Vinayak J Kalas
email: vinayak.kalas@lirmm.fr

Alain Vissi
email: alain.vissiere@lirmm.fr

Thierry Roux
email: thierry.roux@symetrie.fr

Olivier Company
email: olivier.company@lirmm.fr

S Ébastien Krut
email: sebastien.krut@lirmm.fr

Franc ¸ois Pierrot
email: francois.pierrot@lirmm.fr

A NEW EFFICIENT STIFFNESS EVALUATION METHOD TO IMPROVE ACCURACY OF HEXAPODS

Structural compliance of hexapod positioners limits their positioning accuracy. Taking a step towards solving this problem, this paper proposes a new efficient method to evaluate the stiffness of hexapods in order to predict and correct their positioning error due to compliance. The proposed method can be used to predict the six degree of freedom deflection of the platform under load. This method uses a simple lumped stiffness parameter model whose parameters can be estimated using the identification technique presented in this paper. An experimental study with micrometer level measurements performed on a hexapod based micro-positioning system is used to assess the efficiency of the presented method.

INTRODUCTION

Hexapods, also called Gough-Stewart platforms, are increasingly used for applications demanding high accuracy six degree of freedom (DOF) positioning. Some general areas of applications include mirror positioning in telescopes [START_REF] Symétrie | Telescope mirror adjustment[END_REF], posi- * Address all correspondence to this author. tioning components in synchrotrons [START_REF] Symétrie | Hexapods for synchrotrons[END_REF] and research applications demanding high accuracy positioning [START_REF] Symétrie | High payload adjustment with high accuracy[END_REF]. One of the biggest advantages of hexapods is their high inherent stiffness [START_REF] Merlet | Parallel robots[END_REF]. Hence, in most applications, hexapod compliance doesn't considerably degrade the accuracy of positioning. However, as accuracy requirements become more stringent, quantifying the hexapod's compliance error becomes crucial in order to correct it. This work forms a part of a larger project, Posilab [START_REF]Posilab[END_REF], that is focused on improving the performance of the current state of the art hexapod positioners. The work presented in this paper focusses on solving the problem of platform deflection due to hexapod compliance. The developed solution needed to be applicable to existing hexapods. Hence, redesigning the hexapod was not a feasible option. It was, therefore, necessary for the solution developed in this work to be applicable in a position compensation framework. This can be achieved by using a model-based approach to estimate hexapod compliance. This approach would encompass choosing a suitable stiffness parameter model followed by identification of these parameters using appropriate measurements [START_REF] Dumas | A methodology for joint stiffness identification of serial robots[END_REF].

Stiffness modelling of parallel robots has acquired vast attention in research leading to development of very simple stiff-1 ness matrix [START_REF] Lee | Static characteristics of an in-parallel actuated manipulator for clamping and bracing applications[END_REF] to an advanced one [START_REF] Klimchik | Stiffness modeling for perfect and non-perfect parallel manipulators under internal and external loadings[END_REF]. To be included in a position compensation framework, it is crucial to accurately predict platform deflection. This needs accurate prediction of the parameters of the stiffness model. This problem has been mainly approached in research in the following directions: (a) analytical estimation of stiffness model parameters, this option has been widely addressed in literature [START_REF] Corradini | Evaluation of a 4-degree of freedom parallel manipulator stiffness[END_REF][START_REF] Majou | Parametric stiffness analysis of the orthoglide[END_REF][START_REF] Clinton | Stiffness modeling of a stewart platform based milling machine[END_REF][START_REF] Li | Stiffness analysis of a stewart platform-based parallel kinematic machine[END_REF][START_REF] Deblaise | A systematic analytical method for pkm stiffness matrix calculation[END_REF][START_REF] Chen | Instantaneous stiffness analysis and simulation for hexapod machines[END_REF][START_REF] Rebeck | A method for evaluating the stiffness of a hexapod machine tool support structure[END_REF][START_REF] Klimchik | CAD-based approach for identification of elasto-static parameters of robotic manipulators[END_REF], and (b) stiffness parameter identification using measurements, which is a relatively new approach [START_REF] Dumas | A methodology for joint stiffness identification of serial robots[END_REF][START_REF] Carbone | A procedure for experimental evaluation of cartesian stiffness matrix of multibody robotic systems[END_REF][START_REF] Klimchik | Design of calibration experiments for identification of manipulator elastostatic parameters[END_REF][START_REF] Alici | Enhanced stiffness modeling, identification and characterization for robot manipulators[END_REF][START_REF] Klimchik | Identification of the manipulator stiffness model parameters in industrial environment[END_REF][START_REF] Dumas | Joint stiffness identification of industrial serial robots[END_REF][START_REF] Ceccarelli | Numerical and experimental analysis of the stiffness performances of parallel manipulators[END_REF][START_REF] Bonnemains | Stiffness computation and identification of parallel kinematic machine tools[END_REF]. In many cases, analytically estimated stiffness parameters do not lead to accurate deflection predictions of endeffector/platform. Also, the latter option has the advantage of being analytically simple and less time-consuming as it doesn't involve cumbersome calculations. These reasons make option (b) very attractive for implementation in an industrial framework and hence, this method is suitable for our application.

