Jean-François Petiot
email: jean-francois.petiot@irccyn.ec-nantes.fr

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

SUBJECTIVE EVALUATION OF FORMS IN AN IMMERSIVE ENVIRONMENT

Keywords: subjective evaluation, multidimensional scaling, virtual reality, preference mapping

User's perception of product, by essence subjective, is a major topic in marketing and industrial design. Many methods, based on users' tests, are used so as to characterise this perception. Methods like multidimensional scaling, semantic differential method, and preference mapping are well known in sensorial analysis or in the food industry. These methods are used in order to built a perceptual space in order to position a new product, to specify requirements by the study of user's preferences, to evaluate some product attributes, related in particular to style (aesthetic). These early stages of the design are essential for a good orientation of the project. In parallel, virtual reality tools and interfaces are more and more efficient for suggesting to the user complex feelings, and creating in this way various levels of perceptions. In this article, we present on an example an adaptation of multidimensional scaling and preference mapping for the subjective assessment of virtual products. These products, which geometrical form is variable, are defined with a CAD model and are proposed to the user with a spacemouse and stereoscopic glasses. Advantages, uses and limitations of such evaluation is next discussed.

1-Introduction

In today's highly competitive market, developing new products that meet consumers' tastes is a crucial issue in product design. To improve attractiveness, a well-designed product should not only satisfy requirements, but should also satisfy consumers' psychological needs, by essence subjective. For many products, the aesthetic properties are as important as technical functions [START_REF] Wagner | Aesthetics Design Improvement[END_REF]. In order to predict the success of a product, to control and to optimise its performances, it's time now to take into account esteem and aesthetics functions in the beginning of the design. Although industrial and product designers are keenly aware of the importance of design aesthetics [START_REF] Quarante | Eléments de design industriel[END_REF], there is an obvious lack of systematic, scientific, and engineering methods to help them make aesthetic design decisions and conduct aesthetic evaluation. For example, the perception of the shape of a product is often nothing but a style of design, depending much more on the designer's taste than on real customers' trends, as some studies clearly showed [START_REF] Hsu | A semantic differential study of designers' and users' product form perception[END_REF].

In this context, Virtual Reality (VR) seems to offer promising functionalities for the assessment of virtual products [START_REF] Paillot | Revue de projet immersive pour le style automobile[END_REF]. The Virtual reality interfaces available on the market are now mature enough for suggesting to the user relevant feelings and sensations [START_REF] Fischer | La réalité virtuelle pour une conception centrée sur l'utilisateur[END_REF]. The main problem is now to learn how to use it and to define relevant methods for their integration into the design process. How do we use virtual products to study the aesthetics of products and to help make them more aesthetic ? How increase the attractiveness of the shape of a product via virtual reality tools ? In this article, we present the use of VR tools and user-tests for the study of aesthetics. It is assumed that we possess an innate ability to perceive a wide range of qualities in products that shape our response to them [START_REF] Macdonald | Aesthetic intelligence: optimizing usercentered design[END_REF]. But it's often more difficult to detect bad proportions in the design of products, or to explain rules which lead to a feeling of harmony and beauty. A study based on the golden section or "divine proportion", described in [START_REF] Lipovetsky | Generalized golden sections, repeated bisections and aesthetic pleasure[END_REF], shows that it is not always clear whether the artists consciously used the golden section or whether they intuitively approximated the associated ratio. Aesthetics is clearly a multidimensional characteristic, not independent of ergonomics and usability considerations. The concept of ergo-aesthetics, proposed in [START_REF] Yili | Engineering Aesthetics and Ergo-Aesthetics: Theoretical and Methodological Foundations[END_REF], refer to the integrated design approach that is aimed at meeting both ergonomic and aesthetic design objectives. Two main approaches can be used for the study of aesthetics [START_REF] Yili | Engineering Aesthetics and Ergo-Aesthetics: Theoretical and Methodological Foundations[END_REF]:

-The top-down approach attempts to understand aesthetic responses as a whole, -The bottom-up approach attempts to identify the basic pictorial features and compositional patterns that please or displease the senses.

In this article, we propose an illustration of both approaches for the evaluation of forms in an immersive environment. In section 2, we present two classical methods in sensorial analysis which can be used in product design for grasping the subjective response of subjects. In section 3, we propose an adaptation of these methods to the experimental study of the "appeal" of forms. The perception of the aesthetic of forms by a subject is carried out in a VR environment, and with particular products: table glasses. The main results of this experiment are discussed in section 4, and perspectives for the use of VR for the design of forms are discussed.

2-Background

To study users' perceptions, researchers in marketing and sensorial analysis propose various methods [START_REF] Kaul | Research for product positioning and design decision: an integrative review[END_REF][10] [START_REF] Evrard | MARKET : Etudes et Recherches en Marketing[END_REF].

Perceptual maps are commonly used to take perceptions into account and to control the product positioning. The basic idea is to build a multi-attribute perceptual space in which each product is represented by a point. We propose to describe multidimensional scaling (MDS), a classical method for building the perceptual space, and preference mapping techniques, which are relevant to grasp preference assessments. An adaptation of both methods will be used in section 3.

-MDS

Multidimensional scaling uses dissimilarity assessments to create a geometrical representation of a family of "stimuli". This method, developed initially for psychometric analysis [START_REF]Multidimensional scaling: Theory and applications in the behavioral sciences[END_REF], is a process whereby a distance matrix among a set of stimuli is translated into a representation of these stimuli inside of a perceptual space. Taking all the possible pairs of stimuli (here pairs of products) into account, the subject evaluates their degree of similarity on a quantitative scale and fills in this way a dissimilarity matrix. Technically, what MDS does is to find a set of points (X i ) i=1,…,N in a K-dimensional space such that the distances among them correspond as closely as possible to the dissimilarity (or a function of it) in the input matrix, according to a given optimisation criterion. This criterion can be for example a function called stress, (equation 1) which represents the "badness of fit", and the distance can be the Euclidean distance (equation 2), but several different expressions can be used [START_REF] Davison | Multidimensional Scaling[END_REF]. D ij is the perceptual dissimilarity between stimuli i and j, d ij the Euclidean distance, x ik the coordinate of stimuli i on dimension k, K the number of dimensions of the perceptual space, selected by the experimenter. After computation, each stimuli is represented in the Euclidean space by a point X i (x i1 , …,x iK ).

2 / 1 2 2 ) (             - = ∑∑ ∑∑ i j ij i j ij ij d D d stress (1) ∑ = - = K k jk ik ij x x d 1 2 ) ( (2)
The main advantage of this method is that the tests are based on instinctive dissimilarity assessments, which do not impose any criteria or predefined semantic scale. This method provides a space for a visualisation of the relevant dimensions for stimuli's perception. It is used in marketing and sensorial analysis, the stimuli can be sounds, fragrance, products, concepts,…It is well suited to study the relationship between products. We have used in the next section metric MDS, based on the minimisation of the stress, and using a sparse matrix of dissimilarity.

-Preference mapping

Preference mapping constitutes a group of statistical techniques which are also common in sensory analysis and marketing research. Internal (MDPREF) and external (PREFMAP) preference mapping both generate graphic interpretations of individual preferences [START_REF] Caroll | Multidimensional scaling: theory and application in the behavioral sciences[END_REF] [START_REF] Ecrin | Evaluation Subjective : Méthodes, Applications et Enjeux[END_REF]. To construct such preference mappings, multiple regression techniques are used. The principle is to find a correlation between product's preference and product's position in the perceptual space. Technically speaking, one must perform a multiple linear regression of the preference on the perceptual coordinates. These techniques need some strong assumptions. The preference's model is supposed to be linear most of the time, it's not possible to consider threshold phenomena for the preference, and the method needs an absolute evaluation of the preference as input (hedonic scale). The use of a hedonic scale to assess the preference in an absolute manner encounters some difficulties and limitations which can be avoided by the use of pairwise comparison methods [START_REF] David | The method of paired comparisons[END_REF]. In the PREFMAP model, consumers are represented by their ideal point in the perceptual space, which represents the most preferred combination of attributes. Other models with infinite preferred attribute levels are called 'vector models'. We use in the next section preference mapping based on the vector model.

3-Experimental study

-Description of the VR tools

The main objective of the experiment is to show that the study of the aesthetic properties of forms can be conducted on virtual products, and that an immersive environment is particularly helpful for the assessment and the improvement of aesthetics. To illustrate our approach, we propose to study the aesthetics of table glasses, which are very interesting products from an aesthetics and esteem point of view. A study on such products (wine-glass) is proposed in [START_REF] Matsuoka | Method for constructing an inverse reasoning system for form generation from product image objectives[END_REF], where the authors present a method for form generation, and in [START_REF] Petiot | How to comprehend and assess product semantics: a proposal for an integrated methodology[END_REF], where a methodology for a solid assessment of product semantics is proposed. A digital model of glasses with various forms was designed with the CAD and VR software Catia V5R11. A texture with the material "glass" were applied to the models of glass. A Spacemouse and stereoscopic eyeglasses were used by the subject for enhancing his/her perception of the environment, and for facilitating the various assessments.

-Description of the assessment tests

Several forms of table glasses were generated with the CAD software. All glasses are intended to the same usage, for example wine glasses. They all have the same general form, balloon glass, and are made off three parts: base, foot and container. The mathematical model of the curve for the generation of forms is given figure 1. The base, the foot, and the container are modelled with a spline-curve with 15 control nodes. Continuity conditions are added at points P5 and P9. The inner shape of the container is parallel to the outer shape, with a gap of 1 mm. The 3D solid is generated by rotation of the curve. All the glasses used for the experimental study are an "instanciation" of this model. 18 glasses were generated (figure 2), which constitute the stimuli for the assessment tests. VR tools were next used for the evaluation of the aesthetics qualities of the glasses The assessment performed by the subject was made of several stages, presented in the following sections.

-Stage 1: Building of the perceptual space

The aim of this stage is to find how many dimensions can describe the aesthetic, or the "appeal" of the glasses, and to find the location of the glasses according to these dimensions. For that, the subject had to quantify the perceived differences between pairs of products from an aesthetic point of view. This stage belongs to the "top-down" approach. The subject was asked to say to what degree the products were different. He/she chose between "identical", "a little different", "different" and "very different". The answer was coded on an integer scale from 0 (identical) to 3 (very different). In order to reduce the assessment task, the subject was not forced to fill all the dissimilarity matrix. He only had to chose the more obvious comparisons. He had the complete choice of the comparisons to fill in the N.(N-1)/2=153 potential comparisons of the superior half of the dissimilarity matrix. Nevertheless, so as to have computable data, and to reveal the multidimensionality of aesthetics, each product should be involved in at least three comparisons. The pairwise comparison were performed by the subject with the VR software and easy "click and drag" operations. The (sparse) dissimilarity matrix provided by the subject is given table 1. Next, we used MDS for determining how many dimensions can represent the dissimilarities, and for building a perceptual space which expresses these differences. With this sparse matrix as input, an own implementation of metric MDS has been used to calculate the perceptual coordinates of the glasses [START_REF] Petiot | Differentiation of trumpets' sounds : experimental study with an adaptable depth mouthpiece[END_REF]. A 2-dimensional configuration, with a stress value equal to 0.12 (considered as a correct "badness of fit") has been retained (figure 3). These results reveal the two main dimensions (axe1, axe2) which are relevant for describing the differences between the products from an "appeal" point of view.

