Keywords: stiffness modeling, parallel manipulators, passive joints, kinetostatic singularities, recursive computations

The paper focuses on stiffness matrix computation for manipulators with passive joints. It proposes both explicit analytical expressions and an efficient recursive procedure that are applicable in general case and allow obtaining the desired matrix either in analytical or numerical form. Advantages of the developed technique and its ability to produce both singular and non-singular stiffness matrices are illustrated by application examples that deal with stiffness modeling of two Stewart-Gough platforms.

I. INTRODUCTION

N many applications, manipulator stiffness becomes one of the most important performance measures of a robotic system. In particular, for milling, drilling and other types of machining, the stiffness defines the positioning errors due to interaction between the workpiece and the technological tool. Similarly, in industrial pick-and-place automation, the manipulator stiffness defines admissible velocity/acceleration while approaching the target point, in order to avoid undesirable displacements due to inertia forces. Other examples include medical robots, where elastic deformations of mechanical components under the task load are the primary source of positioning errors.

Numerically, this property is usually described by the stiffness matrix C K , which defines a linear relation between the translational/rotational displacement in Cartesian space and the static forces/torques causing this transition (assuming that all of them are small enough). The inverse of C K is usually called the compliance matrix and is denoted as C k . As it follows from related works, for conservative systems, Manuscript received March 7, 2011. The work presented in this paper was partially funded by the Region "Pays de la Loire", France and by the project ANR COROUSSO, France.

A. Pashkevich is with Ecole des Mines de Nantes, 4 rue Alfred-Kastler, Nantes 44307, France and with Institut de Recherches en Communications et en Cybernetique de Nantes, 44321 Nantes, France (phone: Tel. +33-251-85-83-00; fax. +33-251-85-83-49; e-mail: anatol.pashkevich@minesnantes.fr).

A. Klimchik is with Ecole des Mines de Nantes, 4 rue Alfred-Kastler, Nantes 44307, France and with Institut de Recherches en Communications et en Cybernetique de Nantes, 44321 Nantes, France (e-mail: alexandr.klimchik@mines-nantes.fr).

S. Caro is with Institut de Recherches en Communications et en Cybernetique de Nantes, 44321 Nantes, France (e-mail: stephane.caro@irccyn.ec-nantes.fr). D. Chablat is with Institut de Recherches en Communications et en Cybernetique de Nantes, 44321 Nantes, France (e-mail: damien.chablat@irccyn.ec-nantes.fr). C K is 6 6  semi-definite non-negative symmetrical matrix but in general case its structure may be non-diagonal to represent the coupling between the translation and rotation.