The accuracy of robot model parameters identified using measurements is largely dependent on the robot poses used, external forces used and the number of measurements performed. Therefore, the identification procedure needs to be optimized to obtain best set of parameters. The concept of parameter observability addresses this issue. This has been discussed in detail in the 'Stiffness identification method' section.

Stiffness identification technique has been applied very well to serial robots [START_REF] Dumas | A methodology for joint stiffness identification of serial robots[END_REF], [START_REF] Klimchik | Design of calibration experiments for identification of manipulator elastostatic parameters[END_REF][START_REF] Alici | Enhanced stiffness modeling, identification and characterization for robot manipulators[END_REF][START_REF] Klimchik | Identification of the manipulator stiffness model parameters in industrial environment[END_REF][START_REF] Dumas | Joint stiffness identification of industrial serial robots[END_REF], [START_REF] Abele | Modeling and identification of an industrial robot for machining applications[END_REF]. One of the most advanced stiffness identification methods for serial robots has been presented by Dumas et al. [START_REF] Dumas | Joint stiffness identification of industrial serial robots[END_REF]. In this work, the observability of parameters was defined based on an advanced stiffness model. Klimchik et al. [START_REF] Klimchik | Design of calibration experiments for identification of manipulator elastostatic parameters[END_REF] also presented a robust stiffness identification method that used a simple stiffness model. They optimized their identification to improve accuracy at the intended operational configuration of a 6-R serial robot.

Stiffness identification has not been used much for evaluating stiffness of parallel robots yet. Carbone et al. [START_REF] Ceccarelli | Numerical and experimental analysis of the stiffness performances of parallel manipulators[END_REF] evaluated the stiffness of a 3-DOF parallel manipulator CaPaMan by measuring displacements of the platform under load. However, this method was not optimized to obtain the best set of parameters. Bonnemains et al. [START_REF] Bonnemains | Stiffness computation and identification of parallel kinematic machine tools[END_REF] evaluated the stiffness of a 3-DOF tricept parallel robot. This work, too, did not address parameter observability satisfactorily. No work so far, to the best of authors' knowledge, has demonstrated a complete and robust stiffness identification method for parallel robots. Also, no stiffness identification method has been presented for a hexapod system.

The method to be developed for the application concerned to this paper had to be directly applied in the industry. It was, therefore, very important for the stiffness evaluation method to be simple while being accurate enough. It is also desirable to have a less time consuming method in an industrial framework. These factors facilitate the use of a simple stiffness matrix as presented in [START_REF] Lee | Static characteristics of an in-parallel actuated manipulator for clamping and bracing applications[END_REF].

This paper presents a method to identify the components of the Cartesian stiffness matrix of hexapods by means of measurements. This method only requires a small number of tests to be carried out. The method uses a simple lumped stiffness parameter model that uses only one spring per leg to model the overall stiffness. An observability index is then used to find the best set of poses and forces for parameter identification. The choice of this index is also discussed. Finally, the efficiency of the proposed method is validated by means of experiments: the predicted 6-DOF deflection of platform under load is compared to the measured values. The experimental study is performed on a micro-positioning system developed by Symétrie [START_REF] Symétrie | Experts in metrology and positioning[END_REF], a manufacturer of high-precision hexapod positioners. The analysis in this paper is demonstrated using a spherical-prismatic-spherical (SPS) hexapod positioner. However, this method is also valid for other sorts of hexapods: universal-prismatic-universal (UPU), UPS and SPU.

This paper is organised as follows: Firstly, the kinematics and stiffness model of a general SPS hexapod is shown. The method for stiffness identification is then presented. This is followed by a brief description of the measurement technique used to measure the 6-DOF pose parameters and displacement of the hexapod. The implementation and experimental validation of the stiffness identification method is then presented. The results and the intended future work are then discussed followed by conclusions on the work presented in this paper.

KINEMATICS AND STIFFNESS MODEL

Fig. 1 shows the schematic of a general SPS hexapod. Each leg consists of a SPS chain that connects to the base at points A i and to the platform at points B i . Two frames of references are defined for the sake of analysis, the fixed frame of reference R f and the mobile frame of reference R m (fixed to the platform). R f and R m align exactly with each other when the platform is in nominal position. The connection points A i and B i lie on circles with centers O b and O m , respectively.

The platform pose, given by X, can be written as,

X = [a, b, c, α, β , γ] T (1)
The first three elements of X represent the translation along X, Y and Z axes of the mobile frame of reference, R m . α, β and γ are the angular rotations of the platform around X, Y and Z axes of the same frame. The angles are defined as per X-Y-Z extrinsic Euler convention. q contains the lengths of six prismatic links.

q = [q 1 , q 2 , q 3 , q 4 , q 5 , q 6 ] T

A simple lumped stiffness model (as shown in Fig. 2) is used to model the static stiffness characteristics of this mechanism. One spring is used to model the stiffness of each leg (k i=1 to 6 ). This model assumes that the actuators are the main contributors of compliance and that the rest of the structure is rigid. The joints are assumed to be frictionless. Also, the legs of the hexapod are assumed massless and considered to be subjected to axial loads only.