-Stage 2: assessment of the appeal

The aim of this stage is to get an absolute assessment of the appeal, on a continuous scale. The task is the same as previous stage, but the subject is now forced to provide a monodimensional answer. order to reveal the multidimensionality of aesthetic during stage 1, it's important that the subject do not carry out stage 1 from the results of stage 2. In other words, stage 1 must be performed before stage 2. The subject was asked to rank the products in the order of increasing appeal, or into piles of similar appeal. The subject associated to each product (or each pile) an absolute value of the appeal, on a continuous scale from 0 to 10. This scale is similar to the "hedonic" evaluation used in the food industry for preference assessments. The assessment of the glasses, provided by the subject, is given table 2.

-Stage 3: a model of appeal

The main objective is to find how sensitive is the subject in detecting small variations in aesthetics, and to find his abilities to perceive and judge values, changes, and variations in design parameters. This stage belongs now to the "bottom-up" approach [START_REF] Hsiao | Applying the semantic transformation method to product form design[END_REF]. For that, three shape regulating rules were proposed to the subject. These rules correspond to a modification of the three determining design parameters of a glass, proposed in [START_REF] Trocherie | Conception de produits : analyse de la sémantique produit et modélisation des préférences clients[END_REF]. They are expressed as following (figure 4):

-R1: increase the total height of the container, d1, -R2: increase the height of the foot, d2, -R3: increase the diameter of the container, d3. For each glass Gj, the subject has to indicate the effect of the proposed shape regulating rule on his/her assessment of the appeal:

-If rule "Ri" increases the appeal, ∆Pi(Gj) = 1 -If rule "Ri" does not change the appeal, ∆Pi(Gj) = 0 -If rule "Ri" decreases the appeal, ∆Pi(Gj) = -1

This is equivalent to an assessment of the "derivative" of the appeal according to the design parameters.

The virtual reality model of the glass is particularly suitable for the required assessments. Indeed, the subject can even make the modification on the virtual model to visualise and compare the differences, if the assessment of the appeal change is not obvious. The values of the dimensions d1, d2, d3 for each glass and the assessments proposed by the subject are given in the following table 3. For each shape regulating rule, these values ∆Pi(Gj) are proportional to the partial derivative dj P ∂ ∂ , with a positive multiplicative factor kj. [START_REF] Hsu | A semantic differential study of designers' and users' product form perception[END_REF] : Dimensions of the glasses and assessment of the appeal changes according to the shape regulating rules.

Glass # d1 cm d2 cm d3 cm R1 1 1 d P k ∂ ∂ R2 2 2 d P k ∂ ∂ R3 3 3 d P k ∂ ∂ G 1 8 7 8 -1 0 -1 G 2 8 7 9 , 5 -1 1 -1 G 3 8 7 9 , 5 -1 0 -1 G4 8 7 6 -1 -1 0 G 5 8 7 7 -1 -1 -1 G 6 8 7 9 -1 0 -1 G 7 8 5 8 -1 1 -1 G 8 8 5 9 , 5 -1 1 -1 G 9 8 5 9 , 5 -1 1 -1 G10 8 5 6 -1 1 0 G 1 1 8 5 7 -1 1 -1 G 1 2 8 5 9 -1 1 -1 G 1 3 8 3 8 -1 0 -1 G 1 4 8 3 9 , 5 -1 1 -1 G 1 5 8 3 9 , 5 -1 1 -1 G 1 6 8 3 6 -1 1 -1 G 1 7 8 3 7 -1 1 -1 G 1 8 8 3 9 -1 1 -1 Tab.

-Building of an model of appeal

A quadratic model of the appeal P is proposed. The predicted value, denoted P , is given by: 10

+ + + + + + + + + = (3)
The predicted partial derivatives of P are given by: 3 ).

+ + + = ∂ ∂ + + + = ∂ ∂ + + + = ∂ ∂ (4)
The set of coefficients {a1, a2,…, a10} is determined with an optimisation procedure, similar to the least square regression. It's a generalisation of the linear regression of the appeal on the design parameters, taking into account more data concerning the partial derivative. The response surface, corresponding to this model, is plotted in the next section.

) ( ) ( ) ( min 3 2 1 10 2 1 2 1 3 1 2 1 k k k a a a X t r w d P k d P P P X F N i j i j j j i N i i i = ∂ ∂ - ∂ ∂ + - = ∑ ∑ ∑ = = = (5

4-Results and discussions

-Preference mapping 1

The first analysis which can be done is to try to explain the appeal of the glasses by the perceptual dimensions given by MDS.

Like for the PREFMAP model, this is done by a multiple regression using the perceptual axes (axe1, axe2) as independent variables and the appeal P as the dependent variable (equation 4). The regression is significant and the determination coefficient R 2 is close to 1. So the linear model is well adapted for describing the data provided by the subject. This good correlation shows a good coherency of the subject: the data provided in stage 1 are quite compatible with the data provided in stage 2. Furthermore, the VR interfaces proposed to the subject do not introduce incoherence or "noise" in the assessment of the shape. In other words, the good coherence of the data is a sign for the perfect adaptation of the interface to the task proposed to the subject. So as to give a graphical interpretation of the regression, the vector model of the appeal is plotted in the perceptual space. The origin of the vector is located arbitrarily in the origin of the frame, the values of the regression coefficients (a, b) give the orientation of the arrow, the arrowhead points in the direction of increasing appeal (figure 5). It can be shown that this vector is parallel to the steepest slope line of the plane (equation ( 6)). The perpendiculars to the vector are the isoappeal curves. The interest of this graph lies in its ability to study the multidimensionality of aesthetics. What differentiate glass #13 and #17, which have about the same appeal ? May a particular attribute, or a particular design parameter, explain the difference ? Further studies are needed for answering these questions.

-Preference mapping 2

The second analysis which can be done is to try to explain the appeal of the glasses by certain design parameters of the forms.

The response surface of the appeal P , obtained in section 3, can be plotted, according to the design variables (d2, d3) (equation 7)

+ + + + + = (7)
The colormap of this surface is presented figure 6.

Fig. 6 Colormap of the appeal in the design parameter space (d2, d3).

The structure of the appeal seems to be very simple and unidimensional. Increasing d2 and decreasing d3 lead to an increase of the appeal. The perpendicular to the steepest descent line, given by the relation d3=d2/2+c, is about an isoappeal line (figure 7).

2 for all glasses, the dimension d1=8cm The relation d3=d2/2+c is important for the study of the aesthetic of a glass. It's a characteristic of the subject, who has provided data which satisfy this relation. The strong point of our study is that the methodology presented allows the discovery of such relation. The subject is in most of the cases unconscious of this relation and is not able to formulate it explicitly. Both preference mappings are interesting for improving the design of forms, or seeking "good" products because they use two complementary approaches.

5-Conclusions

We have presented in this paper an experimental study of the aesthetic of table glasses. This study of the "appeal" of the shapes has been carried out on virtual models, generated by a CAD/VR software, and with classical VR interface: a spacemouse and stereoscopic eyeglasses. With this tools, the study was based on the assessment by a subject of various characteristics of the glass. Two ways of investigations have been used: (1) a top-down approach, using multidimensional scaling and linear regression of the "appeal" of the shapes ;

(2) a bottom-up approach, with the building of a model of the appeal, based on the assessment of "shape regulating rules ". A new method for establishing the model of appeal, based on an optimisation process, has been proposed. Firstly, the good coherence of the data proposed by the subject shows the perfect adaptation of the interfaces to the proposed task. Furthermore, the subject does not encounter difficulties in the assessment tasks. Secondly, the model of appeal, based on two design parameters of the shape of the glass, reveals how these parameters are linked for the subject's assessments. In perspective, we will conduct a similar study using more design parameters, and based on principal component analysis. The next step will be to use a model of subjective evaluation for forms generation and product synthesis, approach used for example in Kansei engineering type 3

Fig. 1 :Fig. 2 :

 12 Fig. 1 : Model of the outer curve of the glass.

Tab. 1 :

 1 1 * * * * * * 1 * * * 2 0 * * 17 * 2 3 2 * * * * * * 0 * * * 1 * 0 * 18 * 1 * 2 * * * * * * 3 * * * 1 * * 0 Sparse dissimilarity matrix.

Fig. 3 :

 3 Fig. 3 : 2D Perceptual space of the glasses

2 :

 2 Appeal scores of the glasses.

Fig. 4 :

 4 Fig. 4 : Description of the shape regulating rules

  of variables X= (a1, a2,…, a10, k1, k2, k3) is determined by minimising the following function F :

5 :

 5 Coefficients of the model of preference mapping 1

Fig. 5 :

 5 Fig. 5 : vector model of appeal in the perceptual space

Fig. 7 :

 7 Fig. 7 : Iso-appeal line in the design parameter space

  )

	i P is the appeal, dj P i ∂ ∂	the partial derivative, provided by the
	subject, for glass G i ,
	i P is the appeal, dj P i ∂ ∂	the partial derivative, provided by the
	model, for glass G i ,
	The result of the optimisation procedure is given table 2.
	a1	a2	a3	a4	a5	a6	a7	a8	a9 a10
	3.6	0.12 13.6 -1.52 0.02 -0.13 -0.01 -1.71 0.01 82.36
	k1	k2	k3
	-0.01 14.15 0.01
	Tab. 4 : Coefficients of the model of appeal.


A Pashkevich

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

P Wenger

THE ORTHOGLIDE: KINEMATICS AND WORKSPACE ANALYSIS

Keywords: Parallel manipulators, Workspace, Inverse and direct kinematics

The paper addresses kinematic and geometrical aspects of the Orthoglide, a three-DOF parallel mechanism. This machine consists of three fixed linear joints, which are mounted orthogonally, three identical legs and a mobile platform, which moves in the Cartesian x-y-z space with fixed orientation. New solutions to solve inverse/direct kinematics are proposed and a detailed workspace analysis is performed taking into account specific joint limit constraints.

Introduction

For two decades, parallel manipulators attract the attention of more and more researchers who consider them as valuable alternative design for robotic mechanisms (Asada et al, 1986, Fu et al., 1987, Craig, 1989). As stated by a number of authors (Tsai, 1999), conventional serial kinematic machines have already reached their dynamic performance limits, which are bounded by high stiffness of the machine components required to support sequential joints, links and actuators. Thus, while having good operating characteristics (large workspace and high flexibility), serial manipulators have disadvantages of low precision, low stiffness and low power. Also, they are generally operated at low speed to avoid excessive vibration and deflection.

Conversely, parallel kinematic machines offer essential advantages over their serial counterparts (lower moving masses and higher rigidity) that obviously should lead to higher dynamic capabilities. However, most existing parallel manipulators have limited and complicated workspace with singularities, and highly non-isotropic input/output relations (Angeles, 2002). Hence, the performances may significantly vary over the workspace and depend on the direction of the motion, which is a serious disadvantage for machining applications. Research in the field of parallel manipulators began with the Stewart-platform used in flight simulators (Stewart, 1965). Many such structures have been investigated since then, which are composed of six linearly actuated legs with different combinations of link-to-platform connections (Merlet, 2000). In recent years, several new kinematic structures have been proposed that possess higher isotropy. In particular, a 3-dof translational mechanism with gliding foot points was found in three separate works to be fully isotropic throughout the Cartesian workspace (Carricato et all, 2002 andKong et all, 2002). It consists of a mobile platform, which is connected to three orthogonal linear drives through three identical planar 3-revolute jointed serial chains. Although this manipulator behaves like a conventional Cartesian machine, bulky legs are required to assure stiffness because these legs are subject to bending.

In this paper, the Orthoglide manipulator proposed by Wenger et all, 2000, is studied. As follows from previous research, this manipulator has good kinetostatic performances and some technological advantages, such as (i) symmetrical design consisting of similar 1-d.o.f. joints; (ii) regular workspace shape properties with bounded velocity amplification factor; and (iii) low inertia effects (Chablat et all, 2003). This article analyses the kinematics and the workspace of the Orthoglide. Section 2 describes the Orthoglide geometry. Section 3 proposes new solutions for its inverse and direct kinematics. Sections 4, 5 present a detailed analysis of the workspace and jointspace respectively. Finally, Section 6 summarises the main contributions of the paper.

Manipulator geometry

The kinematic architecture of the Orthoglide is shown in Fig. 1. It consists of three identical kinematic chains that are formally described as PRPaR, where P, R and Pa denote the prismatic, revolute, and parallelogram joints respectively. The mechanism input is made up by three actuated orthogonal prismatic joints. The output body is connected to the prismatic joints through a set of three kinematic chains. Each chain includes a parallelogram, so that the output body is restricted to translational movements. To get the Orthoglide kinematic equations, let us locate the reference frame at the intersection of the prismatic joint axes and align the coordinate axis with them (Fig. 2), following the "right-hand" rule. Let us also denote the input vector of the prismatic joints variables as  

,,

      2 22 2 2 2 2 2 2 2 2 2
,,

x x y z x y y z x y z z p p p L p p p L p p p L                (1)
where L is the length of the parallelogram principal links and the "zero" position

  0 0, 0, 0  p corresponds to the joints variables   ,, LLL   
. It should be stressed that the Orthoglide geometry and relevant manufacturing technology impose the following constraints on the joint variables 0 2 ; 0 2 ; 0 2

x y z L L L          , (2)
which essentially influence on the workspace shape. While the upper bound ( 2L   ) is implicit and obvious, the lower one ( 0   ) is caused by practical reasons, since safe mechanical design encourages avoiding risk of simultaneous location of prismatic joints in the same point of the Cartesian workspace (here and in the following sections, while referring to symmetrical constraints are subscript omitted, i.e.

 

,,

x y z      ).

Orthoglide Kinematics

Inverse kinematics

For the inverse kinematics, the position of the end-point ( ,,

x y z p p p ) is treated as known and the goal is to find the joint variables ( ,,

x y z    )
that yield the given location of the tool. Since in the general case the inverse kinematics can produce several solutions corresponding to the same tool location, the solutions must be distinguished with respect to the algorithm "branch". For instance, if the aim is to generate a sequence of points to move the tool along an arc, care must be taken to avoid branch switching during motion, which may cause inefficient (or even impossible) manipulator motions. Moreover, leg singularities may occur at which the manipulator loses degrees of freedom and the joint variables become linearly dependent. Hence, the complete investigation of the Orthoglide kinematics must cover all the above-mentioned topics.

From the Orthoglide geometrical model ( 1), the inverse kinematic equations can be derived in a straightforward way as:

222 x x x y z p s L p p      222 y y y x z p s L p p      222 z z z x y p s L p p      (3)
where sx, sy, sz are the branch (or configuration) indices that are equal to 1. It is obvious that (3) yields eight different branches of the inverse kinematic algorithm, which will be further referred to as PPP, MPP…MMM following the sign of the corresponding index (i.e. the notation MPP corresponds to the indices 1; 1; 1

x y z s s s       ).
The geometrical meaning of these indices is illustrated by Fig. 2, where x, y, z are the angles between the bar links and the corresponding prismatic joint axes. It can be proved that

1 s  if oo (90 ,180 )   and 1 s  if oo (0 ,90 )   . The branch transition ( o 90  
) corresponds to the serial singularity (where the leg is orthogonal to the relevant translational axis and the input joint motion does not produce the end-point displacement). It is obvious that if the inverse kinematic solution exists, then the target point (px, py, pz) belongs to a volume bounded by the intersection of three cylinders

  2 2 2 2 2 2 2 2 2 ;; L x y x z y z C p p L p p L p p L        p (4)
that guarantees non-negative values under the square roots in (3). However, it is not sufficient, since the lower joint limits (2) impose the following additional constraints

222 x x y z p s L p p     ; 222 y y x z p s L p p     ; 222 z z x y p s L p p     (5)
which reduce a potential solution set. For example, it can be easily computed that for the "zero" workspace point

  0 0, 0, 0  p , the inverse kinematic equations (3) give eight solutions   ,, LLL     
but only one of them is feasible. To analyse in details the influence of the joint constraints impact, let us start from separate a study of the inequalities (5) and then summarise results for all possible combinations of the three configuration indices. If 1

x s  , then consideration of two cases, 0

x p  and 0

x p  , yields the following workspace set satisfying the constraint 0

x       2 2 2 2 | 0 | 0; x L L x L x x y z W C p C p p p p L          pp  (6)
which consists of two fractions (½ of the cylinder intersection denoted L C and ½ of the sphere whose geometric center is (0,0,0) and radius is L).

If 1

x s  , then the second case 0

x p  does not give any solution and the joint constraint 0

x   is expressed in the workspace as

  2 2 2 2 | 0; x L L x x y z W C p p p p L        p . ( 7
)
The latter defines a solid bounded by three cylindrical surfaces and the sphere. The remaining constrains 0 y   and 0 z   can be derived similarly, which differ from ( 6), ( 7) by subscripts only.

Then, there can be found intersection of the obtained sets for different combinations of the configuration indices. It can be easily proved that the case "PPP" yields

    2 2 2 2 | , , 0 | PPP L L x y z L x y z W C p p p C p p p L        pp  (8)
while the remaining cases give

    2 2 2 2 ... | , , 0 | MPP MMM L L L x y z L x y z W W C p p p C p p p L          pp  (9)
Expressions ( 8) and ( 9) can be put in the form

;

...

PPP MPP MMM L L L L L L W S G W W G      (10)
where

  2 2 2 2 L L x y z S C p p p L      p ;   2 2 2 2 | , , 0; L L x y z x y z G C p p p p p p L       p ;
LL SG   . Therefore, for the considered positive joint limits (2), the existence of the inverse kinematic solutions may be summarised as follows (i) inside the sphere SL there exist exactly one inverse kinematic solution PPP with positive configuration indices sx, sy, sz, (ii) outside the sphere SL , but within the positive part of cylinder intersection CL, there exist 8 solutions of the inverse kinematics (PPP, MPP, … MMM) corresponding to all possible combinations of the configuration indices sx, sy, sz . These conclusions may be illustrated when 1 L  by numerical examples. If the target point =(-0.5, 0.4, 0.3) p is within the sphere SL, then the joint coordinates must be taken from the sets   0.37,-1.37

x   ,   1.21, -0.41 x   ,   1.07,-0.47 x  
, which allow only one positive combination. In contrast, for the target point =(0.7, 0.7, 0.7) p , which is outside the sphere, the inverse kinematics yields solutions with two positive values

 

, , 0.84, 0.56

x y z    
that allow 8 positive combinations of the joint variables. An interesting feature is that intermediate cases (with 2 or 4 feasible solutions) are not possible.

Direct kinematics

For the direct kinematics, the values of the joint variables (x, y, z) are known and the goal is to find the tool centre point location (px, py, pz) that corresponds to the given joint positions. While, in general, the inverse kinematics of parallel mechanisms is straightforward, the direct kinematics is usually very complex. The Orthoglide has the advantage leave an analytical direct kinematics. Like for the previous section, the solutions must be distinguished with respect to the algorithm "branch" that should be also defined both geometrically and algebraically, via a configuration index.

To solve the system (1) for px, py, pz, first, let us derive linear relations between the unknowns. By subtracting three possible pairs of the equations (1), we leave

2 2 2 2 2 2 2 2 2 2 2 2
, ,

x x y y x y x x z z x z y y z z y z p p p p p p                      (11)
As follows from these expressions, the relation between px, py, pz may be presented as

/ 2 / ; / 2 / ; / 2 / x x x y y y z z z p t p t p t             , ( 12
)
where t is an auxiliary scalar parameter. From a geometrical point of view, the expression (12) defines the set of equidistant points for the prismatic joint centres (Fig. 6). Also, it can be easily proved that the full set of equidistant points is the line perpendicular to  and passing through ( ,,

x y z    )/2, where.   | / / / 1 x x y y z z p p p         p (13)
After substituting (12) into any of the equations (1), the direct kinematic problem is reduced to the solution of a quadratic equation in the auxiliary variable t,

2 0 At Bt C    , ( 14
)
where

2 2 ( ) ( ) ( ) x y x z y z A          , 2 () x y z B     , 2 2 2 2 ( 4 ) / 4 x y z C L B      . The quadratic formula yields two solutions 2 ( 4 ) /(2 ); 1 t B m B AC A m       (15)
that geometrically correspond to different locations of the target point P (see Fig. 6) with respect to the plane passing through the prismatic joint centres (it should be noted that the intersection point of the plane and the set of equidistant point corresponds to

  0 2 t B A 
). Hence, the Orthoglide direct kinematics is solved analytically, via the quadratic formula ( 14) for the auxiliary variable t and its substitution into expressions ( 12). The direct kinematic solution exists if and only if the joint variables satisfy the inequality 2 4 B AC 

, which defines a closed region in the joint variable space

     2 2 2 2 2 2 2 | 4 1 L x y z x y z L                 (16)
Taking into account the joint limits (2), the feasible joint space may be presented as

 

| , , 0

L L x y z          
. Therefore, for the considered positive joint limits (2), the existence of the direct kinematic solutions may be summarised as follows: (i) inside the region L   , there exist exactly two direct kinematic solutions, which differ by the target point location relative to the plane  (Fig. 7a). (ii) On the border of the region L   located inside the first octant, there exist a single direct kinematic solution, which corresponds to the "flat" manipulator configuration, where both the target point and prismatic joint centres belong to the plane  (Fig. 7b).  , the direct kinematics yields a single solution 16

x y z p p p    corresponding to the "flat" configuration (see Fig. 7b).

Configuration indices

As follows from the previous sub-sections, both the inverse and direct kinematics of the Orthoglide may produce several solutions. The problem is how to define numerically the configuration indices, which allow choosing among the corresponding algorithm branches.

For the inverse kinematics, when the configuration is defined by the angle between the leg and the corresponding prismatic joint axis, the decision equations for the configuration indices are trivial:

      sgn ; sgn ; sgn x x x y y y z z z s p s p s p          Geometrically, 0 s  means that (see Fig. 2),   , , / 2 3 / 2 x y z       .
For the direct kinematics, the configuration is defined by the end-point location relative to the plane that passes through the prismatic joint centres (see Figs. 67). Hence, the decision equation may be derived by analysing the dot-product of the plane normal vector   It should be stressed that the feasible solutions for the inverse/direct kinematics, located in the neighbourhood of the "zero" point, have the configuration indices 1

x y z s s s     and 1 m  .

Workspace analysis

As follows from Eq. ( 10), Orthoglide workspace WL is composed of two fractions: (i) the sphere SL of radius L and centre point (0, 0, 0), and (ii) the thin non-convex solid GL, which is located in the first octant and bounded by the surfaces of the sphere SL and the cylinder intersection CL. It can be proved that the volume of CL, SL and WL is defined by the expressions

  3 ( ) 8 2 2 L Vol C L  ,   3 ( ) 2 2 / 6 L Vol G L     ,   3 ( ) 2 7 / 6 2 L Vol W L     (17)
As follows from ( 17), the Orthoglide with the joint limits (2) uses about 53% of the workspace PPP V of its serial counterpart (a Cartesian PPP machine with 222 LLL  workspace). Also, the volume of GL ( 3 0.062 L ) is insignificant in comparison to the volume of sphere SL ( 3 4.189 L ), which is equal to 52% of PPP V . On the other hand, releasing the lower joint limit ( 0   ) leads to an increases the workspace volume of up to 59% of PPP V only, since the volume of the workspace is, then, equal to L C . The mutual location of GL and SL (and their size ratio) may be also evaluated by the intersection points of the first octant bisector. In particular, for the sphere SL the bisector intersection point is located at distance 1 3 0.58  from the origin, while for the solid GL the corresponding distance is 1 2 0.71  (assuming that L=1). Moreover, GL touches the sphere by its circular edges, which are located on the borders of the first octant.

Joint Space Analysis

The properties of the feasible jointspace are essential for the Orthoglide control, in order to avoid impossible combinations of the prismatic joint variables x, y, z, which are generated by the control system and are followed by the actuators. For serial manipulators, this problem does not usually exist because the jointspace is bounded by a parallelepiped and mechanical limitations of the joint values may be verified easily and independently. For parallel manipulators, however, we needs to check both (i) separate input coordinates (to satisfy the joint limits), and (ii) their combinations that must be feasible to produce a direct kinematic solution.

As follows from Sub-Section 3.3, the Orthoglide jointspace L   is located within the first octant and is bounded by a surface, which corresponds to a single solution of the direct kinematics. Therefore, the jointspace boundary is defined by the relation 2 4 B AC  (see equation ( 14)), which may be rewritten as As follows from its analyses, the bounding surface is close to the 1/8th of the sphere S2L. At the edges, which are exactly quarters of the circles of the radius 2L, the surface touches the sphere. However, in the middle, the surface is located out of the sphere. In particular, the intersection point of the first octant bisector is located at the distance 3 2 1.22  from the coordinate system origin for the jointspace border and at the distance 2 3 1.15  for the sphere S2L (assuming L=1).

    2 2 2 2 2 2 2 41 x y z x y z L              (

Conclusions

This article focuses on the kinematics and workspace analysis of the Orthoglide, a 3-DOF parallel mechanism with a kinematic behaviour close to the conventional Cartesian machine taking into account the specific manufacturing constraints in the joint variables. We proposed a formal definition of the configuration indices and developed new simple analytical expressions for the Orthoglide inverse/direct kinematics. It was proved that, for the considered joint limits, the Orthoglide workspace is composed of two fractions, the sphere and a thin non-convex solid in which there are 1 and 8 inverse kinematic solutions, respectively. The total workspace volume comprises about 53% of the corresponding serial machine workspace, where over 52% belongs to the sphere (for comparison, releasing of the joint limits yields to an increase of up to 59% in the workspace volume). It was also shown, that the Orthoglide jointspace is bounded by surface with circular edges, which is more

  Fig. 1. Orthoglide architecture Fig. 2. Orthoglide geometrical model Using this notation, the kinematic equations of the Orthoglide can be written as follows

Fig. 6 .

 6 Fig. 6. Geometrical solution of the direct kinematics

  18)and solved for x assuming that y, z are known,  0 is the length of the vector , and (ex, ey, ez) are the components of the unit direction vector, which are expressed via two angles ,

																			D	42 DE  	0   , E
																			xx
	where		22 D   ;		2 EL  2 2 4   	. However, this equation is non-
									yz					yz
	symmetrical with respect to	,,    and, therefore, is not convenient
																	x	y	z
	the real-time control. An alternative way to obtain the jointspace
	boundary, which is more computationally efficient, is based on the
	conversion	from				Cartesian	to	spherical	coordinates,
		x		x e t	,, y y e t  		z		z e t	, where t  with	e		cos cos 	,	e		cos sin 	,
																			x	y
	z e		sin		, where	,  		 0, / 2 		. For such a notation, the original equation
	(18) is transformed into a linear equation for 2 t , 	F	 22 14 t L F 	,
	222 y z e e  x F e    , with an obvious solution	t	 21  L F F  .

H. Asada and J.J. Slotine, (1986),"Robot Analysis and Control," John Wiley & Sons. K.S. Fu, R. Gonzalez and C.S.G. Lee, (1987), "Robotics: Control, Sensing, Vision, and Intelligence," McGraw-Hill. J.J. Craig, (1989), "Introduction to Robotics: Mechanics and Control," Addison Wesley. L.W. Tsai, (1999), "Robot Analysis, The Mechanics of Serial and Parallel Manipulators," John Wiley & Sons. J. Angeles, (2002), "Fundamentals of Robotic Mechanical Systems," Springer-Verlag. D. Stewart, (1965), "A Platform with Six Degrees of freedom," Proc. of the Institution of Mechanical Enginners, Vol. 180/1, No. 15, 371-386. J.-P. Merlet, (2000), "Parallel Robots," Kluwer Academic Publishers. M. Carricato and V. Parenti-Castelli, (2002), "Singularity-Free Fully-Isotropic Translational Parallel Mechanisms,'' The International Journal of Robotics Research Vol. 21, No. 2, 161-174. X. Kong, and C.M. Gosselin, (2002), "A Class of 3-DOF Translational Parallel Manipulators with Linear I-O Equations," Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, Quebec, Canada, pp. 161-174. Ph. Wenger and D. Chablat, (2002), "Kinematic Analysis of a New Parallel Machine Tool: the Orthoglide", 7th International Symposium on Advances in Robot Kinematics, Piran-Portoroz, Slovenia, pp. 305-314. D. Chablat and Ph. Wenger, (2003), "Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide ", IEEE Trans. on Robotics and Automation, Vol. 19/3, pp. 403-410.

convex than the sphere but is rather close to it. These results can be further used for the optimisation of the Orthoglide parameters, which is the subject of our future research work.


Maher Baili Philippe Wenger
email: philippe.wenger@irccyn.ec-nantes.fr

Damien Chablat

A WORKSPACE BASED CLASSIFICATION OF 3R ORTHOGONAL MANIPULATORS

Keywords: Classification, Workspace, Singularity, Cusp, node, orthogonal manipulator 1

A classification of a family of 3-revolute (3R) positioning manipulators is established. This classification is based on the topology of their workspace. The workspace is characterized in a half-cross section by the singular curves of the manipulator. The workspace topology is defined by the number of cusps and nodes that appear on these singular curves. The design parameters space is shown to be partitioned into nine subspaces of distinct workspace topologies. Each separating surface is given as an explicit expression in the DH-parameters.

Introduction

This paper focuses on positioning 3R manipulators with orthogonal joint axes (orthogonal manipulators). Orthogonal manipulators may have different global kinematic properties according to their link lengths and joint offsets. Unlike usual industrial manipulators, orthogonal manipulators may be cuspidal, that is, they can change their posture without meeting a singularity [START_REF] Parenti | Position Analysis of Robot Manipulators: Regions and Sub-regions[END_REF]Innocenti, 1988, Burdick, 1988). This property was unknown before 1988 (Borrel and Liégeois, 1988). Several years later, some conditions for a manipulator to be noncuspidal were provided, which include simplifying geometric conditions like parallel and intersecting joint axes [START_REF] Burdick | A classification of 3R regional manipulator sin-gularities and geometries[END_REF] and also nonintuitive conditions [START_REF] Wenger | Design of cuspidal and noncuspidal manipulators[END_REF]. A general necessary and sufficient condition for a 3-DOF manipulator to be cuspidal was established in (El [START_REF] Omri | How to recognize simply a non-singular posture changing 3-DOF manipulator[END_REF], namely, the existence of at least one point in the workspace where the inverse kinematics admits three equal solutions. The word "cuspidal manipulator" was defined in accordance to this condition because a point with three equal IKS forms a cusp in a cross section of the workspace [START_REF] Arnold | Singularity Theory[END_REF]. The categorization of all generic 3R manipulators was established in [START_REF] Wenger | Classification of 3R positioning manipulators[END_REF] based on the homotopy class of the singular curves in the joint space. [START_REF] Wenger | Some guidelines for the kinematic design of new Manipulators[END_REF] proposed a procedure to take into account the cuspidality property in the design process of new manipulators. More recently, Corvez and Rouillier, 2002 applied efficient algebraic tools to the classification of 3R orthogonal manipulators with no offset on their last joint. Five surfaces were found to divide the parameters space into 105 cells with the same number of cusps in the workspace. The equations of these five surfaces were derived as polynomials in the DH-parameters using Groebner Bases. A kinematic interpretation of this theoretical work showed that, in fact, only five different domains exist : two domains of noncuspidal manipulators, one domain where manipulators have two cusps and two domains where they have four cusps [START_REF] Baili | Classification of one family of 3R positioning manipulators[END_REF]. However, the authors did not provide the equations of the true separating surfaces in the parameters space. On the other hand, they did not take into account the occurrence of nodes, which play an important role for analyzing the number of IKS in the workspace.

The purpose of this work is to classify a family of 3R positioning manipulators according to the topology of their workspace, which is defined by the number of cusps and nodes that appear on the singular curves. The design parameters space is shown to be divided into nine domains of distinct workspace topologies. In each domain, the distribution of the number of IKS is the same. This study is of interest for the design of new manipulators.

The rest of this article is organized as follows. Next section presents the manipulators under study and recalls some preliminary results. The classification is established in section 3. Section 4 synthesizes the results and section 5 concludes this paper.

Preliminaries

Manipulators under study

The manipulators studied in this paper are orthogonal with their last joint offset equal to zero. The remaining lengths parameters are referred to as d2, d3, d4, and r2 while the angle parameters 2 and 3 are set to -90° and 90°, respectively. The three joint variables are referred to as 1, 2 and 3, respectively. They will be assumed unlimited in this study.

Figure 1 shows the kinematic architecture of the manipulators under study in the zero configuration. The position of the end-tip (or wrist center) is defined by the three Cartesian coordinates x, y and z of the operation point P with respect to a reference frame (O, x, y, z) attached to the manipulator base as shown in Fig. 1.

Singularities and aspects

The determinant of the Jacobian matrix of the orthogonal manipulators under study is det(J) = (d3 + c3d4)(s3d2 + c2(s3d3 -c3r2)), where c i =cos( i ) and s i =sin( i ). A singularity occurs when det(J)=0. The contour plot of det(J)=0 forms a set of curves in

23 ,             . If d3>d4
, the first factor of det(J) cannot vanish and the singularities form two distinct curves S1 and S2 in the joint space [START_REF] Omri | Kinematic analysis of robotic manipulators[END_REF], which divide the joint space into two singularity-free open sets A1 and A2 called aspects (Borrel and Liégeois, 1988). The singularities can also be displayed in the Cartesian space (Kholi andHsu, 1987, Ceccarelli, 1996). Thanks to their symmetry about the first joint axis, a 2-dimensional representation in a half cross-section of the workspace is sufficient. The singularities form two disjoint sets of curves in the workspace. These two sets define the internal boundary WS1 and the external boundary WS2, respectively, with WS1=f(S1) and WS2=f(S2). Figure 2 shows the singularity curves when d2=1, d3=2, d4=1.5 and r2=1. For this manipulator, the internal boundary WS1 has four cusp points. It divides the workspace into one region with two IKS (the outer region) and one region with four IKS (the inner region).

If d3d4, the operation point can meet the second joint axis whenever 3=arccos(-d3/d4) and two horizontal lines appear in the joint space

23 ,            
, which may intersect S1 and S2 depending on d2, d3, d4 and r2 (El [START_REF] Omri | Kinematic analysis of robotic manipulators[END_REF]. The number of aspects depends on these intersections. Note that if d3<d4, no additional curve appears in the workspace cross-section but only two points. This is because, since the operation point meets the second joint axis when 3=arccos(-d3/d4), the location of the operation point does not change when 2 is rotated.

Workspaces classification

Classification criteria

The classification is conducted on the basis of the topology of the singular curves in the workspace, which we characterize by (i) the number of cusps and (ii) the number of nodes. A cusp (resp. a node) is associated with one point with three equal IKS (resp. with two pairs of equal IKS). These singular points are interesting features for characterizing the workspace shape and the distribution of the number of IKS in the workspace.

Number of cusps

For now on and without loss of generality, d2 is set to 1. Thus, only three parameters d3, d4 and r2 need to be handled. [START_REF] Baili | Classification of one family of 3R positioning manipulators[END_REF] showed that one or more surfaces among the five ones found by Corvez and Rouillier, 2002, are not relevant. However, they did not try to find which surfaces are really separating. To derive the equations of the separating surfaces, we investigate the transitions between the five domains. First, let us recall the five different manipulator types associated with the five domains found by Corvez and Rouillier, 2002. The first type is a binary manipulator (i.e. it has only two IKS) with no cusp and a hole (Fig. 3). The remaining four types are quaternary manipulators (i.e. with four IKS). The second type is a manipulator with four cusps on the internal boundary. There are three instances of type 2 according to the number of nodes, as will be shown in section 3.3. The first one is shown in Fig. 4 with two nodes. The second one was shown in Fig. 2 with no node. The last one is such that d3d4 and will be shown in Fig. 10.

Transition between type 1 and type 2 is a manipulator having a pair of points with four equal IKS, where two nodes and one cusp coincide. Proof of this result can be obtained as a straightforward consequence of transitions in quartics root patterns [START_REF] Baili | Classification of 3R Orthogonal positioning manipulators[END_REF]. Deriving the condition for the inverse kinematic polynomial to have four equal roots yields the equation of the separating surface [START_REF] Baili | Classification of 3R Orthogonal positioning manipulators[END_REF]

2 2 2 2 2 22 3 2 3 2 4 3 2 () 1 2 d r d r d d r AB             (1)
where

22 22 3 2 3 2
( 1) and ( 1)

A d r B d r      
The third type is a manipulator with only two cusps on the internal boundary, which looks like a fish with one tail (Fig. 5).

As shown by [START_REF] Baili | Classification of 3R Orthogonal positioning manipulators[END_REF], transition between type 2 and type 3 is characterized by a manipulator for which the singular line given by 3=arccos(-d3/d4) is tangent to the singularity curve S1. Expressing this condition yields the equation of the separating surface The fourth type is a manipulator with four cusps. Unlike type 2, the cusps are not located on the same boundary (Fig. 6).

Transition between type 3 and type 4 is characterized by a manipulator for which the singular line given by 3=-arccos(-d3/d4) is tangent to the singularity curve S2 [START_REF] Baili | Classification of 3R Orthogonal positioning manipulators[END_REF]. Expressing this condition yields the equation of the separating surface

3 43 3 and 1 1 d d B d d     (3)
Last type is a manipulator with no cusp (Fig. 7).

Unlike type 1, the internal boundary does not bound a hole but a region with 4 IKS. The two isolated singular points inside the inner region are associated with the two singularity lines. Transition between type 4 and type 5 is characterized by a manipulator for which the singular line given by 3=+arccos(-d3/d4) is tangent to the singularity curve S1 [START_REF] Baili | Classification of 3R Orthogonal positioning manipulators[END_REF]. We have provided the equations of four surfaces that divide the parameters space into five domains where the number of cusps is constant. Figure 8 shows the plots of these surfaces in a section (d3, d4) of the parameter space for r2=1. Domains 1, 2, 3, 4 and 5 are associated with manipulators of type 1, 2, 3, 4 and 5, respectively. C1, C2, C3 and C4 are the right hand side of ( 1), ( 2), ( 3) and ( 4), respectively. It is interesting to see the correspondence between the equations found with pure algebraic reasoning in (Corvez and Rouillier, 2002) and those provided in this paper. The five equations found by Corvez and Rouillier are:

0 d d r d     (5) 2 2 2 3 4 2 0 d d r    (
               (7) 2 3 4 2 2 2 2 3 2 3 3 3 4 3 4 3 4 2 2 0 d r d d d d d d d d        (8) 2 2 2 3 4 2 2 2 2 3 2 3 3 3 4 3 4 3 4 2 2 0 d r d d d d d d d d        (9)
Equation ( 7) is a second-degree polynomial in d 4 2 . Solving this quadratics for d 4 shows that (7) has the following two branches:

2 2 2 2 2 22 3 2 3 2 4 3 2 () 1 2 d r d r d d r AB               or 2 2 2 2 2 22 3 2 3 2 4 3 2 () 1 2 d r d r d d r AB              
The first branch is the separating surface d 4 =C1 between domains 1 and 2.

Equation ( 8) is a second-degree polynomial in d 4 . Solving this quadratics for d 4 and assuming strictly positive values for d 4 and r2 yields the following two branches for (8): and1) 1 and1) 1

3 43 3 (
d d B d d     or 3 43 3 (
d d B d d    
These two branches are the separating surfaces d 4 =C3 and d 4 =C4, respectively. In the same way, (9) can be rewritten as d 4 =C2.

In conclusion, ( 5) and ( 6) do not define separating surfaces and only one branch of (7) defines a separating surface.

Number of nodes

In this section, we investigate each domain according to the number of nodes in the workspace.

Domain 1: Since all manipulators in this domain are binary, they cannot have any node in their workspace. Thus, all manipulators in domain 1 have the same workspace topology, namely, 0 node, 0 cusp and a hole inside their workspace. This workspace topology is referred to as WT1 (Workspace Topology #1).

Domain 2: Figures 4 and 2 show two distinct workspace topologies of manipulators in domain 2, which feature 2 nodes and 0 node and which we call WT2 and WT3, respectively. Transition between these two workspace topologies is one such that the two lateral segments of the internal boundary meet tangentially (Fig. 9). Equation of this transition can be derived geometrically and the following equation is found [START_REF] Baili | Classification of 3R Orthogonal positioning manipulators[END_REF]:

4 ( ) / 2 d A B  (10)
As noted in section 3.2, a third topology exists in this domain, where the internal boundary exhibits a "2-tail fish". This workspace topology, which we call WT4, features two nodes like in Fig. 4, but these nodes do not play the same role. They coincide with two isolated singular points, which are associated with the two singularity lines defined by 3=arccos(-d3/d4) (the operation point lies on the second joint axis and the inverse kinematics admits infinitely many solutions). Also, the nodes do not bound a hole like in Fig. 4 but a region with four IKS (Fig. 10).

Transition between WT3 and WT4 is a workspace topology such that the upper and lower segments of the internal boundary meet tangentially (Fig. 11). As shown in [START_REF] Baili | Classification of 3R Orthogonal positioning manipulators[END_REF], this transition is the occurrence of the additional singularity d3 + c3d4 = 0. This transition is defined by: Transition between WT5 and WT6 and transition between WT8 and WT9 are such that the internal boundary meets the external boundary tangentially (Fig. 13). This transition can be derived geometrically and the following equation is found [START_REF] Baili | Classification of 3R Orthogonal positioning manipulators[END_REF]:

4 1 () 2 d A B  ( 12
)
Domains 4: Manipulators in domain 4 have four cusps and four nodes. No subcase exists in this domain [START_REF] Baili | Classification of 3R Orthogonal positioning manipulators[END_REF]. Such topologies are referred to as WT7.

Results synthesis

The partition with cusps and nodes is shown in a section (d3, d4) of the parameter space for r2=1(Fig. 14), where E1, E2 and E3 are the right hand side of (10), ( 11) and ( 12), respectively. Plots of the separating curves in sections for different values of r2 show that they deform smoothly with the same intersections when r2 varies. The areas of WT1, WT2, WT7 and WT9 increase when r2 decreases, whereas those of WT3, WT4, WT5 and WT6 decrease. The area of WT4 is very narrow when r2<1 and almost disappears when r2<0.1.

Conclusions

A family of 3R manipulators was classified according to the topology of the workspace, which was defined as the number of cusps and nodes. The design parameters space was shown to be partitioned into nine subspaces of distinct workspace topologies. Each separating surface was given as an explicit expression in the DH-parameters. This study is being extended to manipulators with r3  0. First results show that some may have 6 or 8 cusps (Fig. 15). But for small values of r3, the partition is nearly the same as in Fig. 14. The subspace WT4 does not exist any more. It is replaced by two adjacent tiny subspaces with 6 and 8 cusps. For high values of r3, the partition gets complicated since more workspace topologies exist.

  Figure 1. Orthogonal manipulators under study.

Figure 2 .

 2 Figure 2. Singularity curves in joint space (left) and workspace (right)

Figure 3 .

 3 Figure 3. Manipulator of type 1.Figure 4. Manipulator of type 2.

Figure 4 .

 4 Figure 3. Manipulator of type 1.Figure 4. Manipulator of type 2.

Figure 5 .

 5 Figure 5. Manipulator of type 3.Figure 6. Manipulator of type 4.

Figure 6 .

 6 Figure 5. Manipulator of type 3.Figure 6. Manipulator of type 4.

Figure 7 .Figure 8 .

 78 Figure 7. Manipulator of type 5.

Figure 9 .

 9 Figure 9. Transition WT2-WT3.

Figure 10 .

 10 Figure 10. Workspace topology WT4.

Figure 11 .

 11 Figure 11. Transition WT3-WT4.

Figure 12 .

 12 Figure 12. Workspace topologies WT6 (left) and WT9 (right)

7Figure 14 .

 14 Figure 14. Parameter space partition in a section r2=1.

Figure

  Figure 15. A manipulator with 8 cusps when r30.

2 4


Matthieu Caudron

Fabrice Emeriault

Marwan A L Heib

EFFONDREMENT DE CAVITÉS SOUTERRAINES A FAIBLE PROFONDEUR ET DOMMAGES INDUITS AUX OUVRAGES EN SURFACE. Approche expérimentale sur modèle analogique bidimensionnel

Le sous-sol français est riche en matière premières nécessaires aux développement de l'urbanisme et de l'activité industrielle. L'étude concerne les mouvements de terrains induits par l'effondrement de ces cavités et leurs conséquences sur les structures. Elle aborde ce sujet par des expériences en laboratoire utilisant un matériau analogique et une maquette de structure en modèle réduit. A travers des essais d'effondrement localisé (fontis) en terrain vierge ou en présence d'une maquette de structure, nous mettrons en évidence la nécessité de tenir compte de l'interaction solstructure lors de l'évaluation par l'expert de tels phénomènes.

Introduction

Les mouvements de terrain résultent de l'effondrement de cavités souterraines issues de l'activité industrielle humaine (mines ou carrières) ou formées naturellement par la circulation d'eau dans des massifs de roches solubles (calcaire, gypse). Leur impact sur le bâti existant en surface est généralement très important [START_REF] Al | Effets et conséquences des mouvements du sous-sol sur le bâti. Synthèse des principales méthodes de prévention et de protection[END_REF] comme en attestent les exemples récents des effondrements dans les mines de fer [START_REF] Deck | Taking the soil-structure interaction into account in assessing the loading of a structure in a mining subsidencec area[END_REF]] et des carrières de Paris. Il est donc nécessaire de prévoir les mouvements de sol en surface (tassements et déformation horizontale) résultants de ces phénomènes et surtout de déterminer l'influence que peut avoir la présence de structures en surface sur la forme et l'amplitude de ces mouvements [ABBASS, 2004] et [AL HEIB, 2003].

La pratique actuelle en manière de prévision des effets sur les structures consiste à déterminer les déplacements provoqués par les mouvements de terrains (mise en pente, déformation horizontale, etc.) en l'absence de structure puis d'utiliser ces résultats pour vérifier la capacité de la structure à résister au phénomène [START_REF] Verdel | Evaluation des dommages miniers au bâti par méthodes statistiques et réseaux de neurone[END_REF]. Il s'avère que cette approche peut sous-estimer ou surestimer largement les sollicitations engendrées, ce qui dans les deux cas présente des impacts négatifs (risque subsistant pour la structure dans un cas, surcoût de projet dans l'autre cas). La prise en compte de l'interaction sol-structure est un exercice délicat, nécessitant des compétences dans les domaines de la géotechnique et des structures. Cet article correspond aux premiers pas vers une évaluation réaliste de ce phénomène.

Pour mettre en évidence les phénomènes d'interaction sol-structure, l'approche adoptée utilise des essais expérimentaux sur modèles réduits employant le matériau analogique de Schneebeli [SCHNEEBELI, 1956[START_REF] Kastner R | Excavations profondes en site urbain[END_REF][START_REF] Masrouri | Comportement des rideaux de soutènement semi-flexibles : Etude théorique et expérimentale[END_REF]. Des essais en terrain vierge puis en présence d'une maquette de structure seront réalisés, puis analysés par une technique d'imagerie numérique.

L'aléa que nous étudions est la formation de fontis, vide franc remontant progressivement ou brutalement jusqu'à la surface. Cependant, nous choisissons dans un premier temps, du fait de la nature pulvérulente du matériau de Schneebeli, de simuler des affaissements de terrain. Dans une étude ultérieure, nous introduirons de la cohésion dans ce modèle de sol afin de pouvoir simuler la création d'un fontis.

Le modèle physique

Dispositif expérimental

Le dispositif expérimental utilise un modèle analogique de sol ainsi qu'un modèle réduit de structure pour simuler un cas réel de formation de fontis. Le but est de mettre en évidence les phénomènes d'interaction sol-structure. La cavité que nous simulons présente une largeur de 1.6m pour une ouverture de 3.2m, en présence d'un recouvrement de 3.2m de hauteur.

Le modèle réduit de bâtiment utilisé représente un bâtiment type maison d'habitation R+1 de trois travées employant une structure poutre-poteaux en acier. Les dimensions envisagées sont une longueur de 10m pour le bâtiment (soit environ 3.2m par travée), une hauteur de 2.7m par étage et un vide sanitaire de 1.6m.

Le sol

Le matériau analogique bidimensionnel de Schneebeli (un empilement de rouleaux cylindriques en acier inoxydable de 3, 4 et 5mm de diamètre avec une équirépartition en masse) permet de représenter un sol pulvérulent de manière fiable et pratique à mettre en oeuvre. Le poids volumique du matériau mis en place est en moyenne de 65kN/m 3 soit environ trois fois plus dense qu'un sol granulaire réel. Il présente un angle de frottement interne de 24° et une cohésion nulle.

Un massif de sol de grandes dimensions permet de représenter des situations classiques d'affaissement avec un facteur d'échelle de l'ordre de 1/10 à 1/50 [START_REF] Al | Etude sur modèle réduit bidimensionnel du champ de déplacement induit par le creusement d'un tunnel à faible profondeur[END_REF]]. Le bâti expérimental utilisé présente les dimensions suivantes : 2m de largeur pour 1m de hauteur maximum, ce qui représente environ 200 000 rouleaux.

La structure

Une maquette de bâtiment type maison d'habitation mise en place à la surface de ce massif, bien que nécessairement très simplifiée et ne respectant pas complètement les règles de similitude en terme de rigidité par exemple, permet néanmoins de faire une évaluation qualitative de l'impact de la présence d'une structure en surface. Avant d'utiliser la maquette en présence du massif de sol, elle a tout d'abord été testée sur un banc d'essais en utilisant des conditions aux limites et des chargements bien définis. Cela permet de s'assurer de la répétabilité des résultats et du comportement élastique de celle-ci, ce qui correspond à des régles de similitudes relativement simples.

Le facteur d'échelle géométrique et les règles de similitude

Le facteur d'échelle géométrique utilisé pour établir les dimensions à donner à la maquette de structure et à la cavité a été fixé essentiellement de manière à éviter les effets de bord tout en assurant une faisabilité technique des manipulations. Un facteur d'échelle géométrique de 1/40 a ainsi été retenu.

Le cadre de manipulation et l'utilisation du matériau de Schneebeli (ρ=65kN/m 3 ) nous imposent deux autres facteurs d'échelle : 1 pour g* (la gravité) et 3 pour ρ* (la masse volumique). Ces facteurs fixent la totalité des facteurs d'échelle pour les autres grandeurs.

Les dimensions de la cavité expérimentale sont donc de 0.04m de large pour 0.08m d'ouverture avec un recouvrement de 0.10m.

Représentation de l'effondrement

L'effondrement de la cavité est simulé expérimentalement en enlevant en plusieurs étapes une certaine quantité de matériau à un endroit précis. La somme des petites quantités de rouleaux enlevés initialement repérés par une coloration blanche (Cf. Fig. 2) correspond au volume total de la cavité à créer. La prise de photographies numériques à chaque étape permet, après traitement, de suivre la propagation des déplacements dans le massif de sol jusqu'à la surface en fonction du volume de sol enlevé.

Etape 16 (finale) Etape 13 Etape 7 Etape initiale

Exploitation des résultats

Les différentes expériences réalisées sont analysées du point de vue des déformations induites dans le massif de sol et la structure durant le développement de l'affaissement. Le traitement de la série d'images numériques prises au cours de l'expérience se fait par une technique de corrélation directe (logiciel Sifasoft développé par F. MORESTIN au LAMCOS, INSA de Lyon, [START_REF] Mguil-Touchal | Mesure de champ de déplacements et de déformations par corrélation d'images numériques[END_REF]).

L'Appareil Photo Numérique utilisé dispose d'une résolution de 3 millions de pixels. un ordinateur. Le logiciel de corrélation d'images numériques qui permet d'obtenir le champ de déformations fonctionne sur le principe suivant (Fig. 3 et 4):

• Transformation des photos en niveau de gris (256 niveaux).

• Maillage des images en utilisant des motifs et un pas de maillage (tous deux définis par l'utilisateur) • Calcul des composantes de la déformation pour chaque motif du maillage pour un couple de photos traité. L'ensemble des déformations des motifs forme le champ de déformations du massif de sol. • La précision de la méthode est de l'ordre du 1/100 eme de pixel, soit environ 10µm dans notre cas. Cela est bien inférieur à la précision permise par l'APN qui est de 0.05mm. Ainsi nous pouvons reconstituer l'evolution des déformations étape par étape, à partir du début de l'expérience, ce qui nous permet de suivre les paramètres suivants : la surface du massif de sol, les déformations à certaines profondeurs dans le massif et les caractéristiques géométriques des tassements en surface (tassement maximal, largeur d'influence,...). Deux essais sans la présence d'une structure permettent d'une part d'établir une référence telle que communément faite par la pratique actuelle en terme de prévision des effets sur les structures (on considère le massif de sol seul, puis la structure) et d'autre part d'évaluer la répétitivité de la procédure.

Celle-ci étant satisfaisante, seul un essai est représenté dans la suite, en particulier au travers de l'évolution de la forme et de 'amplitude des déplacements en surface. La figure 5 montre l'évolution de la cuvette d'affaissement obtenue pour les 16 étapes. La forme des cuvettes d'affaissement légèrement dissymétrique peut certainement être expliquée par la méthode utilisée pour créer la cavité. En effet, lorsque l'on cherche à enlever des rouleaux, il est difficile de s'assurer que l'on ne privilégie pas un côté de la cavité plutôt qu'un autre.

16 étapes ont été nécessaires pour créer la totalité de la cavité (Fig. 5). [START_REF] Caudron | Effondrement de cavités souterraines et interaction avec les ouvrages en surface : approche expérimentale sur modèle analogique bidimensionnel[END_REF] ont montré que, pour chaque étape de réalisation d'une cavité dans le massif de sol analogique, la cuvette d'affaissement peut être approximée par une courbe de Gauss (utilisée normalement pour des cavités circulaires) : Les différents paramètres caractéristiques S max , i, D et C, (respectivement le tassement maximal, la distance au point d'inflexion, le diamètre équivalent de la cavité et la hauteur de couverture) sont estimés de manière relativement fiable et ont respectivement les valeurs suivantes : 0,0162m, 0,058m, 0,062m et 0,01m,. Les valeurs numériques sont déterminées selon la méthode explicitée par [START_REF] Caudron | Effondrement de cavités souterraines et interaction avec les ouvrages en surface : approche expérimentale sur modèle analogique bidimensionnel[END_REF].

Les surfaces des deux courbes (théorique et expérimentale) diffèrent peu, en effet l'écart relatif est de 1,4% pour les courbes finales. La perte de volume associé est d'environ 75%, ce qui dénote un foisonnement du matériau effondré important. Le même essai d'affaissement est réalisé avec la maquette de structure présente en surface à l'aplomb de la cavité. Le positionnement de celle-ci est tel que la semelle numérotée 2 est placée sur l'axe de la cavité (Fig. 7).

La même procédure d'effondrement de la cavité est utilisée. En revanche, à la mesure des déformations du massif de sol par imagerie numérique, viennent s'ajouter des mesures de tassements des deux semelles externes (1 et 4) par comparateur ainsi qu'une instrumentation par jauges de déformations des travées et des poteaux de la structure. Cela permet d'avoir un accès aux sollicitations qui se développent dans le modèle durant l'essai.

Deux essais semblables sont réalisés dans cette configuration afin de s'assurer de la répétabilité des résultats obtenus. Aucune différence notable n'apparaît entre les deux essais.

La corrélation d'images numériques fournit les déformations dans le massif de sol : la figure 8 donne l'évolution de la surface du massif de sol durant l'un des essais.

La forme des tassements est fortement perturbée par la présence de la structure en surface durant l'essai. Des bourrelets de sol remontent entre les semelles ainsi qu'aux extrémités de la maquette.

Si l'on calcule le volume des tassements apparus en surface, on obtient une valeur assez éloigné du volume de sol enlevé pour créer la cavité : la perte de volume estimée est de 84%, soit significativement supérieure à celle obtenue lors de l'essai en terrain vierge. Le tassement différentiel maximum se situe entre les semelles 3 et 4 pour la dernière étape et correspond à une pente de 8% environ. C'est la moitié du résultat obtenu en appliquant les résultats obtenus sur l'essai en terrain vierge à la structure.

Les deux comparateurs positionnés sur chacune des semelles externes fournissent des déplacements verticaux cohérents avec les valeurs déterminés par l'imagerie numérique (Fig. 9), ce qui prouve la fiabilité de cette dernière technique. Les différences observables sont faibles et peuvent être imputées aux imprécisions de chaque méthode de mesure et à la nature du matériau analogique employé (les rouleaux de Schneebeli) : localement sous une semelle, un rouleau peut se déplacer de manière importante sans que cela ne soit totalement détecté par l'imagerie numérique (hypothèse de milieu continu). Il apparaît que les différences par rapport aux essais en terrain vierge sont importantes. La présence d'une structure en surface a complètement modifié la forme des tassements ainsi que leurs amplitudes.

L'amplitude maximale, beaucoup plus importante (+30% environ), n'est de plus pas située au même endroit que dans le cas de l'essai en terrain vierge. C'est la semelle 1 qui subit le plus important tassement, alors que l'axe de la cavité est placé en dessous de l'une des semelles centrales.

La perte de volume enregistrée est voisine de 84%, c'est-à-dire supérieure de 10% à celle observée en terrain vierge :cela peut être du à la présence de la surcharge engendrée par la structure. Il semble donc que le foisonnement est plus faible lorsqu'une structure est présente en surface.

De plus, à la fin de l'essai, il apparaît que la maquette de structure a plastifiée de manière importante au niveau de la semelle 4 (semelle la plus éloignée de la cavité) : la plage de sollicitations pour laquelle la structure a été caractérisée a donc été dépassée. Il semblerait ainsi que les efforts les plus importants subis par celle-ci n'étaient pas localisés à proximité de la cavité, comme on aurait pu s'y attendre selon la pratique actuelle de prévision des effets sur les structures.

Si l'on résume les différentes observations faites jusqu'à présent, il apparaît que : • le tassement différentiel le plus important est localisé entre les semelles 2 et 3,

• le tassement maximal se situe sous la semelle 1,

• la sollicitation la plus importante est observée à proximité de la semelle 4, même si macroscopiquement, le mouvement de la structure est voisin de celui d'un corps rigide. En conclusion de cette étude, on peut retenir que les phénomènes d'interaction solstructure ne peuvent pas être négligés lors d'un affaissement de terrain résultant de l'effondrement d'une cavité relativement proche de la surface.

La pratique actuelle de prévision des effets des mouvements de sols sur les structures qui consiste à séparer l'étude de la stabilité de la cavité de celle de la structure, donne des résultats très différents de la réalité où la présence de la structure modifie profondément la forme et l'amplitude des tassements.

Une campagne expérimentale analogue sera menée dans le cas d'une représentation plus pertinente du phénomène de fontis. Ceci nécessite l'utilisation d'un modèle

Figure 1 :Figure 2 :

 12 Figure 1 : Succession de photos prises à différentes étapes

Figure 3 :

 3 Figure 3 : Les différentes phases d'une expérimentation et de l'exploitation des résultats

Figure 5 :

 5 Figure 5 : Evolution de la surface du sol durant un essai en terrain vierge.

  Cavité rectangulaire L*h ~cavitéLa figure 6 montre que ce résultat est toujours valable, même si la cavité est ici verticale.

Figure 6 :

 6 Figure 6 : Comparaison entre les mouvements en surface selon Peck et les résultats expérimentaux

Figure 7 :

 7 Figure 7 : Etat initial de la structure (avec chargement) avant effondrement de la cavité

Figure 8 :

 8 Figure 8 : Evolution des tassements durant la simulation d'un fontis en présence d'une structure en surface.

Figure 9 :

 9 Figure 9 : Comparaison entre les tassements des semelles donnés par les comparateurs et l'imagerie numérique

Figure 9 :

 9 Figure 9 : Comparaison des tassements obtenus entre l'essai en terrain vierge et l'essai en présence d'une structure. 5 Conclusions et Perspectives


Daniel Kanaan
email: daniel.kanaan@irccyn.ec-nantes.fr

Philippe Wenger
email: philippe.wenger@irccyn.ec-nantes.fr

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

Kinematics analysis of the parallel module of the VERNE machine

Keywords: Parallel manipulators, Parallel kinematic machines, inverse kinematics, forward kinematics, complex motion

I. Introduction

Parallel kinematic machines (PKM) are known for their high structural rigidity, better payload-to-weight ratio, high dynamic performances 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] Wenger Ph | A comparative study of serial and parallel mechanism topologies for machine tools[END_REF]. Thus, they are prudently considered as attractive alternatives designs for demanding tasks such as highspeed machining [START_REF] Weck | Parallel Kinematic Machine Tools -Current State and Future Potentials[END_REF].

Most of the existing PKM can be classified into two main families. The PKM of the first family have fixed foot and variable-length struts, while the PKM of the second family have fixed length struts with moveable foot points gliding on fixed linear joints [START_REF] Chablat | Architecture O ptimization of a 3-DO F Parallel Mechanism for Machining Applications, the O rthoglide[END_REF]. In the first family, we distinguish between PKM with six degrees of freedom generally called Hexapods and PKM with three degrees of freedom called Tripods [START_REF] Hervé | Structural synthesis of parallel robots generating spatial translation[END_REF], [START_REF] Kong | Type synthesis of linear translational parallel manipulators[END_REF]. Hexapods have a Stewart-Gough parallel kinematic architecture. Many prototypes and commercial hexapod PKM already exist, including the VARIAX (Gidding and Lewis), the TORNADO 2000 (Hexel). We can also find hybrid architectures like the TRICEPT machine from Neos-robotics [START_REF] Neumann | [END_REF] which is composed of a two-axis wrist mounted in series to a 3-DOF "tripod" positioning structure. In the second family, we find the HEXAGLIDE (ETH Zürich) which features six parallel and coplanar linear joints. The HexaM (Toyoda) is another example with three pairs of adjacent linear joints lying on a vertical cone [START_REF] Toyama | Machine tool having parallel structure[END_REF]. A hybrid parallel/kinematic PKM with three inclined linear joints and a two-axis wrist is the GEORGE V (IFW Uni Hanover). Many three-axis translational PKMs belong to this second family and use architecture close to the linear Delta robot originally designed by Clavel for pick-and-place operations [START_REF] Clavel | DELTA, a fast robot with parallel geometry[END_REF]. The Urane SX (Renault Automation) and the QUICKSTEP (Krause and Mauser) have three noncoplanar horizontal linear joints [START_REF] Company | Modeling and preliminary design issues of a 3-axis parallel machine tool[END_REF].

Many researches have made contributions to the study of the kinematics of these PKMs. Most of these articles focused on the discussion of both the analytical and numerical methods [START_REF] Liu | On the analysis of a new spatial three-degree-of freedom parallel manipulator[END_REF], [START_REF] Nair | On the forward kinematics of parallel manipulators[END_REF].

The purpose of this paper is to formulate analytic expressions in order to find all possible solutions for the inverse and forward kinematics problem of the VERNE machine. Then we identify these solutions in order to find the solution that satisfies the end-user. The VERNE machine is a 5-axis machine-tool that was designed by Fatronik for IRCCyN [START_REF] Martin | A new 5axes hybrid architecture machining center[END_REF], [START_REF] Terrier | VERNE -A five axis Parallel Kinematics Milling Machine[END_REF]. This machine-tool consists of a parallel module and a tilting table as shown in figure 1. The parallel module moves the spindle mostly in translation while the tilting table is used to rotate the workpiece about two orthogonal axes.

In the following three sections, we present the VERNE parallel module, its geometric equations and mobility analysis. Symbolic solutions of the inverse kinematics problem are reported in Section V. Section VI is devoted to the resolution of the forward kinematics problem. Finally a conclusion is given in Section VII. Figure 2 shows a scheme of the parallel module of the VERNE machine. The kinematic architecture can be described by a simple scheme shown in figure 3, where joints are represented by rectangles and links between those joints are represented by lines (P and S indicate prismatic and spherical joint, respectively). The moving platform is rectangular. The vertices of this platform are connected to a fixed-base plate through three legs Ι, ΙΙ and ΙΙΙ. Each leg uses pairs of rods linking a prismatic joint to the moving platform through two pair spherical joints. Legs ΙΙ and ΙΙΙ are two identical parallelograms. Leg Ι differs from the other two legs in that 11 12 [START_REF] Company | Modeling and preliminary design issues of a 3-axis parallel machine tool[END_REF]

12

A A B B 

, that is, it is not an articulated parallelograms. The movement of the moving platform is generated by the slide of three actuators along three vertical guideways. Using the Grubler formula recalled in Equation (1), it can be proved that the mobility m of the platform is equal to three:

  int 1 61 t N p t i i m N N f m       (1)
where m denotes the mobility of the manipulator, p N is the total number of rigid bodies of the manipulator,

III. Kinematic equations

In order to analyse the kinematics of our parallel module, two relative coordinates are assigned as shown in figures 2. A static Cartesian frame xyz is fixed at the base of the machine tool, with the z-axis pointing downward along the vertical direction. The mobile Cartesian frame,

P P P
x y z , is attached to the moving platform at point P and remains parallel to xyz.

In any constrained mechanical system, joints connecting bodies restrict their relative motion and impose constraints on the generalized coordinates, geometric constraints are then formulated as algebraic expressions involving generalized coordinates. Using the parameters defined in figure 4, the constraint equations of the parallel manipulator are expressed as:

      2 2 2 2 0 Bij Aij Bij Aij Bij Aij i x x y y z z L        (2)
where ij A (respectively ij B ) is the center of spherical joint number j on the prismatic joint number i (respectively on the moving platform side), i = 1..3, j = 1..2.

12th IFToMM World Congress, Besançon (France), June18-21, 2007

CK-xxx 3
Leg Ι is represented by two different equations ( 3) and ( 4). This is due to the fact that 11 12

11 12 A A B B  (figure 4).       2 2 1 1 1 1 2 2 1 1 1
cos( ) sin( ) 0

P P P x D d y R r z R L             (3)       2 2 1 1 1 1 2 2 1 1 1
cos( ) sin( ) 0

P P P x D d y R r z R L             (4)
Leg ΙΙ is represented by a single equations [START_REF] Chablat | Architecture O ptimization of a 3-DO F Parallel Mechanism for Machining Applications, the O rthoglide[END_REF].

      22 2 2 2 4 2 2 2 2 2 cos( ) sin( ) - 0 PP P x D d y R r z R L            (5)
The leg ІІІ, which is similar to leg ІІ (figure 4), is also represented by a single equation [START_REF] Hervé | Structural synthesis of parallel robots generating spatial translation[END_REF].

      22 2 2 2 4 2 2 2 3 3 cos( ) sin( ) 0 PP P x D d y R r z R L             (6)

IV. Coupling between the position and the orientation of the platform

The parallel module of the VERNE machine possesses three actuators and three degrees of freedom. However, there is a coupling between the position and the orientation angle of the platform. The object of this section is to study the coupling constraint imposed by leg I.

By eliminating 1

 from equations ( 3) and ( 4), we obtain a relation [START_REF] Kong | Type synthesis of linear translational parallel manipulators[END_REF] between , and

PP xy  independently of z .         2 2 2 2 2 2 1 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1
sin( ) 2 cos( ) sin( ) 2 cos( ) 0

P P R x D d r R r R y R L R r R r               (7)
We notice that for a given  , equation ( 7) represents an ellipse [START_REF] Neumann | [END_REF]. The size of this ellipse is determined by a and b , where a is the length of the semi major axis and b is the length of the semi minor axis.

 

where

          2 2 2 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 22 1 1 1 1 2 cos( ) sin( ) 2 cos( ) 2 cos( ) a L R r R r R L R r R r b r R r R                   
These ellipses define the locus of points reachable with the same orientation . 

V. The Inverse kinematics

The inverse kinematics deals with the determination of the joint coordinates as function of the moving platform position. For the inverse kinematics problem of our spatial parallel manipulator, the position coordinates ( , , To solve the inverse kinematics problem, we first find all the possible orientation angles  for prescribed values of the position of the platform ( , ,

P P P
x y z ). These orientations are determined by solving equation ( 9), a third degree characteristic polynomial in cos( )

 derived from equation (7). 32 1 2 3 4 cos( ) cos( ) cos( ) p p p p     (9)
where

          3 3 2 1 1 1 3 1 1 1 1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 2 2 2 2 2 4 1 1 1 1 1 2 2 2 2 1 1 1 1 2 , 2 2
P P P P p R r p R r R r y p R L R r R x D d p R x D d R r y R L R r                           
However this equation also represents ellipses of isovalues of  . So if we plot all ellipses together by varying  from -to   (figure 5), we notice that every point (defined by , P x P y and P z ) is obtained by the intersection of two ellipses and each ellipse represent two opposite orientations so each point can have a maximum of four different orientations. This conclusion is verified by the fact that we can only find four real solutions to the polynomial (Table I). After finding all the possible orientations, we use the equations derived in section III to calculate the joint coordinates i  for each orientation angle  . To make this task easier, we introduce two new points 1 A and 1 B as the middle of 11 12 AA and 11 12 BB , respectively.

12th IFToMM World Congress, Besançon (France), June18-21, 2007

CK-xxx 4         2 2 2 1 1 1 2 2 2 1 1 1 1 1
2 cos( ) 0

P P P x D d y z L R r R r             (10)
Then, for prescribed values of the position and orientation of the platform, the required actuator inputs can be directly computed from equations ( 10), ( 5) and ( 6):

      2 2 2 1 1 1 1 1 11 2 2 11 2 cos( ) - P PP L R r R r zs x D d y              (11)     2 2 2 2 2 2 2 2 2 24
sin( ) cos( )

P P P L x D d z R s y R r                (12)     2 2 3 2 2 3 2 3 2 24
sin( ) cos( )

P P P L x D d z R s y R r                (13)
where   , respectively. Subtracting equation (3) from equation ( 4), yields: Observing equations ( 11), ( 12), ( 13), Table I and Table II, we conclude that the three legs, with four postures for leg Ι and two postures for leg ΙΙ and ΙΙΙ results in sixteen inverse kinematic solutions (figure 6). From the sixteen theoretical inverse kinematics solutions shown in figure 6, only one is used by the VERNE machine: the one referred to as (m) in figure 6, which characterized by the fact that each leg must have its slider attachment points upper than the moving platform attachment points, i.e. 1 i s  (remember that the z-axis

    P 1 1 1 1 P y R cos( ) r =R sin( ) z     (14)      
is directed downward). (a) (b) (c) (d) (e) (f) (g) (h)
(i) (j) (k) (l)
12th IFToMM World Congress, Besançon (France), June18-21, 2007

CK-xxx 5 (m) (n) (o) (p)
Fig. 6. the sixteen solutions to the inverse kinematics problem when -240 mm, -86 mm and 1000 mm

P P P x y z   
For the remaining 15 solutions one of the sliders leave its joint limits, or the two rods of leg I cross. Most of these solutions are characterized by the fact that at least one of the legs has its slider attachment points lower than the moving platform attachment points. So only , , 1 s s s  in equations ( 11), ( 12) and ( 13) must be selected (remember that the z-axis is directed downward). To prevent rod crossing, we also add a condition on the orientation of the moving platform. This condition is 11 cos( ) .

Rr  

Finally, we check the joint limits of the sliders and the serial singularities [START_REF] Kanaan | Kinematic analysis of the VERNE machine[END_REF].

For the VERNE parallel module, applying the above conditions will always yield to a unique solution for practical applications (solution (m) shown in figure 6).

VI. The forward kinematics

The forward kinematics deals with the determination of the moving platform position as function of the joint coordinates. For the forward kinematics of our spatial parallel manipulator, the values of the joint coordinates S to obtain a new system ( 2) S of three equations ( 15), ( 16) and ( 17) derived from equations (3), ( 5) and ( 6) respectively. We then compute P z as function of ( 1..3) i i   and  by subtracting equation ( 16) from equation (17). We replace this variable in system ( 2)

S

to obtain a new system ( 3) S of two equations ( 18) and ( 19) derived from equations ( 15) and ( 16) respectively. Finally, we compute P x as function of ( 1..3)

i i  
and  by subtracting equation (18) from equation ( 19) and we replace this variable in the system ( 3) S in order to eliminate P x . Equations of system ( ) (i=2..3)

Si

are not reported here because of space limitation. They are available in [START_REF] Kanaan | Kinematic analysis of the VERNE machine[END_REF].

For each step, we determine solutions existence conditions by studying the denominators that appear in the expressions of

     2 3 1 1 4 1 1 2 cos( ) 2sin( ) 0 R r r R r R          (21)
Equation (20) obtained from ( 14) implies that 11 AB is perpendicular to the slider plane of leg І. In this case equation ( 8 5)  ( 6)). To finish the resolution of the system, we perform the tangent-half-angle substitution tan( / 2) t   . As a consequence, the forward kinematics of our parallel manipulator results in a eight degree characteristic polynomial in t , whose coefficients are relatively large expressions in 1  , 2  and 3  . For the VERNE machine, only 4 assembly-modes have been found (figure 7). It was possible to find up to 6 assembly-modes but only for input joint out of the reachable joint space of the machine. Only one assembly-mode is actually reachable by the machine (solution (a) shown in figure 7) because the other ones lead to either rod crossing, collisions, or joint limit violation. The right assembly mode can be recognized, like for the right working mode, by the fact that each leg must have its slider attachment points upper than the moving platform attachment points, i.e. 1 i s  (keep in mind that the z-axis is directed downwards). The proposed method for calculating the various solutions of the forward kinematic problem has been implemented in Maple (Table III).

VII. Conclusion

This paper was devoted to the kinematic analysis of the parallel module of a 5-DOF hybrid machine tool, the VERNE machine. The degrees of freedom, the inverse kinematics and the different assembly modes were derived. The forward kinematics was solved with the substitution method. It was shown that the inverse kinematics has sixteen solutions and the forward kinematics may have six real solutions. Examples were provided to illustrate the results. The forward and inverse kinematics of the full VERNE machine is quite easy to derive [START_REF] Kanaan | Kinematic analysis of the VERNE machine[END_REF]. The controller of the actual VERNE machine resorts to an iterative Newton-Raphson resolution of the kinematics models. A comparative study will be conducted by the authors between the analytical and the iterative approaches. It is expected that the analytical method could decrease the Cpu-time and improve the quality of the control.

Fig. 1 .

 1 Fig. 1. Overall view of the VERNE machine

12th

  Fig. 2. Schematic representation of the parallel module

Fig. 3 .

 3 Fig. 3. Joints and loops graph of VERNE

  -rods, one base, one moving platform and 6 rods. t N is the number of the joints, , 3 prismatic joints P. i f denotes the number of degrees of freedom (DOF) of the th i joint, and m int is the number of internal DOF, which do not the motion of manipulator.Based on equation (1), the mobility of the platform is given by   . Due to the arrangement of the links and joints, as shown in figure2, legs ΙΙ and ΙΙΙ prevent the platform from rotating about y and z axes. Leg Ι prevents the platforms from rotating about z-axis but, because 11 coupled rotation about x-axis exists.

Fig. 4 .

 4 Fig. 4. Dimensions of the parallel kinematic structure in the frame supplied by Fatronik

  ) are given but the joint coordinates ( 1..3) i i   of the actuated prismatic and the orientation angle  of the moving platform are unknown.

Fig. 5 .

 5 Fig. 5. Curves of iso-values of the orientation  from -to   following a constant step of / 90.

  the possible orientations for a fixed position of the platform

 1  1 s 1 

 111 This means that for prescribed values of the position and orientation of the platform, the joint coordinate can take on both values +1 and -1. As a result

  system ( 1) S of four equations ((3), (4), (5) and (6)) to lead to an equation function of the joint coordinates ( orientation angle  of the platform. To do so, we first compute P y as function of P z by subtracting equation (3) from equation (4) and we replace this variable in system[START_REF] Merlet | Parallel Robots[END_REF]

  ) represents a circle because ab  . When 23 =  in equation (21), we have {0obtained from Equations.(

Fig. 7 .

 7 Fig. 7. The four assembly-modes of the VERNE parallel module for 1 250 mm,  

TABLE II .

 II the solution of the joint coordinate 1  according to the values of 

TABLE III .

 III the numerical results of the forward kinematic problem of the example where 1 674 mm,  

	2  	685 mm	and 3 250 mm  

Acknowledgments

This work has been partially funded by the European projects NEXT, acronyms for "Next Generation of Productions Systems", Project no° IP 011815.