration is to use non-linear least square algorithms that aim at minimizing distance equations. The objective function F can thus be written F = N P n=1 f T n f n , with f n the objective function of the n th measurement configuration. In parallel robotics, because of the devices kinematic structures, kinematic calibration can often be performed considering each leg independently. In such cases, the objective function F i can be written for each leg i as

F i = N P n=1 f T i,n f i,n , f i,n
being the objective function for the i th leg in the n th measurement configuration.

In this section the objective functions f n or f i,n and the set of model parameters are given for all models presented in this report. At the end of this section, table 2 resumes the different objective functions. In this table, as for all this section, every parameter that have the n subscript is a measurement or is at least issued from measurements of the n th configuration.

6-PUS model

The objective function 1 f i,n is directly extracted from the kinematic closed-loop equation of section 4.1. Each leg i can be calibrated separately and the set of parameters {a T i , b T i , u T i , l i } describes the complete model.

1 f i,n = l i -p n + R n b * i -a i -ρ n i u i (30)

Jacobian-based model

Since the i th row of the inverse Jacobian matrix J -1 0 also depends on the initial orientation R 0 of the mobile platform, the calibration can not be performed for each leg separately. The set of parameters is set to {a T i , b T i , u T i , l i , ρ 0 i } for each leg, X 0 being derived from ρ 0 through the direct kinematic model of the 6-PUS model.

2 f n = ρ n -ρ 0 + J -1 0 (X n -X 0 ) (31)

Beam theory-based model

The objective function is derived from the kinematic closed-loop of leg i of equation [START_REF] Patel | Calibration of a hexapod machine tool using a redundant leg[END_REF]. Since ∆l n Ai and ∆l n Bi are functions of measurements and design parameters β i and l bi , the design parameters are {a T i , b T i , u T i , l i , β i , l bi } for each leg i.

Stiffness model

The stiffness model considers the static equilibrium of the mobile platform. Thus calibration can not be performed for each leg separately. For each leg i, the geometrical parameters are

{a T i , b T i , u T i , l i , l bi }. The stiffness parameters {k d 1 , k d 2 , k d 3 , k l 1 , k l 2 , k l 3 }
which describe the stiffness matrices of a flexure joint and of a leg respectively are assumed to be the same for each leg i.

4 f n = PE + 6 i=1 M n i Pi,n b ′ (33)
The objective functions of each model are summarized in table 2 below.

n • Model Objective function 1 6-PUS 1 f i,n = l i -p n + R n b * i -a i -ρ n i u i 2 Jacobian 2 f n = ρ n -ρ 0 + J -1 0 (X n -X 0 ) 3 Beam-theory 3 f i,n = (l i -l bi + ∆l n Ai + ∆l n Bi ) -p n + R n b * i -a i -ρ n i u i 4 Stiffness 4 f n = PE + 6 i=1 M n i Pi,n b ′
Table 2: Objective functions used for calibration

Photogrammetry process choice

Photogrammetry is chosen as measurement system because of its high accuracy (3 µm of standard deviation) over a wide number of object points. Thus, a final accuracy of some micrometers can be reached after calibration. In order to minimize the influence of non-geometric parameters not considered in the telescope modelling [START_REF] Niaritsiry | Study of the sources of inaccuracy of a 3 DOF flexure hinge-based parallel manipulator[END_REF], the photogrammetry process will be performed in a clean room under controlled pressure, temperature and humidity conditions. However, the desired accuracy of 10 µm is only three times the standard deviation of the photogrammetry process. Thus the sensitivity of the calibration results to measurement noise has to be set to its minimum. This can be done through a proper choice of the measurement configurations as explained in section 6.3.

Measurement configurations

The last step of the calibration setup is to define the measurement configurations. It was shown that a proper choice of the measurement configurations increase the robustness of calibration with respect to measurement noise [START_REF] Daney | Choosing measurement poses for robot calibration with the local convergence method and tabu search[END_REF]. Algorithms of configuration choice are always based on observability indices [START_REF] Nahvi | The noise amplification index for optimal pose selection in robot calibration[END_REF][START_REF] Sun | Observability index selection for robot calibration[END_REF], based on properties of the observability Jacobian matrix J P [START_REF] Mooring | Fundamentals of Manipulator Calibration[END_REF].

To perform the measurement configuration choice, we choose to consider the observability index O 1 , the determinant of J T P J P , since it considers all the singular values of J T P J P which allows to obtain an optimum over a reasonable computation time.

The structure of the used algorithm is presented in figure 13, where X n stands for one configuration of measurement with n = 1 . . . N P , with N P the total number of measurement configurations for calibration and W the workspace of the telescope.

Inria

Find X opt = {X 1 , . . . , X N P } such as Max det J T P (X)J P (X) with respect to X n ∈ W , for n = 1...N P Figure 13: Algorithm of the measurement configurations choice

Observability matrix

The measurement configuration choice is performed considering the telescope as a perfect 6-PUS, with objective function 1 f i,n of equation [START_REF] Zhang | The constantjacobian method for kinematics of a three-DOF planar micro-motion stage[END_REF]. Simulations of calibration show that parameters a z i , b z i and l i are linearly related in the equations and thus cannot be both identifiable. Indeed, differentiating 1 f i,n with respect to only a z i , b z i and l i leads to the observability matrix J 3 P whose i th row for the n th configuration is given in equation ( 34):

(J 3 P ) i,n =   g T i,n 2 g T i,n g i,n   0 0 -z n E 1   , 1   (34)
We have to notice that

g i,n = p n + R n b * i -a i -ρ n i u i ≈ l i z n i .
Thus 1 f i,n can be rewritten considering the parameters uncertainties δa z i , δb z i and δl i as:

2( 1 f i,n ) = (z n i ) T z 0 δa z i -(z n i ) T z n E δb z i + δl i (35)
Because of the small orientation workspace of the telescope, z n E does not vary too much and is almost equal to z 0 considering the measurement noise. Moreover, the legs cannot be bent at an angle more than 0.6 • , which implies that z n i has only small variations: (z n i ) T z 0 ≈ λ, with λ a constant. Thus equation ( 35) can be rewritten as:

2( 1 f i,n ) = λδa z i -λδb z i + δl i (36)
It is now clear from equation ( 36) that a z i , b z i and l i are linearly related in the equations of calibration. Thus among those three parameters only one can be identified. Considering the assembly and manufacturing tolerances, it was found that l i supports the maximal uncertainties among those three parameters. Thus, we choose to identify l i . Note that u i is a unit vector and thus can be fully described by only two of its components u x i and u y i . The set of identifiable parameters is consequently reduced to {a x i , a y i , b x i , b y i , u x i , u y i , l i } for each leg i. Differentiating 1 f i,n with respect to the 7 identifiable parameters gives the observability Jacobian matrix J P . The i th row of J P for the n th configuration is given in equation (37):

(J P ) i,n =   g T i,n 2 g T i,n g i,n   1 0 ρ n i 0 0 1 -x n E -y n E 0 ρ n i 0 0 0 0   , 1   (37)

Algorithm validation

In order to validate results of the algorithm presented in figure 13, a comparison between three different algorithms of measurement configurations choice is performed through simulations of calibration.

The first algorithm choose N P random configurations in the device workspace and is used as a reference. The second one is the IOOPS (refer to [START_REF] Daney | Choosing measurement poses for robot calibration with the local convergence method and tabu search[END_REF] for details) which considers an initial set of calibration poses and then eliminates and adds poses for maximizing O 1 . The third one is the algorithm of figure 13 which directly optimizes the entire set of poses using the fmincon function of Matlab.

The comparison is performed regarding the final accuracy reached after calibration. First, an ideal 6-PUS with exact geometric parameters is considered. Then initial guesses of the geometrical parameters are obtained by adding random uncertainties with a normal distribution of variance 0.5 mm to the exact parameters. For each considered measurement configuration, the exact parameters are used to calculate the position and orientation of the mobile platform, and a measurement noise of variance 2 µm is then added. Finally, implicit calibration is performed considering the N P measurement configurations.

Results are given in figure 14. For each number of calibration poses N P , each algorithm is repeated 17 times. The mean and the standard deviation of the obtained positioning accuracy for the 17 configuration sets are represented as an error bar. Figure 14 validates the proposed algorithm of figure 13 since it allows to obtain the best accuracy results after calibration. However, its computation time is about five times higher to the IOOPS algorithm that performs only "local" optimizations. With both the IOOPS and the proposed algorithm, optimal configurations are found to be at or very close to the boundary of the telescope workspace.

Results

Algorithm of figure 13 is used to find the N P optimal measurement configurations for calibration. Even if those configurations are optimal for calibration of a 6-PUS, they will also be used for calibration of the other models of section 4 and 5. Indeed, considering the same measurement configurations for all models allows to compare results of calibration while reducing the total number of measurements. Moreover, the identification of the model parameters of the joints depends on the bending of the flexure joints, and the maximal bending of the flexure joints is obtained when the telescope is at its workspace boundary. Since the optimal configurations are found to be at or very close to the boundary of the telescope workspace, we made the assumption that the optimal configuration poses for a 6-PUS allow calibration of more complete models of the flexure joints. Considering the stiffness model, 10 parameters per leg and 6 stiffness parameters can be identified. Thus, if the position and orientation of the platform are extracted from the photogrammetry data, at least 12 measurement configurations are needed. For safety reasons, we choose to perform calibration on N P = 40 poses. Results of the optimization algorithm for the 40 measurement configurations are shown on table 3, with ∆ρ i the maximum range of the i th actuator. For verification purpose, 6 verification poses are also considered.

Inria n • 1 2 3 4 5 6 7 8 9 10 ∆ρ 1 1 1 0 1 1 0 1 1 -1 -1 ∆ρ 2 -1 0 1 1 1 0 1 -1 -1 1 ∆ρ 3 -1 1 -1 0 1 -1 0 -1 -1 -1 ∆ρ 4 1 1 -1 -1 0 -1 0 0 0 0 ∆ρ 5 1 1 0 -1 -1 1 0 -1 -1 -1 ∆ρ 6 -1 -1 1 1 1 1 -1 -1 -1 -1 n •
11 12 13 14 15 16 17 18 19 20 ∆ρ 1 -1

1 -1 1 -1 -1 1 -1 0 -1 ∆ρ 2 -1 -1 1 1 -1 0 1 -1 -1 1 ∆ρ 3 -1 0 -1 0 0 1 1 1 1 1 ∆ρ 4 1 1 -1 0 -1 -1 1 0 -1 0 ∆ρ 5 -1 1 1 0 0 1 -1 -1 1 -1 ∆ρ 6 0 -1 -1 -1 -1 1 0 -1 -1 -1 n •
21 22 23 24 25 26 27 28 29 30 ∆ρ 1

1 -1

1 0 1 -1 -1 -1 -1 1 ∆ρ 2 -1 -1 1 -1 -1 1 1 1 1 -1 ∆ρ 3 0 0 1 1 -1 1 -1 1 1 0 ∆ρ 4 1 0 -1 -1 1 0 -1 0 -1 -1 ∆ρ 5 1 1 -1 0 1 -1 1 -1 0 1 ∆ρ 6 1 1 -1 1 1 0 1 -1 1 1 n • 31
1 0 1 -1 -1 0 1 1 -1 1 ∆ρ 2 1 1 1 1 1 0 0 1 0 -1 ∆ρ 3 0 -1 1 -1 0 1 -1 0 1 -1 ∆ρ 4 -1 -1 -1 1 1 -1 1 1 -1 -1 ∆ρ 5 0 -1 -1 -1 1 -1 1 1 -1 -1 ∆ρ 6 1 1 0 -1 0 -1 1 -1 -1 1
Table 3: 40 optimized measurement configurations

n • 1 2 3 4 5

EXPERIMENTAL RESULTS

This section gives the experimental results of calibration of the space telescope using photogrammetry data. For each measurement configuration, the position and orientation of the mobile platform are extracted from the photogrammetry object points using Haralick method. Calibration minimizes each objective function of section 6.1 without considering the mobile platform deformation, with the linear model and with the matrix model respectively. The accuracy reached after calibration is compared for all models in section 7.4, which allows the choice of the most suitable model for the telescope. Every calibration is performed considering the objective functions presented in section 6.1 and utilizes the lsqnonlin function of Matlab. Same results are obtained utilizing either the Levenberg-Marquardt or the trust-region reflective algorithms. Experiments showed that in our case, those algorithms are non sensitive to changes on the initial parameters.

Without considering the mobile platform deformation

For all this section, the mobile platform deformation is not taken into account for comparison purpose. Thus in all the objective functions of section 6.1, the following relation is considered:

b * i = b i .
For the kinematic and stiffness models, first the parameter identifiability is analysed and calibration results are given as distances between nominal and identified values of the model parameters in table 5. Then, results of final positioning and orientation accuracies are given in figure 15.

with the 6-PUS model

The experimental calibration confirms that only one parameter among {a z i , b z i , l i } can be identified. Regarding the actuator orientation u i , only one of its components can be experimentally identified, whatever the used parametrization. We choose to identify u x i while setting u y i to zero. Thanks to the knowledge of those two components, u z i can be easily calculated since u i is a unit vector. Note that the choice of which component of u i is identified does not change the experimental calibration results. The set of identifiable parameters is thus reduced to {a x i , a y i , b x i , b y i , u x i , l i } while a z i , b z i and u y i are set to zero. Calibration is performed and results are given in table 5 as distances between the parameter identified values and their nominal ones.

with the Jacobian-based model

Since the Jacobian matrix is derived from the 6-PUS model, the same identifiability problems appear. However, since the small device workspace is already considered in the definition of the Jacobian-based model, explanations of the non-identifiable parameters can be directly given from the shape analyse of the Jacobian matrix. Indeed, each row of the Jacobian matrix only takes into account the leg orientation z 0 i , the position of point b * i and the scalar product u T i z 0 i between the actuator and leg orientations of the initial pose X 0 of the device. Thus for each leg, if b * i is known, only a x i and a y i can be identified considering the initial legs orientation z 0 i . Then, only one more information is not yet utilized which is the scalar product u T i z 0 i . So only one component of u i can be identified.

