Fahad R Golra
email: fahad.golra@telecom-bretagne.eu

Fabien Dagnat
email: fabien.dagnat@telecom-bretagne.eu

The Lazy Initialization Multilayered Modeling Framework (NIER Track)

Keywords: D.2.2 [Software Engineering]: Design tools and techniques, D.1.5 [Programming Techniques]: Object-Oriented Programming, H.1 [Information Systems]: Models and principles Design, Standardization Metamodeling, Strict metamodeling, Instantiation, LIMM

Lazy Initialization Multilayer Modeling (LIMM) is an object oriented modeling language targeted to the declarative definition of Domain Specific Languages (DSLs) for Model Driven Engineering. It focuses on the precise definition of modeling frameworks spanning over multiple layers. In particular, it follows a two dimensional architecture instead of the linear architecture followed by many other modeling frameworks. The novelty of our approach is to use lazy initialization for the definition of mapping between different modeling abstractions, within and across multiple layers, hence providing the basis for exploiting the potential of metamodeling.

INTRODUCTION

Modeling was once considered a tool to understand the System Under Study (SUS) but recently it has emerged out as a key activity of system development. This led to the rapid and incremental development of UML. However UML restricts modeling to 1) a 4 layers hierarchy, 2) 'strict' metamodeling (a model element must be an instance of exactly one meta-model element), and 3) the exclusive use of instance-of relationship amongst adjacent modeling layers. Holding to backward compatibility, UML2.0 could not unleash the complete potential of metamodeling. Different researchers have been using its evolution mechanism (UML Profiles) to add up to its capabilities [START_REF] Rumbaugh | The Unified Modeling Language Reference Manual[END_REF]. Some recent approaches of metamodeling (deep instantiation, power type based modeling, etc.) question its shortcomings and present their own methodologies [START_REF] Gonzalez-Perez | A Powertype-based Metamodelling framework[END_REF][START_REF] Atkinson | Rearchitecting the UML Infrastructure[END_REF]. However none of these approaches succeeded in replacing or influencing UML.

In order to exploit the potential of metamodeling both at micro level (inside a model) and macro level (between models), LIMM is presented as an object oriented multilayered modeling language that aims to provide a better support for the definition of different DSLs. The usage of lazy initialization in modeling is a novel idea that adds a lot of flexibility for the modelers, without introducing any functional limitations. In our proposal, the designer of a model can specify for each model element if it is at meta level, at application level or it is a data. To achieve this goal, flags are associated to model elements to control the way they can be used in the subsequent layers. A flag can take three different values to restrict, allow or force the initialization of a model element in the next adjacent layer.

This paper is structured as follows. Section 2 explains the multilayered modeling framework and describes the recent associated endeavors. Section 3 describes the LIMM framework. Section 4 presents the metamodel for LIMM framework. Finally Section 5 outlines the conclusions and future work.

MULTILAYERED MODELING

When modeling started to mature, modeling languages themselves needed to be modeled (language-ware). This led to a shift from traditional bi-layered modeling towards multilayered modeling hierarchy. UML modeling framework as defined in MOF rests on a 4 layers hierarchy. Though UML holds the industry, still some modelers (specially the meta-CASE tool designers) face some problems using it. The well known problems of this framework are Ambiguous Classification, Replication of Concepts, Structure redundancies and problems with metalevels [START_REF] Kuhne | Understanding Metamodeling[END_REF][START_REF] Atkinson | Rearchitecting the UML Infrastructure[END_REF][START_REF] Gonzalez-Perez | A Powertype-based Metamodelling framework[END_REF][START_REF] Varró | VPM: A visual, precise and multilevel metamodeling framework for describing mathematical domains and UML (the mathematics of metamodeling is metamodeling mathematics)[END_REF].

One of most discussed problems in literature regarding the UML modeling framework is its incompetence to differentiate between the logical and the physical flavors of the instance-of relationship. As illustrated in Figure 1(a), the document AR2010 is an instance-of class Document, which is an instance of both ResourceType and Class, where the two instantiations are not the same. The first instantiation relationship is a logical relationship mapping an instance to a concept of the system under study (SUS), whereas the second relationship indicates its physical existence as a class. Thus the concept of object/type facets is addressed in MOF, but is not very well defined. The term clabject coined by Atkinson refers to this concept [START_REF] Atkinson | Rearchitecting the UML Infrastructure[END_REF].