The problem of stiffness matrix computing for different types of manipulators is the focus of robotic experts for several decades [START_REF] Salisbury | Active Stiffness Control of a Manipulator in Cartesian Coordinates[END_REF][START_REF] Gosselin | Stiffness mapping for parallel manipulators[END_REF][START_REF] Yi | Geometric analysis antagonistic stiffness redundantly actuated parallel mechanism[END_REF][START_REF] Griffis | Global stiffness modeling of a class of simple compliant couplings[END_REF][START_REF] Ciblak | Asymmetric Cartesian stiffness for the modeling of compliant robotic systems[END_REF][START_REF] Pigoski | Stiffness mappings employing different frames of reference[END_REF][START_REF] Howard | On the 6 x 6 Cartesian stiffness matrix for three-dimensional motions[END_REF][START_REF] Chen | Conservative Congruence Transformation for Joint and Cartesian Stiffness Matrices of Robotic Hands and Fingers[END_REF][START_REF] Svinin | On the stiffness and stability of Gough-Stewart platforms[END_REF][START_REF] Gosselin | Stiffness analysis of parallel mechanisms using a lumped model[END_REF][START_REF] Company | A Method for Modeling Analytical Stiffness of a Lower Mobility Parallel Manipulator[END_REF][START_REF] Alici | Enhanced stiffness modeling, identification and characterization for robot manipulators[END_REF][START_REF] Kövecses | The stiffness matrix in elastically articulated rigid-body systems[END_REF][START_REF] Quennouelle | Stiffness Matrix of Compliant Parallel Mechanisms[END_REF][START_REF] Merlet | Parallel mechanisms and robots[END_REF][START_REF] Merlet | Analysis of Wire Elasticity for Wire-driven Parallel Robots[END_REF][START_REF] Bonnemains | Definition of a new static model of parallel kinematic machines: highlighting of overconstraint influence[END_REF][START_REF] Tyapin | Kinematic and elastostatic design optimization of the 3-DOF Gantry-Tau parallel kinamatic manipulator[END_REF]. The existing approaches may be roughly divided into three main groups: (i) the Finite Element Analysis (FEA) [START_REF] Piras | Dynamic finite-element analysis of a planar high-speed, high-precision parallel manipulator with flexible links[END_REF][START_REF] Hu | Finite Element Analysis of a Six-Component Force Sensor for the Trans-Femoral Prosthesis[END_REF][START_REF] Nagai | A Systematic Approach to Stiffness Analysis of Parallel Mechanisms and its Comparison with FEM[END_REF][START_REF] Bouzgarrou | Rigidity analysis of T3R1 parallel robot with uncoupled kinematics[END_REF][START_REF] Rizk | A comparative stiffness analysis of a reconfigurable parallel machine with three or four degrees of mobility[END_REF], (ii) the Matrix Structural Analysis (SMA) [START_REF] Deblaise | A systematic analytical method for PKM stiffness matrix calculation[END_REF][START_REF] Martin | Introduction to matrix methods of structural analysis[END_REF], and (iii) the Virtual Joint Method (VJM) [START_REF] Salisbury | Active Stiffness Control of a Manipulator in Cartesian Coordinates[END_REF][START_REF] Gosselin | Stiffness mapping for parallel manipulators[END_REF][START_REF] Chen | Conservative Congruence Transformation for Joint and Cartesian Stiffness Matrices of Robotic Hands and Fingers[END_REF][START_REF] Quennouelle | Stiffness Matrix of Compliant Parallel Mechanisms[END_REF][START_REF] Sh | Geometrical Approach to The Conservative Congruence Transformation (CCT) for Robotic Stiffness Control[END_REF][START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF]. The most accurate of them is obviously the FEA-based technique but it requires rather high computational expenses. The SMA is less computationally hard due to fairly large structural elements employed (3D flexible beams instead of numerous tiny tetrahedrals and hexahedrals of FEA) but it nevertheless is not convenient for the parametric analysis. And finally, the VJM method is the most attractive in robotic domain since it operates with an extension of the traditional rigid model that is completed by a set of compliant virtual joints (localized springs), which describe elastic properties of the links, joints and actuators. This paper contributes to the VJM technique and focuses on some particularities of the manipulators with passive joints.

For conventional serial manipulators (without passive joints), the VJM approach yields rather simple analytical presentation of the desired stiffness matrix

C K . Relevant expression θ -1 C θ -T θ ••  K J K J
can be found in the work of Salisbury [START_REF] Salisbury | Active Stiffness Control of a Manipulator in Cartesian Coordinates[END_REF] who assumed that the mechanical elasticity is concentrated in actuators and the deflections are small enough to apply linear approximation of the force-deflection relation. Here the matrix θ K aggregates the stiffness coefficients of all elastic joints, and θ J is the corresponding kinematic Jacobian. Further, this result was extended by Gosselin for the case of parallel manipulators taking into account elasticity of other mechanical elements [START_REF] Gosselin | Stiffness mapping for parallel manipulators[END_REF]. More recent publications present VJM-based stiffness analysis for particular case studies, such as various variants of the Stewart-Gough platform, manipulators with US/UPS legs, CaPAMan, Orthoglide, H4 etc. [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF][START_REF] Ceccarelli | A stiffness analysis for CaPaMan (Cassino Parallel Manipulator)[END_REF][START_REF] Company | Modelling and preliminary design issues of a 4-axis parallel machine for heavy parts handling[END_REF][START_REF] Zhang | Analysis of parallel kinematic machine with kinetostatic modeling method[END_REF][START_REF] Vertechy | Static and stiffness analyses of a class of over-constrained parallel manipulators with legs of type US and UPS[END_REF][START_REF] Li | Stiffness analysis for a 3-PUU parallel kinematic machine[END_REF][START_REF] Majou | Parametric stiffness analysis of the Orthoglide[END_REF][START_REF] Company | Modelling and preliminary design issues of a 4-axis parallel machine for heavy parts handling[END_REF].

It should be noted that in majority of related works, the presence of passive joints does not cause any specific computational problems, since these joints are eliminated via geometrical constraints describing assembling of the relevant parallel architecture [START_REF] Gosselin | Stiffness mapping for parallel manipulators[END_REF]. Besides, in most of publications, it is implicitly assumed that the Jacobian θ

J describing

Cartesian stiffness matrix of manipulators with passive joints: analytical approach

Pashkevich A., Klimchik A., Caro S., and Chablat D.

I

influence of the elastic joints on the end-location is nonsingular 1 , i.e. θ ( ) 6 rank  J , to ensure inversion of the related matrix in the modified expression

- θθ T 1 θ T C • () •   K J K J
that always produce non-singular C K . It is obvious that the assumption concerning θ J is completely realistic if the VJM model includes at least a single 6dimensional virtual spring of a general type (see [START_REF] Pashkevich | Accuracy Improvement for Stiffness Modeling of Parallel Manipulators[END_REF] for details), while it is not realistic that the manipulator stiffness matrix is always non-singular. Hence, common stiffness modelling techniques must be revised with respect to influence of passive joints, which in certain cases can not be straightforwardly eliminated from the kinetostatic equations and, consequently, may cause singularity of C K . In this paper, it is applied another approach that originates from our publication [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF] where the desired stiffness matrix C K of size 6 6  is extracted from the inverse of a larger matrix, of size qq ( 6) (6 ) nn   , which additionally includes the passive joint Jacobian q J ( q n is the passive joint number). Advantages of this approach and its ability to produce singular stiffness matrices were confirmed by a number of examples, but explicit analytical solution was not presented. Hence, this work concentrates on analytical computations of the stiffness matrix and also on influence of the passive joints on particular elements of C K . It is also worth mentioning that some previous works [START_REF] Taghaeipour | Online computation of the stiffness matrix in robotic structures using finite element analysis[END_REF] propose (or at least discuss) a trivial solution of the considered problem, which deals with a straightforward modification of the matrix θ K , in accordance with the passive joint type and geometry (some rows and colons are simply set to zero). However, as it will be shown below, this straightforward approach gives true results if (and only if) the matrix θ K is diagonal, but it is not valid in general case where there is a coupling between different types of the elementary virtual springs presented by non-diagonal coefficients.

The remainder of this paper is organized as follows. Section 2 presents a simple motivating example that confirms the problem non-triviality. Then, Sections 3 and 4 propose relevant analytical solutions for a serial kinematic chain and a parallel robot respectively. Section 5 focuses on computational issues and proposes recursive procedure and a set of corresponding analytical rules. Section 6 contains application examples that demonstrate the developed technique advantages. And finally, Section 7 summarizes the main results and gives prospective for future work. 1 It is important to distinguish the conventional kinematic Jacobian J , which is computed with respect to actuated coordinates and may be both singular and non-singular, and the Jacobian θ J that is computed with respect to the virtual springs coordinates and is always non-singular. Besides, they differ in sizes, which for a standard serial 6-d.o.f. manipulator are respectively 6x6 and 6 36  .

II. MOTIVATING EXAMPLE

Let us present first a simple example that demonstrates non-trivial transformation of the stiffness matrix due to the presence of passive joints. For the purpose of simplicity, let us limit our study to 2D Cartesian space and consider a single manipulator link, which is assumed to be fixed at the left end. It is also assumed that the external loading (the forces , y x FF and the torque z M ) is applied to the right end; either directly or via a passive joint.