The relationship between the platform velocity vector and the joint velocity vector is given by,

q = J -1 V (3)
Here, V contains the components of velocity of the platform. J is the jacobian matrix of the hexapod which is evaluated as per method outlined in [START_REF] Dombre | Robot manipulators: modeling, performance analysis and control[END_REF]. Now, the relationship between differ-ential leg lengths and the differential platform pose (∆q and ∆X) can be written as,

∆q = J -1 ∆X (4)
Here, J -1 is given by,

J -1 = J -1 I 0 0 M (5)
In equation ( 5), I is a 3 × 3 identity matrix and matrix M is given by,

M = R ×   1 0 -s β 0 c α c β .s α 0 -s α c β .c α   (6)
Here, c • = cos(•) and s • = sin(•). R in equation ( 6) is the rotation matrix given by,

R =   c γ .c β -s γ .c α + c γ .s β .s α s γ .s α + c γ .s β .c α s γ .c β c γ .c α + s γ .s β .s α -c γ .s α + s γ .s β .c α -s β c β .s α c β .c α   (7)
The operation shown in equation ( 5) is necessary because J relates the derivatives of components of X to the derivatives of components of q. Hence, J -1 from equation (3) needs to be altered to take into account the relationship between angular velocities and angle derivatives as per X-Y-Z extrinsic Euler convention as shown in [START_REF] Ardakani | Review of the 3-2-1 euler angles: a yaw-pitch-roll sequence[END_REF]. Now, the Cartesian stiffness matrix K C of this parallel system can be expressed as:

K C = J -T K J -1 ( 8
)
Matrix K is diagonal matrix with leg stiffnesses forming its diagonal elements as shown in equation ( 9).

K =         k 1 0 0 0 0 0 0 k 2 0 0 0 0 0 0 k 3 0 0 0 0 0 0 k 4 0 0 0 0 0 0 k 5 0 0 0 0 0 0 k 6         (9)
The matrix K C relates the 6 × 1 wrench vector ∆F applied at the platform and the 6 × 1 platform displacement vector ∆X as,

∆F = K C ∆X (10)
Equation ( 10) can be reorganized to get the following equation:

Ax = ∆X (11)
Here, x is a 6 × 1 vector that contains joint compliances 1/k i .

x = [1/k 1 , 1/k 2 , 1/k 3 , 1/k 4 , 1/k 5 , 1/k 6 ] T (12)
The 6 × 6 matrix A is a function of ∆F, J and J. The elements of A are given by,

A ij = J ij ( 6 ∑ k=1 J kj ∆F k ) (13)
Here, i, j and k are the subscripts for the row and column of respective matrices. ∆F k=1 to 6 refer to the components of wrench applied to the platform.

STIFFNESS IDENTIFICATION METHOD Method description

When we substitute the values of estimated lumped stiffnesses (k i=1 to 6 ), wrench ∆F, hexapod jacobian J and J in equation [START_REF] Clinton | Stiffness modeling of a stewart platform based milling machine[END_REF], we get the expected platform deflection ∆X. The same equation can be used to identify lumped stiffness values when we know everything in equation [START_REF] Clinton | Stiffness modeling of a stewart platform based milling machine[END_REF] except x. This implies loading the platform, measuring the platform deflection and solving equation [START_REF] Clinton | Stiffness modeling of a stewart platform based milling machine[END_REF]. If we increase the number of poses to be measured, we get a 6n × 1 x vector and a 6n × 6 A matrix, where n is the number of poses.

Fig. 3 shows the procedure for stiffness identification. It consists of the following steps:

1. Define experimental constraints that restrict the identification process. For example: workspace limits, loading limitations, etc. 2. Choose the number of poses and the number of measurements per pose. More number of measurements per pose help to reduce the uncertainty of identified parameters at a given pose due to measurement errors. 3. Identify the best set of poses (considering workspace constraints) and corresponding external forces (considering loading constraints) to identify the stiffness parameters. The next sub-section elaborates on the need and method for choosing the optimal set of poses and forces.

4. Load the platform suitably at identified best poses and measure the 6-DOF platform deflection. 5. Solve equation ( 11) to obtain the vector x. This can be accomplished by minimizing the Euclidean norm of (Ax -∆X). The MATLAB function lsqminnorm can be used for this purpose.

Algorithm to identify optimal set of poses and forces

The idea behind having measurements at more number of poses is the same as that for robot geometric calibration. In geometric calibration, measurements of large number of poses spread across the robot's workspace are used. These measurements are then used to get the best geometric parameter set that minimizes the error between commanded and attained robot pose throughout the workspace. Large number of poses are needed for this to reduce the influence of measurement errors. However, practicality concerns demand reduction of these measurements. It has been shown that choosing the right set of measurement poses improves identification accuracy [START_REF] Borm | Experimental study of observability of parameter errors in robot calibration[END_REF].

In our application, matrix A decides the quality of identification of parameters. This is because it relates the measurements done to the identified parameters. Since A is a function of X and ∆F, the best parameters can be identified by judicious choice of robot poses X and platform forces ∆F. Hence, the parameters need to be optimized for all reachable poses and allowable forces.

Parameter identification observability is a well-researched topic in robot calibration literature. However, stiffness parameter identification observability is a relatively new area of research. Klimchik et al. [START_REF] Klimchik | Design of calibration experiments for identification of manipulator elastostatic parameters[END_REF] have presented a method to optimize the pose set for stiffness parameter identification to increase the accuracy of displacement prediction at just one particular pose. This is not suitable for us since we seek to increase the accuracy of displacement prediction throughout the workspace of the hexapod.

Many observability indices can be found in the calibration literature: O 1 [START_REF] Borm | Experimental study of observability of parameter errors in robot calibration[END_REF], O 2 [START_REF] Driels | Significance of observation strategy on the design of robot calibration experiments[END_REF], O 3 [START_REF] Nahvi | Calibration of a parallel robot using multiple kinematic closed loops[END_REF], O 4 [START_REF] Nahvi | The noise amplification index for optimal pose selection in robot calibration[END_REF] and O 5 [START_REF] Sun | Observability index selection for robot calibration[END_REF]. The reliability and performance of these observability indices have been studied very well too [START_REF] Sun | Observability index selection for robot calibration[END_REF][START_REF] Horne | Comparison of pose selection criteria for kinematic calibration through simulation[END_REF][START_REF] Joubair | Comparison of the efficiency of five observability indices for robot calibration[END_REF][START_REF] Zhou | Comparison of the observability indices for robot calibration considering joint stiffness parameters[END_REF][START_REF] Joubair | Performances of observability indices for industrial robot calibration[END_REF][START_REF] Joubair | Calibration efficiency analysis based on five observability indices and two calibration models for a six-axis industrial robot[END_REF]. There is no absolute consensus on the best observability index and also their performances have been noted to depend on the type of robot being calibrated [START_REF] Joubair | Performances of observability indices for industrial robot calibration[END_REF]. Joubair et al. [START_REF] Joubair | Comparison of the efficiency of five observability indices for robot calibration[END_REF] have also shown that all these observability indices yield good result when the measurement noise is low.

The observability indices also depend on the properties of robot models. Matrix scaling needs to be employed when parameters for different variables with different units need to be identified in calibration. Also, different scaling approaches produce different results. Only O 1 is suitable for optimization of parameter estimation with unscaled robot models. The mathematical proof for this can be found in [START_REF] Sun | Observability index selection for robot calibration[END_REF]. If this stiffness identification technique is to be later included in a complete calibration technique that also includes geometric parameters, the issue of matrix scaling can be more complex. Therefore, to avoid complications related to matrix scaling, it is advantageous to select O 1 as the observability index.

For stiffness parameter identification using equation ( 11), O 1 is obtained by singular value decomposition of matrix A. It can be calculated as shown in [START_REF] Borm | Experimental study of observability of parameter errors in robot calibration[END_REF],

O 1 = (σ 1 σ 2 σ 3 ....σ p ) 1/p √ N (14)
Here, p is the number of parameters to be identified which in our case is 6 and N is the number of poses. σ 1 to p are the singular values of matrix A. The best set of poses and corresponding external forces maximize the index O 1 . An optimization routine then needs to be used to find the best set of poses and external forces that maximize O 1 .

MEASUREMENT TECHNIQUE

This section elaborates on the measurement technique used to measure the hexapod pose parameters in the experiments presented in this paper. The measurement setup is as shown in Fig. 4. The measurement system consists of a LK-METRIS coordinate measuring machine (CMM) with RENISHAW SP25M scanning probe. Precision balls are glued to the hexapod for measurement of poses using the CMM. The procedure can be divided into two steps: (i) identifying the coordinate frame fixed to the mobile platform, and (ii) measurement of poses using precision balls. The interest here is to measure the 6-DOF parameters, X, of the platform pose by measuring the positions of precision balls in space.

The identification of reference frame fixed to the mobile platform is done in the manner similar to the one described in [START_REF] Zhang | Calibration of a small size hexapod machine tool using coordinate measuring machine[END_REF]. The reference holes on the hexapod are used for this purpose. These reference holes are machined into the platform of the hexapod in the production phase. This is followed by measuring the positions of three precision balls with reference to the mobile frame. These three balls' positions can then be used as three points in space to identify the mobile frame.

The measurement technique can be demonstrated using Fig. 5. Let Pose 1 be the pose in which the precision balls are identified in the mobile frame. Let P A1 , P B1 and P C1 be the positions of the three precision balls associated with this pose and expressed in the mobile frame of reference. The pose vector X 1 associated with this pose is [0, 0, 0, 0, 0, 0] T . Let's call this measurement M1. Now, when the platform moves to Pose 2, we carry out measurement M2 of the precision balls' positions: P A2 , P B2 and P C2 . The precision balls' positions of measurement M2 are also expressed in mobile frame. A least square fitting algorithm is then used to superimpose the relative positions of the balls measured in M1 on to the ball positions measured in M2. The mobile frame associated with Pose 2 can then be easily identified since the pose of mobile frame relative to precision balls' positions (P A1 , P B1 and P C1 ) of Pose 1 is known. We now obtain Pose 2 (X 2 ) with respect to Pose 1. Subsequent poses can then be measured with respect to Pose 1. All this is done automatically using dedicated software tools developed by Symétrie. The 6-DOF deflection between the poses can be obtained by subtracting X 1 from X 2 .

METHOD IMPLEMENTATION AND EXPERIMENTAL VALIDATION

This section elaborates on how the stiffness identification method was used to evaluate stiffness of a hexapod positioner from Symétrie. The product details cannot be disclosed due to confidentiality. The test setup is as shown in Fig. 6. The experimental constraints included: (i) the allowable workspace of the robot, and (ii) the loading constraint. Concerning the workspace constraints for identification, the platform could not be rotated around its X and Y axes in addition to allowable workspace of the hexapod. This constraint was incorporated to make sure that the mounted mass doesn't slide off the platform. The loading constraint in this case was that the platform could only be loaded by placing mass on it. The load was, therefore, along the Z axis of the platform. Now, the number of poses and number of measurements per pose had to be chosen. For any system of equations, the number of equations should at least be equal to the number of unknowns. In this case, we have 6 unknowns and each pose measurement gives us six equations (6-DOF measurement). Hence, just one measurement is analytically enough to identify all parameters. For efficiency, it is also desirable to keep the number of poses for identification less. Hence, compromising between efficiency and accuracy, 3 poses and 3 measurements per pose were chosen. This choice would lead to a total of 9 measurements for identification. Given the nature of measurements in our case, this result in 54 equations for 6 unknowns.

These choices and constraints lead to 13 design variables for the optimization: twelve pose variables and one force variable. Concerning the load variable, ∆F 3 , further analysis revealed that the index O 1 increased with increased magnitude of ∆F 3 for any random set of poses (Fig. 7). It was also observed that the hexapod has linear stiffness characteristics. Fig. 8 shows the results of an experiment that shows the hexapod's linear stiffness behavior. In this case, the hexapod platform was loaded only along its Z-axis at pose [0, 0, 0, 0, 0, 0] T . Due to these reasons, a choice was made to use just one load vector of maximum possible magnitude to induce maximum platform deflection. This was 34.5 kg in the given case. For displacement calculation, the reference condition was a pre-load of 12.2 kg. This was necessary to suppress the play in actuators. Hence, the effective load for which the platform deflection was measured was 22.3 kg along the Z-axis of the platform. It must be noted that in cases where stiffness is considerably dependent on applied force, more loading conditions will have to be considered for this optimization.

Since the force variable was already chosen, the number of design variables reduced to 12. MATLAB's f mincon function was then used to find the best set of 3 configurations that gave the highest O 1 . These results are very sensitive to the starting point supplied to the optimization algorithm. To tackle this, par- allel computing was used along with large number of starting points to find the best solution. The best pose set for identification generated by this optimization is shown in table 1. The hexapod was loaded in these poses and the platform deflections were measured. Equation [START_REF] Clinton | Stiffness modeling of a stewart platform based milling machine[END_REF] was then solved to get the stiffness parameters which are tabulated in table 2. The efficiency of this stiffness identification method was then validated using platform deflection measurements done at different poses along the X and Y axes of the hexapod. The detailed description of poses is tabulated in table 3. The load used was same as that for stiffness identification. Figures 9 and10 show the comparison between the predicted and measured 6-DOF deflections of the platform at different poses along the X will also reduce the positioning inaccuracy due to compliance of the hexapod to the same level. This is due to the fact that the model prediction accuracy is the only source of error in position compensation technique. This implies that the positioning error due to compliance deflection will reduce, for example, along Zaxis translational coordinate of the hexapod from 13.5 µm (RMS value of measured ∆c) to 2.2 µm.

The small difference that exists between the prediction and the measured values could be attributed to the following:

1. Uncertainty of measurement of the measurement system:

All measurements have an uncertainty bound and this limits both stiffness parameter identification and the experimental validation. 2. Geometric parameter error: The geometric parameters of the hexapod used for analysis was not calibrated. This is a source of error in parameter identification and experimental validation. This is because of the use of an erroneous geometric model to identify stiffness parameters from the measurements and for predicting the displacements. 3. Incomplete stiffness model: The simple stiffness model used cannot capture the complete compliance behavior of the hexapod. Consequently, there will be some residual error due to unmodelled compliance.

For future works, the method presented in this paper will, firstly, be validated for other loading conditions such as loading along X and Y axes of the platform. It will also be validated for other hexapod mounting conditions in which some or all legs are subjected to tensile loading. These studies are of interest to validate the efficacy of the presented method to improve accuracy of positioning in applications such as the one shown in [START_REF] Symétrie | Synchrotron Diffractometer[END_REF]. This method will then be extended to include geometric and thermal parameter models to further improve the positioning accuracy of the hexapod.

CONCLUSION

This paper proposed a new efficient method to experimentally evaluate the stiffness of hexapods. This can be used to accurately predict and correct its positioning error due to structural compliance. The method employed a simple lumped stiffness parameter model which uses one spring per leg to mimic its compliance behavior. A stiffness identification procedure was outlined to estimate the parameters of the proposed stiffness model. This method required small number of measurements. An optimization based on an observability index was used to generate the best set of poses and external forces for stiffness parameter identification. The efficiency of the presented method was evaluated using an experimental study performed on a hexapod based micro-positioner. This was done by comparing the predicted 6-DOF deflections of the platform with the measured deflections. Results showed that the chosen model with identified parame-ters could predict and consequently allow to correct the platform deflections of the tested hexapod very well. The RMS values of prediction error in the validation experiments is under 3.1 µm for translational deflections and 8.8 µrad for rotational deflections. The method is shown to reduce the RMS of positioning error due to compliance deflection, for example, along the Z-axis translational coordinate of the hexapod from 13.5 µm to 2.2 µm.

ACKNOWLEDGMENT

This work has been supported by Agence nationale de la recherche (ANR), France (Projet ANR-15-LCV3-0005). The authors hereby express their gratitude for the support. The authors would like to thank Marielle Baud from Symétrie for the help provided in using CMM. The authors would also like to acknowledge the valuable inputs from Olivier Lapierre, Gilles Diolez, Pierre Noire and Tristan Caritey of Symétrie.

FIGURE 1 .FIGURE 2 .

 12 FIGURE 1. Schematic of a SPS hexapod

FIGURE 3 .

 3 FIGURE 3. Procedure for stiffness identification

FIGURE 4 .FIGURE 5 .

 45 FIGURE 4. Measurement setup

FIGURE 6 .

 6 FIGURE 6. Experimental setup

3 FIGURE 7 . 3 FIGURE 8 .

 3738 FIGURE 7. Trend of O 1 with respect to ∆F 3

  ) b(mm) c(mm) α(deg) β (deg) γ(deg)

FIGURE 9 .

 9 FIGURE 9. Plot of predicted and measured 6-DOF deflection of hexapod under load at poses along X-axis

FIGURE 10 .

 10 FIGURE 10. Plot of predicted and measured 6-DOF deflection of hexapod under load at poses along Y-axis

TABLE 1 .

 1 Best pose set for stiffness identification

		a(mm)	b(mm) c(mm) α(deg) β (deg)	γ(deg)
	Pose1	2.616	0.832	-0.666	0	0	15.756
	Pose2 10.017 -3.434 -0.764	0	0	-11.007
	Pose3	-7.784	-3.210	1.760	0	0	-11.266

TABLE 2 .

 2 Identified stiffness parameters

		Stiffness parameter value (N/µm)
	k 1	k 2	k 3	k 4	k 5	k 6
	4.1981 3.7898 2.7503 4.3200 3.6716 3.7288
	and Y axes of the hexapod, respectively. Table 4 shows the RMS
	values of error in prediction of the 6-DOF deflections.

TABLE 3 .

 3 Pose set for experimental validation

TABLE 4 .

 4 Error in deflection prediction (RMS values) Validation results show that the predicted and measured deflections of the platform are very close. As seen from table 4, the RMS values of prediction error is under 3.1 µm for translational deflections and 8.8 µrad for rotational deflections. This method

	ε ∆a	2.7 µm
	ε ∆b	3.1 µm
	ε ∆c	2.2 µm
	ε ∆α 6.4 µrad
	ε ∆β 8.3 µrad
	ε ∆γ 8.8 µrad
	DISCUSSION AND FUTURE WORK


S Barbanotti

N Panzeri

P Pierini

J L Biarrotte

P Blache

C Commeaux

P Duthil

IPNO E Rampnoux

M Souli

DESIGN OF THE PROTOTYPICAL CRYOMODULE FOR THE EUROTRANS SUPERCONDUCTING LINAC FOR NUCLEAR WASTE TRANSMUTATION

One task of the accelerator workpackage of the EUROTRANS program for the design of a nuclear waste transmutation system is dedicated to the engineering and realization of a prototype cryomodule of the high energy section of the linac, equipped with elliptical superconducting niobium cavities. We review here the present status of the design and the planned program that foresees the experimental characterization of the fully equipped cavity and RF system under its nominal operating conditions.

INTRODUCTION

The EUROTRANS program objective is to develop an advanced design for an experimental facility demonstrating the technical feasibility of Transmutation in an Accelerator Driven System (XT-ADS) and a generic conceptual design of a modular European Facility for Industrial Transmutation (EFIT). Its objective is to provide also an experimental evaluation of the reliability figures for the main modular components of the accelerator configuration. One of the tasks in the accelerator working package of the Eurotrans project is dedicated to building and testing a full prototypical cryomodule of the high energy section of the superconducting proton linac.

A cryomodule is the basic building block of the superconducting accelerator section and has the main role to provide to the cavities both a mechanical support and their cryogenic environment for operation. The XT-ADS linac, which main parameters have been briefly reviewed in Reference [START_REF] Biarrotte | Status of the Eurotrans R&D activities for ADS Accelerator Development[END_REF], on the basis of the high availability/reliability requirements and maintenance considerations, needs a cryomodule designed for easy disconnection both from the beam line and the cryogenic fluids distribution plant [START_REF] Pierini | Task 1.3.3. Preliminary Report: Specifications of the Prototype Cryomodule[END_REF]. The aim of the work here briefly presented is to deliver an operational prototype cryomodule, which can be extensively tested (without beam, but at high RF power levels as in its realistic operating condition), to assess its main reliability characteristics.

The test cryomodule will be a prototypical module of the beta 0.5 section containing one single elliptical multicell superconducting niobium cavity with all its auxiliary equipments.

Besides the development of the bare superconducting cavities, it is important to prototype each auxiliary system needed for the cavity operation in a real environment (frequency tuner, power coupler, RF source, power supply, RF control system, cryogenic system, cryostat…), and relative procedures of assembly and alignment.

The development of the cryomodule relies heavily on prior R&D results, existing infrastructures and investments of INFN, CNRS and CEA. INFN will make available its two beta 0.47 (geometrical length) TRASCO [START_REF] Napolitano | Status Of The High Current Proton Accelerator For The TRASCO Program[END_REF] cavities equipped with the cold tuning system [START_REF] Pierini | Report on Tuner Design[END_REF], which have outperformed the EUROTRANS specifications during vertical tests [START_REF] Bosotti | RF Tests Of The Beta=0.5 Five Cell TRASCO Cavities[END_REF]. CNRS will contribute with clean room for cryomodule assembly, cryogenic infrastructure for test at 2 K, manpower for the cold box study and integration, assembly and tests. CEA will contribute with the cavities chemical treatments.

The final installation and testing of the cryomodule will be done at the Supratech infrastructure under preparation at IPN/Orsay, where the following hardware and facilities will be available:

• A 350 MHz and a 700 MHz high power RF sources.

• A clean room for cryomodule assembly (85 m 2 clean room, with 45 m 2 of class 10/100). • Ultra-pure water production system and HPR facility.

• An Helium liquefier.

• A cavity chemistry facility.

LAYOUT OF THE MODULE

The actual layout of the module is shown in figure 1. The vessel of the module for a single cavity has a diameter of about 1.4 meters (similar to the SNS vessel) and a length of about 1.5 meters. The design has been performed on the following reliability-based considerations:

• As a general request, easy and reliable connection interfaces to the cryogenic and RF system and fast and reliable cold mass alignment strategies are needed to guarantee a short mean time to recover in the case of a module exchange in the linac.

• As in the SNS experience, in order not to produce mechanical stresses on the warm ceramic RF window the fundamental power coupler is positioned vertically. Furthermore, in order to avoid the possibility of contamination of the inner cavity surface during the coupler assembly with dust particles, the warm RF window is positioned below the cavity. • The actual layout simplifies the handling of the subassembly coming out of the clean room. The suspension of the dressed cavity to a room temperature "spaceframe" with tension tie rods does not require any vertical movement of the cavity during assembly. In more details the cavity will be supported by 8 tension rods in a symmetric X pattern to a room temperature "spaceframe" support cage similarly to the SNS concept. The spaceframe acts also as the support for the thermal shield that protects the cold mass at 2 K from the room temperature surfaces of the vacuum vessel, intercepting the thermal radiation at higher temperatures. • This design allows the connection to the cold box providing the liquid cryogenic circuits at different temperature levels already developed by IPN Orsay and similar to the CM0 coldbox used for the spoke cryomodule. The coldbox will be located above the module, connected on the top of the vacuum vessel. As a last consideration, it seems important to borrow, as extensively as possible, proven technologies from existing state-of-art cryomodule designs, again with the perspectives of reaching the ADS reliability goals. Most of the technical solutions, and the underlying superconducting RF technology, outlined for the module layout have been derived from the huge experience accumulated by the TESLA Test Facility (TTF) and the Spallation Neutron Source (SNS) experience.

The Dressed Cavity

The module will be equipped with one of the two TRASCO cavities, fully "dressed" with the components required for their operation in a linac cryomodule.

In particular, the cavity will be equipped with a titanium helium reservoir (which provides the low pressure bath liquid He for the operation at 2 K) and with an internal magnetic shield: a layer of a high-permittivity (at low temperatures) material (Cryoperm) that provides the required shielding from the earth magnetic field in order to guarantee low values of surface resistance for RF losses minimization.

A coaxial blade-tuner [START_REF] Pierini | Report on Tuner Design[END_REF], based on the prototype tested on the TTF linac, provides the slow tuning and fast piezoassisted actions. The mechanism has been fully engineered and fabricated.

THERMAL DESIGN

One of the main issues driving the technical design of a cryomodule is the heat load budget at low temperatures. This either comes from a static contribution (thermal radiation, heat conduction or convective heat transfer from the room temperature environment) or from the dynamic contribution driven by the presence of the RF fields sustained by the accelerating cavities.

Static Heat Loads

Static losses are handled in the modules by minimizing the heat flows towards the 2 K helium bath and intercepting it at higher temperatures, thus gaining in thermodynamic efficiency. The thermal radiation flowing from surfaces at room temperature is intercepted with a thermal shield fixed at intermediate temperature, and minimized by using multilayer insulating (MLI) blankets (layers of doubly aluminized mylar® sheets separated by a low thermal-conducting spacer material). Direct heat conduction paths from the room temperature environment to the 2 K circuit need to be carefully intercepted at intermediate temperatures. Finally, convective effects are prevented by providing a good insulating vacuum between the cold mass and the external vessel at room temperature.

For the estimation of the static heat load budget and its proper handling, a thermal analysis of each direct path from the room temperature vessel to the cold mass environment has been performed by taking into account the proper temperature-dependent material properties. Estimation based on simplified geometries and tabulated material data has been verified with finite element calculations on the model geometry.

Dynamic Loads:RF Losses

For the TRASCO cavity operation at its design parameters (E acc =8.5 MV/m, Q 0 =10 10 , with L active =0.5 m and R/Q=160 Ω), the estimation of the dynamic loads gives 11.3 W of dissipated power at each cavity. For comparison, in the SNS case, the two cryomodules types (medium beta and high beta) have a nominal RF dynamic load of 10 and 13 W, respectively. Even if the vertical tests performed on the existing structures were able to exceed these limits, for the heat load computations, a conservative value of Q 0 =5 10 9 has been assumed.

Heat Load Budget

Table 1 reviews the overall heat load budget on the module. The overall heat load estimates seems to be well within the capacity of the coldbox (~ 50 W).

On the basis of the heat load assessment reported above were calculated the required mass flow for the shield and cavity circuits. For these calculations, it has been assumed that the shield circuit is cooled by nitrogen gas. Even using a conservative estimate for the dynamic heat loads and a 30% overcapacity budget for the coolant mass flow, a total requirement of 1.7 g/s at 2 K is needed for the liquid helium circuit. The Cryogenic Cold Box

The cryogenic cold valves box is derived from the IPN Orsay Eurisol design, which has a nominal cooling capacity of 50 W at 2 K. The cold box supplies 3.3 g/s of liquid helium for the cavity cooling, 2.5 g/s of liquid helium at 4.3K and a Joule Thomson valve for the 2K operation (it can produce a maximum flow rate of 2.2 g/s at 1.9K, ~50W). A ~ 20 litre helium buffer is placed inside the valve box to prevent from pressure perturbations on the cavity when feeding liquid helium. A thermal shield, able to evacuate around 60 W at around 80K, made of copper and covered with multi layer insulation, surrounds the cold internal parts of the valve box.

THE COUPLER

The 704.4 MHz power coupler is under development at IPN Orsay and is based on the SNS design, adapted and optimized for the needs of this design (different frequency, more CW power, additional cooling provisions included).

The main characteristics of the coupler are the following:

• capacitive coupling to the cavity • coaxial geometry of the antenna • ceramic window geometry: disk with chokes • doorknob to perform the transition from the coaxial to the waveguide geometry • capable to transmit up to 80 kW of RF power to the cavity • inner cooling of the antenna with r.t. water • window cooling by water @ 300 K • cooling of the outer conductor between the ceramic and the cavity by supercritical liquid helium @4.5 K.

FUTURE WORK

For the experimental characterization of the fully equipped cavity and RF system the following steps are foreseen:

• finalization of the cryomodule design and production • production of the coupler system • cold box design finalization for production. The cryomodule assembly is expected in summer 2009, while the final tests should take place up to December 2009.

Figure 1 :

 1 Figure 1: Module layout.

Figure 2 :

 2 Figure 2: The main cryogenic scheme.

Table 1 :

 1 Heat load budget on the module

		Circuit 4.3K	Circuit 1.9K	Thermal shield
		(W)	Static (W)	Dynamic (W)	(W)
	Transfer line	0.5
	Cold Box	2.0			20.0
	Cryomodule		4.0	25.0	50.0
	Coupler	1.2	0.0
	Total	3.7	4.0	25.0	70.0
				32.7
	Mass flow (g/s)		1.70		0.45

Proceedings of EPAC08, Genoa, Italy 07 Accelerator Technology Main Systems

T07 Superconducting RF

Proceedings of EPAC08, Genoa, Italy MOPP11407 Accelerator Technology Main Systems T07 Superconducting RF

ACKNOWLEDGMENTS

This work is supported by the EURATOM 6 th Framework Program of the EC under contract FI6KW-CT-2004-516520.