A unique notion of instance-of relationship between two layers seems unable to portray the concept of class as an instance. Power type based modeling proposed to use instantiation to carry on the instance facet to the subsequent modeling layer, and inheritance to carry the class facet [START_REF] Gonzalez-Perez | A Powertype-based Metamodelling framework[END_REF]. In this framework, a class (such as Class in Figure 1 two instances in the subsequent modeling layer. First, the power type, ResTypeKind, carries the object facet to the next level using the instance-of relationship. Second, the partitioned type, ResourceType, assumes the class facet and is intended to be the super class of the power type's instance (Document in the example). Instead of having a simple typeinstance relationship like in traditional modeling, it creates a triplet of the power type, the partitioned type and the class at a lower modeling layer, as depicted in Figure 1(b).

The drawback of power type based modeling is its inability to define associations in terms of clabjects. All the complexity of partitioning the power type from partitioned type in order to classify the different facets is transferred to the semantics of the relationship isClassifiedAs, which lacks concrete explanation. Furthermore, the inheritance relationship across different layers in the architecture makes it quite complex to draw the boundaries of a modeling layer.

In contrast to power type based modeling, deep instantiation based modeling consents to defer the instantiation for a "known" number of layers [START_REF] Atkinson | Rearchitecting the UML Infrastructure[END_REF][START_REF] Kuhne | Understanding Metamodeling[END_REF]. Deep instantiation uses the concept of potency, where each association, attribute and class is assigned a potency. The potency amounts to the number of levels an artifact can deny instantiation. Thus if an attribute is assigned a potency of 2, and it resides at layer Mn, then it would not be instantiated till Mn-2. As it gets down to Mn-2, it will get instantiated and become a slot. Deep instantiation addresses the dual nature (class and instance facets) of the classes using this potency. An example of deep instantiation can be viewed in Figure 1(c).

One of the major drawbacks of deep instantiation is that one has to know in advance, the number of layers each artifact should resist instantiation. The metamodeler must anticipate the uses of its metamodel and fix the potency of its elements. This fixed potency eventually takes the flexibility out of the reach of the application modeler, who is constrained to follow it.

Álvarez et al. suggested that instead of a linear modeling hierarchy, one should follow the nested hierarchy [START_REF] Álvarez | Mapping between levels in the metamodel architecture[END_REF]. In such a hierarchy, the level M0 is adjacent to both M1 and M2 layers. So the instance at level M0 can have instance-of relationships both from M1 and M2. The essence of nested layers is that it is recursive in nature and has a conforming mapping structure called the G-mapping. The drawback of using such a multi layer structure is that, though it works fine for 3 layers, but as it progresses to more layers, the original benefit being sought is lost. Thus it is restricted to a maximum of 4 layers. And the problem of ambiguous classification still remains; as the only relationship allowed is instance-of and no well defined differentiation among the different flavors of instance-of relationship is discussed.

A detailed study of the existing multilayered modeling frameworks shows that each of these techniques tries to resolve a part of the explored shortcomings of UML. None of these techniques, other than UML, gained popularity in the software industry because of the respective drawbacks detailed above. For these reasons, we propose a general multilayered modeling framework influenced by the previously presented approaches but relying on a lazy initialization mechanism inspired from functional programming.

LAZY INITIALIZATION

The Lazy Initialization Multilayer Modeling (LIMM) takes its inspiration from lazy functional programming, where evaluation may be delayed until required. Using lazy initialization helps clarify the instance-of relationships between the layers and their classification.

In UML, the links in the data models are the instances of the associations in the model at upper layer; same goes for the slots which are the instances of the attributes in the model at upper layer. But the instance-of relationship which bypasses the layers is not well explained. LIMM uses a special sort of abstraction known as interconnection to specify the different relationships that bypass the model boundaries. This framework gives a proper justification and sound basis to various relationships within a model or amongst different models, residing either on same or different abstraction levels.