Under these assumptions, the elastostatic properties of the link can be described by a symmetrical stiffness matrix , ,

• • • 1 2 K K K x x y x y K K K y K K K E                               (1)
If the link is equipped with a passive joint, the energy of this mechanical system (link with passive joint) must be minimised with respect to the joint variable. For instance, in the case of the rotational passive joint z R allowing free rotation around the z-axis at the reference point, the potential energy should be rewritten as

    mi , , , n p xy E y E x        (2)
and the passive joint variable  may be expressed via the remaining coordinates as

13 23 33
( ) / K x K y K        .
Then, after relevant transformations and computations of the second-order derivatives

K E x K E x y K E            (3)
the desired stiffness matrix of links with passive joint may be expressed as •• 0

•• 0 000 p KK KK K KK KK K K KK KK K                  K (4)
This expression clearly shows that, if the matrix K is non-diagonal, a trivial transformation that was proposed in some previous works (i.e., simple setting to zero of the third raw and column) does not produce a truthful result. Moreover, the elements of the upper-left 2 2

 block must be modified taking into account the elements of K that are located outside of this block. This conclusion motivates development of a general methodology of the stiffness matrix transformation, which is presented below.

III. PASSIVE JOINTS IN A SERIAL CHAIN

In contrast to conventional serial manipulators, whose kinematics does not include passive joints and assures full controllability of the end-effector, parallel manipulators include a number of under-actuated serial chains that are mutually constrained by special connection to the base and to the end-platform. Let us derive an analytical expression for the stiffness matrix of such kinematic chain taking into account influence of the passive joints.

The kinematic chain under study (Fig. 1) consists of a fixed base, a series of flexible links, a moving platform, and a number of actuated or passive joints separating these elements. Following the methodology proposed in our previous work [START_REF] Pashkevich | Stiffness analysis of overconstrained parallel manipulators[END_REF], a relevant VJM model may be presented as a sequence of rigid links separated by passive joints and six-dimensional virtual springs describing elasticity of the links and actuators. For this VJM representation, the direct kinematics is defined by a product of homogeneous transformations that after extraction of the end-platform position and orientation is transformed into the vector function ( , , ..., ) q q q  q contains passive joint coordinates, the vector T 2n θ 1 ( , , ..., )

    θ
collects coordinates of all virtual springs; q n and n  are the sizes of q and θ respectively. It can be proved that the static equilibrium equations of this mechanical system may be written as

θq TT •• ; •  θ J F K θ J F 0 (6)
where q θ , ( , ) ( , )       JJ gq θ q g q θ θ are kinematic Jacobians with respect to the passive and virtual joint coordinates respectively, F is the external loading (force and torque), and θ K the aggregated stiffness matrix of the virtual springs. Using these equations simultaneously with [START_REF] Ciblak | Asymmetric Cartesian stiffness for the modeling of compliant robotic systems[END_REF] and applying the first-order linear approximation under assumption that corresponding values of the external force F and the coordinate variations ,,    q θt are small enough, one can derive the matrix expression [START_REF] Howard | On the 6 x 6 Cartesian stiffness matrix for three-dimensional motions[END_REF] that allows obtaining the desired Cartesian stiffness matrix C K numerically. Corresponding procedure includes inversion of qq ( 6) ( 6) nn   matrix in the right-hand side of ( 7) and extracting from it the upper-left sub-matrix of the size 66  that defines a liner force-deflection relation in Cartesian space:

1 1T θ θ θ q T q ••                      J K J J Ft J0 q0
C •  F K t (8)
In spite of computational simplicity, the above procedure is not convenient for the parametric stiffness analysis that usually relies on analytical expressions. To derive such expression for the matrix C K , let us apply the blockwise inversion based on the Frobenius formula [START_REF] Gantmacher | Theory of matrices[END_REF], that allows (after some transformations) to present the desired stiffness matrix as

  1 0 0 T 0 T 0 C C C q q C q q C • • • • • •   K K J J K J J K K (9)
where the first term

0 1 T 1 C θ θ θ •• ()   K J K J
is the stiffness matrix of the corresponding serial chain without passive joints and the second term defines the stiffness decrease due to the passive joints. It is worth mentioning that this result is in good agreement with other relevant works [START_REF] Quennouelle | Stiffness Matrix of Compliant Parallel Mechanisms[END_REF], [START_REF] Chen | Instantaneous stiffness analysis and simulation for hexapod machines[END_REF], [START_REF] Quennouelle | Instantaneous Kinemato-Static Model of Planar Compliant Parallel Mechanisms[END_REF] where C K was presented as the difference of two similar components but the second one was computed in a different way.

Analyzing the latter expression, one can get to the following conclusion concerning computational singularities: Hence, in practice, expression [START_REF] Ciblak | Asymmetric Cartesian stiffness for the modeling of compliant robotic systems[END_REF] does not cause any computational difficulties and always produce a singular stiffness matrix of rank q 6 n 

. In analytical computations, the following proposition can be also useful that allows us to modify the original stiffness matrix 0 C K sequentially: Proposition. If the chain does not include redundant passive joints, expression (5) allows recursive presentation   [START_REF] Gosselin | Stiffness analysis of parallel mechanisms using a lumped model[END_REF] in which the sub-Jacobians qq i  JJ are extracted from 12 q q q , , ...    J J J

1 1 T T C C C q q C q q C ; 1, 2, ... • • • • • • i i i i i i i i i i      K K K J J K J J K
in arbitrary order (column-by-column, or by groups of columns). Here 0

1 T 1 C θ θ θ •• ()   K J K J
Corollary. The desired stiffness matrix

C K can be computed in q n
steps, by sequential application of expression ( 6) for each single column of the Jacobian q J (i.e. for each passive joint separately). These results give convenient analytical and numerical computational techniques that are presented in details in Section 6.

IV. PASSIVE JOINTS IN A PARALLEL MANIPULATOR

Let us consider now a parallel manipulator, which may be presented as a strictly parallel system of the actuated serial legs connecting the base and the end-platform (Fig. 2) [START_REF] Merlet | Parallel Robots[END_REF]. Using the methodology described in previous section and applying it to each leg, there can be computed a set of m Cartesian stiffness matrices () C i K , 1, ...,  im expressed with respect to the same coordinate system but corresponding to different platform points. If initially the chain stiffness matrices were computed in local coordinate systems, their transformation is performed in standard way [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms[END_REF], as

T CC T glob loc          R0 R0 KK 0R 0R ( 11
)
where R is a 33  rotation matrix describing orientation of the local coordinate system with respect to the global one.

To aggregate these matrices () C i K , they must be also recomputed with respect to same reference point of the platform. Assuming that the platform is rigid enough (compared to the legs), this conversion can be performed by extending the legs by a virtual rigid link connecting the endpoint of the leg and the reference point of the platform (see Fig. 2 where these extensions are defined by the vectors i v ). After such extension, an equivalent stiffness matrix of the leg may be expressed using relevant expression for a usual serial chain, i.e. as T1 ( )

( ) ( ) C •• i i i vv  J K J
, where the Jacobian () i v J defines differential relation between the coordinates of the ith virtual spring and the reference frame of the end-platform. Hence, the final expression for the stiffness matrix of the considered parallel manipulator can be written as

T1 ( ) ( ) ( ) ( ) СС 1 •• m m i i i vv i     K J K J ( 12
)
where m is the number of serial kinematic chains in the manipulator architecture. Besides, it is implicitly assumed here that all stiffness matrices (both for the legs and for the whole manipulator) are expressed in the same global coordinate system. Hence, the axes of all virtual springs are parallel to the axes x, y, z of this system and corresponding Jacobians and their inverses can be easily computed analytically as 33 ( ) ( ) 1

33 6 6 6 6 ( ) ( ) , ii ii vv                   I v I v JJ 0 I 0 I (13)
where 3 I is a identity matrix of size 33  , i v is the vector from the leg end-point to the platform reference point (see V. COMPUTATIONAL TECHNIQUES Explicit expressions ( 9), [START_REF] Company | A Method for Modeling Analytical Stiffness of a Lower Mobility Parallel Manipulator[END_REF] derived in previous sections allow obtaining the Cartesian stiffness matrix instantly, for any Jacobian q J describing special location of the passive joints. However, recursive equation [START_REF] Gosselin | Stiffness analysis of parallel mechanisms using a lumped model[END_REF] allows essentially simplify the computational procedure by sequential modification of the original stiffness matrix 0 C K for each passive joint independently, using separate columns of q q1, q 2 [ , .

..]  J J J

. Moreover, for some typical cases, relevant computations may be easily performed analytically. This section presents some useful techniques related to this approach.

A. Recursive computations: single-joint decomposition

Let us assume that a current recursion deals with a single passive joint corresponding to the i-th column of the Jacobian q J , which is denoted as q i J and has size 61  . In this case, the matrix expression

T1 q C q • ) • ( i i i  J K J
is reduced to the size of 11  and the matrix inversion is replaced by a simple scalar division. Besides, the term Cq

•

ii KJ has size 61  , so the recursion ( 10) is simplified to

1 T ( 1) ( ) ( ) ( ) CC 11 i i i i i i i i jk jk j k or K K u u                    K K u u (15)
where

Cq • ii i  u K J is a 61  vector and T q C q •• i i i   J K J is a
scalar. It can be also proved that each recursions reduces the rank of the stiffness matrix by 1 [START_REF] Merlet | Analysis of Wire Elasticity for Wire-driven Parallel Robots[END_REF] provided that the current Jacobian q i J is independent of the previous ones 12 qq , ... JJ (i.e. the i-th passive joints is not redundant relatively to the joints 1, ..., 1 i  ). Since in practice any combination of passive joints can be decomposed into elementary translational and rotational ones, it is enough to consider only two types of the Jacobian columns qi J :

    1 CC 1 ii rank rank   KK
    Hence, in general case, the recursion [START_REF] Gosselin | Stiffness analysis of parallel mechanisms using a lumped model[END_REF] involves rather intricate matrix transformation, different from simple setting to zero a row and/or a column. Let us consider now several specific (but rather typical) cases where the transformation rules are more simple and elegant.

B. Analytical computations: trivial passive joints

In practice, many parallel robots include kinematic chains for which the passive joint axes are collinear to the axes x, y or z of the Cartesian coordinate system. For such architectures, the vector-columns of the Jacobian q J include a number of zero elements, so the expressions (13) can be essentially simplified. Let us consider a set of trivial cases where q i J are created from the columns of the identity matrix 66  I :

Corresponding passive joints will be further referred to as the "trivial" ones. It can be easily proved that they cover the following range of the joint geometry: (i) translational passive joint with arbitrary spatial position (but with the joint axis directed along x, y or z); (ii) rotational passive joints positioned at the reference point (and with the joint axis directed along x, y or z). Besides, it is worth to consider additional case-study corresponding to (iii) rotational passive joints shifted by a distance L with respect to the reference point in the direction either x, y or z (and with the joint axis directed along x, y or z), which will be further referred to as the "quasi-trivial" and gives the Jacobian columns of the following structure: [START_REF] Griffis | Global stiffness modeling of a class of simple compliant couplings[END_REF] ( 5 ) ( 6 ) q q q 0 0 0 ,, 1 0 0

0 1 0 0 0 1 xx yy zz dd dd dd                                                                   J J J (18)
where ,,

x y z d d d denote the elements of the vector d , which are equal here either L  or 0. For the trivial passive joints, assuming that () q p J denotes the vector-column with a single non-zero element in the p-th position, a straightforward substitution yields

() i j jp uK  ; () i pp K  
. So, the recursive expression (9) for the Cartesian stiffness matrix is simplified to

( ) ( ) ( 1) ( ) (
)

ii jk kj ii jk jk i pp KK KK K               (19)
that is very similar to those presented in the motivating example (see Section 2). Also, here the p-th row and column of the matrix

1 i C  K become equal to zero ( 1) ( 1) 0
; 0; 1, ...6 ii pk kp K K p      (20)
and the recursive computations are easily performed analytically.

VI. APPLICATION EXAMPLES

Let apply now the developed technique to computing of the stiffness matrix for two versions of a general Stewart-Gough platform presented in Fig. 3 [42]- [START_REF] Arumugam | Stiffness analysis of a class of parallel mechanisms for micro-positioning applications[END_REF]. It is assumed that in both cases the manipulator base and the moving plate (platform) are connected by six similar extensible legs (Fig. 4) but their spatial arrangements are different:

Case A: the legs are regularly connected to the base and platform, with the same angular distance 60° (it is obviously a degenerate design, where the stiffness matrix should be singular) Case B: the legs are connected to the base and platform in three pairs, with the angular distance of 120° between the mounting points (it is a classical design of Stewart-Gough where the stiffness matrix should be non-singular).

For both designs, the original leg stiffness (i.e. without the passive joints) can be described by the sparse matrix corresponding to the symmetric beam. Further, to take into account the passive joints influence, the procedure (10) should be applied recursively, using the elementary Jacobians

1) ( 2 ) ( 3 ) ( 4 ) ( 5 ) R R R R R 0 0 0 0 0 0 0 0 0 0 0 0 0 ; ; ; ; 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 x y z y z L L                                                                                                            J J J J J ( 21
)
where L is the leg length. It is obvious that, due to trivial structure of q i J , the recursive computations can be easily performed analytically.

Hence, in final form, the derived matrix includes only the traction/compression term (and not bending, torsion, etc.) what perfectly agrees with other results on Stewart-Gough platforms.

Further, to be applied to each leg, the obtained matrix must be transformed from the local to the global coordinate system. In this specific case, due to the special structure of

1 i ii ii K                       0 R 0 R 0 K 00 0 R 0 R (22)
expressed via the orthogonal rotation matrix

() i i i i i K                            I0 u u 0 I v K vI 0 0 0 I )
where i v is the vector from the leg end-point to the platform reference point (see Fig. 2). So, after relevant transformations, one can get the final expression of the manipulator stiffness matrix [START_REF] Martin | Introduction to matrix methods of structural analysis[END_REF] where the vector i u , i v describing spatial locations of the legs and computed via the direct kinematics, and 0 ii  vu denotes the vector product which is referred to the corresponding skew-symmetric matrix).

0 6 0 T 0 T 0 С 11 1 () () i i i i i ii K              u K u v u vu
The derived equation was applied to both case studies, assuming that the manipulators are in their "home" configurations when the platform is parallel to the base and it is symmetrical with respect to the vertical axis. Corresponding expressions for the leg vectors are cos( ) cos( ) cos( ) sin( ) sin( ) , sin( ) L is the leg length, and the superscripts "A" and "B" define the relevant case study. As follows from these expressions, in "home" location, the matrix

0 i i i i i i i i r R r r R r h                                uv (26)

VII. CONCLUSION

For robotic manipulators with passive joints, the stiffness matrices of separate kinematic chains are singular. So, the most of existing stiffness analysis methods can not be applied directly and this problem is usually solved by elimination the passive joint coordinates via geometrical constraints describing the manipulator assembly. However, such techniques degenerate if the number of passive joints is redundant and/or the resulting matrix is inherently singular.

To deal with such architectures in more efficient way, this paper proposes an analytical approach that allows obtaining both singular and non-singular stiffness matrices and which is appropriate for a general case, independent of the type and spatial location of the passive joints. The developed approach is based on the extension of the virtual-joint modelling technique and includes two basic steps which sequentially produce stiffness matrices of separate chains and then aggregate them in a common matrix.

In contrast to previous works, the desired stiffness matrix is presented in an explicit analytical form, as a sum of two terms. The first of them has traditional structure and describes manipulator elasticity due to the link/joint flexibility, while the second one directly takes into account influence of the passive joints. It is proved that, for each chain, the rank-deficiency of the resulting matrix is equal to the number of independent passive joints. To simplify analytical computations, it is proposed a recursive procedure that sequentially modifies the original matrix in accordance with the geometry of each passive joint. For the trivial cases, for which the passive joint axes are collinear to the axes of the base coordinate system, this modification is presented in the form of simple analytical rules.

Advantages of the developed technique are illustrated by application examples that deal with stiffness modelling of two Stewart-Gough platforms. They demonstrate its ability to produce both singular and non-singular stiffness matrices, and also show its feasibility for analytical computations. These examples give also some prospective for future work that include development of the dedicated techniques for the stiffness matrix aggregation in the case of non-rigid platform and an extension of these results for the case of manipulators with external loading.

Fig. 1 .

 1 Fig. 1. The VJM model of a general serial chain (Pspassive joint, Acactuated joint)

Remark 3 .Remark 4 .

 34 If both terms of (9) are non-singular, their difference produces a symmetrical stiffness matrix, which always singular If the matrix 0 C K of the chain without passive joints is symmetrical and positive-definite, the stiffness matrix of the chain with passive joints C K is also symmetrical but positive-semidefinite.

Fig. 2

 2 [START_REF] Alici | Enhanced stiffness modeling, identification and characterization for robot manipulators[END_REF] allows explicit aggregation of the leg stiffness matrices with respect to any given reference point of the platform. It is worth mentioning that in practice, while there aggregation usually produce non-singular singular matrix. Relevant examples are presented in following sections.

Fig. 2 .

 2 Fig. 2. Typical parallel manipulator (a) and transformation of its VJM models (b, c)

  the passive joint axis (both for translational and rotational ones) and the vector the rotational passive joints on the linear velocity at the reference point, i.e.  d e r where r is a vector from the joint centre point to the reference point.

Fig. 3 Fig. 4 .

 34 Fig. 3 Geometry of the Stewart-Gough platforms under study

(

uK

  of the i-th local coordinate system with respect to the global one, is easily reduced to is the unit vector directed along the leg axis i u (see Fig.2). Besides, before aggregation, the stiffness matrices of separate legs i C must be re-computed with respect to same reference point in accordance with expressions[START_REF] Alici | Enhanced stiffness modeling, identification and characterization for robot manipulators[END_REF],[START_REF] Kövecses | The stiffness matrix in elastically articulated rigid-body systems[END_REF]



  , h is the vertical distance between the base and the platform. Substitution of these vectors to the expression (25) leads to the following stiffness r denote the circle radius which comprise the leg connection point at the base and moving platform respectively,

K

  is singular and allows "free" rotation of the end-platform around the vertical axis. In contrast, for the same location, the matrix () C B K is nonsingular and the manipulator resists to all external forces/torques applied to the platform. These results are in good agreement with previous research on the Stewart-Gough platforms and confirm efficiency of the developed computational technique for manipulator stiffness modeling.

ACKNOWLEDGEMENTS

The work presented in this paper was partially funded by the Region "Pays de la Loire" (France), by the EU commission (project NEXT) and by the ANR, France (project COROUSSO).


Frédéric Gruy

Michèle Pijolat

Michel Soustelle

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Etude de l'influence des modes de chauffage sur l'évolution d'une poudre de zircone Alain Méthivier, Michèle Pijolat, Michel Soustelle To cite this version: Alain Méthivier, Michèle Pijolat, Michel Soustelle. Etude de l'influence des modes de chauffage sur l'évolution d'une poudre de zircone. AFCAT 1990 (Journées de l'Association Française d'Analyse Thermique et de Calorimétrie), May 1990, Clermont-Ferrand, France. emse-00610466