Sébastien Briot

Anatol Pashkevich
email: anatol.pashkevich@emn.fr.

Damien Chablat

TECHNOLOGY-ORIENTED OPTIMIZATION OF THE SECONDARY DESIGN PARAMETERS OF ROBOTS FOR HIGH-SPEED MACHINING APPLICATIONS

In this paper, a new methodology for the optimal design of the secondary geometric parameters (shape of links, size of the platform, etc.) of parallel kinematic machine tools is proposed. This approach aims at minimizing the total mass of the robot under position accuracy constraints. This methodology is applied to two translational parallel robots with three degreesof-freedom (DOF): the Y-STAR and the UraneSX. The proposed approach is able to speed up the design process and to help the designer to find more quickly a set of design parameters.

1

INTRODUCTION

Parallel kinematic machines (PKM) are commonly claimed to offer several advantages over their serial counterparts, such as high structural rigidity, better payload-to-weight ratio, high dynamic capacities and high accuracy [START_REF] Merlet | Parallel Robots[END_REF][START_REF] Tlusty | Fundamental Comparison of the use of Serial and Parallel Kinematics for Machine Tools[END_REF][START_REF] Chablat | A Comparative Study between Two Three-DOF Parallel Kinematic Machines using Kinetostatic Criteria and Interval Analysis[END_REF]. Therefore, they are prudently considered as promising alternatives for many modern material processing operations, especially in automotive and aerospace industry, in which high accuracy positioning and high-speed motions of a work tool are required. Thus, PKM have gained essential attention of a number of companies and researchers. However, most of the existing PKM still suffer from two major drawbacks, namely, a complex workspace and highly non-linear input/output relations [START_REF] Tsai | Robot Analysis: the Mechanics of Serial and Parallel Manipulators[END_REF][START_REF] Bonev | The parallel mechanism information center[END_REF].

For most of PKM, the performances vary considerably for different points in the workspace and for different directions at one given point. This is a serious disadvantage for machining applications [START_REF] Kim | Performance Analysis of Parallel Manipulator Architectures for CNC Machining Applications[END_REF][START_REF] Wenger | A Comparative Study of Parallel Kinematic Architectures for Machining Applications[END_REF], which require regular workspace shape and acceptable performances throughout. In milling applications, for instance, the machining conditions must remain constant along the whole tool path [START_REF] Rehsteiner | Putting parallel kinematics machines (PKM) to productive work[END_REF]. Nevertheless, in many research papers, this criterion is not taken into account in the algorithmic methods used for the optimization of robots [START_REF] Luh | Working capability analysis of Stewart platforms[END_REF][START_REF] Merlet | Determination of 6D Workspace of Gough-Type Parallel Manipulator and Comparison between Different Geometries[END_REF].

Our previous work [START_REF] Briot | Optimal Technology-Oriented Design of Parallel Robots for High-Speed Machining Applications[END_REF] was focused on the optimal design of the primary geometric parameters (length of links, shape of the base, etc.) of robots for given geometric, kinematic and kinetostatic properties (e.g. size of the workspace, maximum speeds, forces transmission, accuracy). Contrary to many works on optimal design of parallel robots (see for example [START_REF] Pashkevich | Design Optimization of Parallel Manipulators for High-Speed Precision Machining Applications[END_REF][START_REF] Merlet | Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots[END_REF]), it was proposed to use technology oriented indices in order to define the optimal design parameters. This work was the first step of an optimization process. Indeed, the previously proposed algorithm does not allow obtaining any information about the value of the secondary design parameters (e.g. shape of links, size of the platform, etc.). For finding these values, other constraints should be considered, such as constraints derived from the robot dynamics (acceleration capacities), its elastostatic behaviour or its natural frequencies.

Thus, this paper is the continuity of work [START_REF] Briot | Optimal Technology-Oriented Design of Parallel Robots for High-Speed Machining Applications[END_REF]. Here, a new methodology for the optimal design of the secondary design parameters of parallel kinematic machine tools is proposed. This approach aims at minimizing the total mass of the robot under position accuracy constraints. The main contribution of this paper is in the area of CAD methodology and application of the operation research methods to the integrated design optimization of complex mechanical structures, such as parallel robots.

The paper will be divided as follows. In the second part, the design problem and methodologies are explained. In the third section, the performance measures, design constraints and objectives are presented and the optimization procedure is described. It is applied to an industrial case study in part four. Finally, in the last section, conclusions are drawn.

2

DESIGN PROBLEM AND METHODOLOGY Design of a robot involves simultaneous optimisation of many types of criteria that may evaluate the kinematic, the kinetostatic or also the dynamic properties. However, the design of a manipulator, once its architecture is known, may be decomposed into two steps: -Step 1: find the values of the primary geometric parameters (length of links, base radii, joint limits, etc.) using geometric, kinematic and kinetostatic constraints and objectives (workspace, velocity, effort transmission, etc.); -Step 2: find the values of the secondary geometric parameters (cross-section, shape of links, etc.) using dynamic, elastostatic and/or elastostatic constraints and objectives (acceleration capabilities, maximal deformations of the tool, natural frequencies of the structure, etc.).

Step 1 has been studied in many works [START_REF] Luh | Working capability analysis of Stewart platforms[END_REF][START_REF] Merlet | Determination of 6D Workspace of Gough-Type Parallel Manipulator and Comparison between Different Geometries[END_REF][START_REF] Briot | Optimal Technology-Oriented Design of Parallel Robots for High-Speed Machining Applications[END_REF][START_REF] Pashkevich | Design Optimization of Parallel Manipulators for High-Speed Precision Machining Applications[END_REF][START_REF] Merlet | Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots[END_REF]. Indeed, as it is based on simple geometric, kinematic and kinetostatic models, this step is mostly the simplest to achieve. This is not the case of the second optimization level which requires more complicated and time-consuming models. For example, in order to obtain a very accurate elastostatic model, finite element analysis (FEA) should be used. However, because of the huge number of elements used for the meshing of the robot, such model cannot be integrated in a design optimization process.

Here, a design methodology that uses more efficient models is proposed. This will allow obtaining near-optimal results in a reasonable computational time. Then, these results may be introduced in more accurate, but more time-consuming models, in order to verify that they respect the design specifications. If they do not, two solutions are possible: (i) to improve the accuracy of the model used in the optimization loop or (ii) to modify the design constraints (with safety coefficient) in order to obtain acceptable results. It should be mentioned that, even if several optimization loops are necessary, this approach will considerably speed up the design process.

To formulate the design problem, let us define the manipulator geometry by the mapping g:   W, where  =  1 × …  n and = p 1 × … p n denote respectively the configuration space and the workspace.  i are the joint coordinates and p i are the coordinates of the end-effector. n is the number of degrees of freedom (DOF). Besides, for each workspace point P  W, let us define the matrices K d (P, ), K s (P, ), K v (P, ), that describe various mechanical properties of the manipulator (dynamics, stiffness, etc.) for any given set of the design parameters . Let us also assume that for each type of the matrices K  ,   {d, s, …}, there are defined physically consistent scalar measures   (K  ),   {i, a, …} that may be directly included in the design objectives or constraints. Some examples of such measures (input efforts, accuracy of the tool, etc.) are presented in the following subsections.

Similarly, for the global evaluation of the manipulator, let us introduce the performance measures   (g, ),   {m, e, n, …}, that depend both on the adopted geometrical structure g and the physical parameters of the links . Examples of the global measures include the total mass of the manipulator, the maximal admissible input effort, the minimal admissible natural frequency, etc.

Then, following the general methodology adopted for the considered application area (high-speed machining), the design optimisation problem can be stated as achieving the best value of the performance indices

     , min ) , ( π π g (1)
subject to the constraints

        , , ) , (   S P π K
(2) that must be satisfied for all points of the workspace W which includes the manufacturing task. Since in practice this problem cannot be solved by the direct search methods, in the following subsections, there will be presented the discretisation scheme and relevant optimisation algorithms allowing to obtain desired solutions in reasonable time.

PERFORMANCE MEASURES AND DESIGN CONSTRAINTS

In the following of this paper, it is assumed that the primary design parameters are already known. Thus, let us now present the procedure for optimization of the secondary design parameters.

Design constraints and objectives

Several indices may be used in order to define the secondary geometric parameters of the robots. As an example, it is possible to choose the maximal deformations under a load (variable or constant), the first natural frequency, which indicates the way a mechanism tends to vibrate, the input efforts, that are needed in order to observe the dynamic capacities of the mechanism, etc. All these indices are strongly related to the shape of the links of the robot and their material properties (density, stiffness, etc.). However, some of them seem to be more important, such as the required input efforts and the maximal deformations. Indeed, if manufacturers have no problems to define a fixed strong value for a maximal admissible accuracy or for the dynamics capacities of the robot (acceleration), this is not the case with the natural frequencies, for which the minimal admissible value may vary up to 50%, depending on the manufacturer's opinion. Therefore, in the following of this paper, proposed indices are only based on the computation of the maximal deformations and maximal input efforts. Now, we have to choose if the maximal deformations and the maximal input efforts will be used as constraints or as objectives. Indeed, it seems quite obvious to use the maximal deformations as constraints because, mostly, the desired accuracy of the mechanism is given. For the input efforts, the question is a bit more complex. The input efforts allow characterizing the dynamic properties of the robot (acceleration capabilities) and also to choose the wellappropriated actuator during the design process. If the actuator is powerful enough to accept the desired input efforts, the mechanism will be able to accelerate properly. If not, the mechanism will accelerate more slowly. Of course, most powerful actuators will be more expensive. So, the input efforts should not be considered as mechanical constraints of the mechanism, but as design objectives in order to minimize the cost of the robot and also its power consumption.

In the next parts, the design constraints and the design objectives are defined.

Design constraints: the deformations

The modelisation of the deformations used in this part has been presented in [START_REF] Pashkevich | Stiffness Analysis of Overconstrained Parallel Manipulators[END_REF]. This model, which combines advantages of the traditional methods (the finite element analysis [START_REF] El-Khasawneh | Computation of Stiffness and Stiffness Bounds for Parallel Link Manipulators[END_REF][START_REF] Bouzgarrou | Rigidity Analysis of T3R1 Parallel Robot Uncoupled Kinematics[END_REF], the matrix structural analysis [START_REF] Deblaise | Systematic Analytical Method for PKM Stiffness Matrix Calculation[END_REF][START_REF] Ghali | Structural Analysis: A Unified Classical and Matrix Approach[END_REF] and the virtual joint method [START_REF] Gosselin | Stiffness Analysis of Parallel Mechanisms Using a Lumped Model[END_REF][START_REF] Majou | Parametric Stiffness Analysis of the Orthoglide[END_REF]) is based on a multidimensional lumped-parameter model that replaces the link flexibility by localized 6-DOF virtual springs that describe both the linear/rotational deflections and the coupling between them. In addition, it employs a new solution strategy of the kinetostatic equations, which allows computing the stiffness matrix for the overconstrained architectures, including the singular manipulator postures. This gives almost the same accuracy as FEA but with essentially lower computational effort because it eliminates the model re-meshing through the workspace.

This model states that the deformations t i of the extremity of the leg i of the manipulator are related to the efforts f i applied to its extremity via the relation:

                     0 t q f 0 J J S i i i T i q i q i δ δ  ,   T i i i i     J K J S 1   ( 3
)
where q i represents the passive joints displacements of the leg i, i  K is the stiffness matrix describing the rigidity of all the elements of the leg i and i  J , i q J are the Jacobian matrices relating the displacements of the extremity of the leg i to the spring deflections  i and passive joint displacements q i , such as

i i i i q i θ J q J t δ δ δ    ,          i i i q q t J ,          i i i θ t J  . ( 4
)
After the stiffness matrices K i for all kinematic chains are computed, the stiffness of the entire manipulator can be found by simple addition:

   n i i m 1 K K . (5)
Then, once K m is known, the deformations t of the tool under a load f may be computed as:

f K t 1 δ   m .
(6) Here, it is clear that the matrix K m depends on the position of the end-effector, and as a consequence, that the deformations vary with the position of the robot in its workspace.

We would like to mention that the vector t is a six dimensional vector, of which three first components, denoted by the vector t t , represent the translational deformations of the tool of the robot, and the three last components, denoted by the vector t r , its rotational deformations.

So, using the proposed elastostatic modelisation, let us express the design constraints. The vector containing all or some components of the vector t t (t r , resp.) is denoted as p (, resp.). p max ( max , resp.) is the maximal admissible value for the norm of the vector p (, resp.) wherever in the workspace W. Thus, the design constraints may be written as:

max δ ) δ ( max p  p W , ( 7a
) max δ ) δ ( max   Φ W .
(7b)

Design objective: minimisation of the input efforts

The input efforts of a manipulator depend on its position, velocity and acceleration, as well as the shape and material density of its links. However, it is clear that, at a design stage, one cannot consider the velocity and the acceleration of the manipulator, as it depends on the required application. So the number of sets of admissible velocities and accelerations is infinite. Moreover, these parameters may be also optimized in a next step [START_REF] Angeles | Fundamentals of Robotics Mechanical Systems[END_REF], but only when the type of application is given, which is not the case now.

From the shape S i and material density  i of a link i, one may found its mass and matrix of inertia. It could be shown that the parameters that have the most influence on the input efforts of a manipulator (for given positions, velocities and accelerations) are the masses of the links. The influence of the axial moments of inertia is negligible compared to the influence of the masses. Therefore, the problem of minimizing the input efforts may be reduced to the minimization of the total mass m robot of the robot: ,

ρ S  robot m where S = [S 1 ,… , S n ] T ,  = [ 1 ,… ,  n ] T (8) (n being the total number of links) subject to the constraints max δ ) δ ( max p  p W and max δ ) δ ( max   Φ W . min ) , ( ,ρ S
In the next part, the algorithm used for the design procedure is presented.

Design procedure

The simplest way to find the design parameters is to apply the following straightforward algorithm: The main drawback of this algorithm is that, as it computes the deformations of the robot for all workspace points P, it is very time-consuming. Thus, it should be simplified. Two ways may be followed in order to accelerate the computing process. The first one consists in assuming that the largest deformations will appear at the boundary of the studied workspace. Thus, only a short number of points are tested. The assumption is realistic because, kinematically speaking, i.e. analyzing the Jacobian matrix, the properties of most of robots are better in the centre of the workspace than on its boundary. As the stiffness matrix of the robot depends partially on the Jacobian matrix, the assumption is quite realistic. However, there are no direct proofs of that. But this considerably speeds up the routine process.

for  =  min to  max compute
The second way is to use the goal attainment programming. The goal attainment optimization allows generating specific Pareto-optimal solutions. Let us apply the goal-attainment technique that yields the following nonlinear programming formulation:

π , min    (10) subject to i h h f w f i i i i     ; ) ( ; ) ( 0 0 π π  (11)
Here,  is an unrestricted scalar variable, 0  i w are designer selected weighting coefficients, and 0 i f are the goal to be realized for each design objective. For computational conveniences, the design constraints have been transformed in the scalar form 0

) ( i h h  π .
In this formulation, minimization of  tends to force the specifications to meet their goal. If, at the solution point,  is negative, the goals have been overattained; if  is positive, then the goals have been underattained. The method is appealing since it is possible for the designer to specify unrealizable objectives and still obtain a solution which represents a compromise. For more information about the goal-attainment optimization, the reader may refer to [START_REF] Fleming | Computer Aided Control System Design Using a Multi-Objective Optimisation Approach[END_REF].

APPLICATION EXAMPLES

Industrial problem

Let us demonstrate the efficiency of our design approach on a concrete problem coming from the industrial sector of the region of Nantes (France).

One of the most important activity areas of this region is the manufacturing of bathroom components (shower cabin, washbasin, bathtub, etc. as shown in Fig. 1). Most of parts used during the assembly process of the bathroom component are flat and made of thermosetting materials. The main operations achieved on these parts is trimming, i.e. the suppression of the edges of the parts in order to obtain a good surface roughness.

The machines tools that are used for the trimming of these bathroom components must be designed as they attain the following characteristics:

-workspace W abc of size {2. Looking at these requirements, and from our industrial experience, several types of robots may be envisaged, such as the Y-STAR [START_REF] Hervé | Group Mathematics and Parallel Link Mechanisms[END_REF], the UraneSX [START_REF] Company | Modelling and Design Issues of a 3-Axis Parallel Machine-Tool[END_REF], the Orthoglide [START_REF] Chablat | A Comparative Study between Two Three-DOF Parallel Kinematic Machines using Kinetostatic Criteria and Interval Analysis[END_REF], the Hybridglide [START_REF] Company | Machine-Outils Rapides à Structure Parallèle[END_REF], the 3-UPU [START_REF] Tsai | Kinematics and Optimization of a Spatial 3-UPU Parallel Manipulator[END_REF], etc. However, because the workspace is not a cube, but a flat parallelepiped, it appears that only two kinds of architectures, that have non-orthogonal arrangement of legs and that are already used in machining process, seem to be best adapted: the Y-STAR (Fig. 2) and the UraneSX (Fig. 3). It should be mentioned that, in our study, the prismatic guides of the UraneSX are vertical. In our previous work [START_REF] Briot | Optimal Technology-Oriented Design of Parallel Robots for High-Speed Machining Applications[END_REF], the optimal primary geometric parameters have been found, taking into account the above requirements. The values of the optimal lengths of legs for the two robots were 2.16 m for the Y-STAR and 4.62 m for the UraneSX. The value of the optimal radius of the base circumcircle for the UraneSX was 2.77 m.

Starting from these parameters, it is proposed to achieve the optimization of the secondary geometric parameters of these two robots. The assumption is made that the base, the platform and the feet are designed such as they may be considered rigid. The only deformable links are the bars of the parallelograms. This assertion is quite realistic when considering the huge length of the legs. So, the only parameters to optimize are: -the radius of platform circumcircle (i.e. the circle passing through the points B i of Figs. 2a and3a), denoted as R Tool ; -the length 2d of the small links of the parallelograms (see Figs. 2a and3a); -the cross-sections S Bar of the long bars of parallelograms (see Figs. 2a and3a) Let us first present the elastostatic modelisation of the two robots.

Elastostatic modelisation

As presented in Figs. 2 and3, both manipulators have similar architectures (Fig. 4 1 ). They may only be differentiated by the position and orientation of the axes of the actuated prismatic pairs. Therefore, their elastostatic models may be obtained in similar ways.

As the feet, base and platform are considered rigid, each chain of Fig. 4 may be decomposed into two subchains (Fig. 5) which are linked to the same foot. The subchains may be described as: (a) a rigid link between the manipulator base and the i th actuator (part of the base platform) described by the constant homogeneous matrix i Base T ; (b) a 1-DOF actuating joint which is defined by the homogeneous function matrix

  i a q 0 V
, where i q 0 is the actuated coordinate; (c) a rigid foot linking the actuating joint and the axis of the parallelogram, which is described by the constant homogeneous matrix i Foot T ; (d) a 2-DOF passive U-joint at the beginning of the bar of the parallelogram, allowing two independent rotations with angle i q 1 and i q 2 , which is described by the homogeneous matrix function

  i i U q q 2 1 , V
; (e) a rigid bar linking the two axis of the parallelogram, which is described by the constant homogeneous matrix Bar T ; (f) a 6-DOF virtual spring describing the bar mechanical stiffness, which is defined by the homogeneous matrix function

  i i s 5 0 ,...,  V where   i i i 2 1 0 , ,    ,   i i i 5 4 3
, ,    are the virtual spring coordinates corresponding to the spring translational and rotational deflections, respectively; matrix V s is composed of six elementary transformations; (g) a 2-DOF passive U-joint at the end of the bar of the parallelogram, allowing two independant rotations with angle i q 3 and i q 4 , which is described by the homogeneous matrix function

  i i U q q 4
3 , V ; (h) a rigid link from the manipulator leg to the end-effector described by the constant homogeneous matrix transformation i Tool T . In the following simulations, the controlled point of the end-effector is considered to be located at the extremity of a milling cutter of length equal to 20 cm. The corresponding mathematical expression defining the end-effector location subject to variations of all above defined coordinates of a single kinematic chain i may be written as follows:

        i Tool i i U i i s Bar i i U i Foot i a i Base i q q q q q T V V T V T V T T 4 3 5 0 2 1 0 , ,..., ... ... ,      , i = 1 to 6 (12)
In the rigid case, the virtual joint coordinates i i 5 0 ,...,  are equal to zero, while the passive and active coordinates i i q q 4 0 ,..., are obtained through the inverse kinematics, ensuring that all matrices T i are equal to the prescribed one that characterizes the desired spatial location of the moving platform (kinematic loop-closure equations). Particular expressions for all components of the product (12) may be easily derived using standard techniques for homogeneous transformation matrix. Now, matrices i  J , i q J presented at Eq. ( 3) have to be found. The matrix i  J may be obtained from the differentiation of the matrix T i with respect to the spring parameters i j  (j = 0 to 5), at the point

0  i j  , considering that                        0 0 0 0 ' 0 ' ' ' ' 0 ' ' ' ' 0 iz ix iy iy ix iz ix iy iz R ij i j i j L ij i j i p p p j           H V H T ( 13
)
where the first and the third multipliers are the constant homogenous matrices which do not include the displacement The Jacobians i q J can be computed in a similar manner, but the derivatives are evaluated in the neighbourhood of the "nominal" values of the passive joint coordinates i j q (i = 1, to 4) corresponding to the rigid case (these values are obtained from the inverse kinematics.

Once these matrices are obtained, the stiffness matrix of the robots may be easily obtained using expressions (3) to (5).

Computation of the total mass

The total mass of the robot is obviously the sum of the masses of each of its links, i.e. the masses of the three feet plus the six axes and bars of the parallelograms plus the platform.

The length of the bars of the parallelograms is already known here. But this is not the case of their shape, which should be properly defined by an analysis of the effort transmission and deformations of the mechanism. Indeed, looking at the two robots under study, it appears that the bars of the parallelograms have only to support traction/ compression and torsion solicitations.

It is known (and it could also be easily demonstrated) that the best appropriated shape of bars able to best resist to these types of solicitations is the hollow cylinder. Thus, once this shape is fixed, one can compute the mass of one bar of a parallelogram:

  Bar Bar Bar Bar Bar Bar Bar Bar S l r R l m       2 2 . ( 14
)
where 

Bar is the density of the material, l Bar is the length of the bars, R Bar is the external radius of the cross-section, and r Bar the internal radius (R Bar and r Bar are parameters that should be defined during the optimization process). S Bar is the value of the cross-section. It should be mentioned that, for any fixed value of S Bar , the cross-section that has the best torsional stiffness is the cross-section with the largest value of R Bar . Therefore, for one given maximal value max

Bar

R

of R Bar , r Bar should be equal to

   / 2 max Bar Bar Bar S R r   . ( 15
)
The mass of the feet is directly linked to their shape and length (which is considered equal to 2 d, where d is a parameter that should be defined during the optimization process). In a first approximation, it is considered that these feet should have the shape of cylinders. This shape is not the best one, but at this design stage, it is sufficient to give an approximation of their mass. It will be refined in another step using CAD/FEA software. So, their mass is equal to

d k R d m Foot Foot Foot 1 2 2 2     . ( 16
)
where 

Foot is the density of the material, and R Foot is the external radius of the cross-section. The parameter k 1 regroup all constant terms of expression [START_REF] Bouzgarrou | Rigidity Analysis of T3R1 Parallel Robot Uncoupled Kinematics[END_REF].

Finally, the mass of the platform will also vary with its shape. It is commonly considered that the platform may be kinematically represented by a circle. Therefore, it will be considered here that its shape is a cylinder. Once again, this is not the best solution, but at this design stage, it is sufficient to give an approximation of their mass. It will be also refined in another step using CAD/FEA software. So, its mass is equal to

2 2 2 Tool Tool Tool Tool R k R h m     . ( 17
)
where 

Tool is the density of the material, h is the height of the cylinder and R Tool is the platform radius (which is a parameter that should be defined during the optimization process). The parameter k 2 regroup all constant terms of expression [START_REF] Deblaise | Systematic Analytical Method for PKM Stiffness Matrix Calculation[END_REF].

Finally, the total mass of the platform is equal to

2 2 1 6 6 Tool Bar Bar Bar R k d k S l m     . ( 18
)
It should be mentioned here that, to avoid interferences between two adjacent parallelograms links, the radius R Tool of the platform should be superior to 3 / 2d (Fig. 6). In the next subsection, a design procedure that allows finding the optimal design parameters S Bar , d and R Tool will be proposed.

Optimization results

Before using the optimization algorithm proposed in section 3.4, the lower and upper bounds of the search intervals for the parameters d, S Bar and R Tool have to be fixed. These intervals are: -d  [0.1 m, 0.7 m]; the lower limit of 0.1 m is set because of design issues. Indeed it is difficult to design parallelograms with small axes and a lower limit for d of 0.1 m is considered as a minimal acceptable value;

-

R Tool  [ 3 / 2d , 0.8 m]; -S Bar  [10 -5 m², 0.01 m²].
The obtained results are presented below.

Comparison between the alternative solutions

At this step, as the exact form of the feet and of the platform is not known, one way to proceed is to make the parameters k 1 and k 2 vary. This will allow the designer to test several potential designs with CAD/FEA software and to find, in the last step of the optimization process, the most adapted one.

Here, the used values of parameter k 1 are 25, 100 and 200. These values correspond to axes of parallelogram made of aluminium, of which radius is equal 5 cm, 10 cm and 15 cm respectively.

The lower value of 5 cm is chosen because, in our previous work [START_REF] Briot | Optimal Technology-Oriented Design of Parallel Robots for High-Speed Machining Applications[END_REF], it was shown that the maximal rate between the force applied to the platform and the force applied to the feet was about 10. It means that, if the force applied to the platform is 300 N, the maximal force applied to the feet will be of 3000 N. Let us consider a beam of length 0.4 m. The value of 0.4 m is chosen here as it corresponds to the medium value of the search interval for parameter d. It be shown that the deformation of a beam under a load of 3000 N, of which the circular section's radius is 5 cm, is a bit less than 0.25 mm (0.25 mm being the maximal desired value of the end-effector deformation). The other values for the radius (10 cm and 15 cm) are used because the exact form of the feet is not known. Thus, it is preferable to make the radius vary, in order to test several potential design with CAD software and to find the most adapted one.

The used values of parameter k 2 are 250, 500, 750 and 1000. These values correspond to platform of height comprised 3 cm, 6 cm, 9 cm and 12 cm. As it is difficult to know the real shape of the platform, the lower value of 3 cm is chosen because it is approximately the width of the platform of the Orthoglide robot [START_REF] Chablat | A Comparative Study between Two Three-DOF Parallel Kinematic Machines using Kinetostatic Criteria and Interval Analysis[END_REF] manufactured by Symetrie® and located at the IRCCyN (France). The other values are used because the exact form of the platform is not known. Thus, it is preferable to make the height of the platform vary, in order to test several potential design with CAD software and to find the most adapted one.

In a first step, let us analyze the evolution of the design parameters as a function of the desired accuracy of the platform. It is proposed to vary the value of p max from 0.01 mm to 0.5 mm. For any set of parameters k 1 , k 2 and p max , the values of the optimal design variables are computed. The less time-consuming algorithms proposed in section 3 are used. The results are presented at tables 1 and 2. It is only presented one table per robot for the two algorithms proposed above, because the obtained results in the two cases are identical. Thus, the assumption made in section 3 for accelerating the computing process was correct, i.e. in this case, the worst points for accuracy are located at the boundary of the workspace.

It may be observed for the UraneSX (Y-STAR, resp.) that, for any fixed value p max ≤ 0.1 mm (p max = 0.01 mm, resp.), the values of design variables S Bar , d and R Tool evolve as a function of the parameters k 1 and k 2 . This is not the case for p max = 0.5 mm (p max ≥ 0.05 mm, resp.). In this case, the design parameters remain constant whatever is the values of k 1 and k 2 . The optimal value of d is 0.1 m, which was the lower bound fixed in our optimization algorithm. As previously said in section 4.4, this lower limit was fixed because of design issues. The reason for which all the values of the design variables S Bar , d and R Tool are constant for p max = 0.5 mm (p max ≥ 0.05 mm, resp.) is that optimal solutions will appear for d < 0.1 m, but it is not possible to pass though this limit. Thus, because of the huge impact of the feet platform masses on the total mass of the robot, the algorithm converge to the sizes of platform and feet that will keep the total mass as small as possible.

It could also be observed that, when the parameters k 1 and k 2 increase, i.e. when the cross-section of the feet and the height of the platform cylinder increase, the optimal robots will be obtained when the values of the length d of the axes of the parallelograms and of the platform radius R Tool decrease. This is due to the fact that the mass of the feet and of the platform largely increase with the increase in the parameters d and R Tool . When this appears, it may also be observed that the cross-section of the bars of the parallelograms increases. This phenomenon is due to the fact that, the smaller the length d of the axes, the poorer the torsional stiffness of the parallelograms. Therefore, it is necessary to increase the crosssection of the bars in order to assure the good stiffness of the entire mechanism.

Finally, it should be mentioned that, for identical sets of parameters k 1 , k 2 and p max , the mass of the Y-STAR is lower than that of the UraneSX.

Optimization results for bathroom components milling

In order to be sure that the obtained robot will not suffer from a lack of accuracy due to non modelized phenomena (such as clearance, elasticity of joints, etc), some safety coefficient is applied to the value of ||p xy ||. The corrected value is fixed to 0.1 mm.

First, the height of the platform was fixed to 3 cm, i.e. k 2 = 250. For p max = 0.1 mm, the maximal value of d is 0.22 m. It could be shown that the maximal deformation of the foot, represented by a aluminium beam of length 0.22 m, under a load of 3000 N (see section 4.4.1), of which cross-section's radius is 10 cm, is about 0.002 mm, i.e. it is insignificant compared with the deformations of the bars of the parallelogram. Thus, parameter k 1 is fixed to 100. Taking into account these parameters and extracting data from tables 1 and 2, the optimal design parameters are presented in table 3.

Our simulations show that manufacturing a Y-STAR for high-speed machining of composite will allow creating robots with a lower mass.

Discussion

From the obtained results, it should be mentioned several comments. For reason of simplicity and reduction of computational time, it has been considered that the only deformable parts were the links of the parallelograms and that all other elements where rigid. This assumption about the feet and platform is quite realistic because, generally, they are designed such as they have very little deformations. However, the deformations of the bearings and axes that will be used in order to create the universal joints have not been taken into consideration. These parts have generally smaller crosssections than the others; therefore they may lead to non negligible deformations of the robot. Thus, the proposed model is not sufficient to fix the desired cross-section of the links, and more refined FEA model should be created in order to validate the results. However, these models are complicated and an optimization algorithm using them will be much more time consuming than that proposed here.

So, it should be concluded that the proposed design procedure is not sufficient to definitely fix the design of a robot. However, the algorithm will give the designer an idea of the near-optimal solution that should be confirmed by a FEA analysis. This will be used in order to readjust the values of parameters k i and p max . After several loops between the FEA software and the optimization algorithm, the designer will converge to an optimal solution. This optimal solution could be obtained using only FEA model, but the proposed algorithm will considerably accelerate the design process.

CONCLUSIONS

In this paper, a new methodology for the optimal design of the secondary geometric parameters of parallel kinematic machine tools was proposed. This approach aims at minimizing the total mass of the robot under position accuracy constraints. This methodology was applied to two translational parallel robots with three DOF: the Y-STAR and the UraneSX. The proposed approach will help the designer to find more quickly an optimal set of design parameters and they will allow considerably accelerating the design process.

To conclude, we would like to mention that our next task in the design optimization of robots will be the analysis and comparison of the results obtained during this optimization process with several other architectures. We also would like to work on the definition of more meaningful criteria (different from the total mass) for the characterization of the robots dynamic properties. Finally, another research perspective is to more deeply focus on the improvement of the elastic models accuracy, especially in the neighbourhood of singular configurations. Indeed, it appears that near singularities, the elastic behaviour of robots becomes nonlinear. As a result, the classical deformation models are no more accurate enough. Therefore, it is necessary to solve this problem.

6

ACKNOWLEDGMENT This work has been funded by the French région Pays de la Loire (RoboComposite project).