We argue against deep instantiation because one cannot decide before hand, the number of layers after which a model element is to be initialized. Indeed, we believe that the number of layers cannot be fixed and must result from the choice of the designer. Furthermore, each layer is usually built by a different designer, making it impossible to predict when a model element will reach its object layer (where it is initialized). Our proposal is that at a certain layer, the designer must specify the initialization scheme of model elements in the subsequent layer. The initialization scheme can either forbid the initialization or devolve the decision, to choose the number of layers before initialization, to the designer of the subsequent layer.

The flag value of the associated model element specifies whether it is going to be initialized in the subsequent layer or not. This flag can be null or have three different values: 2 for Restrict, 1 for Relax or 0 for Force. A null flag describes an instantiated model element. While developing a model, the flags of the meta-elements in the reference model are tested to determine which elements must or may be initialized at current layer. A flag value 2 (Restrict) specifies a model element that cannot be initialized in the subsequent layer (where the flag value can shift to 2, 1 or 0). A flag value 1 (Relax ) allows the initialization of a model element in the subsequent layer (where its flag value can become 1, 0 or null). Lastly, a flag value 0 (Force) forces the designer of the subsequent layer to initialize the model element (moving its flag to null).

Models being model elements maintain their own flags to specify their abstraction level. Models having a flag value 2 are at some metalevels whereas the models having the flag value 0 are the user models. Models having a flag value 1 can be intermediate level models that can/cannot be initialized. A model contains classes and these classes contain attributes. This containment hierarchy constrains flags: the flag value of a container must be greater than or equal to the flag value of its contained elements. Data models or object models do not carry a flag, so they have a null value. The flag values of a model can either be specified by the designer or calculated automatically.

LIMM uses a two dimensional modeling framework with a horizontal division between the core multilayering metamodel and the linear modeling hierarchy, which is further divided into multiple modeling layers. The core multilayering metamodel is the LIMM metamodel that defines the language in which the models placed in the various layers are defined. This is a top level meta-metamodel that defines itself. Our numbering of the linear modeling hierarchy, in contrast to UML, keeps augmenting from the topmost layer, M0, to the subsequent layers up to Mn.

LIMM METAMODEL

LIMM metamodel presents two common models; a user model and a reference model, which are connected through interconnections, as shown in Figure 2. A reference model can be referenced by another reference model or by a data model. A data model is a model which is not referenced by any other model. The core multilayering metamodel does not have a flag because all other models in its linear hierarchy are instance-of this model, thus all the physical instanceof relationships are implicit in this model. The dual nature (type/instance) of a model element is described in terms of initialized and non-initialized contained model elements. The common model is itself a model element to support abstraction hierarchies. All the meta-metamodels for defining new DSLs are presented as an extension of the core multilayering metamodel. This allows them to utilize multiple layers for model definitions.

Channels and Items are both model elements, which are used to relate a model to the graph semantics. They are kept open to extension for new abstractions in order to support the definition of new DSLs. One of the hallmarks of UML was the visual representation through adding semantics to the graphs. LIMM uses the same methodology but maintains a distinction between the channel and interconnection. A channel is a relationship between two model elements within a model, whereas an interconnection is a relationship between elements of different models. Thus the semantics of relationships crossing the boundaries of a model is different from those who do not. The classification of in- A DSL for metamodel specification, Kernel MetaMeta-Model (KM3) [START_REF] Jouault | KM3: A DSL for Metamodel Specification[END_REF], defines the structure of a model using directed multigraphs. We follow their definition and extend it further for describing our specification. Let GM = (NM , EM , ΓM ) be the directed multigraph for the common model M , where • NM is the set of nodes (representing the Items);

• EM is the set of edges (representing the Channels);

• ΓM is the function from EM to NM × NM associating an edge to its extremities.

A common model M is represented as M = (GM , R, κM ), where

• R is the reference model of M . The associated graph of this reference model is GR = (NR, ER, ΓR).

• κM is a function from NM ∪ EM to NR ∪ ER. It relates the elements of the model M to the elements of the reference model R (meta-elements), thus representing the explicit logical relationships.

Having the common model and reference model in hand, we are able to represent the multilayer metamodel ML = (GML, CMM , κML), such that ER associating this edge to its extremities.

• GML = (NML,
• CMM is the core multilayering metamodel which defines itself. It is the topmost level of hierarchy, where the meta-metamodel is defined recursively by itself.

• κML is a function from NML ∪ EML to NCMM , relating all the model elements and relationships of the multilayer model to the nodes of core multilayering metamodel representing the physical relationships. These relationships are implicit in the visual representation. A reference model is referenced by a data model from a lower layer but at the same time it may refer to some other reference model (acting itself as data model ). The elements of the reference model R are hence meta-elements for the elements of the models residing in the subsequent modeling layer. The CLIs (instantiation and conformity) can be defined when M and R are both at different layers, whereas SLIs can be defined when M and R are both at the same layers. For a model M , a partial function fo M gives the flag value of its model elements if it is not null. Thus, ∀e ∈ NM ∪ EM , fo M (e) ∈ {0, 1, 2}.

Conforms-to relationship (µ) is defined for LIMM as a Cross Level Interconnection (CLI) when,

µ = {i ∈ κM |dom(i) ⊆ dom(fo M )} Instance-of relationship (η) is defined for LIMM as a Cross Level Interconnection (CLI) when, η = {i ∈ κM |dom(i) ∩ dom(fo M ) = ∅}
The example used to compare other multilayered modeling frameworks is continued in Figure 3 where a part of object oriented metamodel lies at M0 layer. The Resource-Type class at M1 layer conforms to the class Class at M0. The flag value of the attribute name is not initialized till M3, whereas category is initialized at M2. In this example, AR2010 has an instance-of relationship to Document, which in turn has a conforms-to relationship with ResourceType. The usedby link at M3 is initialized from the usedby association at M2. The flag values on the models and model elements show that they belong to meta layer, application layer or data layer. The instance facet of the class Document can be viewed from category (slot), whereas the class facet from name (attribute). All the relationships explicitly defined are the logical relationships, whereas the physical relationships are implicit in the visual representation, from each model to the core multilayering metamodel, adjacent to all the the layers of the multilayer model.

CONCLUSION

Different multilayered modeling frameworks have been explored to stress their deficiencies and strengths. We proposed to capture the semantics of a model using graphs, where the semantics of the instance-of and conforms-to relationships can be clarified. A metamodel for the lazy initialization multilayered modeling framework is presented, where lazy initialization schemes are used to resolve the issues faced by multilayered models. LIMM framework supports the definition of new DSLs that need the usage of multiple abstraction levels. In addition, this novel framework improves the level of expression in multilayered modeling by adding semantics to the mappings across layers, which would in turn facilitate model validations. A vital impact on the designer's approach would be the enhanced control over his model to guide the development, without forcing any unnecessary restrictions to the subsequent layer designers. Having models as model elements helps add semantics to the mappings amongst the models also. This inter-model mapping would have a considerable impact in assisting model management.

The focus of MDE is to exploit domain modeling by improving compatibility and mappings amongst models. We are looking forward to add semantics to the mapping functions between the model elements. This would further assist the automation of software development processes for multimetamodel applications. For the future work, this multilayered modeling framework would also help to build transformation hierarchies. Having both the model hierarchies and transformation hierarchies would allow LIMM framework to exploit the potential of MDE.

Figure 1 :

 1 Figure 1: Comparative Example in different Multilayered modeling frameworks

Figure 2 :

 2 Figure 2: Minimal LIMM metamodel terconnection is done in terms of abstraction levels. A Cross Layer Interconnection (CLI) relates two model elements residing on different layers, whereas a Same Layer Interconnection (SLI) relates the model elements residing on the same layer. Instance-of and Conformity are CLIs that contravene the strict metamodeling of UML, where the only relationship across layers can be instance-of.A DSL for metamodel specification, Kernel MetaMeta-Model (KM3)[START_REF] Jouault | KM3: A DSL for Metamodel Specification[END_REF], defines the structure of a model using directed multigraphs. We follow their definition and extend it further for describing our specification. Let GM = (NM , EM , ΓM ) be the directed multigraph for the common model M , where

  EML, ΓML) is a directed multigraph for the multilayered model, where -NML is the set of nodes of the graph representing the elements of multilayer model, NM ∪ NR. -EML is the set of edges of the graph representing the associations in multilayer model, EM ∪ ER. -ΓML is the function from EML to NM ∪EM ×NR ∪

Figure 3 :

 3 Figure 3: Example in LIMM


Félix Majou
email: felix.majou@irccyn.ec-nantes.fr

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

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

DESIGN OF A 3 AXIS PARALLEL MACHINE TOOL FOR HIGH SPEED MACHINING : THE ORTHOGLIDE

Keywords: Parallel Machine tool Design, Isotropic Design, Singularity

The Orthoglide project aims at designing a new 3-axis machine tool for High Speed Machining. Basis kinematics is a 3 degree-of-freedom translational parallel mechanism. This basis was submitted to isotropic and manipulability constraints that allowed the optmization of its kinematic architecture and legs architecture. Thus, several leg morphologies are convenient for the chosen mechanism. We explain the process that led us to the choice we made for the Orthoglide. A static study is presented to show how singular configurations of the legs can cause stiffness problems.

Introduction

Serial machine tools problems

Most industrial machine tools have a serial kinematic architecture, which means that each axis has to carry the following one, including its actuators and joints (Fig. 1). High Speed Machining (HSM) highlights some drawbacks of such architectures: heavy moving parts require from the machine structure high stiffness to limit bending problems that lower the machine accuracy, and limit the dynamic performances of the feed axes. In a PKM, the tool is connected to the base through several kinematic chains or legs that are mounted in parallel. Figure 2 shows a 3-degree-of-freedom (3-DOF) planar mechanism ( 2 translations, 1 rotation) mounted on 3 RPR legs (Revolute, Prismatic and Revolute joints). PKMs attract more and more researchers and companies, because they are claimed to offer several advantages over their serial counterparts, like high structural rigidity and high dynamic capacities. Indeed, the parallel kinematic arrangement of the links provides higher stiffness and lower moving masses that reduce inertia effects. Thus, PKMs have better dynamic performances, which is interesting for HSM. Most existing PKMs can be classified into two main families: (i) PKMs with fixed foot points and variable strut lengths and (ii) PKM with fixed length struts and moveable foot points.

The first family is mostly composed of the so-called hexapod machines which, in fact, feature a Gough-Stewart platform architecture. Many prototypes and commercial hexapod PKMs already exist like the VARIAX-Hexacenter (Gidding&Lewis), the CMW300 (Compagnie Mécanique des Vosges), the TORNADO 2000 (Hexel), the MIKROMAT 6X (Mikromat/IWU), the hexapod OKUMA (Okuma), the hexapod G500 (GEODETIC). In this first family, we find also hybrid architectures : the TRICEPT 845 from Neos Robotics which is a 2-axis wrist serially mounted with a 3-DOF parallel structure, and a 3-DOF hybrid high speed machine tool designed by LARAMA and PCI [START_REF] Bouzgarrou | Static and Dynamic Characterization of New Parallel High Speed Machine Tool[END_REF].

The second family (ii) of PKM has been more recently investigated. In this category we find the HEXAGLIDE (ETH Zürich) which features six parallel (also in the geometrical sense) and coplanar linear joints. The HexaM (Toyota) is another example with non coplanar linear joints. A 3-axis translational version of the hexaglide is the TRIGLIDE (Mikron), which has three coplanar and parallel linear joints. Another 3-axis translational PKM is proposed by the ISW Uni Stuttgart with the LINAPOD. This PKM has with three vertical (non coplanar) linear joints. The URANE SX (Renault Automation Comau) and the QUICKSTEP (Krause & Mauser) are 3-axis PKM with three non coplanar horizontal linear joints. A hybrid parallel/serial PKM with three parallel inclined linear joints and a two-axis wrist is the GEORGE V (IFW Uni Hanover). H4, a family of 4-DOF parallel robots was presented in [START_REF] Pierrot | H4 : a New Family of 4-dof Parallel Robots[END_REF].

Most PKMs suffer from the presence of singular configurations in their workspace that limit the machine performances.

Singular configurations

The singular configurations (also called singularities) of a PKM may appear inside the workspace or at its boundaries. There are two main types of singularities [START_REF] Gosselin | