N Roca Filella

A Koessler

B C Bouzgarrou

J.-A Corrales Ramon

3D visual-based tension control in strip-like deformable objects using a catenary model

In recent years, there has been a growing interest in robotic manipulation of deformable objects. In order to perform certain tasks, the robot must control the shape of the object while taking care not to apply excessive stresses so as not to deform it irreversibly. This is the case when extracting elasto-plastic objects in strips from an industrial reel. In order to control the mechanical stresses within the object, we propose a vision-based control scheme to minimize tension by regulating the angular velocity of a motorized reel on which they are wound. In this paper, we propose a method, based on a catenary model and visual feedback from a low-cost RGB-D camera, to estimate the tension distribution along a rubber strip. Thus, the control strategy aims to achieve a desired tension value by varying the length of the suspended portion of the manipulated strip. Simulation and experimental results validate the proposed approach for strip-like objects of various dimensions.

I. INTRODUCTION

Deformation control of soft objects like fabrics, foam and other elastic bodies is now a matter of ongoing research in the field of robotic manipulation [START_REF] Sanchez | Robotic manipulation and sensing of deformable objects in domestic and industrial applications: a survey[END_REF]. Their deformation is generally modeled by introducing new degrees of freedom (DOF), which makes control more challenging. Perception based on vision, tactile or multi-modal sensing is hence used to estimate these additional DOFs, which represent the state of deformation of the object [START_REF] Nadon | Multi-modal sensing and robotic manipulation of non-rigid objects: A survey[END_REF]. That allows closing the control loop and ensures task achievement. Some methods, known as model-free methods, do not require a model and are based on perceived data, like position, angle, or shape, decorrelated from the object parameters. In [START_REF] Navarro-Alarcón | Modelfree visually servoed deformation control of elastic objects by robot manipulators[END_REF] visual data are collected to estimate locally the deformation Jacobian matrix, avoiding any model identification step. The work of [START_REF] Long | Robotic cutting of soft materials using force control and image moments[END_REF] uses both visual and force sensor for robotic cutting of deformable objects. These methods avoid modeling the dynamics of the objects in its environment but often need a lot of input data if they are based on supervised learning and are not very generalizable. Another solution is to use a model that facilitates the description of the object. These models can be classified in two main categories based on their complexity.

Physical-based models: Based on mechanical equations, they are as close as possible to reality. The two main models are Finite Element (FE) and Mass-Spring-Damper models. For instance, we can mention the research of [START_REF] Sanchez | Online shape estimation based on tactile sensing and deformation modeling for robot manipulation[END_REF] and [START_REF] Das | Autonomous shape control of a deformable object by multiple manipulators[END_REF], which use such models in generic manipulation cases.In [START_REF] Ficuciello | Fem-based deformation control for dexterous manipulation of 3d soft objects[END_REF] the desired actuator motions are computed in real-time as an inverse solution of a FEM problem. The gripper's configuration is coupled to a FE model in [START_REF] Duenser | Interactive robotic manipulation of elastic objects[END_REF] to perform an open-loop shape control. These methods are physically relevant and useful for changeable object shapes and material behaviours. However, these physical models are not suitable for varying mass objects (such as a strip that is unrolled from a reel) and often requires a lot of nodes which is resourceconsuming. Recently, the work of [START_REF] Koessler | An efficient approach to closed-loop shape control of deformable objects using finite element models[END_REF] used low node count FE models for shape control to reduce the computational cost.

Geometry-based models: To overcome these drawbacks, non-physical models can be used. In these approaches, deformation is generally modeled using explicit mathematical formulations of object shape, which allow fast computations within a control loop. The deformation is estimated using geometrical features like positions, curvatures or surfaces. In [START_REF] Zea | Tracking elongated extended objects using splines[END_REF] a simple approach to track elongated extended objects using Bézier curve is introduced. This algorithm is fast and robust even with partially occluded target. The deformation of the surface object is estimated in [START_REF] Jordt | Fast tracking of deformable objects in depth and colour video[END_REF] using non-uniform rational basis splines approximation with a RGB-D camera. A shape control of isometrically deforming objects with 2D camera is proposed in [START_REF] Aranda | Monocular visual shape tracking and servoing for isometrically deforming objects[END_REF]. These methods show a good balance between computational cost and accuracy. However, most of these approaches do not take material properties into account and only guarantee good results for the controlled features.

For a deformable object, this is a crucial point in several applications because stress must not increase beyond the elastic limit, so that the material does not undergo a plastic deformation. In this paper, we explore the field of tension control which consists in setting and maintaining a desired tension in an object to prevent damaging it while manipulating it. The research of [START_REF] Flixeder | Force-based cooperative handling and lay-up of deformable materials: Mechatronic design, modeling, and control of a demonstrator[END_REF] and [START_REF] Xiao | Indoor uav localization using a tether[END_REF] exploits and validates the use of a model for strip-like deformable objects, based on a catenary curve. This geometry-based model can be considered at the intersection of the two previous methods, as it gives information on the tension in the object. These works could be coupled to those of [START_REF] Laranjeira | Embedded visual detection and shape identification of underwater umbilical for vehicle positioning[END_REF] to replace expensive force sensor by a vision sensor in [START_REF] Flixeder | Force-based cooperative handling and lay-up of deformable materials: Mechatronic design, modeling, and control of a demonstrator[END_REF], but also in [START_REF] Qin | Take a long deformable belt out of a bobbin by humanoid robot[END_REF] where a robot takes a deformable belt out of a bobbin using a F/T sensor.

Proposed contributions: In this work, we introduce a visual-based tension control scheme for strip-like deformable objects, whose one end is grasped by a robot and the other is wound onto a reel. Using a low-cost RGB-D camera and a 3D catenary model, the deformation state is estimated without any contact by partial observation of the object. A closed-loop tension control scheme managing the rotation of the reel is then proposed to vary the length of linear objects in real-time. Experiments regarding our use case are performed in simulation, then on the real robotic system. A simplified control logic block diagram is shown in Fig. 1.

II. TENSION ESTIMATION BY VISION

In this section, a catenary-based model is introduced. We use a method to fit strip-like deformable objects in 3D space and estimate the tension in the object. We evaluate the quality of tension estimation in comparison with a force sensor.

A. Catenary-based model

A catenary is the curve that an idealized hanging chain or cable assumes under its own weight (uniformly distributed) when supported only at its ends [START_REF] Yates | A Handbook on Curves and Their Properties[END_REF]. The use of catenaries was encouraged by the fact that this model gives useful information on tension in the object. The model is well suited for strip-like deformable objects, considering that the thickness d ob j and the width of a strip-like deformable object are very small when compared to its length L, and assuming zero bending stiffness. It is also assumed that no torsional stress is applied, which allows us to reduce the representation in a plane.

Let O(⃗ x,⃗ z) be the global frame in 2D space, E 1 and E 2 the ends of the catenary, P i (x i , z i ) any point in the curve and P low (x low , z low ) the lowest point of the curve, as depicted in Fig. 2.

The height h i = z iz low represents the sag at point P i , and l i is the length of the segment of curve, from P low to P i . At any point, the following catenary equation is verified [START_REF] Sugden | Construction of transmission line catenary from survey data[END_REF]:

z = z low + c cosh x -x low c -1 (1)
where c = T H,i /µg is the catenary constant depending on the shape of the strip. µ is the mass per unit length and g = 9.81 m•s -2 is the acceleration of gravity. This introduces T H,i , the horizontal component of the total tension T T,i at point P i , as follows:

T H,i = µgc. (2)
According to Fig. 1, the vertical component is defined as T V,i = T H,i tan Φ i where Φ i is the tangential angle formed by the catenary at point P i . We assume that for a infinitely small change in x, dx, the infinitely small change in z is given by dz=dx tan Φ i and we have

T V,i = T H,i sinh x i -x low c = µgl i (3)
where, by definition and for x i < x low ,

l i = x low x i 1 + dz dx 2 dx = -c sinh x i -x low c . (4)
Considering that T T,i = T H,i 2 + T V,i 2 and (1) , we have As we can see, the total tension depends only on three parameters of the catenary, µ, c and z low , no matter the value of x low . From now, we consider changing it to a 3D object being projected.

T T,i = µgc 1 + sinh 2 x i -x low c = µgc z i -z low + c c = µg (c + h i ) (5)
A model of 3D catenary is proposed in [START_REF] Chan | 3d catenary curve fitting for geometric calibration[END_REF]. Its expression at any point in the 3D camera frame C( ⃗ x′, ⃗ y′,⃗ z) is as follows:

z = z low + c cosh x ′ cos θ -y ′ sin θ -ρ sin θ c -1 ( 6
)
where θ is the azimuth and ρ is the distance from the origin when the 3D catenary is projected into the x ′y ′ plane as a straight line. For a transformation from 3D camera frame to 2D object frame, we use the variable substitution introduced in [19]: x = x ′ cos θy ′ sin θ and x low = ρ sin θ .

B. Object detection and 3D curve fitting

An RGB-D camera is positioned above the target object. After the acquisition of the image, a depth thresholding is processed to extract the foreground. A color segmentation is then applied, and the relevant point cloud is extracted. To reduce the computing cost, a 3D down-sampling is performed. Finally, to eliminate noise in the detection, a 3D clustering is applied using DB-SCAN algorithm [START_REF] Ester | A density-based algorithm for discovering clusters in large spatial databases with noise[END_REF].

A 3D curve fitting method using non-linear least squares is then implemented to fit the function to the point cloud. It uses a Trust Region Reflective algorithm [START_REF] Branch | A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems[END_REF] which is ideal for a bound-constrained minimization problem. Fig. 3 (a) shows that the strip is well detected in color image and Fig. 3 (b) shows the point cloud before down-sampling and clustering. Despite the significant depth noise, the identified curve correctly fits to the point cloud in Fig. 3 (c). In Fig. 3 (d) the identified parameters are shown. Finally, the results highlight that the proposed method does not need full object perception, as seen in Fig. 3 (a) where both ends are not in the color image.

C. Validation of the visual estimation

Once the parameters z low , c, ρ and θ are estimated, the strip-like object is reduced to a curve with a negligible width, so (2), ( 3) and ( 5) can be used to estimate the tension.

We now make a comparison between the tension estimated using the visual method and the one measured by two accurate force sensors (s-type load cells) with an announced accuracy of ±0.1 percent for 10 N. We use the three objects of Table I of length l = 0.90 and in four different cases: |x 1x 2 | = {0.74, 0.79, 0.84, 0.89}. We compute the mean values and standard deviations (SD) and the results are shown in Table II. Even though small differences can be observed between the two methods, our method is suitable for low tension estimation.

For case 4, the object remains flat, even after being more stretched and our method reaches its limit of validity. In that case, a Force/Torque sensor can be used as an add-on tool for the robot. Fig. 4 now shows a comparison between the horizontal tension measured by a Robotiq FT300 F/T sensor and the tension estimated with our method. We manually rotate the reel to vary the length of the object during 50 acquisitions. Our proposed visual method seems to be more stable thanks to the fitting step which smooth the values of tension. When the object is flat, it becomes impossible to estimate the correct value of parameters c with our visual method, therefore we impose a maximum threshold for this value and the F/T sensor gives better results.

In conclusion, the catenary model is well adapted especially when the object does not become flat. The precision and efficiency of this novel perception method enables loop closure for tension control. We will synthesize such a controller, with an approach based on the formerly introduced model.

III. CLOSED-LOOP TENSION CONTROL SCHEME

In this section, we present the control scheme to manipulate the strip-like object using the previous visual feedback. The purpose is to control the angular velocity of the reel ω to vary the length l of the object.

A. Tension control law synthesis

We assume that positions at both ends are known and that E 1 is fixed. It is, and stays, the highest point of the curve during all the manipulation. This assumption implies that, according to [START_REF] Sanchez | Online shape estimation based on tactile sensing and deformation modeling for robot manipulation[END_REF], the tension has a global maximum at this point, as h 1 = max(h i ). E 2 is grasped by the end-effector of the robot. To simplify the notation in the following, both T H,x 1 and T T,x 1 are represented by T . We denote T d the desired tension and define Ṫ as:

Ṫ = dT dl l ( 7
)
where l = l 1l 2 . The aim is to exponentially decrease the error e = T d -T towards zero with the following control law based on a first-order dynamic behavior:

ė + Ke = 0 (8)
with K > 0. Merging [START_REF] Ficuciello | Fem-based deformation control for dexterous manipulation of 3d soft objects[END_REF] and [START_REF] Duenser | Interactive robotic manipulation of elastic objects[END_REF], we obtain the following control law on l to ensure that the tension converges towards the desired tension:

l = dl dT u + Ṫ d (9)
with u = Ke. This leads to a rotational control of the reel though ω = l/R, where R is the radius of the reel. After that, the length of the strip-like object is modified, and so the catenary too. We consider the desired tension remaining constant over time, hence Ṫ d = 0. By now, the aim is to define the derivative of T with respect to l, i.e. determine the influence of a small variation of the length on the tension.

B. Control of the horizontal component

Controlling the horizontal component of the tension at E 1 allows a control at any point of the object, as dT H,i /dx = 0. The derivative of T H,1 with respect to l is defined as follows:

dT H,1 dl = µg dc dl . ( 10
)
We now want to express dc/dl. By subtracting (1) at points E 1 and E 2 , we obtain

z 2 -z 1 = c cosh x 2 -x low c -cosh x 1 -x low c = 2c sinh x 1 + x 2 -2x low 2c sinh (A) (11)
where A = (x 2x 1 )/2c. Using (4), we have

l = c sinh x 2 -x low c -sinh x 1 -x low c = 2c cosh x 1 + x 2 -2x low 2c sinh (A) . ( 12
)
If we divide ( 11) by ( 12) for l ̸ = 0, we obtain

x 1 + x 2 -2x low 2c = arctanh z 2 -z 1 l . ( 13
)
Integrating ( 13) into ( 12), we finally get

l cosh arctanh z 2 -z 1 l = 2c sinh (A) . ( 14
)
One can verify that the left side of ( 14), denoted f 1 , depends on l and the right side, denoted f 2 , depends on c. We hence have the following derivative of c with respect to l:

dc dl = d f 1 dl dc d f 2 (15)
where, by derivation,

d f 1 dl = l l -(z 2 -z 1 ) l + (z 2 -z 1 ) (16)
and

d f 2 dc = 2 sinh (A) -(2A) cosh (A) . (17)
From equation ( 3), we understand that the longer the object is, the higher the vertical tension is. That means that even if the object is sagging, it can be stretched at its ends under its weight. The vertical tension hence becomes impossible to ignore in our computations and we will now focus on the control of the total tension.

C. Control of the total tension

We now express the derivative of T T,1 with regards to l. From (5), we obtain:

dT T,1 dl = µg dc dl + dh 1 dl . ( 18
)
To express the derivative of h 1 with respect to l, we use the following decomposition:

dh 1 dl = dh 1 dc dc dl (19)
Using ( 1) and ( 14) into h 1 , we obtain:

h 1 = z 1 -z low = c (cosh (B) -1) (20)
where

B = x 1 -x 2 2c + arctanh z 2 -z 1 l
. By derivation, we have: Finally, replacing ( 21) into ( 19), we have an equation for [START_REF] Sugden | Construction of transmission line catenary from survey data[END_REF] and we can integrate it into (9) to formulate a law that controls the total tension, and not only the horizontal component. From now on, we consider the total tension as the servoed variable and drop the subscript, so that T T,1 is now referred to as T 1 in the following experimental analysis.

dh 1 dc = cosh (B) -1 -c sinh (B) . dl dc z 2 -z 1 l 2 -(z 2 -z 1 ) 2 + x 1 -x 2 2c 2 (21)

IV. SETUP OF THE APPLICATION

The simulation and the experimentation will be firstly focused on the rubber strip-like object 2. As shown in Fig. 5, an Intel RealSense D435 RGB-D camera is placed at C(0, 0). The point E 2 is grasped and driven by a Universal Robots UR10 robot while E 1 (-0.570, -0.120) is held fixed during the manipulation. The rotating reel is motorized to adjust T 1 . This first experiment Exp. 1 is divided in three phases:

• Phase 1: For 5 s, the robot stays in A so that the tension settles around the initial desired tension. • Phase 2: The robot drives E 2 from A(-0.018, -0.258) to B(0.197, -0.511) into the xz plane in 5 s seconds at an average velocity of v = 0.062 m.s -1 . • Phase 3: The robot stops for 5 seconds so that the tension settles around the final desired tension

V. SIMULATION RESULTS

As the control law was already established, the choice of the desired tension remains to be determined. But looking at the equations, the choice of a reachable tension is not trivial. In this section, we perform simulation to find adequate values for desired tensions. Once this is done, we introduce the tuning process and evaluate the impact of disturbances on the control law, such as robot motion and radius variation.

A. Trajectory planning of the desired tension

For a given position of E 1 and E 2 , it is possible to find c given the length l by solving numerically the folowing equation:

z 2 -z 1 -2c sinh (A + B) sinh (A) = 0 (22)
found using [START_REF] Jordt | Fast tracking of deformable objects in depth and colour video[END_REF] and [START_REF] Flixeder | Force-based cooperative handling and lay-up of deformable materials: Mechatronic design, modeling, and control of a demonstrator[END_REF]. We likewise find x low solving numerically [START_REF] Jordt | Fast tracking of deformable objects in depth and colour video[END_REF]. Now that c and x low are known, the tension components according to l are presented in Fig. 6 (a) for a given position and the evolution of the catenary can be seen in Fig. 6 (b). We notice that the horizontal tension (in orange) decreases with l increasing, while the vertical tension (in green) increases. They hence have opposite behaviours while changing the length. It is then difficult to predict that of the total tension. To overcome this, we propose an online trajectory planning method for T d 1 . The Algorithm 1 is dedicated to minimize T 1 (l) (i.e. to find the minimum of the blue curve) so that, at each time, T d 1 = min(T 1 (l)). As the robot has soft acceleration and deceleration, a smooth variation of the desired tension is planned.

Algorithm 1: find T d 1 , the minimum of T 1 1 Initialisation: T d 1 ← 10; dl ← 0.05; l ← || ⃗ E 1 E 2 || + dl; c 0 ← 0.1; x low,0 ← (x 1 + x 2 )/2; 2 while l ≤ 3 do 3 c ← f indc(l, x 1 , x 2 , z 1 , z 2 , c 0 ) solving numerically (22); 4 x low ← f indx low (x 1 , x 2 , z 1 , z 2 , c, x low,0 ) solving numerically (11); 5
T 1 ← computeTension(c, x 1 , x low ) using (5);

6 if T 1 ≤ T d 1 then 7 T d 1 ← T 1 ; 8 end 9 l ← l + dl; c 0 ← c; x low,0 ← x low ; 10 end

B. Proportional gain tuning

To evaluate the impact of K in the control law, we choose four different values: K = {1, 3, 6, 9}. The results of tension estimation and output control are presented in Fig. 7. The error exponentially converges toward zero during phases 1 and 3, as expected. Regarding phase 2, the robot motion causes a delay, that K = 1 cannot compensate. The high gain K = 9 involves undesired oscillations. Finally, K = 3 and K = 6 seems to be appropriate solutions.

C. Influence of a time-dependent radius

One can see that a constant radius has been chosen in the control law. However the more the object is unwound, the smaller the radius is. The radius is indeed a time-dependent variable.

To include the thickness in the simulation, we multiply l by (Rvar)/R with var = β .d ob j /2π the variation of radius and where β is the rotational angle of the reel (initially, β = 0). In Fig. 8, we compare the tension with different values for thickness of the object: d ob j = {0, 0.01, 0.05, 0.1} and for R = 0.148. The error due to the radius variation is taken into account in the controller error when d ob j ̸ = 0. The final target tension is reached shortly after, because l is smaller than it should be. Despite that small difference in terms of speed of convergence, we assume that the radius could be measured off-line before each experiment.

Positive results achieved during this simulation phase encourage us to evaluate the fitting method and the proposed control law in real settings.

VI. EXPERIMENTAL RESULTS

This section shows the tension control on the real system described in Section IV. We present here a comparison with the simulation use case, as well as novel experiments with more varied robot arm trajectories and objects.

A. Comparison between simulation and experimental results

We performed experimental tension control for K = {1, 3, 6} and we compare the results with the simulation results in Fig. 9. Given the measurement accuracy of our visual tension estimator, we decided to limit the output control when the robot is not moving if |e| is smaller than ε = 0.01 N. This solution helped our system to become more stable and to converge towards the final desired tension for K = 3 and K = 6 and the results are nearly the same as the simulation ones. As expected, K = 1 is not sufficient for a fast response. One can notice that the output for K = 6 presents some considerable oscillations. (5 repetitions). Probably, the robot does not have time to reach this velocity but this experiment was designed to see how the controller reacts when faced to oscillations. One can find the trajectory of each experiments and the results of the tension control in Fig. 10. The root mean square percentage error (RMSPE) and the maximal percentage error e p,max while the robot is moving are presented in Table III. The choice of a percentage error is motivated by the difference in scale between two objects. From Exp. 2, it can be seen that increasing the velocity of the robot also increases the tension overshoot. However, T 1 still converges towards the desired value once the motion is stopped. This behaviour is also observed in Exp. 3, where the initial mismatch in tension has no visible effect on the quality of tension servoing. From this, we can conclude on the efficiency of the proposed method, provided the dynamics of the robot leading the motion are not too high.

C. Instability due to dynamics

The experimental analysis has shown the viability of the proposed method. However, as the length increases a novel problem appears: the object becomes more vulnerable to oscillations caused by the variation of the output control. For Exp. 1-3, there was no notable difference in RMSPE depending on the object. However, in Exp. 4, the system performs way worse on the heavier object than on the lighter one. We conclude that taking dynamics into account would be a mandatory step in the future in order to deal with lengthier and heavier objects. In that precise case, the use of a FE model might be useful.

VII. CONCLUSION

In this work, we have proposed a closed-loop tension control for strip-like deformable objects. A 3D catenarybased model was used to estimate the internal tension of a linear deformable object pulled by a robot. The control law regulates the angular velocity of a reel onto which the object is wound, in order to vary the object length and to ensure the object is not stretched. In that way, we prevent the object from an irreversible deformation. The proposed work has been validated in simulation first. An algorithm of trajectory planning for the desired tension has been implemented and a gain tuning has been made. After that, we evaluated our controller with several trajectories of the robot and several parameters of the objects.

The proposed method gives a deformation state in realtime using a low-cost RGB-D sensor instead of a dedicated force sensor. The 3D fitting method allows an estimation despite partial observation and results in smoothed feedback. Finally, the catenary model is easy to identify, since the only relevant parameter is the mass per unit length. Experimental results have proven the capacity of the method to handle various materials successfully.

Future works will focus on strip-like deformable objects with higher thickness and significant bending stiffness. Moreover, the study of inertial effects would be necessary to control the oscillations of the strips, especially for longer objects and faster task execution.

Fig. 1 .

 1 Fig. 1. Control logic block diagram of our proposed method for visual-based tension control scheme.

Fig. 3 .

 3 Fig. 3. (a) object detection in color image, (b) down-sampled 3D point cloud, (c) fitted catenary curve and (d) identified parameters.

  Strip-like object µ kg • m -

Fig. 4 .

 4 Fig. 4. Comparison of horizontal tension measured using the F/T sensor (green) and the visual sensors (red).

Fig. 5 .

 5 Fig. 5. Picture of the experimental setup.

Fig. 6 .

 6 Fig. 6. (a) tensions according to l, (b) evolution of the catenary while l is increasing.

Fig. 7 .

 7 Fig. 7. Estimated tension vs. computed velocity output.

Fig. 8 .

 8 Fig. 8. Estimated tension with different thickness of the object.

Fig. 9 .

 9 Fig. 9. Estimated tension vs. computed velocity output and real velocity.

Fig. 10 .

 10 Fig. 10. Trajectory of each experiment and results of the tension control.

TABLE I MASS

 I PER UNIT LENGTH OF THE MANIPULATED OBJECTS.

		case 1	case 2	case 3	case 4
	Object	mean	SD	mean	SD	mean	SD	mean	SD
	1 (force)	0.77 0.001 0.88 0.001 1.12 0.001 3.48 0.011
	1 (vision) 0.76 0.004 0.86 0.006 1.16 0.014 3.22 0.212
	2 (force)	0.61 0.001 0.71 0.001 0.98 0.001 2.41 0.009
	2 (vision) 0.61 0.003 0.70 0.008 0.99 0.019 2.45 0.159
	3 (force)	0.22 0.001 0.26 0.001 0.36 0.001 1.20 0.011
	3 (vision) 0.21 0.001 0.24 0.001 0.30 0.004 0.82 0.090

TABLE II COMPARISON

 II OF MEAN VALUE AND SD (IN N).

TABLE III RMSPE

 III AND e p,max FOR EACH EXPERIMENTS.

Université Clermont Auvergne, Clermont Auvergne INP, CNRS, Institut Pascal, F-63000 Clermont-Ferrand, France {firstname.lastname}@sigmaclermont.fr

[START_REF] Nadon | Multi-modal sensing and robotic manipulation of non-rigid objects: A survey[END_REF] Université de Lorraine, Arts et Metiers Institute of Technology, LCFC, HESAM Université, F-57070 Metz,

France[START_REF] Navarro-Alarcón | Modelfree visually servoed deformation control of elastic objects by robot manipulators[END_REF] Centro Singular de Investigación en Tecnoloxías Intelixentes (CiTIUS), Universidade de Santiago de Compostela, 15782, Santiago de Compostela, SPAIN {firstname.lastname}@usc.es

ACKNOWLEDGEMENTS

This research made in the FACTOLAB framework has been sponsored by Michelin Tyres Manufacturer, by the French government research programs "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-001 (CAP 20-25) and the IMobS3 Laboratory of Excellence (ANR-10-LABX-16-01). CIFRE 2019/0917. JACR was funded by the Spanish Ministry of Universities through a 'Beatriz Galindo' fellowship (Ref. BG20/00143) and by the Spanish Ministry of Science and Innovation through the research project PID2020-119367RB-I00.


Julien E Bois

Philippe G Sarrazin

Robert J Brustad

Julien P Chanal

David O Trouilloud

Reflected appraisal process 1 Running head: REFLECTED APPRAISAL PROCESS Parents' appraisals, reflected appraisals, and children's self-appraisals of sport competence: A yearlong study

Keywords: reflected appraisal, parental influence, perceived competence, physical activity, mediation, motivation

This study investigated the contribution of the reflected appraisal process to the ontogeny of children's self-perceptions of physical competence. Emanating from symbolic interactionist thought, reflected appraisals refer to individuals' perceptions of others' evaluations of their abilities. The influence of parents' appraisals of their child's competence on the child's self-appraisals was hypothesized to be mediated by parents' reflected appraisals. Data were collected from a sample of 147 sixth graders and their parents in three waves over one school year. Structural equation modeling revealed that (1) parents' appraisals at Wave 1 influenced children's reflected appraisals of parents at Wave 2; (2) reflected appraisals at Wave 2 influenced children's self-appraisals at Wave 3, controlling for both the child's actual level in sport (as assessed by the child's grade in physical education) and the child's self-appraisals at Wave 1; and (3) the influence of parents' appraisals on their child's self-appraisals was nonsignificant when reflected appraisals were controlled for. This study supported the mediational role of the reflected appraisal process. The findings from this study provide a unique contribution to the sport socialization knowledge base by highlighting the role of the reflected appraisal process in the formation of children's self-appraisals of ability.

Parents' appraisals, reflected appraisals, and children's self-appraisals of sport competence: A yearlong study.

Numerous theoretical models in contemporary psychology have highlighted the role of perceived competence, or ability, as a central antecedent of achievement motivation and behavior (e.g., [START_REF] Deci | The "what" and "why" of goal pursuits: human needs and the Self-Determination of behaviour[END_REF]Eccles, Wigfield & Schiefele, 1998;[START_REF] Elliot | A hierarchical model of approach and avoidance achievement motivation[END_REF][START_REF] Harter | A model of mastery motivation in children: Individual differences and developmental change[END_REF]. Although these frameworks differ from each other in numerous ways, each proposes that individuals who have high perceptions of their ability in a particular achievement domain will demonstrate greater motivation, as demonstrated through their decision to become involved, as well as by their levels of effort, perseverance, and affective characteristics in a given context. In the sport and physical activity setting, considerable support has been found for the role of perceived competence in contributing to motivation (for reviews, see [START_REF] Weiss | Self-esteem and perceptions of competence in youth sport: Theory, research, and enhancement strategies[END_REF][START_REF] Weiss | Motivational orientations and sport behavior[END_REF].

Given the influence of competence perceptions on motivation, an important area of study involves understanding the processes individuals use to evaluate or judge their personal competence or ability in achievement contexts. Although much of this research has been conducted in academic contexts (e.g., [START_REF] Stipek | Developmental change in children's assessment of intellectual competence[END_REF], a handful of studies have also examined these issues in the physical activity domain (see for reviews, [START_REF] Horn | Sources of competence information[END_REF][START_REF] Horn | Perceived competence in young athletes: Research findings and recommendations for coaches and parents[END_REF][START_REF] Weiss | Children's self-perceptions and sources of competence information: A cluster analysis[END_REF]. Several theoretical frameworks have highlighted the importance of significant others in shaping one's self-perceptions. Contemporary developmental and social psychologists (e.g., [START_REF] Harter | The development of self-representations[END_REF][START_REF] Eccles | Motivation to succeed[END_REF] have continued in the line of symbolic interactionists such as [START_REF] Cooley | Human, nature and the social order[END_REF] and [START_REF] Mead | Mind, self, and society[END_REF] in asserting that our self-perceptions develop and change in relation to other people in the social environment. Specifically, our self-perceptions tend to reflect the appraisals of others. This process, whereby perceptions of others' evaluations function as a mirror in which one sees oneself, has been referred to as the reflected appraisal process (see [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF][START_REF] Kinch | A formalized theory of self-concept[END_REF][START_REF] Shrauger | Symbolic interactionist view of self-concept: Through the looking glass darkly[END_REF][START_REF] Tice | The reflected self: Creating yourself as (you think) others see you[END_REF].

In essence, the reflected appraisal process consists of three elements [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF][START_REF] Kinch | A formalized theory of self-concept[END_REF][START_REF] Shrauger | Symbolic interactionist view of self-concept: Through the looking glass darkly[END_REF], including the individual's self-appraisals or self-perceptions, the actual appraisals of significant others, and the person's perceptions of those appraisals, called reflected appraisals. Moreover, theory suggests three basic postulates [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF]. The first postulate asserts that people are generally accurate to some extent in perceiving what others think of them [START_REF] Cooley | Human, nature and the social order[END_REF][START_REF] Mead | Mind, self, and society[END_REF]. In sport, for example, a child may believe that his/her father perceives him/her as having high, moderate, or low ability through the use of varied indices such as the father's enthusiasm level, the amount of attention provided, the feedback and encouragement that the father displays, and the participatory opportunities that he offers to the child. The second postulate is that reflected appraisals are generally internalized by the child, whether they are positive or negative, accurate or not. In this regard, if the child believes that his or her parent has a positive appraisal of his or her sport competence he or she is likely to develop a similarly positive perception of sport competence. In contrast, if the child believes that the parent has a less favorable view of their competence, he or she is likely to perceive himself or herself as less competent in sport. In the ontogeny of self-appraisals, the symbolic interactionist perspective gives critical importance to the internalization of the reflected appraisal. The third postulate is that the actual appraisals of others influence selfappraisals indirectly, as mediated by reflected appraisals. That is, actual appraisals of others have no effect on self-appraisals when reflected appraisals have been taken into account.

Few researchers have investigated the issue raised by the first postulate (see for reviews, [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF][START_REF] Shrauger | Symbolic interactionist view of self-concept: Through the looking glass darkly[END_REF]. Generally, the studies that have been conducted have found weak correlations between reflected appraisals and the actual appraisals of others. In other words, respondents are not very accurate in guessing what significant others think of them, probably because others often conceal their actual opinions, particularly when these opinions are negative [START_REF] Felson | Parents and the reflected appraisal process: A longitudinal analysis[END_REF][START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF][START_REF] Tice | The reflected self: Creating yourself as (you think) others see you[END_REF]. That is why [START_REF] Shrauger | Symbolic interactionist view of self-concept: Through the looking glass darkly[END_REF] have changed the original metaphor somewhat by suggesting that people see "through the looking glass darkly."

Nevertheless, studies (e.g., [START_REF] Malloy | Interpersonal perception and metaperception in nonoverlapping social groups[END_REF] have found that the degree of accuracy depends on the significant other with whom one is interacting. In this research, people were most accurately aware of the perceptions of family members rather than other people such as friends or coworkers.

Many investigations have provided support for the second postulate, namely the relationship between individuals' self-appraisals and the reflected appraisals of various significant others (see for reviews, [START_REF] Harter | The development of self-representations[END_REF]Schrauger & Schoeneman, 1979). In the sport and physical activity domain, some studies have revealed a correspondence between the reflected appraisals of parents and the physical self-appraisals of children and adolescents (e.g., [START_REF] Babkes | Parental influence on children's cognitive and affective responses to competitive soccer participation[END_REF][START_REF] Brustad | Who will go out and play? Parental and psychological influences on children's attraction to physical activity[END_REF][START_REF] Brustad | Attraction to physical activity in urban schoolchildren: Parental socialization and gender influences[END_REF][START_REF] Kimiecik | Relationships among children's beliefs, perceptions of their parents' beliefs and their moderate to vigorous physical activity[END_REF]. Two recent studies by [START_REF] Amorose | The influence of reflected appraisals on middle school and high school athletes' self-perceptions of sport competence[END_REF][START_REF] Amorose | Reflected appraisals and perceived importance of significant others' appraisals as predictors of college athletes self-perceptions of competence[END_REF] found that athletes' self-appraisals of competence could be predicted by the reflected appraisals of mothers, fathers, coaches, and teammates.

Very few studies have attempted to test the third postulate of the theory, namely that reflected appraisals mediate the influence of another person's actual appraisal on the individual's self-appraisal. Support for the two first postulates is, by itself, not sufficient to prove the mediation effect of the reflected appraisal process [START_REF] Shrauger | Symbolic interactionist view of self-concept: Through the looking glass darkly[END_REF]. It is also necessary to demonstrate the existence of a relationship between the actual appraisals of others and individuals' own self-appraisals as well as a diminishment or a disappearance of the strength of this relationship when reflected appraisals are controlled for (see [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: conceptual, strategic and statistical considerations[END_REF].

A number of studies have examined the relationship between the actual appraisals of others and individuals' self-appraisals (see for a review, [START_REF] Eccles | Motivation to succeed[END_REF][START_REF] Shrauger | Symbolic interactionist view of self-concept: Through the looking glass darkly[END_REF]. The majority of these studies have found significant relationships between the two, although the magnitude of the relationship tends to be moderate. Studies have found that children's self-appraisals are linked to their parents' appraisals of the child's capacity in both academic (e.g., [START_REF] Frome | Parents' influence on children's achievement-related perceptions[END_REF][START_REF] Parsons | Socialization of achievement attitudes and perceptions: Parental influences[END_REF][START_REF] Phillips | Socialization of perceived academic competence among highly competent children[END_REF] and sport (e.g., Bois, Sarrazin, Brustad, Trouilloud, & Cury, in press, 2002;[START_REF] Dempsey | Parental influence on children's moderate to vigorous physical activity participation: An expectancy-value approach[END_REF][START_REF] Felson | The effect of parents on the self-appraisals of children[END_REF][START_REF] Jacobs | The impact of mothers' gender role stereotypic beliefs on mothers' and children's ability perceptions[END_REF] domains. For example, using structural equation modeling on a sample of third through fifth grade children, [START_REF] Bois | Mothers' expectancies and young adolescents' perceived physical competence: a yearlong study[END_REF] demonstrated that mothers' appraisals of her child's sport competence predicted the child's self-appraisal of sport competence one year later controlling for both the child's initial self-appraisal and their actual physical performance. Nevertheless, none of these studies mentioned has directly tested whether such influences occur through a process of internalization of reflected appraisals.

Few studies have specifically tested if actual appraisals have an effect on selfappraisals when reflected appraisals are controlled for. Three studies carried out with adults have supported the mediating role of the reflected appraisal on several self-attributes [START_REF] Ichiyama | The reflected appraisal process in small-group interaction[END_REF][START_REF] Jussim | Understanding reactions to feedback by integrating ideas from symbolic interactionism and cognitive evaluation theory[END_REF][START_REF] Swann | Should we create a niche or fall in line? Identity negotiation and small group effectiveness[END_REF]. Only one study to our knowledge has been conducted with children [START_REF] Felson | Parents and the reflected appraisal process: A longitudinal analysis[END_REF]) and this study did not support the mediation hypothesis. More precisely, using a sample of fourth through seventh grade children and longitudinal data, Felson investigated the role of reflected appraisals in mediating the influence of parents' appraisals on children's self-appraisals in relation to children's self-evaluations in the academic and athletic domains as well as to their self-evaluations of appearance and popularity over a one-year period. Findings revealed that the reflected appraisals of mothers and fathers were very similar, and could be combined into a single variable representing the reflected appraisals of "parents." The main findings were consistent across the four domains and indicated that parents' actual appraisals affected both reflected appraisals and children's self-appraisals and that reflected appraisals had substantial effects on self-appraisals, after controlling for the self-perceptions of children one year earlier.

However, the magnitude of the effect of parents' appraisals on children's self-appraisals was not significantly reduced after controlling for the reflected appraisals. This finding, which casts doubt on a basic component of the reflected appraisal process, deserves further examination.

The present study

The purpose of this study was to examine the reflected appraisal process as it applies to the influence of parents on sixth graders' self-appraisals of sport competence.

For a number of reasons it is generally assumed that parents have a substantial influence on their children's self-appraisals in school (e.g., [START_REF] Eccles | Motivation to succeed[END_REF][START_REF] Jacobs | The impact of mothers' gender role stereotypic beliefs on mothers' and children's ability perceptions[END_REF] and sport (e.g., [START_REF] Brustad | Youth in sport: Psychological considerations[END_REF][START_REF] Greendorfer | Sport socialization[END_REF]. First, parents are typically responsible for providing children with their initial opportunities to play sport and parents are integrally involved in the youth sport experience [START_REF] Brustad | Youth in sport: Psychological considerations[END_REF][START_REF] Greendorfer | Sport socialization[END_REF]. Second, parental feedback is highly salient to youngsters during childhood and the early adolescent years. Research indicates that youngsters are likely to use adult sources of feedback in assessing their competence in various achievement domains throughout childhood and into early adolescence [START_REF] Frieze | Developmental trends in cue utilization for attributional judgments[END_REF][START_REF] Horn | A developmental analysis of children's self-ability judgments in the physical domain[END_REF]. As children enter adolescence, they also tend to become increasingly reliant on peer sources of feedback about their competence (e.g., [START_REF] Horn | A developmental analysis of children's self-ability judgments in the physical domain[END_REF]. However, parental appraisals still have influence through adolescence, particularly in the sport domain, because parents are likely to be extensively involved in their children's sport experiences at this age in terms of transporting children, attending games and practices, coaching teams, etc. In contrast to other achievement domains, such as academics, parents have extensive opportunities to convey competence information in the sport and physical activity domain and thus should remain an important significant other. Socialization research to date has focused upon how parents may influence children's self-appraisals but minimal attention has been devoted to understanding the role of reflected appraisals in contributing to children's self-appraisals in the physical domain.

To date, only one study has completely tested the reflected appraisal process by assessing the three elements of the model simultaneously [START_REF] Felson | Parents and the reflected appraisal process: A longitudinal analysis[END_REF] and the findings from this study did not support the mediation hypothesis. The present study was designed to examine these relationships more thoroughly and special attention was devoted to methodological concerns because previous studies suffer from limitations identified by [START_REF] Felson | Parents and the reflected appraisal process: A longitudinal analysis[END_REF][START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF]. A major problem with most previous work is that investigations have relied on data collected at a single point in time, making directional interpretation problematic. For example, the correlation between actual appraisal of a significant other and the reflected appraisal or self-appraisal can be the result of an influence of the first on the second (i.e., the first postulate of the reflected appraisal process) as much as it could be through the reverse influence. In this regard, a child who has a high self-appraisal of their sport ability can carry out superior sport performance likely to convince other people that he/she is good at sport.

Similarly, if the correlation between reflected appraisals and self-appraisals is in accordance with the reflected appraisal postulates, one could not rule out the alternative interpretation that self-appraisals "cause" reflected appraisals. Individuals may project their own self-perceptions onto others, especially if there is insufficient information to know how others view them. This alternative explanation has been named a "false consensus effect" (e.g., [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF]Ross, Greene, & Hoise, 1977), or "projection effect" (e.g., [START_REF] Jussim | Understanding reactions to feedback by integrating ideas from symbolic interactionism and cognitive evaluation theory[END_REF]. To overcome these alternative explanations, longitudinal designs including auto-regressive influence are particularly useful [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF][START_REF] Maccallum | Applications of structural equation modeling in psychological research[END_REF].

In addition, an omitted variable, such as the individual's performance level, could produce spurious relationships between self-appraisals, reflected appraisals, and the appraisals of significant others. It may be that successful people evaluate themselves more favorably, believe that others evaluate them more favorably, and are actually evaluated by others more favorably. For example, more highly skilled young athletes may believe that they are athletically able and that others think they are good, and others may logically think they are gifted. Thus, if one variable predicts the other ones (for example, if the others' actual appraisals predict others' reflected appraisal and/or individual's self-appraisal) it is perhaps for reasons of a shared standard. For this reason, as recommended by [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF], it is necessary to control for objective measures of ability (like grades or standardized test scores) when one is examining the reflected appraisal process with correlational data.

The main hypothesis of this study was that parents' appraisals of their child's sport competence would predict the child's self-appraisals indirectly, as mediated by the reflected appraisal. To test for this mediation hypothesis, three models were compared (see Figure 1) in order to thoroughly examine the three requirements for mediation defined by [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: conceptual, strategic and statistical considerations[END_REF]. The first model reflected by Figure 1 represents a fully mediated (FM) model in which it is hypothesized that parents' appraisals affect reflected appraisals (i.e., Path A) that, in turn, influence children's self-appraisals (i.e., Path B) after controlling for children initial self-appraisals and actual level of ability (as assessed by child's grades in sport and physical education). The second model represents a partially mediated (PM) model in which an additional path was added to the FM model between the parents' appraisals of their child's competence and the child's physical self-appraisals (i.e., Path C).

Finally, Model 3 represents an unmediated (NM) model in which parents' appraisals directly predict children's self-appraisal when paths A and B have been deleted. The PM and NM models allow for examination of the third requirement for the mediation effect, namely "when Paths A and B are controlled, a previously significant relation between the independent and dependent variables is no longer significant" [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: conceptual, strategic and statistical considerations[END_REF], p. 1176). The NM model allows us to see if Path C is significant, and the PM model allows us to see if this path is reduced, or is near zero, when Paths A and B have been controlled.

We tested these models using a three wave longitudinal design over one school year, in order to allow for a more sensitive test of our specific directional hypotheses. We also used more sophisticated data analytic techniques, specifically structural equation modeling (SEM), than those which have typically been used in previous studies. SEM allows for the examination of hypothesized relations among all of the constructs involved in a model, using a latent representation of those constructs that is less vulnerable to measurement errors such as those that can be encountered in research with child populations (see [START_REF] Maccallum | Applications of structural equation modeling in psychological research[END_REF]. It might be that the lack of support for the mediation hypothesis found in Felson's study (1989) was attributable to measurement limitations, to the extent that each variable was only assessed with a single item.

A related focus of this study addresses possible gender differences in the functioning of the reflected appraisal process. Some researchers (e.g., [START_REF] Eccles | Age and gender differences in children's self and task perceptions during elementary school[END_REF] have found that mothers have a stronger impact than fathers on children's behavior, beliefs, and attitudes. Other authors (e.g., [START_REF] Bussey | Social cognitive theory of gender development and differentiation[END_REF] have suggested that a parent's influence may be stronger for a child of the same gender in that sons might be more sensitive to what their fathers think of them and daughters might be more sensitive to what their mothers think of them. Therefore, it is possible that the RA process is different for sons compared to daughters. Thus, a related objective of this study was to test the invariance of the reflected appraisal process across gender. children whose parents agreed to participate were given a questionnaire to complete and were asked to return them a week later. Conditions (i.e. groups and location) for completing the questionnaire in Wave 1, 2 and 3 were the same for the children.

Method

Measures

Child's perceived physical competence. Child's self-appraisals of competence in physical activity were assessed in Wave 1 and Wave 3 with a 3-item questionnaire, similar to the one developed by Nicholls and colleagues (e.g., [START_REF] Duda | Dimension of achievement motivation in schoolwork and sport[END_REF].

Participants were asked to answer questions such as "when you are playing sport and physical activity 1 , you consider yourself…" on 7-point scales anchored by "very bad" (1) and "very good" (7). This scale has demonstrated good construct validity, internal consistency and predictive validity in previous studies (e.g., [START_REF] Sarrazin | Exerted effort and performance in climbing among boys: The influence of achievement goals, Reflected appraisal process 30 perceived ability, and task difficulty[END_REF][START_REF] Trouilloud | The influence of teacher's expectations on students' achievement in physical education classes: Pygmalion revisited[END_REF].

Parents' appraisals of their child's physical competence. During Wave 1, mothers and fathers completed a French version of the Jacobs and Eccles' (1992) questionnaire on parents' appraisals of their child's competence. The back-translation method (see [START_REF] Brislin | The wording and translation of research instruments[END_REF]) was used to translate the original scale into French. This 3-item scale is comprised of questions such as "In general how good is your child in sports?" Each item was rated on a scale from 1 = "not at all good" through 7 = "very good." In previous research, this scale has been found to be valid and reliable (e.g., [START_REF] Bois | Mothers' expectancies and young adolescents' perceived physical competence: a yearlong study[END_REF].

Reflected appraisal of parents. Children's reflected appraisal of parents was assessed distinctly for mothers and fathers at Wave 2, with a six-item questionnaire similar to the one developed by Amorose (e.g., 2002). Specifically, three items for each parent tapped the extent to which children thought that their mother and their father considered them physically competent. The items were designed to match the ones used in assessing parents' appraisals of their child's competence (e.g., "How good does your mother/father think you are at sports"). The child responded according to a 7-point scale with higher scores reflecting more positive reflected appraisals.

Child's physical education grade. In France, all students from the sixth through the twelfth grades take mandatory sport and physical education classes. Each student is graded at the end of each term on a scale ranging from 0 to 20. We collected the teacher's evaluation of the participating students for the first term (i.e., from September to December, before the assessment of Wave 2 variables).

Data Analysis

Preliminary data analyses and descriptive statistics were performed before testing the hypotheses. First, a confirmatory factor analysis (CFA) was conducted to examine the construct validity of the variables in the study. Then, SEM was used to test relations among the variables as hypothesized in the three models of Figure 1. To investigate the issue of mediation, the requirements advocated by [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: conceptual, strategic and statistical considerations[END_REF] were carefully examined. A chi-square difference test between the PM and FM models was carried out to see whether the global fit is improved by including the path C. If this were the case, the chi-square difference between the two models should be significant and would mean that the mediation of the reflected appraisal is not complete. Subsequently multi-sample analysis was used to assess the invariance of the model across gender of the child.

Results

Descriptive Statistics and Preliminary Analyses

Results from descriptive statistics of all indices under study revealed that the data were normally distributed with values for skewness and kurtosis within an acceptable range of -1.0 to +1.0. Seven participants (less than 5% of the sample) did not provide complete responses to questionnaires and their responses were not included in the data analysis.

Table 1 presents the Cronbach alphas reflecting internal reliability of the scales as well as the means and standard deviations of the variables included in the study. A MANOVA was conducted to investigate possible gender differences on variables of interest. Results revealed a marginally significant main effect, Wilks' lambda = .92, R (6, 140) = 2.06, p =.062. Follow-up univariate ANOVAS indicated only two gender differences in that boys had higher scores than girls on self-appraisal at Time 1 (F(1, 145) = 9.21, p < .01, η² = .06) and at Time 3 (F(1, 145) = 5.31, p < .05; η² = .04, see group means in Table 1). To test whether parents' appraisals of their child's competence varied as a function of their own or their child's gender, a 2 (child's sex) × 2 (parent's sex) analysis of variance was conducted, with repeated measures on the last variable. Neither main effects nor significant interactions were present in this analysis.

Measurement Model

To examine the construct validity of the measures included in the study, a confirmatory factor analysis was conducted. On the first step of this analysis a model based on six latent constructs (i.e., child perceived competence at Wave 1 and Wave 3, mothers' appraisals, fathers' appraisals, reflected appraisals of mother and father) and 18 observed variables (individual items from each questionnaire) was tested. The analysis was conducted with LISREL 8.3 [START_REF] Jöreskog | Lisrel 8 : User's reference guide[END_REF] using a covariance matrix 2 and maximum likelihood method of estimation. In view of the current controversy regarding measures of overall goodness of fit, it is generally considered appropriate to report multiple indices of fit [START_REF] Bollen | Structural equations with latent variables[END_REF]. Thus, the chi-square statistic, the non-normed fit index (NNFI), the comparative fit index (CFI) and the standardized root mean square residual (SRMR) all were used to evaluate the adequacy of the fit of the model to the data. For the NNFI and CFI, values above .90 are considered satisfactory. For SRMR, values below .10 indicate a good fit of the model to the data. This first model presented a poor fit to the data: χ²(120, N = 147) = 364.34, p < .001, NNFI = .72, CFI = .78, SRMR = .08. Analysis of the output revealed large residuals between the three indicators of mothers' and fathers' reflected appraisals. As the questions for reflected appraisals of mothers and fathers were the same (only the word "mother" or "father" differed) some children may have answered the same way for mothers and fathers on this variable. Hence we decided to add three covariance errors between the three indicators of mothers' reflected appraisals and the three indicators of fathers' reflected appraisal. This modification resulted in a significant improvement of the fit of the model: χ²(117, N = 147) = 185.08, p < .001, NNFI = .94, CFI = .96, SRMR = .05. The correlations between all the variables were low to moderate except for the correlation between mothers' and fathers' appraisals of their child's competence r = . 79) on one hand and between reflected appraisals of mothers and fathers ( r = . 99) on the other hand.

Therefore, we decided to test the discriminant validity of these constructs. We examined whether each of those two pairs of latent factors could be treated as a single construct by setting each correlation to 1.0 and comparing the constrained model to the original one in which the correlation was free to be estimated (see [START_REF] Anderson | Structural equation modeling in practice: A review and recommended two-step approach[END_REF]. This test was performed separately for each pair of latent construct. Chi-square difference tests indicated that the models with the correlation set to 1.0 were not different from the model freely estimated (ps > .10). Therefore mothers' and fathers' appraisals of their child's ability on one hand, and the child's reflected appraisals of mothers and fathers on the other hand were combined to form two factors (in place of four) based on six observed variables each. The first term was labeled "parents' appraisals of their child's perceived competence" and the second was labeled "reflected appraisals of parents." The new four-factor model presented an adequate fit to the data: χ²(126, N = 147) = 222.7, p < .001, NNFI = .93, CFI = .94, SRMR = .05 and was used in the subsequent analyses 4 . The construct reliability 3 of the four factors were ρ = .82, .87, .84, .82, respectively for children's self-appraisals at Wave 1, parents' perceptions of their child competence, parents' reflected appraisal and children's self-appraisals at Wave 3.

Comparison of Structural Models

Since the validity of the factor structure was confirmed, the hypothesized relationships between the four latent variables and the child's grade (considered as a manifest variable) were estimated and the three proposed models (see Figure 1) were tested.

The FM model presented an adequate fit to the data: χ²(141, N = 147) = 229.25, p < .001, NNFI = .93, CFI = .95, SRMR = .05. The structural and measurement coefficients of this model are presented in Figure 2. Parents' appraisals significantly predicted reflected appraisals (γ = .52, p < .001) independent of the effects of children's initial self-appraisals (γ = .17, p > .05) and grades (γ = .05, p > .05). In turn, reflected appraisals affected children self-appraisals at Time 3 (β = .44, p < .001), independent of children initial selfappraisals (γ = .39, p < .001) and grades (γ = .12, p > .05) 5 . This model accounted for 59 percent of the variance of the child's self-appraisals at Time 3. PM model presented an adequate fit to the data: χ²(140, N = 147) = 229.24, p < .001, NNFI = .93, CFI = .94, SRMR = .05. Nevertheless, the effect of parents' appraisals on child's self-appraisals at Time 3 (i.e., Path C) was nonsignificant (γ = .01, p >.05). This model account for 59 percent of the variance of the child's self-appraisals at Time 3. The comparison between the PM and FM models suggested that the addition of the Path C did not significantly improve the fit [∆χ 2 (1) = .01, p > .05], hence the simplest model (i.e., the FM model) should be preferred.

The NM model presented a poorer, albeit adequate, fit to the data: χ²(141, N = 147) = 244.06, p < .001, NNFI = .92, CFI = .94, SRMR = .064. Parents' appraisals significantly predicted their child's self-appraisals at Time 3 (γ = .26, p< .05) independent of the effects of children's initial self-appraisals (γ = .47, p <.001) and grades (γ = .12, p >.05). This model accounted for 50 percent of the variance of the child's self-appraisals at Time 3.

Inspection of the relevant parameters indicated that the three criteria for mediation listed by [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: conceptual, strategic and statistical considerations[END_REF] were fully achieved: (1) parents' appraisals significantly influenced reflected appraisals (i.e., Path A; γ = .52, p < .001); (2) reflected appraisals of parents significantly predicted the child's final self-appraisals (i.e., Path B; γ = .44, p <.001); and (3) the direct path from parents' appraisals to child's self-appraisals (i.e., Path C) was nonsignificant when Paths A and B were controlled for (γ = .01, p > .05, in the PM model), whereas this path was significant (γ = .26, p < .05) when the Paths A and B were removed (in the NM model).

Multigroup analyses

A multi-group analysis was conducted to test invariance of the FM model across gender. This involved testing to determine if setting the parameters to be equal across gender produced a significant change in chi-square compared to the free-constraint model.

The free-constraint model represents a model in which parameters of the boys' and girls' models are evaluated without any between-group constraints. The results demonstrated that constraining the parameters did not significantly decrease the fit of the model [∆χ2(6) = 8.73, p > .05], which suggests that the processes involved in the model appear to be similar across gender.

Discussion

The goal of this study was to examine the role of the reflected appraisal process between parents and sixth graders in the ontogeny of children's self-appraisals of physical competence. Reflected appraisals may constitute an important dimension of the sport and physical activity socialization process and a primary purpose of this study was to extend the current knowledge base on sport socialization influence by systematically evaluating the role of reflected appraisals on children's self-appraisal of sport competence.

Preliminary findings and issues dealing with the validity of our model will be discussed before addressing the question of the reflected appraisal process per se.

Preliminary Findings

Preliminary analyses revealed marginally significant gender differences, with boys having greater perceived competence than girls, both at Wave 1 and at Wave 3. Those findings are consistent with numerous previous studies (e.g., Bois et al., in press, 2002;[START_REF] Dempsey | Parental influence on children's moderate to vigorous physical activity participation: An expectancy-value approach[END_REF][START_REF] Eccles | Gender differences in sport involvement: Applying the Eccles' expectancy-value model[END_REF], which have found that boys tend to be more confident in their physical abilities than are girls. On the other hand, parents' appraisals of their child's competence did not vary in relation to the child's gender. This finding is consistent with some recent studies (e.g., Bois et al., in press, 2002;[START_REF] Kimiecik | Parental beliefs and children's moderate-to-vigorous physical activity[END_REF] but not with some others (e.g., [START_REF] Eccles | Gender role stereotypes, expectancy effects, and parents' socialization of gender differences[END_REF][START_REF] Jacobs | The impact of mothers' gender role stereotypic beliefs on mothers' and children's ability perceptions[END_REF] which have found a differentiation in parents' expectancies in relation to the child's gender.

A confirmatory factor analysis indicated that mothers' and fathers' appraisals of their child's competence were strongly related. Although some authors (e.g., [START_REF] Brustad | Youth in sport: Psychological considerations[END_REF] have argued that the two parents might not share similar appraisals of their child's ability, we are not aware of studies that have found substantial parental differences in ability appraisals. It was also found through the CFA that the reflected appraisals of parents were not appreciably different between mothers and fathers. This finding suggests that children of this age do not distinguish between their mother's and their father's appraisals of them. This finding is in accordance with other studies [START_REF] Amorose | Reflected appraisals and perceived importance of significant others' appraisals as predictors of college athletes self-perceptions of competence[END_REF][START_REF] Felson | Parents and the reflected appraisal process: A longitudinal analysis[END_REF] that have found that children were more influenced by a generalized other (e.g., parents or sport others) than by a distinctly perceived significant other. However this finding was not surprising given that mothers' and fathers' ability perceptions were not highly differentiated. Hence, children in this sample may not distinguish between mothers' and fathers' evaluations of their competence because those appraisals are not actually different. These findings considered together seem to be consistent with the hypothesis from symbolic interactionist theory [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF][START_REF] Mead | Mind, self, and society[END_REF] that individuals have a better idea of how they are viewed by collective others than how they are viewed by individual others alone.

Reflected Appraisal Process

The main hypothesis of this study was that reflected appraisals mediate the effect of parents' appraisals on children's self-appraisals of physical competence. According to [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: conceptual, strategic and statistical considerations[END_REF], providing support for this mediation effect required: (1) a link between parents' appraisals and reflected appraisals; (2) a link between reflected appraisals and children's self-appraisals; (3) a significant decrease in the path between parents' appraisals and children self-appraisals after the first two links had been controlled.

In relation to the first point it was hypothesized that parents' appraisals would have positive effects on the child's reflected appraisals of parents, controlling for actual level of ability in sport as well as prior level of self-appraisals. The effect of initial level of selfappraisals on later reflected appraisals is called the projection effect (e.g., [START_REF] Jussim | Understanding reactions to feedback by integrating ideas from symbolic interactionism and cognitive evaluation theory[END_REF] or "false consensus effect" (e.g., [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF][START_REF] Ross | The "false consensus effect": An egocentric bias in social perception and attribution processes[END_REF] and refers to the processes whereby individuals can base their reflected appraisals from others on their own self-views rather than on what others think of them (e.g. [START_REF] Jussim | Understanding reactions to feedback by integrating ideas from symbolic interactionism and cognitive evaluation theory[END_REF][START_REF] Tice | The reflected self: Creating yourself as (you think) others see you[END_REF]. In other words, people's inferences on how they are appraised by others may be based on their self-views rather than on an accurate understanding of how others actually see them especially if there is insufficient information to know how others view them.

Our results (see Figure 2) lend support to an effect of parents' appraisals on children's reflected appraisal (γ = .52, p < .001) and don't provide evidence for the projection effect (γ = .17, p > .05). The significant influence of parents' appraisals on children's reflected appraisals indicates that children are rather accurate in perceiving their parents' appraisals. This finding is controversial to the extent that it does not support most of the recent work in this line of research that has found that people are not very accurate in identifying what significant others think of them [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF][START_REF] Shrauger | Symbolic interactionist view of self-concept: Through the looking glass darkly[END_REF][START_REF] Tice | The reflected self: Creating yourself as (you think) others see you[END_REF]. But how can one explain that the youngsters in this study were more accurate than participants in previous studies in identifying significant others' appraisals of their abilities? Two explanations can be provided. First, we investigated the accuracy of the appraisal between children and parents. Some investigators [START_REF] Malloy | Interpersonal perception and metaperception in nonoverlapping social groups[END_REF] have found that people are more accurate in guessing how they are viewed by family members than by other people (i.e. friends or coworkers). Moreover, as children in our sample are relatively young they are likely to be more attuned to their parents' judgments than to other significant others, such as peers. Hence the close relationship between parents and children at this age may explain those results.

However, it can also be emphasized that the methodology used in this study with data collected at three points in time combined with structural equation modeling may contribute to the explanation of these results. In particular, the fact that measurement error is partly controlled for by SEM may have provided a more thorough test of the hypotheses in this study than in previous studies. This sort of methodology (i.e. combining SEM and longitudinal design we can be sure that the variables assessed at the start of the year were not caused by the ones assessed ten months later, but it is always possible that an omitted variable is the origin of a link found between a predictor variable and an outcome variable.

Nevertheless, including the variables of the child's physical education grade and the child's initial level of perceived competence limits the likelihood that this occurred.

Secondly, the use of grades as an indication of the child's actual physical ability may constitute another limitation of the study. As emphasized by [START_REF] Jussim | Grades may reflect more than performance: Comment on Wentzel (1989)[END_REF], marks can be biased by the teacher's beliefs or a priori opinion. Furthermore, the fact that grades are often specific to a limited range of physical activities may reflect that they do not entirely capture the overall physical competence of the child. These two points can explain why grades did not predict reflected appraisals and self-appraisals. However using grades as an objective indicator of ability also enhanced the quality of the study as it enabled us to control for, to some extent, the actual physical ability of the child. A third limitation of the study pertains to the independence of mothers' and fathers' responses. As no specific mechanism was used to ensure that each parent completed his/her questionnaire independently from the other, it is possible that some of the parents completed the questionnaires together.

In sum, this study has demonstrated a link between parents' appraisals and the reflected appraisals of ability in the sport domain held by their children. Secondly, a link was found between reflected appraisals and self-appraisals that is consistent with symbolic interactionist perspectives [START_REF] Cooley | Human, nature and the social order[END_REF][START_REF] Mead | Mind, self, and society[END_REF] and certain empirical studies [START_REF] Amorose | The influence of reflected appraisals on middle school and high school athletes' self-perceptions of sport competence[END_REF][START_REF] Amorose | Reflected appraisals and perceived importance of significant others' appraisals as predictors of college athletes self-perceptions of competence[END_REF][START_REF] Babkes | Parental influence on children's cognitive and affective responses to competitive soccer participation[END_REF]. This result cannot be explained by a projection effect, a concurrent hypothesis to the reflected appraisal process that has been mentioned by some authors (e.g. [START_REF] Felson | Parents and the reflected appraisal process: A longitudinal analysis[END_REF][START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF][START_REF] Tice | The reflected self: Creating yourself as (you think) others see you[END_REF], since the projection effect was controlled for. Finally, this study supports the mediation of parents' appraisal effects on child's self-appraisals by reflected appraisals although this relation has not been previously supported [START_REF] Felson | Parents and the reflected appraisal process: A longitudinal analysis[END_REF].

From a pragmatic view, these results emphasize the importance of the interpretation that children can make about their parents' beliefs and behaviors and helps to further the knowledge base on parental socialization effects in the physical domain. Parents should be aware not only of what they do and say to their children, but also of the way children understand, perceive and interpret those beliefs and behaviors as it affects their subsequent motivation. In terms of positive outcomes, parents may promote positive reflected appraisals and self-appraisals, positive expectancies, and encouragement and thus contribute to their children's successful involvement in physical activity and sport. .17

.05

.59 .18

.47

  Participants A sample of 147 sixth graders (73 boys and 74 girls) enrolled in two different French schools, their mothers (M age = 40.34 ± 4.61 years) and fathers (M age = 42.99 ± 6.12 years) were the participants in this study. The child participants ranged in age from 10 to 13 years old (M age = 11.73 ± 0.58 years). The schools were located in a medium size city (100,000 inhabitants) in the southeast of France. The participant sample was comprised primarily of middle to upper class families. Procedure Parents and children completed a consent form prior to participating in the study. Data were obtained from questionnaires in three waves during one school year. The first wave (in September 2001) was collected from parents and children. In the second (February 2002) and third (in June 2002) waves, we only collected data from children. Data were collected during the PE class by a Master's degree student familiar with the precautions needed for data collection. Children completed questionnaires in groups of about 10 individuals in the classroom. After completion of the first-wave questionnaire,

Figure 1 .

 1 Figure Caption.Figure 1. Series of tested models. From top to bottom: fully mediated model (FM), partially mediated model (PM), and non-mediated model (NM).Figure 2. Structural equation model of the fully mediated model. Standardized solutions are presented. ** p < .01 *** p < .001

Figure 2 .

 2 Figure Caption.Figure 1. Series of tested models. From top to bottom: fully mediated model (FM), partially mediated model (PM), and non-mediated model (NM).Figure 2. Structural equation model of the fully mediated model. Standardized solutions are presented. ** p < .01 *** p < .001

  Cronbach alpha, mean ratings, standard deviations and correlations of children and Child's self-appraisals (Wave 1) .82 5.03 1.05 5.59 1.16 .29 -3. Mothers' appraisals .79 4.67 1.00 4.71 1.10 .42 .44 -4. Fathers' appraisals .81 4.67 1.04 4.70 1.13 .39 .40 .66 -5. Reflected appraisal of mothers .75 4.78 1.14 4.96 1.10 .30 .35 .47 .50 -6. Reflected appraisal of fathers .74 4.73 1.16 4.92 1.16 .32 .38 .50 .53 .94 -7. Child's self-appraisals (Wave 3) .81 5.01 1.24 5.45 1.04 .35 .53 .42 .43 .55 .53 -Notes. N = 147. All correlations are significant: p < .05 Reflected appraisal process 34 Figure Caption.

	parents variables.
	Variables 1. Grade 2.	Girls M SD -13.05 2.11 13.4 2.75 -Boys 1 2 3 4 5 6 7 M SD α

longitudinal data) had not been used in the previously cited studies. Hence the difference in the pattern of results might arise from this analytical strategy. In comparison to previous studies, our results demonstrated a relatively high accuracy of reflected appraisals, in contrast to what has been found in the past [START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF][START_REF] Tice | The reflected self: Creating yourself as (you think) others see you[END_REF] but has casts doubt on the presence of the projection effect. Further longitudinal studies are needed to disentangle the relationships among those variables.

The second point hypothesized that the reflected appraisals of parents would have positive effects on children's self-appraisals, independent of the child's initial selfappraisals and the child's actual level of ability in sport. The results support this hypothesis. The fact that reflected appraisals and self-appraisals are significantly related is acknowledged in the literature (e.g. [START_REF] Babkes | Parental influence on children's cognitive and affective responses to competitive soccer participation[END_REF][START_REF] Felson | The (somewhat) social self: How others affect self-appraisals[END_REF]. However our results show only a moderate relationship between reflected appraisals and self-appraisals.

Hence, even though children are rather accurate in perceiving what their parents think of them, they don't completely base their self-views on these perceptions. It is likely that numerous significant others, such as peers, coaches or teammates are involved in this process as has been hypothesized by [START_REF] Amorose | Reflected appraisals and perceived importance of significant others' appraisals as predictors of college athletes self-perceptions of competence[END_REF]. Further studies that extend the number of significant others investigated are needed to get a better understanding of the role of social influences in the development of children's self-appraisals.

It was also proposed that the magnitude of the influence of parents' appraisals on children's self-appraisals would be reduced when reflected appraisals had been controlled for. The results supported this expectation. In the NM model, the path from parents' appraisals to children's self-appraisals was significant (γ = .26, p< .05) although it was nonsignificant in the PM model (γ = .01, p> .05) where the influence of reflected appraisals on self-appraisals was controlled for and significant (β = .43, p <.01).

Subsequent analyses led us to prefer the FM model (see Figure 2) which was more Reflected appraisal process 22 parsimonious and powerful in terms of variance explained in the dependent variable. To our knowledge, this study is the first conducted in a natural setting that fully demonstrates the three hypothesized steps in the reflected appraisal process. Previous studies either only partially tested this process (e.g. [START_REF] Amorose | The influence of reflected appraisals on middle school and high school athletes' self-perceptions of sport competence[END_REF][START_REF] Amorose | Reflected appraisals and perceived importance of significant others' appraisals as predictors of college athletes self-perceptions of competence[END_REF], or failed to demonstrate the mediation effect (e.g. [START_REF] Felson | Parents and the reflected appraisal process: A longitudinal analysis[END_REF]. The fact that our results are not consistent with Felson's study is intriguing. However two methodological points might explain those discrepancies. First, all of the constructs involved in Felson's study (1989) were assessed using only a single item. This could have caused some measurement difficulties. Second, the sample used in the Felson study (1989) covered a broader age range (fourth through seventh grade children) than our sample of sixth grade children. We are aware that as children get older they become increasingly able to use others' appraisals to estimate their physical competence (e.g., [START_REF] Horn | Sources of competence information[END_REF]. Therefore, the mediational process identified in our study might have been masked in Felson's study because younger and older children were included in the same sample.

We also investigated the invariance of the reflected appraisal process across gender.

The goal of this analysis was to test whether parental influence through the reflected appraisal process could vary as a function of the child's gender. The multigroup analysis revealed no significant differences in the model between boys and girls. This seems to indicate that the process does not differ between boys and girls. However, other individual difference factors such as age or level of involvement in sport may influence the reflected appraisal process and could constitute a new area of investigation. Special attention was dedicated to the methodology used in this study; however some limitations have to be acknowledged. First, although we used a longitudinal design to enhance directional inferences that can be drawn from our data, the findings remain correlational in nature and therefore must be interpreted cautiously. Due to the use of a Footnotes 1. We used alternatively the terms "sport" or "physical activity", because in France the two have a very similar meaning and designate all forms of exercise and leisure-time physical activity.

2. The covariance matrix used for this analysis is available upon request to the second author.

3. Reliability estimates for the total scales are obtained by [START_REF] Bollen | Structural equations with latent variables[END_REF]) : ρ = (Σλi)² / ((Σλi)² + Σδii) where λi are the factor loading and δii the error variances.

4. Omission of the three error covariance between measurement error of indicators of reflected appraisal resulted in a poorer fit of the model: χ² (129, N = 147) = 425,24, p < .001, NNFI = .71, CFI = .76, SRMR = .09, ∆χ² (df= 3) = 202,5, p < .001. To the extent that the same person (i.e. the child) answers to questions where only the word "father" or "mother" change, the error covariance has to be seen as method variance due to the similarity of the items. 5. Omission of the three insignificant paths did not significantly alter the fit of this model: χ²(144, N = 147) = 231.92, p < .001, NNFI = .93, CFI = .94, SRMR = .06.