Since b z i and l i are linearly related, we choose to identify l i instead of b z i for comparison purpose with other models. Moreover, the same conclusion on the parametrization of u i was observed: changing its parametrization does not influence the calibration results. As for the Inria initial position ρ 0 of the actuators, first their identification do not change the final accuracy reach after calibration, and then, they are found to be low-observable. Indeed, identifying ρ 0 leads to 6 singular values of J T P J P being under 10 -14 . On the other hand, when ρ 0 are not identified, the smallest singular value of J T P J P is 10 -4 . Thus, considering the measurement noise, it is preferable to not identify ρ 0 . So the identifiable parameters are setted to {a x i , a y i , b x i , b y i , u x i , l i } for each leg i.

with the beam theory-based model

Because of the small joints orientation δ ≤ 1 • , variations of ∆l Ai and ∆l Bi are less than 0.3 µm. Considering a measurement noise of 5 µm on the object points, it is clear that ∆l Ai and ∆l Bi can not be properly identified. Thus the identified parameters are still the same as the 6-PUS model and β i and l bi are fixed to their nominal values.

with the stiffness model

The stiffness equation considers the loads Pi b ′ acting on points b ′ i . Those loads are functions of the displacements ũi a ′ and ũi b ′ as shown in equation [START_REF] Haralick | Pose estimation from corresponding point data[END_REF]. Points b ′ i and a ′ i being functions of b i and (a i + ρ i u i ) respectively, the positions of b i and a i can be thought to be identifiable. However, since a z i , b z i and l i are linearly related, only l i will be identified. The extra information in (a i + ρ i u i ) will be used to identify u x i . The stiffness model also considers stiffness parameters. However, experiment shows that none of them can be properly identified. Hence, the set of identifiable parameters for each leg becomes

{a x i , a y i , b x i , b y i , u x i , l i }.

Results

For both the models, the calibration results are given as distances between the nominal and identified parameter values in table 5. Experimental results are also given in term of final accuracy for each identified model in figure 15. The position error refers to the distance between the calculated and the measured position of the center of the mobile platform. The orientation error is the angle of the axis-angle representation between the calculated and the measured orientation of the mobile frame F E . Each bar has the size of the maximal error and the additional line stands for the average error. For the verification poses, the mean and maximal values are provided.

Figure 15 shows that the same final positioning accuracy can be reached for both models. As for the orientation accuracy, better results are obtained while not considering the static equilibrium of the mobile platform. Thanks to the small workspace of the telescope, the Jacobianbased model gives similar results as the 6-PUS model. Moreover, considering the parasitic shift of the flexure joints over this small workspace does not improve the accuracy results since their deflections are less than 0.3 µm with the beam theory-based model.

However, as shown in table 5, the identified values of the model parameters are quite "far" from their nominal values, which does not correspond to the manufacturing tolerances. The most important deviations are obtained in the identification of u x for both the models. Those deviations can be explained regarding the linear model of the mobile platform deformation as done in section 7. In this section, the linear model for the mobile platform deformation is taken into account. Thus, the following relation is considered: b * i = b i + κ i ∆ρ i z o . For both models, the identifiability of the model parameters does not change from section 7.1. However, κ i also has to be identified for each leg. Experiments show that κ i and u x i cannot be both identified. Indeed, those two parameters are found to be linearly related, as explained below. Let consider the objective function 1 f i,n of the 6-PUS model with the linear model for the mobile platform deformation:

1 f i,n = l i -p n + R n (b i + κ i ∆ρ i z o ) -a i -ρ n i u i (38)
Because of the small workspace of the telescope, the platform orientation does not vary too much and approximating R n (κ i ∆ρ n i z o ) by (κ i ∆ρ n i z o ) leads to maximal errors of less than 1 µm for n = 1...N p . Let also consider that u i is set to its nominal value z o , then equation (38) can be rewritten as:

f i,n = l i -p n + R n b i -a i -(ρ n i -κ i ∆ρ n i )u i (39)
Considering ρ n i = ρ 0 i + ∆ρ n i leads to:

f i,n = l i -p n + R n b i -a i -ρ 0 i u i -(1 -κ i )∆ρ n i u i (40)
Analysing the term (1κ i )∆ρ n i u i shows that when the mobile platform deformation is not considered, the identification of κ i is accounted in the identification of u i . This results in values of u x i being far from their nominal values as shown on table 5. This also explains why only one component of u i can be identified while not considering the mobile platform deformation. Indeed, the identification process automatically considers the platform deformation in a linear approximation and distributes the model errors on the parameter identification. Since the linear approximation is only one degree of freedom, only one component of u i can be identified.

Thus, in order to take into account the mobile platform deformation, κ i will be identified for both models while setting u i to its nominal value z o . Moreover, setting u i to its nominal value allows us to consider the mobile platform deformation directly in the actuator positions ρ i such as: ρ i = ρ 0 i + (1κ i )∆ρ i . Results are provided in table 6 and figure 16.

For both models, it is interesting to note that the same accuracy results are obtained considering or not the mobile platform deformation in a linear model. This is due to the calibration that accounts the mobile platform deformation in the parameter identification. Indeed, instead of identifying κ i , the linear model of the mobile platform deformation is automatically taken into account in the identification of u x i as explained above. As for the identified values of the other model parameters, they are quite the same comparing tables 5 and 6, which is as expected since the mobile platform deformation is considered in a linear model in both cases. However, their identified values stay quite "far" from their a priori. Considering the matrix model in section 7.3, we will see that those deviations are not only due to geometric uncertainties but mainly due to the mobile platform deformation that is accounted in the parameter identification. In this section the matrix model of deformation is taken into account. Thus, the following relation is considered: b * i = b i + (k i ∆ρ)z o , with k i the i th row of the coefficient matrix K. As for the linear model, k i and u x i cannot be both identified. Moreover, setting u x i to its nominal value allows the identification of only one term in k i . Experiments show that the best accuracy results are obtained by identifying all the diagonal terms of K, which actually relate to the coefficients κ i of the linear model. The diagonal terms of matrix K are identified and are referred as κ i in table 7.

The final accuracy reached after calibration is given in figure 17. It shows that a better orientation accuracy is obtained considering the matrix model for the mobile platform deformation. However, regarding the telescope requirements, the desired orientation accuracy of 180 arcmin is already reached by the linear model. Moreover, no significant improvement on the positioning accuracy are observed for the matrix model. Since the matrix model needs information on the mobile platform deformation in order to identify the coefficient matrix K and since our objective is to find the most suitable model to calibrate the telescope in space with only measurements of the position and orientation of the mobile platform, the linear model of deformation will be preferred.

Results analysis

With or without considering the mobile platform deformation, the Jacobian-based, the beam theory-based and the stiffness models give similar accuracy results as the 6-PUS model. This is due to the small workspace of the telescope. Indeed, because of the small orientation of the flexure joints over the entire workspace, their parasitic shift stay negligible and their generated stresses do not influence the mobile platform positioning. Thus they can be modelled by pure spherical joints as it is done in the 6-PUS model.

Regarding the developed models of deformation, both of them give quite the same accuracy results. However, without considering the deformation or with the linear model, the identified model parameters are found to be "far" from their a priori. The deviation of u x i was already explained in section 7.2. However, the mobile platform deformation is also accounted in the other identified parameters as shown on figure 18. In order to highlight those deviations, errors between the nominal values and the identified parameters are amplified in the figure. Figure 18 shows that the minimum deviations from the nominal parameters are obtained for the points marked by triangles, when the mobile platform deformation is considered with the matrix model. This result is as expected since the matrix model is a good approximation of the mobile platform deformation. Thus, this inaccuracy source does not need to be accounted in the parameter identification, which allows the identified parameters to be closer to their nominal values when considering the matrix approximation.

Without considering the mobile platform deformation or with the linear model, we remark that the identified positions of points a i and b i do not correspond to the initial design of the telescope. For example, the identified geometry of the base platform considers a smaller design parameter α 0 as its initial design. Those modifications must again be due to the platform deformation: errors that cannot be taken into account in the identification of u i are accounted in the other identified parameters.

However, the goal of calibration is to obtain the most suitable accuracy considering the robot requirements. In our case, the positioning accuracy is more critical than the orientation one as regards to the accuracy results, and both the developed models for the mobile platform deformation give similar results for the positioning accuracy. Thus, the choice of the most suitable model of deformation cannot be done regarding the final accuracy. However, our objective is to develop a model of the telescope that will be used for calibration in space. Thus we will only have access to measurements of position and orientation of the mobile platform, which means that the identification of the coefficient matrix K cannot be made in space. So we made the choice to use the linear model in order to take into account the mobile platform deformation.

CONCLUSIONS

The accurate calibration of the active space telescope allows us to choose the most suitable model for the telescope. Two sources of inaccuracy were considered and modelled: the mobile platform deformation and the flexure joints behaviour.

Concerning the behaviour of the flexure joints, the models developed based on equations of the beam theory and those developed considering the static equilibrium of the mobile platform give the same accuracy results as the 6-PUS model. Indeed, because of the small deflection of the flexure joints over the small workspace of the telescope, their parasitic shift, just as the influence of their generated stresses on the mobile platform equilibrium, are negligible. Thus the flexure joints can be modelled by perfect spherical joints.

The mobile platform deformation was considered with two models of deformation. The matrix model seems to be a good approximation of those deformations since it allows the identified model parameters to be close to their nominal values. However, good positioning accuracy results are also obtained either with the linear model or by ignoring the mobile platform deformation. This is due to calibration that accounts for the model errors in the parameter identification. This leads to identified parameter values being "far" from their a priori. This report explains that those deviations are not only due to geometric uncertainties on the telescope parameters but mainly due to the platform deformation that is accounted in the parameter identification.

However, it is not always possible to directly use the identified parameter values in the control loop of a robot, namely if calibration is not performed regarding the control requirements. For example, it can be problematic to consider unreal values of the geometric model parameters for dynamic control of a robot. The space telescope only has slow motions over a small workspace, which allows the use of the kinematic Jacobian in the control loop. Thus, having identified parameter values being "far" from their a priori does not influence the telescope functioning in our case as long as it allows a good positioning accuracy. So the linear model of the mobile platform deformation can be considered, which enables calibration of the telescope in space where only the position and orientation of the mobile platform can be measured.

Inria

ACKNOWLEDGMENT

This work was supported by Thales Alenia Space and the Region "Provence-Alpes-Côte-d'Azur".

Figure 1 :

 1 Figure 1: Space telescope for observation

Figure 2 :

 2 Figure 2: Kinematic structure of the space telescope

Figure 3 :

 3 Figure 3: Geometry of the base and the mobile platforms

Figure 4 :

 4 Figure 4: Photo of the telescope mobile platform and photogrammetry data

Figure 5 :

 5 Figure 5: Comparison between the first algorithm (red squares) and Haralick method (black circles)

Figure 6 :

 6 Figure 6: Measured deformation of the mobile platform for one measurement configuration. Blue lines represent the amplified error between the measured object points and their rigid body position estimation (red circle points). The observed deformations mostly impact the position b z i of points b i (b x i , b y i and b z i being

Figure 7 :

 7 Figure 7: Linear approximation of the platform deformation

Figure 8 :

 8 Figure 8: Beam theory model of a flexure joint and its deformations

Figure 9 :Figure 10 :

 910 Figure 9: Displacement of a flexure joint considering ideal pivot and beam theory-based model kinematics for δ ∈ [0; 1 • ]

Figure 11 :

 11 Figure 11: Local coordinate system of a flexure joint

Figure 12 :

 12 Figure 12: Stiffness model of one chain of the device

Figure 14 :

 14 Figure 14: Comparison between algorithms of poses choice

Table 5 :Figure 15 :

 515 Figure 15: Positioning and orientation errors on calibration and verification poses for the 4 models without considering the mobile platform deformation, the average and maximal errors are provided for the verification poses

Table 6 :Figure 16 : 7 . 3

 61673 Figure 16: Positioning and orientation errors on calibration and verification poses for the 4 models with the linear model of the mobile platform deformation, the average and maximal errors are provided for the verification poses

Figure 17 :

 17 Figure 17: Positioning and orientation errors on calibration and verification poses for the 4 models with the matrix model of the mobile platform deformation, the average and maximal errors are provided for the verification poses

  i

Figure 18 :

 18 Figure 18: Representation of errors between the nominal and identified parameters values of the 6-PUS model without considering the mobile platform deformation (circle) and with the linear (cross) and the matrix models (triangle) of the mobile platform deformation

Table 1 :

 1 Requirements of the telescope

	2	1	1

  32 33 34 35 36 37 38 39 40 ∆ρ 1

Table 4 :

 4

	6 verification configurations
	RR n°8096

  2.

	i	1	2	3	4	5	6
	∆a x i ∆a y i ∆b x i ∆b y i ∆u x i ∆l i	0.98 -0.23 -2.72 -2.31 2.62 2.57 -2.55 -1.97 -1.76 -2.42 0.37 4.42 3.59 0.73 -0.09 2.45 -3.67 -0.49 -1.38 -0.88 4.22 3.04 1.39 -0.02 -0.07 0.05 0.10 -0.06 0.05 -0.15 0.95 0.07 1.53 0.50 0.63 1.22
			(a) For the 6-PUS model
	i	1	2	3	4	5	6
	∆a x i ∆a y i ∆b x i ∆b y i ∆u x i ∆l i	1.33 3.25 3.75 -0.79 -0.08 0.09 -2.46 -3.32 3.36 -2.18 -2.40 -2.56 -2.53 0.90 3.31 0.48 -0.85 1.79 -3.15 0.13 5.35 2.73 2.38 -0.26 -0.09 0.07 0.12 -0.07 0.07 -0.17 0.95 0.24 1.78 0.38 0.58 1.29
			(b) For the Jacobian-based model
	i	1	2	3	4	5	6
	∆a x i ∆a y i ∆b x i ∆b y i ∆u x i ∆l i	0.99 -0.25 -2.73 -2.31 2.60 2.57 -2.57 -1.98 -1.78 -2.44 0.36 4.43 3.58 0.73 -0.09 2.45 -3.66 -0.50 4.25 1.41 0.00 3.06 -1.36 -0.86 -0.07 0.05 0.10 -0.06 0.05 -0.15 0.94 0.06 1.54 0.50 0.62 1.23
		(c) For the beam theory-based model
	i	1	2	3	4	5	6
	∆a x i ∆a y i ∆b x i ∆b y i ∆u x i ∆l i	-0.40 2.14 3.05 -2.72 -1.23 0.22 2.45 -2.73 -0.20 0.87 -2.64 0.14 1.46 4.02 -4.85 0.55 0.29 -2.97 4.18 0.31 5.91 3.57 0.50 3.95 -0.05 0.05 -0.05 -0.02 0.03 -0.17 1.15 0.31 1.95 0.68 1.05 1.64
			(d) For the stiffness model

Table 7 :

 7 Distances between nominal and identified parameter values for the fourth models considering the matrix model of the mobile platform deformation (in mm except for ∆κ i which is dimensionless)

	Inria

RR n°8096

f i,n = (l il bi + ∆l n Ai + ∆l n Bi )p n + R n b * ia iρ n i u i(32)RR n°8096

route des Lucioles -BP 93 06902 Sophia Antipolis Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

This work was supported by Thales Alenia Space and the Region "Provence-Alpes-Côte-d'Azur".


Orianne Dumas
email: orianne.dumas@inserm.fr

L A M Smit

I Pin

H Kromhout

V Siroux

R Nadif

R Vermeulen

D Heederik

M Hery

D Choudat

F Kauffmann

N Le Moual

Do young adults with childhood asthma avoid occupational exposures at first hire?

Keywords: Asthma, healthy worker hire effect, longitudinal study, occupational exposure, selection bias

Information on the healthy worker hire effect in relation to asthma is scanty. We aimed to assess whether and how childhood asthma-related characteristics (before hire) relate to occupational exposures at first hire.

Analyses were conducted in 298 children examined at the first survey of the Epidemiological study on the Genetics and Environment of Asthma (1991-95), who reported a training period or a job at follow-up in 2003-07 (aged 17-29 years, 53% men). Exposure likelihood to dust, gases and/or fumes at the first occupation was estimated by the ALOHA job-exposure matrix.

Asthma before the first occupation and two asthma classifications for severity (GINA2002 guidelines) and symptoms were defined by questionnaire.

At the first occupation, 47% of subjects were exposed. After adjustment (age, gender, diploma), pre-hire-onset asthmatics (59%) were non-significantly less likely to be exposed (OR[95%CI]: 0.67[0.41-1.11]). Associations were stronger considering those with severe asthma or high symptom score in childhood (0.27[0.11-0.63], 0.49[0.25-0.99]). The association was observed in those who completed a university degree level (0.55[0.29-1.04])

but not in the others (0.98[0.44-2.22]), with consistent results for all asthma characteristics.

Results suggest a healthy worker hire effect in subjects with more severe or more symptomatic asthma in childhood. Diploma level may modulate self-selection.

Introduction

Recent studies estimated that 10 to 15% of adult-onset asthma is attributable to occupational exposure [START_REF] Toren | Asthma caused by occupational exposures is common -a systematic analysis of estimates of the population-attributable fraction[END_REF][START_REF] Kogevinas | Exposure to substances in the workplace and new-onset asthma: an international prospective populationbased study (ECRHS-II)[END_REF]. Work-exacerbated asthma, i.e. worsening of a pre-existing asthma induced by exposure at the workplace, has been less studied but might represent 45% of all work-related asthma cases [START_REF] Henneberger | Work-exacerbated asthma[END_REF][START_REF] Tarlo | Diagnosis and management of work-related asthma: American College Of Chest Physicians Consensus Statement[END_REF]. However, associations between occupational exposures and asthma are probably under-estimated as a result of the healthy worker effect selection phenomenon [START_REF] Moual | The healthy worker effect in asthma: work may cause asthma, but asthma may also influence work[END_REF].

New-onset asthma or exacerbation of pre-existing asthma may lead to avoidance of occupational exposures by less healthy workers who may decide to move to a low-or nonexposed job. This so-called healthy worker survivor effect has been suggested in relation to asthma in several studies [START_REF] Moual | The healthy worker effect in asthma: work may cause asthma, but asthma may also influence work[END_REF].

The healthy worker hire effect, i.e. an early selection mechanism arising from career choice through the selection at hire of a work environment without hazardous exposures by individuals with a pre-existing asthma, has been less studied. Recently, a study from the European Community Respiratory Health Survey (ECRHS) showed that adults with asthma onset prior to entering the workforce were less likely to report current or previously held jobs involving inhalation exposures [START_REF] Olivieri | Healthy hire effect, job selection and inhalation exposure among young adults with asthma[END_REF]. Whether these associations reflect initial (hire effect) or continuing (survivor effect) selection processes could not be determined. In a study that evaluated the job that teenagers would like to have in the future, teenagers with respiratory symptoms did not significantly more often indicate a job with no asthma risk than teenagers without respiratory symptoms [START_REF] Radon | Do respiratory symptoms predict job choices in teenagers?[END_REF]. The magnitude and the determinants of health-based selfselection at hire due to asthma remain to be ascertained. In particular, the role of the clinical expression and the history of the disease, and of socioeconomic factors needs to be studied [START_REF] Olivieri | Healthy hire effect, job selection and inhalation exposure among young adults with asthma[END_REF].

The longitudinal design of the French Epidemiological Study on the Genetics and Environment of Asthma (EGEA) enabled us to follow subjects from childhood to the beginning of their working life. In the current analysis, we aimed to assess: (1) whether subjects with childhood-onset or pre-hire-onset asthma will avoid occupational exposure at their first hire, (2) which clinical characteristics are involved in this self-selection process, with a particular attention to asthma severity, asthma symptoms, as well as to allergic sensitization and lung function in both asthmatics and non asthmatics, and (3) whether sociodemographic characteristics such as gender, diploma level and smoking modify asthma-based self-selection at hire.

MaterialsandMethods

Population

The present analysis used the longitudinal data from the case-control and family-based EGEA study [START_REF] Kauffmann | Epidemiological study of the genetics and environment of asthma, bronchial hyperresponsiveness, and atopy: phenotype issues[END_REF]. The present study was conducted in subjects included as a child (below 16 years of age) at baseline (EGEA1, 19911995) and who had ever worked at follow-up (EGEA2, 2003(EGEA2, -2007)). Of the 604 children enrolled at baseline (including asthmatic cases, their first-degree relatives and population-based controls), 547 took part in the follow-up phase. A total of 141 completed only a brief postal questionnaire with no information about occupation (or only for the 2 last jobs). A total of 393 completed a detailed questionnaire comprising complete occupational history (supplementary figure E1), and among them 318 had ever worked (i.e. reported at least one occupation). Subjects who had only done short-term part-time or seasonal work irrelevant to a career choice (n=8), or with unclear occupational history (n=1) were excluded. Finally, analyses were performed for 298 subjects, after excluding subjects with missing data for exposure (n=1), diploma level (n=2) or parental asthma (n=9, in order to have a constant number for the non asthmatics reference category).

Asthmacharacteristics

Information on asthma and related clinical characteristics at baseline (including allergic rhinitis) were recorded by questionnaire (based on standardized questionnaire designed for children [START_REF] Ferris | Epidemiology Standardization Project (American Thoracic Society)[END_REF] with further questions from the ECRHS questionnaire for asthma-like symptoms), administered by an interviewer to the parents (usually the mother). Methods used to ascertain asthma and related characteristics have been described before [START_REF] Kauffmann | Epidemiological study of the genetics and environment of asthma, bronchial hyperresponsiveness, and atopy: phenotype issues[END_REF][START_REF] Siroux | Relationships of allergic sensitization, total immunoglobulin E and blood eosinophils to asthma severity in children of the EGEA Study[END_REF] and details are provided in online supplementary material.

Subjects with pre-hire-onset asthma, i.e. asthma onset before the first occupation, comprised asthmatics at EGEA1 (asthma onset in childhood, n=150), and asthmatics at EGEA2 with age at onset younger than the age at the first occupation (with at least one year between asthma onset and first hire, n=25) (Figure 1). To evaluate whether asthma was still present just before the hire, asthmatics were classified according to the presence of asthma attacks in the year before the first occupation, using EGEA2 retrospective questions on asthma history: Have you ever had a period of several years without attacks? If yes: At what age did attacks disappear / reappear?. Parental asthma was taken into account only if parental asthma onset occurred before the first hire of the child.

To assess asthma severity in childhood (available among the 150 subjects with asthma at baseline, see figure 1), a 4-level classification was constructed in close accordance with the principles of the Global Initiative for Asthma (GINA 2002) guidelines [START_REF]lung, and Blood Institute. Global strategy for asthma management and prevention[END_REF], as previously in EGEA [START_REF] Bouzigon | Scores of asthma and asthma severity reveal new regions of linkage in EGEA study families[END_REF]. The classification was based on clinical characteristics and type of treatment in the past year (see online supplementary material). A score of 1-2 was classified as mild asthma and a score of 3-4 as moderate to severe asthma. Additionally, a 5-level asthmasymptom score has been constructed for children, adapted from that described for adults by Pekkanen et al [START_REF] Pekkanen | Operational definitions of asthma in studies on its aetiology[END_REF][START_REF] Sunyer | Asthma score: predictive ability and risk factors[END_REF] and already used in EGEA [START_REF] Rage | Are asymptomatic airway hyperresponsiveness and allergy risk factors for asthma? A longitudinal study[END_REF]. Examination at baseline included total serum IgE, skin prick tests (SPT) to 11 aeroallergens, and lung function tests including methacholine challenge. Bronchial responsiveness was assessed by the dose-response slope according to Chinn et al [START_REF] Chinn | Sensitization to individual allergens and bronchial responsiveness in the ECRHS. European Community Respiratory Health Survey[END_REF], which decreases with hyperresponsiveness.

Occupationalexposure

At follow-up, diploma level was recorded, and each training period and job performed was precisely recorded (position and industry title, job task), and subsequently coded according to the ISCO-88 4-digit coding system [START_REF]International standard classification of occupations[END_REF] by an experienced coder as recommended [START_REF] Kennedy | Development of an asthma specific job exposure matrix and its application in the epidemiological study of genetics and environment in asthma (EGEA)[END_REF]. For the current study, the first occupation, i.e. the first job or the first training period performed was selected.

For the main analysis, exposure to dust (biological and mineral), vapours, gases, fumes, pesticides, solvents and metals was assessed using an updated version (HK, RV) of the socalled ALOHA job-exposure matrix [START_REF] Matheson | Biological dust exposure in the workplace is a risk factor for chronic obstructive pulmonary disease[END_REF][START_REF] Sunyer | Pulmonary ventilatory defects and occupational exposures in a population-based study in Spain. Spanish Group of the European Community Respiratory Health Survey[END_REF], which assign to each job code a probable exposure category (none, low, or high) to each agent. The outcome of interest was low or high exposure to at least one agent vs. no exposure. All subjects classified as exposed were at least exposed to dust, vapours, gases and/or fumes, except one (exposed to solvents and metals only).

Additional analyses were performed using an asthma-specific job-exposure matrix [START_REF] Kennedy | Development of an asthma specific job exposure matrix and its application in the epidemiological study of genetics and environment in asthma (EGEA)[END_REF] containing 22 exposure groups classified as high risk agents (moderate or high probability of exposure to: high or low molecular weight agents, mixed environment), or low risk agents (possible exposure to irritants or antigens). The application of this matrix included an occupational hygiene expert re-evaluation step (DH, MH, NLM).

Statisticalanalysis

Associations between exposure at the first occupation (outcome) and asthma before hire or related clinical characteristics (predictors) were evaluated by logistic regression analysis.

Analyses were adjusted for gender, age and diploma level (university diploma level refers to completion of at least 2 years of post-secondary education) at follow-up. Further adjustment on smoking led to similar results (not shown). Gender, smoking and diploma level have also been tested as potential effect modifiers for the relation between asthma and exposure by adding interaction terms in multivariate models. The familial structure of the data was taken into account by using Generalized Estimating Equations approach, which allows the specification of a correlation structure (the correlation between siblings has been assumed constant) in a generalized linear model. The analysis was performed using SAS 9.1.

Results

Among the 393 subjects who completed the detailed adult questionnaire at follow-up, having ever worked was not associated with any studied clinical characteristic (data not shown). The 298 selected subjects were aged 17-29 years at follow-up and 175 (59%) had asthma before their first occupation (table 1). There were marked differences according to asthma for childhood symptom score, bronchial hyperresponsiveness (BHR), SPT, IgE, and allergic rhinitis (table 1). A total of 36% of pre-hire-onset asthmatics had a high (4-5) symptom score, and 28% of asthmatics at baseline had moderate to severe asthma. Subjects with pre-hireonset asthma were significantly more often male, younger at EGEA2 and younger at their first occupation than non-asthmatics. The slight difference in diploma level according to asthma (p=0.11) disappeared after adjusting for age (p=0.44). Among pre-hire-onset asthmatics, 21 (12%) declared having received advice regarding job choices related to asthma or respiratory health. These advices were most often (33%) provided by a physician, and regarded mostly a specific job or exposure (animals, diving, flying, etc).

The first occupation was a training period for 226 subjects and a job for 72. Beginning with a training period was unrelated to asthma status, exposure, gender and age, but associated with a higher diploma level (65%) than beginning with a job (46%, p=0.003).

The 140 subjects (47%) classified as exposed according to the ALOHA job-exposure matrix (to dust, gases, fumes, pesticides, solvents and/or metals) were more likely to be male (54% men exposed vs. 39% women, p=0.01) and to have a lower diploma level (38% exposed in subjects with university diploma level vs. 60% for a lower level, p<0.001) and were similar regarding age or smoking status. According to the asthma-specific job-exposure matrix, 30% of subjects were exposed, with consistent results regarding gender, age, diploma level and smoking status.

Subjects with pre-hire-onset asthma were less frequently exposed (ALOHA job-exposure matrix, table 2) at the first occupation (45%) than non-asthmatics before hire (50%), but the association did not reach significance (adjusted p=0.12). Associations were similar in men and women (0.62 [0.31 1.25] and 0.73 [0.34 1.57] respectively), and in smokers and nonsmokers (0.63 [0.30 1.30] and 0.71 [0.36 1.38] respectively). Among pre-hire-onset asthmatics with a high symptom score in childhood, 39% were exposed (compared to nonasthmatics before hire, p=0.12). This association was more pronounced and significant after adjustment (0.49 [0.25 0.99]). No difference was observed among asthmatics according to the presence or not of attacks the year before the first occupation. The analysis of exposures estimated by the asthma-specific job-exposure matrix provided similar associations (table 2).

Among asthmatics at baseline with moderate to severe asthma (table 3), 27% were exposed at the first occupation (compared to non-asthmatics before hire, p=0.01), resulting in adjusted OR [95% CI]: 0.27 [0.11 0.63] (ALOHA job-exposure matrix) and 0.31 [0.12 0.82] (asthma-specific job-exposure matrix). In tables 2 and 3, differences between crude and adjusted ORs were mainly explained by adjusting for gender and diploma level.

Pre-hire-onset asthma combined with parental asthma was associated with a lower risk of exposure at the first occupation compared to subjects without asthma and parental asthma (0.48 [0.25 0.89], ALOHA job-exposure matrix). Intermediate associations were observed for the effect of parental asthma or asthma alone (0.62 [0.30 1.29] and 0.67 [0.33 1.37] respectively). According to the asthma-specific job-exposure matrix, no significant association was observed.

In analyses of the specific exposures estimated by both matrices, significant associations were found for most of the specific exposures, and the lowest OR was obtained for exposure to high molecular weight agents (supplementary table E2).

Associations between exposure at first hire (ALOHA job-exposure matrix) and lung function, allergic sensitization and triggers for respiratory symptoms in childhood were studied in both non-asthmatics and asthmatics (supplementary table E1). In both groups, lung function, BHR and allergic rhinitis were unrelated with exposure. Regarding total IgE and positive SPT, there were no difference according to exposure in those with a history of asthma, whereas among non asthmatics exposed subjects tended to have lower IgE level (p=0.07) and less often positive SPT (p=0.02). Subjects with asthma at baseline who reported respiratory symptoms triggered by hay, flowers or pets were significantly less often exposed.

The association between pre-hire-onset asthma and both estimates of exposure was observed in subjects with university diploma level but not in subjects with a lower diploma level: 0.55 All associations between exposure and asthma characteristics stratified by diploma level are presented in supplementary table E3.

Discussion

The current prospective study shows that young adults with an asthma history before their first hire may preferentially choose a first occupation with low probability of exposure that may be hazardous to respiratory health. This self-selection phenomenon varies according to clinical characteristics of asthmatics: more severe or more symptomatic asthma in childhood was associated with a stronger selection at hire. Allergic sensitization was also involved in this self-selection process. Socio-demographic characteristics such as a higher diploma level may increase asthma-based self-selection. This analysis was conducted in subjects followed from childhood to the beginning of their working life. The knowledge of the complete job history allowed us to study for the first time the first training period or job. In France, training periods are in principle done within the framework of studies or vocational education, and thus represent an involvement in a vocational field and an initial career choice. Two studies have suggested a healthy worker survivor effect in populations of young apprentices, who can be compared to trainees [START_REF] Iwatsubo | Healthy worker effect and changes in respiratory symptoms and lung function in hairdressing apprentices[END_REF][START_REF] Monso | Individual characteristics and quitting in apprentices exposed to highmolecular-weight agents[END_REF].

Training periods must be taken into account to evaluate a potential early selection phenomenon. Exposures were assessed using job-exposure matrices rather than self-reported exposure that may be influenced by health status [START_REF] De Vocht | Comparison of self-reported occupational exposure with a job exposure matrix in an international community-based study on asthma[END_REF]. The ALOHA job-exposure matrix estimates a broad range of exposures (dust, gases, fumes, pesticides, solvents and/or metals), which might better correspond to the subjects awareness or perception of occupational risks than a more accurate assessment of asthma-specific agents. Consistent results have been observed with the asthma-specific job-exposure matrix, and the analysis of specific exposures suggested a stronger avoidance of occupations with likely exposure to high molecular weight agents, even if the low numbers of subjects in each exposure group limited the interpretation of the results. We studied a young population, describing a recent asthma history (mostly at baseline survey) and job history (at follow-up), and therefore recall bias is unlikely to have affected our results. However, the population was only of moderate size, and consequently a lack of power limited some investigations (for instance taking into account exposure level) and the interpretation of some results.

Our study confirms and extends recent results regarding the healthy worker effect in asthma.

Childhood asthma has been reported as a factor influencing the first occupational social class (not being blue-collar worker) in a French men cohort [START_REF] Thaon | Long-term occupational consequences of asthma in a large French cohort of male workers followed up for 5 years[END_REF]. Furthermore, findings from the ECRHS [START_REF] Olivieri | Healthy hire effect, job selection and inhalation exposure among young adults with asthma[END_REF] showed that asthma status prior to entering the workforce was related to jobs less exposed based on retrospective information on asthma and exposures, assessed for the current or most recent job, or a previously held job (if a previous occupational exposure was suspected). Associations observed in ECRHS might reflect both initial selection in career choice (hire effect) and continuing selection processes (survivor effect). Our results are consistent with these findings as they support an influence of asthma characteristics prior to entering the workforce on the career choice. Furthermore, we have shown that a substantial part of job selection would take place at first hire.

Our results suggest for the first time that childhood health history and socioeconomic parameters modify self-selection at hire. More severe asthma and more symptomatic asthma in childhood were associated with a strong selection at hire. Asthma severity in adults has previously been related to job changes [START_REF] Blanc | Work disability among adults with asthma[END_REF]. Our results suggest that the severity and symptoms of the disease experienced in childhood may have an impact on job choice in young adulthood. Despite a good characterization of asthma in childhood, detailed data on asthma severity and symptoms at the precise moment of hire were not available. In the current analysis, the presence of asthma attacks the year before hire did not increase self-selection, suggesting that job choices may be based on a longer disease history and a larger range of clinical characteristics.

Parental asthma strengthened selection at hire, possibly related to parental knowledge or work experience. Genetic factors may also be considered, as they may be associated to specific asthma characteristics which would influence job choice. Familial resemblance in asthma severity has been previously evidenced in the EGEA population [START_REF] Pin | Familial resemblance of asthma severity in the EGEA study[END_REF].

Our results suggest that allergic sensitization might be involved in health-based selection mechanisms at hire. Positive SPT in childhood predicted less exposure at first occupation, particularly among non-asthmatics (few asthmatics had negative SPT), with consistent results regarding total IgE. Allergic rhinitis did not significantly influence job choice. This result partly differs from a previous study in a cohort of Swedish men born in 1949-1951 [START_REF] Wiebert | Mortality, morbidity and occupational exposure to airway-irritating agents among men with a respiratory diagnosis in adolescence[END_REF], reporting job selection in the early period of the working life similarly in men with allergic rhinitis and in men with asthma (diagnosed at military service).

Our results suggest that diploma level might modify healthy worker hire effect, as only asthmatics with a university diploma level avoided jobs with likely exposures (for both exposure estimates), with consistent results for all asthma characteristics studied. Low socioeconomic level has been previously mentioned as a factor leading to less job selection in mortality studies [START_REF] Chen | The influence of study characteristics on the healthy worker effect: a multiple regression analysis[END_REF], as well as on removal from exposure in red cedar asthmatics [START_REF] Marabini | Clinical and socioeconomic features of subjects with red cedar asthma. A followup study[END_REF].

Diploma level reflects both education and socioeconomic conditions, two different aspects that might influence selection at work, through the knowledge of occupational risks [START_REF] Bhinder | Perception of asthma as a factor in career choice among young adults with asthma[END_REF] and the larger amount of career opportunities. As underlined recently [START_REF] Olivieri | Healthy hire effect, job selection and inhalation exposure among young adults with asthma[END_REF] more comprehensive information on socioeconomic conditions are warranted to better understand the various components of the healthy worker effect. In a study that investigated the preferred future job choice in teenagers [START_REF] Radon | Do respiratory symptoms predict job choices in teenagers?[END_REF], inconclusive associations were found between respiratory symptoms and the choice of a job without an increased risk for asthma. This study was conducted in vocational trainees, likely with a lower education, and probably less prone to self-selection, as was shown in our study. We did not evidence gender differences in selection (in agreement with the study of Olivieri et al. [START_REF] Olivieri | Healthy hire effect, job selection and inhalation exposure among young adults with asthma[END_REF]), nor difference between smokers and non-smokers.

Work-related asthma has an important individual socioeconomic impact [START_REF] Vandenplas | Socioeconomic outcomes in work-exacerbated asthma[END_REF][START_REF] Vandenplas | Health and socioeconomic impact of work-related asthma[END_REF], that might be reduced by health-based selection in the early period of the working life, but career choices also depend on socioeconomic conditions. Selection might have arisen from a personal choice, an advice from a physician or another person, or from the employers choice.

However, no health-based pre-employment selection policy exists in France. Very few asthmatics declared to have received advice regarding job choice, and this advice concerned very specific occupational fields. Considering public health surveillance, this study raises the importance of taking into account this selection bias in studies of occupational asthma.

Further research is needed to better understand how current estimates of the effect of specific exposures on respiratory health may be affected by the healthy worker effect.

In conclusion, the current study shows in a population of young adults that asthma history plays a role in the choice of the first occupation, and that this choice is further modified by clinical and socio-demographical characteristics. This phenomenon probably leads to an under-estimation of associations between occupational exposures and asthma.

[ 0 .

 0 29 1.04] vs. 0.98 [0.44 2.22] (p interaction = 0.19, ALOHA job-exposure matrix); 0.47 [0.23 0.97] vs. 1.38 [0.61 3.11] (p interaction=0.02, asthma-specific job-exposure matrix).

TABLE 1 .

 1 Description of the population according to asthma status before hire Asthma severity was recorded only among the 150 subjects with asthma at baseline, and was missing for 16 of them.

		Asthma before hire
	All	No	Yes	p
	(n=298)	(n=123)	(n=175)

*

Acknowledgements

EGEA cooperative group:

Hospital program of clinical research (PHRC)-Paris; National Research Agency -Health environment, health-work program (ANR-SEST 2005); Merck Sharp & Dohme (MSD); Isere committee against respiratory diseases (COMARES); French Agency of health . research (NWO) Van Gogh program for French Dutch cooperation; and the University Paris Sud 11 -ED420 doctoral grant.

Adjusted for age at follow-up, gender and diploma level. § Analysis performed in 282 subjects (data missing for symptom score in 16 pre-hire-onset asthmatics). || Analysis performed in 286 subjects (data missing for attacks the year before the first occupation in 12 pre-hire-onset asthmatics). Exposed (moderate or high probability of exposure to asthmogens, or possible exposure to low risk irritants or antigens) vs. non-exposed.

Adjusted for age at follow-up, gender and diploma level. § Analysis performed in 257 subjects (data missing for asthma severity in 16 asthmatics at baseline).