  5 m × 2.5 m × 0.5 m}; -||v xy || = 60 m/min (v xy contains the components of the platform velocity vector v in the xy plane); -||f xy || = 300 N (f xy contains the components of the external effort vector f in the xy plane); -||p xy || = 0.25 mm (p xy contains the components of the platform deformations vector p in the xy plane).

Figure 1 .

 1 Figure 1. Typical examples of bathroom components.

Figure 2 .Figure 3 .

 23 Figure 2. The robot Y-STAR: (a) kinematic chain and (b) prototype.

Figure 4 .

 4 Figure 4. Architecture of the Y-STAR and the UraneSX.

Figure 5 .

 5 Figure 5. Modelisation of the subchains.



  , and the second multiplier corresponds to the derivative of the elementary translation or rotation corresponding to i j  . In the right-hand term, symbol " ' " stands for the derivation of the variables with respect to i j ) correspond to the small translations along (resp. rotations about) x, y and z axes of the extremity of the leg i due to the variation of the parameter i j  .

Figure 6 .

 6 Figure 6. Attachment of the bars of the parallelogram onto the platform.

Table 1 .

 1 Values of the secondary geometric parameters of the UraneSX, for bars of the parallelograms made of aluminum and for max

			Bar R = 7.5 cm.
	p max (mm)	k 1	k 2	m (kg)	S (cm²) Bar	d (m)	R (m) Tool
			250	168 17,93 0,5 0,29
		25	500 750	185 19,29 0,42 0,24 197 20,76 0,36 0,21
			1000 208 21,37 0,34 0,20
			250	202 19,73 0,4 0,23
	0.01	100	500 750	214 20,76 0,36 0,21 223 22,07 0,32 0,18
			1000 232 22,88 0,3 0,17
			250	238 21,37 0,34 0,20
		200	500 750	247 22,88 0,3 0,17 254 22,88 0,3 0,17
			1000 261 23,80 0,28 0,16
			250	49	4,75 0,28 0,16
		25	500 750	55 59	5,21 0,24 0,14 5,51 0,22 0,13
			1000	62	5,87	0,2 0,12
			250	67	5,87	0,2 0,12
	0.05	100	500 750	71 73	5,87 6,30 0,18 0,10 0,2 0,12
			1000	76	6,84 0,16 0,09
			250	86	6,84 0,16 0,09
		200	500 750	88 89	6,84 0,16 0,09 7,51 0,14 0,08
			1000	91	7,51 0,14 0,08
			250	30	2,75 0,22 0,13
		25	500 750	34 36	3,15 0,18 0,10 3,42 0,16 0,09
			1000	38	3,42 0,16 0,09
			250	44	3,42 0,16 0,09
	0.1	100	500 750	45 47	3,75 0,14 0,08 4,17 0,12 0,07
			1000	48	4,17 0,12 0,07
			250	56	4,70	0,1 0,06
		200	500 750	57 58	4,70 4,70	0,1 0,06 0,1 0,06
			1000	59	4,70	0,1 0,06
			250	10	0,94	0,1 0,06
		25	500 750	11 12	0,94 0,94	0,1 0,06 0,1 0,06
			1000	13	0,94	0,1 0,06
			250	18	0,94	0,1 0,06
	0.5	100	500 750	19 20	0,94 0,94	0,1 0,06 0,1 0,06
			1000	20	0,94	0,1 0,06
			250	28	0,94	0,1 0,06
		200	500 750	29 30	0,94 0,94	0,1 0,06 0,1 0,06
			1000	30	0,94	0,1 0,06

Table 2 .

 2 Values of the secondary geometric parameters of the Y-STAR, for bars of the parallelograms made of aluminum and for max

			R = 7.5 cm.
			Bar
	p max (mm)	k 1	k 2	m (kg)	S (cm²) Bar	d (m)	R (m) Tool
			250	163 40,22 0,38 0,22
		25	500 750	173 42,06 0,32 0,18 180 43,63 0,28 0,16
			1000 185 45,57 0,24 0,14
			250	188 43,63 0,28 0,16
	0.01	100	500 750	194 45,57 0,24 0,14 198 46,71 0,22 0,13
			1000 202 47,97 0,2 0,12
			250	212 47,97 0,2 0,12
		200	500 750	215 49,36 0,18 0,10 217 50,86 0,16 0,09
			1000 218 55,45 0,1 0,06
			250	40	10,53 0,1 0,06
		25	500 750	41 42	10,53 0,1 0,06 10,53 0,1 0,06
			1000	43	10,53 0,1 0,06
			250	48	10,53 0,1 0,06
	0.05	100	500 750	49 49	10,53 0,1 0,06 10,53 0,1 0,06
			1000	50	10,53 0,1 0,06
			250	58	10,53 0,1 0,06
		200	500 750	59 59	10,53 0,1 0,06 10,53 0,1 0,06
			1000	60	10,53 0,1 0,06
			250	22	5,23	0,1 0,06
		25	500 750	23 23	5,23 5,23	0,1 0,06 0,1 0,06
			1000	24	5,23	0,1 0,06
			250	29	5,23	0,1 0,06
	0.1	100	500 750	30 31	5,23 5,23	0,1 0,06 0,1 0,06
			1000	32	5,23	0,1 0,06
			250	39	5,23	0,1 0,06
		200	500 750	40 41	5,23 5,23	0,1 0,06 0,1 0,06
			1000	42	5,23	0,1 0,06
			250	7	1,04	0,1 0,06
		25	500 750	8 9	1,04 1,04	0,1 0,06 0,1 0,06
			1000	9	1,04	0,1 0,06
			250	14	1,04	0,1 0,06
	0.5	100	500 750	15 16	1,04 1,04	0,1 0,06 0,1 0,06
			1000	17	1,04	0,1 0,06
			250	24	1,04	0,1 0,06
		200	500 750	25 26	1,04 1,04	0,1 0,06 0,1 0,06
			1000	27	1,04	0,1 0,06

Table 3 .

 3 Values of the optimal secondary geometric parameters for the UraneSX and Y-STAR robots.

		UraneSX	Y-STAR
	S Bar (cm²)	3.42	5.23
	d (m)	0.16	0.1
	R Tool (m)	0.09	0.06
	m (kg)	44	29

On this picture, P stands for an actuated prismatic joint, and U for a passive universal joint.


S Briot

A Pashkevich
email: anatol.pashkevich@emn.fr

D Chablat
email: damien.chablat@irccyn.ec-nantes.fr

Reduced Elastodynamic Modelling of Parallel Robots for the Computation of their Natural Frequencies

Keywords: elastodynamic modelling, model dimension reduction, natural frequencies, parallel robots

In this paper is presented an approach for computing the first natural frequencies of parallel robots. This approach, based on the Rayleigh-Ritz approximation, aims at reducing the model dimension by decreasing the size of the matrices describing the robot mass and stiffness, saving computational time and avoiding inaccuracy due to ill-conditioning. Thus it can be used in the design procedure as well as in real-time control. Simulations are carried out on a PRRRP robot modelled in 3D and comparisons with FEA software are presented.

I Introduction

Parallel robots have increasingly been used in industry in the last few years, mainly for pick-and-place applications or high-speed machining [START_REF] Brogardh | Present and future robot control development -an industrial perspective[END_REF], [START_REF] Chanal | A study of the impact of machine tool structure on machining processes[END_REF]. This interest is due to their main properties, i.e. their higher rigidity and dynamic capacities compared with their serial manipulator counterparts.

Having a good knowledge of the elastodynamic behaviour of a manipulator (especially its natural frequencies) is a crucial point. In this sense, accurate elastodynamic models are necessary at both the control stage [START_REF] Singer | Residual vibration reduction in computer controlled machines[END_REF]- [START_REF] Pelaez | Input shaping reference commands for trajectory following Cartesian machines[END_REF] and design stage [START_REF] Voglewede | Overatching framework for measuring the closeness to singularities of parallel manipulators[END_REF]- [START_REF] Briot | On the optimal design of parallel robots taking into account their deformations and natural frequencies[END_REF], in order to optimize the geometry, as well as the shape of the elements of the manipulator. This will lead to the creation of a mechanism in which vibrations will be minimized.

Several models have been proposed and used in the literature in order to compute the natural frequencies of a mechanism during a preliminary design. Three main general methods can be distinguished:  Finite element analysis (FEA); the FEA method is proved to be the most accurate and reliable, since the links/joints are modeled with its true dimension and shape [START_REF] Bouzgarrou | Rigidity analysis of T3R1 parallel robot uncoupled kinematics[END_REF], [START_REF] Shabana | Dynamics of multibody systems[END_REF]- [START_REF] El-Khasawneh | Computation of stiffness and stiffness bounds for parallel link manipulators[END_REF]. Its accuracy is limited by the discretization step only. However, because of high computational expenses required for the repeated remeshing, this method is usually applied at the final design stage for the verification and component dimensioning.  Matrix structural analysis (MSA) method is a common technique in mechanical engineering [START_REF] Deblaise | Systematic analytical method for PKM stiffness matrix calculation[END_REF], [START_REF] Imbert | Analyse des structures par éléments finis[END_REF]; it incorporates the main ideas of the FEA but operates with rather large flexible elements (beams, arcs, cables, etc.). This obviously yields reduction of the computational expense and, in some cases, allows analytical stiffness matrix to be obtained. However, this method can only be applied to links with simple shapes and requires improved skills in FEA.  Virtual joint methods (VJM) [START_REF] Briot | On the optimal design of parallel robots taking into account their deformations and natural frequencies[END_REF], [START_REF] Wittbrodt | Dynamics of flexible multibody systems[END_REF], which is also referred to as ''lumped modelling", is based on the expansion of the traditional rigid model by adding virtual joints (localized springs), which describe the elastic deformations of the manipulator components (links, joints and actuators). Such kind of modelling is well adapted for links with complex shapes [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF]. Generally, lumped modelling is simpler to use than MSA and provides acceptable accuracy in reduced computational time. It is widely used at the pre-design stage, especially for the analytical parametric analysis. However, due to the large number of virtual joints that have to be modelled in order to obtain good accuracy, such an approach is time-consuming. Moreover, for closed-loop mechanisms with passive joints, the way to assemble the matrices and to model the passive joints in kinematic chains is not straightforward. The purpose of this paper is to propose a reduced elastodynamic modelling approach on parallel robots for the computation of the natural frequencies only, based on VJM. By keeping the simplicity of use of the VJM, this modelling considerably decreases the dimension of the problem by inverting only 6×6 matrices. It also proposes a straightforward procedure for the modelling of passive joints on parallel robots and for the assembly of the mass and stiffness matrices of closed-loop structures. This paper will be organised as follows. Section 2 presents the theoretical background necessary in order to reduce the size of the elastodynamic model for the computation of the natural frequencies. In section 3, a numerical example is proposed in order to show the efficiency of our method. Finally, conclusions are drawn in section 4.

II. Theoretical background A. Problem statement

Let us consider a general parallel manipulator made of n legs, each leg being composed of m links, m passive and one active joints, considered as ideal, i.e. without friction (Fig. 1a). Using the basic idea of VJM [START_REF] Wittbrodt | Dynamics of flexible multibody systems[END_REF], let us decompose the j-th link of the i-th leg (denoted as the link ij where j = 1, 2, …, m and i = 1, 2, …, n) into p ij rigid elements and virtual springs (Fig. 1b). The mass and inertia matrix of the k-th rigid element of link ij (denoted as the element ijk) expressed in the local frame attached to this element, are denoted as m ijk and I ijk , respectively. The displacement of its centre of mass S ijk expressed in the base frame, resulting from the robot deformations, is denoted as

1 2 6 [ , ,..., ] T ijk ijk ijk ijk q q q  0 0 0 0 q
, where the first three components of this vector correspond to the translational displacements along the x, y and z axes of the base frame, respectively, and the last three components to the rotational displacements along the same axes. It should be mentioned that in the remainder of the paper, the left superscript "0" will stand for the coordinates expressed in the global frame. If no left superscript is mentioned, the vector is expressed in the local frame attached to the link ij.

The deformations of the k-th spring of link ij (denoted as the spring ijk) are denoted as

1 2 6 [ , ,..., ] T ijk ijk ijk ijk     θ
, where the three first components of this vector correspond to the translational deformations along the x, y and z axes of the base frame, respectively, and the last three components to the rotational deformations along the same axes. The kinetic energy of the system is equal to:  0 0 0 q q q q     is the vector regrouping all vectors ijk 0 q  (i.e. the velocity of the centre of masses S ijk ) and M tot is the global mass matrix of the system.

1 1 1 1 1 2 2 ij p n m T T ijk ijk ijk tot i j k T       0 0 0 q M q q M q     (1)
The potential (elastic) energy of the system is equal to:

1 1 1 1 2 ij p n m T e ijk ijk ijk i j k V      θ K θ (2)
where K ijk is the stiffness matrix of the virtual spring ijk.

It is shown in [START_REF] Briot | On the optimal design of parallel robots taking into account their deformations and natural frequencies[END_REF], [START_REF] Wittbrodt | Dynamics of flexible multibody systems[END_REF] that the deformations ijk θ can be related to the displacements of the centres of masses of elements ijk and ij(k+1) by the relation:

2, 1, ( 1) ( 1)
ijk ijk ijk ij k ij k                0 0 q θ C C q (3)
where C 2,ijk and C 1,ij(k+1) are two 6 by 6 matrices. Therefore, (2) can be rewritten as: q q q q is the vector regrouping all vectors ijk 0 q and K tot is the global stiffness matrix of the system.

2, 2, ( 1) ( 1) 1, 1 1, 1 , , 1 1 2 2 T T T T ijk ijk k k T e i j k t o t T T T ij k ij k k k i j k V                                        0 0 0 0 q q C C K q K q q q C C (4)
Starting from these definitions, and considering that the robot is in an equilibrium position and no external forces are applied, the authors of [START_REF] Wittbrodt | Dynamics of flexible multibody systems[END_REF] demonstrate that the system is governed by the relation:

tot tot   M q K q 0  . ( 5
) A solution q l of this equation can be found by solving the system:

  2 l t o t t o t l    M K q 0, with 2 l l f    ( 6
)
where q l represents the vectors of the shape of free vibrations of the system for the l-th natural mode, f l and  l are their corresponding natural frequency and pulsation, respectively. If the matrix

2 l t o t t o t   M K
is singular (which is always the case when  l is one of the modal pulsation of the robot), q l becomes non-null and is the eigenvector corresponding to the pulsation  l . There is an infinity of vectors q l validating (6) (for a given  l ), but all are proportional to the others. Vector q l is not necessarily dependent of time, but almost represents the amplitude of the vibrations. Therefore, when only the l-th mode of the system is excited, the displacements of all springs may be written under the form: * sin( ) where  l is a phase difference corresponding to the mode l. When all modes are excited, the displacement q of all spring centres may be written in the form:

l l l l t     q q (7)
6 6 * 1 1
sin( )

d d l l l l l l t          q q q (8)
The most common way to find the values of the pulsation  l is to solve the eigenvalue problem

  2 1 det 0 l t o t t o t     I M K . ( 9
)
where I is an identity matrix of dimension 6 d,

with d = , ij i j n m p  .
However, solving this problem involves finding the 6 d eigenvalues of the problem and also inverting the d matrices ijk M constituting the matrix M tot , which is highly time-consuming and can lead to few accurate results because of the problem dimension.

In order to avoid such kind of drawbacks [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF] has recently proposed (for elastostatic modelling only) a new procedure to compute the deformations of the robot when a force is applied at the end-effector. This procedure computes a stiffness (and also compliance) matrix K r (and S r , resp.) of dimension 6 that represents the behaviour of the robot in terms of deformations. Moreover, during the procedure, only inversions of 6 dimensional matrices are involved, which considerably reduces computational time and avoids accuracy problems due to ill-conditioning of the large global stiffness matrix. Thus, the global 6 d dimensional problem defined with respect to all variables q has been reduced to a 6 dimensional problem defined with respect to platform deflections t only. As a result, the entire robot can locally be seen as a virtual spring of dimension 6 that deforms when applying a wrench on the endeffector.

Starting from these considerations, it would be interesting to reduce the dimension of the problem by expressing Eq. ( 6) as a function of the reduced stiffness matrix K r , of the platform deflections t and, also, as a function of a reduced mass matrix that will be denoted as M r , which could represent the global behaviour of the robot in terms of natural frequencies. The remainder of the paper will focus on this problem.

B. Rayleigh-Ritz approximation

Another way to find the natural pulsation  l of the mode l would be to know the exact amplitude  l of the displacements q l for this mode. For the natural mode l, the potential and kinetic energies of the system are given by:

  * * 2 1 1 sin 2 2 T T e l t o t l l t o t l l l V t      q K q q K q (10a)   * * 2 2 1 1 cos 2 2 T T l tot l l l tot l l l T t       q M q q M q   (10b)
From the principle of energy conservation, it follows that

    max max 0 e V T   , i.e   2 0 T l l t o t t o t l    q M K q (11)
which is an univariate equation in  l .

It is obvious that the exact knowledge of the amplitude q l is an impossible task without a direct measure on the system of all the displacements of the robot nodes. However, this vector may be approximated by another denoted as ˆl q that is close to the exact amplitude q l . Introducing this approximated vector ˆl q into (10) will allow us to find a corresponding value of ˆl  and, as a result, ˆl f which is the approximated natural frequency of the system. Such kind of elastodynamic problem resolution is called the Rayleigh-Ritz method [START_REF] Birkhoff | Rayleigh-Ritz approximation by piecewise cubic polynomials[END_REF].

The better the approximation, the more accurate the value of ˆl  . The designer's skills in terms of understanding and analysis of robot physical behaviour here are of the utmost importance. In this sense, let us recall that the first natural frequencies are associated with the highest level of energy due to vibrations, and represent the highest displacements of the structure. Therefore, in design optimization loops, it is essential to maximize the value of the first frequency.

Using the Rayleigh-Ritz approximation in order to compute the first natural frequencies, the stresses for which the maximal displacements appear have to be found. From our experience in elastic behaviour of robots, it is assumed that a good approximation of these maximal displacements will be the deformation of the robot with a load applied at the end-effector, and it can be shown in the following that this hypothesis is valid. Using this assumption, the displacements q l of all springs can be computed as a function of the end-effector displacements t, i.e. it is possible to define a matrix J q such as:

l  q q J δt . ( 12
)
As a result, introducing ( 12) into (11) will lead to

  2 T T T l t o t t o t    q q q q δt J M J J K J δt 0 ( 13
)
where the matrices T tot q q J M J and T tot q q J K J are now of dimension 6. It should be mentioned here that, in the case where an external load is applied at the end-effector only, the term T tot q q J K J is equal to the reduced stiffness matrix K r of the robot. Therefore, (13) can be rewritten as:

  2 T l    r r δt M K δt 0 , with T tot  r q q M J M J (14)
In the following sections it is explained how to obtain expressions [START_REF] El-Khasawneh | Computation of stiffness and stiffness bounds for parallel link manipulators[END_REF] and [START_REF] Imbert | Analyse des structures par éléments finis[END_REF].

C. Reduction of the link mass matrix

It is possible to decompose the previously cited task into two sub-problems. First, the displacements of each spring in the beam ij can be expressed as a function of the displacement of its extremities. Then, one can express the beam extremity displacements as a function of the platform displacements t. Using this approach will allow for a reduction in the size of the link mass matrices, and thus avoiding creating global mass matrices M tot with very large dimension.

Two main ways can be followed to reduce the size of the link mass matrices. The first one consists in discretizing the link ij into p ij rigid links and springs and to express their displacements as a function of the beam extremity displacements. However, such numerical method must be repeated for each link and, thus, increases the size of the algorithm and decreases its efficiency. As a result, it is prefered to use the following procedure which allows analytical expressions to be obtained for the reduced link mass matrices.

Let us consider the link ij, modelled as a beam (Fig. 2). At this beam is attached a local frame represented by the vectors x ij , y ij and z ij . Before any deformation of the system, the beam ij is linked (rigidly or by a passive joint) to beams i(j-1) and i(j+1) at points O ij and O i(j+1) , respectively (Fig. 2). After deformation of the robot, the beam extremity located at O ij is displaced from

1 2 6 1 1 1 [ , ,..., ] T ij ij ij q q q  1 ij q
and the one located at O i(j+1) is displaced from

1 2 6 2 2 2 [ , ,..., ] T ij ij ij q q q  2 ij q
, where the three first components of each vector correspond to the translational displacements along local x ij , y ij and z ij axes, respectively, and the three last components to the rotational displacements along the same axes.

The general formula for the kinetic energy of an elastic Bernoulli beam is equal to [START_REF] Imbert | Analyse des structures par éléments finis[END_REF]:

0 1 2 ij L T ij ij T d x    ij ij ij q Q q   (15)
with diag( , , , , , )

p y z ij ij ij ij ij ij A A A I I I  ij Q
. In this expression, ij q  represents the velocity of the beam cross-section located at position x from the local reference frame (Fig. 2), L ij is the length of the beam ij,  ij the mass density at cross-section x, A ij its area, p ij I its torsional constant and

y ij I , z ij
I the quadratic momentums along y ij and z ij , respectively.

For the l-th natural mode, and from ( 7), the kinetic energy can be rewritten as:

  2 2 0 1 cos 2 ij L T ij l l l ij T t d x        ij ij ij q Q q (16)
q ij being the amplitude of the displacement of the beam cross-section located at position x from the local reference frame (Fig. 2).

In the Rayleigh-Ritz approximation, considering that the deformations due to the natural vibrations are similar to those obtained when an external load is applied at the robot end-effector only, each link of the structure will deform due to the stresses transmitted through the robot joints at points O ij . As a result, the deformations  ij of the beam cross-section (Fig. 2) can be approximated by the deformations of a tip-loaded beam, given by [START_REF] Imbert | Analyse des structures par éléments finis[END_REF]: diag( , , , , , )

ij ij ij ij ij ij f g g f h h  2 ij ij δ δ (17)
where ( )

ij x L   2 ij ij δ δ
represents the deformation of the beam at its tip and ( ) /

ij ij f x x L  , (18a) 2 3
( ) 0.5 (3 )/ ij ij ij g x x L x L   , (18b) 2 ( ) ( 0.5 )
/ ij ij ij h x x L x L   . ( 18c
)
As a result, the global displacement q ij of the beam cross-section at x can be expressed as a sum of two terms:

( ) x         3 1 ij ij ij 3 3 I D q q δ 0 I where ( ) 0 0 0 0 0 0 0 x x x             D (19)
In this sum, the left terms corresponds to the displacement of the undeformed beam due to the displacement of the node located at O ij .

In the hypothesis that the beam cross-section is constant, introducing [START_REF] Birkhoff | Rayleigh-Ritz approximation by piecewise cubic polynomials[END_REF] and ( 19) into [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF], and integrating, leads to:

      2 2
1 cos 2

T T ij l l l T t                            1 ij 1 2 ij ij ij 2 ij q q δ M δ (20)
where

  T          11 12 ij ij ij 12 22 ij ij M M M M M with 2 2 0 0 0 0 0 0 0 0 0 / 2 0 0 0 /2 0 0 0 0 0 0 0 0 /2 0 /3 0 0 /2 0 0 0 /3 ij ij ij ij ij ij ij ij p ij ij ij ij y ij ij ij ij ij ij ij ij z ij ij ij ij ij ij ij m m m L m m L I L m L m L I L m L m L I L                             11 ij M (21a) / 2 0 0 0 0 0 0 3 /8 0 0 0 0 0 0 3 / 8 0 0 0 0 0 0 / 2 0 0 0 0 1 1 / 4 0 0 2 / 3 0 0 11 /40 0 0 0 2 /3 ij ij ij ij p ij ij ij ij y ij ij ij ij ij ij z ij ij ij ij ij m m m I L m L I L m L I L                          12 ij M (
p ij ij ij ij ij ij y z ij ij ij ij ij ij m m m I L I L I L             22 ij M (21c)
It should be mentioned that, in the case where the section of the beam is not constant, ( 18) to (21) are no more correct. In this case, the deformation of the beam and its kinetic energy should be obtained numerically.

From [START_REF] Birkhoff | Rayleigh-Ritz approximation by piecewise cubic polynomials[END_REF], it can be shown that:

( ) ij L           3 2 2 1 ij ij ij 3 3 I D δ q q 0 I . ( 22
)
which leads to:

                                1 1 1 6 6 ij ij ij 21 ij 2 2 2 ij 6 ij ij ij I 0 q q q E E I δ q q , ( ) ij L           3 21 ij 3 3 I D E 0 I (23)
Introducing ( 23) into (20) leads to:

  2 2
1 cos 2

T T ij l l l T t                              1 1 ij ij ij ij ij 2 2 ij ij q q E M E q q (24)
Expressing vectors v ij q (v = 1, 2) in the base frame, (24) becomes:

  2 2 1 cos 2 T red ij l l l T t                              0 1 0 1 ij ij 0 ij 0 2 0 2 ij ij q q M q q ( 25
)
where

  0 0 T red T ij ij  0 ij ij ij ij M R E M E R and R 0ij is the rotation matrix from the local frame of beam ij to the global frame, i.e. 0ij                  0 1 1 ij ij 0 2 2 ij ij q q R q q . ( 26
)
To show the validity of the approach, let us compare the obtained equations when they are applied on a single beam fixed at one extremity O 11 , i.e.  0 1 11 q 0. For the numerical example, it is considered that this beam has the following characteristics: length L = 1 m -hollow circular cross-section of external diameter 40 mm and internal diameter 30 mm material: steel (Young modulus E = 204 GPa, Poisson coefficient  = 0.3, density  = 8020 kg/m 3 ) For such an element, beam theory gives the following expressions for the natural frequencies [START_REF] Birkhoff | Rayleigh-Ritz approximation by piecewise cubic polynomials[END_REF]: -1 st natural frequency due to transverse displacements:

   ( 27a
)
where m is the mass of the beam and I z its quadratic momentum along z axis.

-1 st natural frequency due to longitudinal displacements: / (4 ) 1260.86 Hz

l f E L    (27b)
-1 st natural frequency due to torsional displacements: /( ) (4 ) 781.95 Hz

p y z to f I G I I L    ( 27c
)
where G is the shear modulus, I y its quadratic momentum along y axis and I p its torsional constant.

Applying the previously developed equations to this case by considering that the reduced mass matrix M r = 22 11

M is given at (21c) and the stiffness matrix K r is expressed in [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF] as:

                       r K , (28)
it is found: -1 st natural frequency due to transversal displacements: 35.78 Hz (error of 1.4%) -1 st natural frequency due to longitudinal displacements: 1390.30 Hz (error of 10.27%) -1 st natural frequency due to torsional displacements: 862.23 Hz (error of 10.27%) As a conclusion, the proposed reduced model is accurate enough for the computation of the first natural frequency due to transversal displacements, which is also the first natural frequency of the beam.

Let us now reduce the size of the total mass matrix of the robot.

D. Reduction of the robot mass matrix

Using the results of the previous section, the total kinetic energy of the system is given by:

    2 2 , , 2 2
1 cos 2

1 cos 2 T red ij l l l i j i j T l l l t o t T T t t                                      0 1 0 1 ij ij 0 ij 0 2 0 2 ij ij q q M q q q M q (29) with ( , ,...,
)

red red red tot diag  0 0 0 11 12 nm M M M M and             , , , ,..., ,
T T T T T T T        0 1 0 2 0 1 0 2 0 1 0 2 11 11 12
12 nm nm q q q q q q q . It is necessary to express the relationship linking all vectors v 0 ij q (v = 1, 2) to the end-effector displacements t. From [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF], this displacement is equal to:

i i i i   θ p δt J θ J p , (30)
where  i represents the deformations of all virtual springs of the leg i and p i the displacements of its passive joints, and J i and J pi are Jacobian matrices relying these displacements to the displacement t. These matrices can be obtained by the differentiation of the global transformation matrix T i of the chain i including rigid, passive and elastic coordinates [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF]. The way to obtain these matrices is explained in more depth in the next section. It should be mentioned that the dimension of J pi is 6 by m, m being the number of passive joints of the leg, with m < 6 (because one leg of a parallel robot cannot have more than five passive degrees of freedom (DOF)).

From [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF], it can also be shown that: where i θ S is a block-diagonal matrix regrouping all the compliance matrices of each virtual springs, and f i is the force applied on the i that involves the displacement t. The expression of f i is such as

T i i i i i i   θ θ θ p δt J S J f J p , (31)
i i  f K δt
(32) K i being the global stiffness matrix of the leg i. Introducing (32) into (31) leads to:

  T i i i i i i   θ θ θ p I J S J K δt J p . (33)
Therefore, the displacements p i of the robot passive joints due to the displacement t are equal to:

    1 T T T i i i i i i i i i     p p p θ θ θ tp p J J J I J S J K δt J δt . ( 34
)
Let us now compute the displacements 02 ij q of the virtual spring located at the extremity of the link ij. This spring ij is preceded by j-1 other virtual springs and passive joints. The deformations of the j springs (including the one at the link extremity) will be denoted as  ij , and the displacements of the j-1 passive joints p ij , respectively. They can be related to the displacement 0 2 ij q by the Jacobian matrices J ij and J pij as

ij ij ij ij   0 2 ij θ p q J θ J p , (35)
These matrices may be computed in a similar way to the matrices J i and J pi , but by considering the truncated chain i going from the base to the virtual spring ij. The deformations  ij involve constraints into the j virtual springs of the truncated chain that are denoted as  ij . Their expressions may be found from the relationship [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF]:

ij ij ij  θ S τ θ , (36)
where S ij is a block-diagonal matrix regrouping all the compliance matrices of the j virtual springs of the truncated chain. Introducing (34) into (33) leads to:

ij ij ij ij ij   0 2 ij θ θ p q J S τ J p . ( 37
)
Let us consider now the entire leg i of the mechanism. The constraints  i applied to the springs due to the force f i applied on the platform are given by [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF],

T ij j i T i i i i T ir j i                  θ θ θ τ J τ J f f τ J , (38)
where J ji corresponds to the j first columns of matrix J i . Introducing (38) into (37),

T T ij ij j i i ij ij ij ij j i i ij ij
Moreover, the passive displacements p ij are known from expression (32). Considering that J jtpi represents the matrix composed of the j first lines of J tpi given at (32), it follows that

  T ij ij j i i ij j ij    2 q 0 2 ij θ θ θ p t p i j q J S J K J J δt J δt . ( 40
)
Once all 0 2 ij q are found, the vectors 0 1 ij q can be computed. Indeed, displacement 0 1 ij q is nothing none than displacement ( 1)  0 2 i j q plus the small deflection of the passive joint located at O ij , i.e.:

( 1)

ij p     0 1 0 2 0 ij i j ij q q s (41)
where p ij is the value of the displacement of the passive joint located at O ij and 0 s ij is a unit screw corresponding to its twist. Introducing (34) and ( 40) into (41):

  * * j i j i      2 2 1 q q q 0 1 0 0 ij ij ij tp ij ij tp ij q J δt s J δt J s J δt J δt (42)
where * j i tp J is the j-th line of matrix J tpi given at (32).

Thus, from (40) and (42), one can express vector q given at (29) as a function t as follows:

            , , , ,. . . , , T T T T T T T nm nm         1 2 1 2 1 2
q q q q q q 11 11 12 12 q q J J J J J J δt

J δt (43)
Introducing this relation into (29) will lead to:

  2 2 1 cos 2 T l l l T t      r δt M δt ( 44
)
where the expression of M r is given at [START_REF] Imbert | Analyse des structures par éléments finis[END_REF]. As a result, from ( 14), finding the robot natural frequencies relies on solving the 6 dimensional eigenvalues problem

  2 1 det 0 l     6 r r I M K . (45)
Let us now apply this method using an example.

III. Case study

In this section, the example of the planar PRRRP 2 manipulator with two parallel prismatic axes (Fig. 3) will be treated. Let us first compute the global 3D stiffness matrix K r of the robot. The method presented in [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF] states that each leg may be decomposed into a sequence of rigid links and virtual 6-DOF springs, which includes: (f) a rigid leg linking the foot and the end-effector;

(g) a 6-DOF virtual spring describing the leg stiffness;

(h) a 1-DOF passive R-joint at the end of the leg 2 (not for leg 1) allowing one rotation with angle 22  ; (i) a rigid link from the manipulator leg to the endeffector (part of the movable platform). From these assumptions, the matrix regrouping the stiffness matrices of all segments of the leg i has the following form [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF]:

Foot elt Leg i i i            6 6 6 6 K 0 K 0 K . ( 46
)
and by extracting the 6 by 6 sub-matrix with indices corresponding to i θ S . After the stiffness matrices K i for all kinematic chains are computed, the stiffness of the entire manipulator can be found by simple addition:

1 n i i    r K K . ( 51
)
Using expressions (46) to (51), all relations of sections II.C and II.D can be computed. Let us now compare the results obtained by the presented procedure with a more general method.

For this simulation, each link is modelled as a beam of constant cross-section. The stiffness matrix expressions for beams are given in [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF]. Then, the following numerical parameters are used: -length of segments: AB = AC = 0. Our model will be compared with one created using the FEA software Castem [19]. In the FEA model, all links are modelled using 3D beam finite elements (10 elements by beam). As for one fixed position along x axis of the robot, the configuration is invariant for any position along y axis, only the position along x axis will vary. The results are presented in Table 1.

In this table, it is shown that, for our model, the value of the first natural frequency is obtained with at most 4% error. This is almost sufficient for using such algorithm in 13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 A12-358 8 design procedure. Moreover, this percentage decreases to 0.9% for the second natural frequency. However, the value of the third frequency has quite a large percentage of uncertainty. Indeed, this is due to the fact that the displacements due to the two first frequencies are very close from the displacement of the robot when a load is applied at the end effector, which is not the case for the third frequency.

These results confirm the validity of our hypotheses and of our approach. It should be also mentioned that the Castem software computes the results in 0.09s while our approach takes 0.006 s, i.e. our procedure runs 15 times faster (for a Pentium 2.53 GHz, 4 Gb of RAM). Even if both computation times are very small in this example, in design optimization loops, for which thousands of possible configurations are tested, this saving of time is of the utmost importance.

IV. Conclusions

This paper has presented an approach for computing the first natural frequencies of parallel robots. The approach, based on the Rayleigh-Ritz approximation, aims at reducing the model dimension by decreasing the size of the matrices describing the robot mass and stiffness. This saves computational time, avoiding inaccuracy due to illconditioning and thus can be used in design procedure as well as in real-time control. Simulations have been carried out on a PRRRP robot modelled in 3D and comparisons have been achieved with FEA software. The obtained results confirm the validity of the proposed procedure and shows that the algorithm runs 15 times faster than a classical FEA method.

13th

  World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 A12-358 2

  Fig. 1. A general parallel manipulator.

  13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 A12-358 4

Fig. 2 .

 2 Fig. 2. Displacements and elastic deformations of a beam.

13th

  World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 A12-358 6

  (a) a rigid link between the manipulator base and the ith actuating joint (part of the base platform); (b) a 1-DOF actuating joint allowing one translation 0 i  ; (c) a rigid foot; (d) a 6-DOF virtual spring describing the foot stiffness; (e) a 1-DOF passive R-joint at the beginning of the leg allowing one rotation with angle 1 i  ;

Fig. 3 .

 3 Fig. 3. Description of the PRRRP manipulator.

  5 m, a = 0.33 m, d = 0.1 m; -Young modulus: E foot = E Leg = 210 GPa; Poisson coefficients:  foot =  Leg = 0.3; -Each beam has the same cross-section represented by a circle of radius 0.05 m; -Material density:  foot =  Leg = 7800 kg/m 3 .

TABLE 1 .

 1 Values of the three first natural frequencies of the PRRRP robot under study computed with two different methods: (i) using a FEA model and (ii) using the proposed method.

	Position	First natural frequency (Hz)	Second natural frequency (Hz)	Third natural frequency (Hz)
	along x axis	FEA	Our model % error	FEA	Our model % error	FEA	Our model % error
	x = 0 m	204.65	210.48	2.85	555.51	556.58	0.19	787.09	903.57	14.80
	x = 0.05 m	204.85	210.71	2.86	558.10	558.64	0.10	786.97	906.63	15.21
	x = 0.1 m	205.46	211.44	2.91	565.98	566.85	0.15	786.60	910.46	15.75
	x = 0.15 m	206.53	212.75	3.01	579.52	581.39	0.32	785.98	916.76	16.64
	x = 0.2 m	208.19	214.80	3.17	599.36	602.32	0.49	785.12	929.35	18.37
	x = 0.25 m	210.75	218.02	3.45	626.64	628.86	0.35	784.06	956.93	22.05
	x = 0.3 m	215.25	223.88	4.01	664.27	658.27	0.90	782.99	1023.64	30.73

In the paper, P stands for an actuated prismatic pair and R for a passive revolute joint.

Acknowledgements

This work has been partially funded by the French region Pays de la Loire.

A12-358 7

For obtaining matrices i p J and i θ J of (31), let us now consider the corresponding mathematical expression defining the end-effector location subject to variations of all the above defined coordinates of a single kinematic chain i may be written as follows:

where the matrix function ( ) a  V is an elementary translation along y, the matrix functions ( ) rj  V (j = 1, 2) are elementary rotations around z, the spring matrix ( ) s  V is composed of six elementary transformations defined by the coordinates 1 6

( ,..., )

and Leg i T represents the rigid displacements of the base, foot and leg, respectively [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF] (Fig. 3).

The matrix i θ J may be obtained from the derivation of the matrix T i with respect to the spring parameters v ij

where the first and the third multipliers are the constant homogenous matrices which do not include the displacement v ij  , and the second multiplier corresponds to the derivative of the elementary translation or rotation corresponding to v ij  . In the right-hand term, symbol " ' " denotes the first derivative of the variables with respect to v ij  . Therefore, ix p' , iy p' and iz p' (resp. ix '  , iy '  and iz '  ) correspond to the small translations along (resp.

rotations about) x, y and z axes of the extremity of the leg i due to the variation of the parameter v ij  . The Jacobians i p J can be computed in a similar manner, but the derivatives are evaluated in the neighborhood of the "nominal" values of the passive joint coordinates ij  corresponding to the rigid case (these values are obtained from the inverse kinematics).

Finally, the stiffness matrix K i of the leg i, which relates the deformations t to the force f i can be computed by direct inversion of relevant 7 by 7 (8 by 8 in the case of the leg 2) matrix in the left hand side of the following equation [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF]


Sébastien Briot

Vigen Arakelian

Damien Chablat

Philippe Wenger

Optimal Force Generation of 3-RRR Decoupled Planar Robots for Ensuring Unlimited Platform Rotation

This paper deals with the optimal dynamics of the 3-RRR decoupled robot in which the linear displacements of the platform are decoupled from its orientation. The particularity of the 3-RRR decoupled robot is that an optimal choice of the leg's geometric parameters allows an unlimited platform rotation, which can be attractive for many industrial applications. However, a singular configuration is necessarily encountered during full platform rotation. In order to generate a stable motion in the presence of singularities, optimal dynamic conditions are disclosed. The suggested optimal conditions ensuring unlimited platform rotation are illustrated and validated by numerical simulations with ADAMS.

Introduction

In the last few decades, parallel manipulators have been the subject of various studies. Many problems devoted to their singularity, kinematics and dynamics properties were disclosed. Parallel manipulators are used in more and more applications nowadays. Consequently, new research problems are continuously appearing. One of the structural disadvantages of parallel manipulators is the low platform rotation capability due to the multi-branch architecture. In the last years this problem has been studied by several researchers. In [START_REF] Lessard | A new family of spatial 3-dof fullyparallel manipulators with high rotational capability[END_REF] a new family of 3-DOF parallel manipulators with high rotational capability was proposed. The platform orientation range was increased by non symmetric placement of three non-identical legs. In [START_REF] Sacludean | 3-dof parallel manipulator[END_REF][START_REF] Company | Schoenflies motion generator: A new non redundant parallel manipulator with unlimited rotation capability[END_REF]Nabat et al., 2006b) new architectures of 3-DOF parallel manipulators with unlimited rotation of the end-effectors were developed. They consist of a pair of connected five-bar mechanisms. A similar approach was applied in a parallel robot for medical 3D-ultrasound imaging [START_REF] Lessard | A new family of spatial 3-dof fullyparallel manipulators with high rotational capability[END_REF][START_REF] Lessard | A new medical parallel robot and its static balancing optimization[END_REF]. In the parallel manipulator H4, a complementary gear transmission [START_REF] Kokikabushiki | 4-dof parallel robot[END_REF] together with a belt transmission (Nabat et al., 2006a) were used to amplify the orientation angle of the end-effector. A new family of spatial parallel mechanisms with decoupled and unlimited rotation of the moving platform was proposed in [START_REF] Gogu | Parallel mechanisms with decoupled rotation of the moving platform in planar motion[END_REF]. In the suggested architectures the moving platform performs two independent translations and one independent unlimited rotation whose axis is perpendicular to the plane of translation.

A modified mobile platform design (Fig. 1a) was proposed in a recent study [START_REF] Arakelian | A new 3-dof planar parallel manipulator with unlimited rotation capability[END_REF], thus decoupling the position and the orientation of the platform. In the 3-RRR decoupled robot, the loop (O 1 , A 1 , C, A 2 , O 2 ) moves point C and the leg (O 3 , A 3 , B 3 ) adjusts the orientation φ according to the position. If the position of point C is given, this third leg is equivalent to a four-bar linkage, which ensures a full rotation of the platform. For the generation of a full rotation of the platform, however, the four-bar linkage must cross a singularity.

In this paper, the optimal dynamic conditions are found in order that the 3-RRR decoupled robot can cross a singularity during a full rotation motion of its platform.

Description of the 3-RRR Decoupled Robot

Structural features and singularities

Let us consider the architecture of the suggested manipulator (Fig. 1a). As shown in Fig. 1a, axis x 0 is along vector

---→ O 1 O 2 . The lengths of link O i A i (resp. A i B i ) are denoted as L 1i (resp. L 2i
). The length of the platform CB 3 is denoted as R. The positions of the base joints O i along x 0 and y 0 are denoted as (x Oi , y Oi ), with x O1 = y O1 = y O2 = 0.

The aim of such a modification of the 3-RRR robot is to obtain an unlimited platform rotation. Indeed, the four-bar linkage, which generates the rotation of the platform, is equivalent to a rocker-crank mechanism. In this case, it is able to transfer the oscillation of the input link O 3 A 3 to the full rotation of the platform CB 3 . The robot should be designed so that this condition is satisfied in the whole workspace. However, a singularity must be encountered during a full platform rotation motion. This singularity occurs when CB 3 and A 3 B 3 are aligned (Fig. 1b). As a result, the optimal generation of the input link O 3 A 3 for passing through singular configurations must be achieved [START_REF] Briot | Optimal force generation of parallel manipulators for passing through the singular positions[END_REF]. This approach is based on the study of the robot dynamic model.

O O 1 O 2 O 3 A 1 A 2 A 3 B 3 C B 1 B 2 q 12 q 11 q 13 φ y 0 x 0 platform base q 22 q 21 q 23 S
y 0 x 0 O O 1 O 2 O 3 A 1 A 2 A 3 B 3 C B 1 B 2 uncontrollable motion (b) Example of singularity
Figure 1: Schematics of the 3-RRR decoupled parallel robot and its singular configurations encountered during a full platform rotation.

Dynamics of the robot

In this part, the robot inverse dynamic model is computed using the Lagrange equations. To simplify the computation, it is preferable to replace the masses of moving links 2i by concentrated masses [START_REF] Seyferth | Massenersatz duch punktmassen in rumlichen getrieben[END_REF]. For a link ji with mass m ji and its axial moment of inertia I ji , we have:

⎡ ⎣ 1 1 1 r ji 0 r ji -1 r 2 ji L 2 ji 0 (r ji -1) 2 L 2 ji ⎤ ⎦ ⎡ ⎣ m ji1 m ji2 m ji3 ⎤ ⎦ = ⎡ ⎣ m ji 0 I ji ⎤ ⎦ (1)
where:

• r ji is the non-dimensional position of the centre of masses S ji of element ji, i.e. for example for link 11, OS 11 = r 11 L 11 , • m jik (k = 1, 2, 3) are the values of the three point masses placed at the centers of the revolute joints and at the center of masses of the link ji. The robot kinetic energy can be expressed as:

2T = i,j m ji v 2 Sj i + I ji q2 ji + m p ẋ2 + ẏ2 + I p φ2 = 3 i=1 I r1i q2 1i + 2I r2i q1i (-ẋ sin q 1i + ẏ cos q 1i ) + m rp ẋ2 + ẏ2 + I rp φ2 + 2I r3 q 13 φ cos(q 13 -φ) + 2I r4 φ(-ẋ sin φ + ẏ cos φ) (2)
where

I r1i = I 1i + (m 1i r 2 1i + m 2i1 + m 2i2 (1 -r 2i ) 2 )L 2 1i , I r2i = m 2i2 r 2i (1 -r 2i ))L 2 1i , m rp = m p + m 213 + m 223 + m 233 + m 212 r 2 21 + m 222 r 2 22 + m 232 r 2 23 , I rp = I p + (m 233 + m 232 r 2 23 )R 2 , I r3 = m 232 r 23 (1 -r 23 )RL 13 , I r4 = (m p r p + m 233 + m 232 r 2 23
)R, and ẋ, ẏ are the platform translational velocities along x 0 and y 0 axes and φ is the platform rotation velocity, q1i is the joint rate of actuator i, v Sji is the velocity of the centre of masses S ji , qji the rotational velocity of link ji, m p is the platform mass, I p its axial moment of inertia expressed at C and r p the non-dimensional position of the centre of masses S p of the platform, i.e. CS p = r p R.

The gravity field is directed along the z 0 axis. As a result, the dynamic model can be computed as [START_REF] Moon | Applied dynamics[END_REF]:

τ = w b -B T λ, with A T λ = w p (3)
where λ are the Lagrange multipliers, matrices A and B are defined such that At + B q = 0 for t = [ ẋ, ẏ, φ] T and q = [ q11 , q12 , q13 ] T and

w b = d dt ∂T ∂ q - ∂T ∂q , w p = d dt ∂T ∂t - ∂T ∂x , x = [x, y, φ] T (4)
3 Optimal Motion Generation

Condition for crossing the singularities

As disclosed in [START_REF] Briot | Optimal force generation of parallel manipulators for passing through the singular positions[END_REF], the singularities considered in this work can be crossed if and only if:

t T s w p = 0, ( 5
)
where t s is the unit screw describing the gained degree of freedom when the robot encounters a singularity. For the robot at hand, for the 'orientation' singularity (Fig. 1b), it can be proven that t s is defined by t s = [0, 0, 1] T throughout the workspace. Therefore, the condition for crossing this type of singularity remains to cancel the last line of w p , i.e. from ( 2) and ( 4)

I rp φ + I r3 (q 13 cos(q 13 -φ) -q2 13 sin(q 13 -φ)) + I r3 (ÿ cos φ -ẍ sin φ) = 0, (6)
To lower the computation burden, it is preferable to express equation ( 6) as a function of ẋ, ẏ, φ, ẍ, ÿ and φ only. The terms q13 and q13 can be removed using the following kinematic relationships:

q = -B -1 At q = -B -1 (A ṫ + Ȧt + Ḃ q) = -B -1 (A ṫ + ( Ȧ -ḂB -1 A)t) (7)
The values of q13 and q13 can be extracted from the last line of vectors q and q.

Illustrative example

In order to validate the previous approach, an ADAMS model is developed with the following robot parameters extracted from [START_REF] Chablat | Solution regions in the parameter space of a 3-RRR decoupled robot for a prescribed workspace[END_REF]: R = 0.1 m; L 1i = L 2i = 0.33 m; x O2 = 0.1 m; x O3 = 0.05 m; y O3 = 0.6 m; m 1i = m 2i = 1 kg; m p = 0.5 kg; I 1i = I 2i = 0.009 kg.m 2 ; I p = 0.0017 kg.m 2 ; r ji = r p = 0.5.

The robot is located at x(t 0 ) = x in = [-0.1 m, 0.45 m, 90 deg] T for t 0 = 0 s. Its task is to maintain the position of point C while achieving a full rotation of the platform in t f = 1.5 s. Singularities will appear for φ s1 ≈ 269 deg and φ s2 ≈ 328 deg. Two cases will be treated: (i) Case 1: a motion without taking into account the conditions for passing through the singularities, and (ii) Case 2: a motion that considers the conditions for passing through the singularities.

For Case 1, the motion planning is achieved with a 7th degree polynomial law in order to cancel the velocity, acceleration and jerk at the beginning and at the end of the trajectory:

φ(t) = 360s(t)+90, s(t) = 35(t/t f ) 4 -84(t/t f ) 5 +70(t/t f ) 6 -20(t/t f ) 7 (8)
where the following initial and final conditions are respected:

s(t 0 ) = ṡ(t 0 ) = ṡ(t f ) = s(t 0 ) = s(t f ) = ... s (t 0 ) = ... s (t f ) = 0, s(t f ) = 1.
The input torque of actuator 3 is plotted in Fig. 2a (it can be proven that if the platform position is not changed, the torques of actuators 1 and 2 are cancelled). It is apparent that the input torque tends to infinity at each singularity.

For Case 2, the motion is planned using three different polynomials s k . Each part of the trajectory has to be achieved in 0.5s.

s 1 is defined for φ ∈ [90deg, φ s1 ] on the time interval [t 0 , t f 1 = 0.5s], s 2 for φ ∈ [φ s1 , φ s2 ] on the time interval [t f 1 , t f 2 = 1s], and s 3 for φ ∈ [φ s2 , 450deg] on the time interval [t f 2 , t f 3 = 1.5s]: φ(t) = (φ s1 -90)s 1 (t) + 90, for t ∈ [t 0 , t f 1 ] φ(t) = (φ s2 -φ s1 )s 2 (t) + φ s1 , for t ∈ [t f 1 , t f 2 ] φ(t) = (450 -φ s2 )s 3 (t) + φ s2 , for t ∈ [t f 2 , t f 3 ] (9) with s 1 (t) =441.8819t 4 -2074.5386t 5 + 3411.5717t 6 -1932.0441t 7 , s 2 (t) =0.1690t + 0.0159(t -t f 1 ) 2 + 443.8291(t -t f 1 ) 4 -2116.1982(t -t f 1 ) 5 + 3521.1306(t -t f 1 ) 6 -2011.4236(t -t f 1 ) 7 , s 3 (t) =2.4400(t -t f 2 ) + 3.8429(t -t f 2 ) 2 + 15.8838(t -t f 2 ) 4 -316.3353(t -t f 2 ) 5 + 746.8231(t -t f 2 ) 6 -506.5084(t -t f 2 ) 7 ,
where the following initial and final conditions are respected in order to ensure the velocity, acceleration and jerk continuity all along the trajectory: It should be mentioned that the value of s1 (t f 1 ) (s 2 (t f 2 ), resp.) is calculated so that it respects the condition for crossing the singularities expressed at (6), taking into account that ṡ1 (t f 1 ) = 1 ( ṡ2 (t f 2 ) = 1, resp.). The value of φ all along this trajectory is plotted at Fig. 3a. Figure 3b presents the resulting displacement of actuator 3. It can be observed that this actuator has an oscillatory behaviour while the platform is ensuring a full rotation. The input torque of actuator 3 is plotted in Fig. 2b. It appears now that when crossing the singularities, the input torques remain finite.

s 1 (t 0 ) = ṡ1 (t 0 ) = s1 (t 0 ) = ... s 1 (t 0 ) = ... s 1 (t f 1 ) = 0, s 1 (t f 1 ) = ṡ1 (t f 1 ) = 1, s1 (t f 1 ) =0.0318, s 2 (t f 1 ) = ... s 2 (t f 1 ) = ... s 2 (t f 2 ) = 0, s 2 (t f 2 ) = ṡ2 (t f 2 ) = 1, s2 (t f 2 ) = 7.6858, ṡ2 (t f 1 ) = ṡ1 (t f 1 )d 1 , s2 (t f 1 ) = s1 (t f 1 )d 1 , d 1 = (φ s1 -90)/(φ s2 -φ s1 ) s 3 (t f 2 ) = ṡ3 (t f 3 ) = s3 (t f 3 ) = ... s 3 (t f 2 ) = ... s 3 (t f 3 ) = 0, s 3 (t f 3 ) = 1, ṡ3 (t f 2 ) = ṡ2 (t f 2 )d 2 , s3 (t f 2 ) = s2 (t f 2 )d 2 , d 2 = (φ s2 -φ s1 )/(450 -φ s2 ).
It should be noted that in this example, only one turn has been achieved with a fixed position of point C, but it is obvious that the number of platform turns may be increased and that point C can move as soon as the global trajectory respects equation ( 6) at each singular configuration.

Conclusions

It is known that the singularity leads to a loss of robot controllability and it is a serious obstacle for reproduction of stable motions. This drawback can be overcome by means of the optimum force control. In this paper, the optimal force generation of 3-RRR decoupled robots for ensuring unlimited platform rotation has been studied. The particularity of these robots is that an optimal choice of the legs' geometric parameters allows an unlimited platform rotation. However, a singular configuration is necessarily encountered during a full platform rotation. In order to generate a stable motion in the presence of singularities, optimal dynamic conditions were found. To illustrate the fruitfulness of the obtained results two different types of control have been compared: a motion generated via a 7th degree polynomial law without taking into account the conditions for passing through the singularity and a motion that considers the optimal dynamic conditions for passing through the singularity. For the second case, a full rotation of the platform without perturbation of motion has been achieved. The obtained results showed that the generation of stable unlimited platform rotation of 3-RRR decoupled robots is possible. It can be attractive for many industrial

Figure 2 :

 2 Figure 2: Torque of actuator 3.

Figure 3 :

 3 Figure 3: The trajectory defined at (9).

applications.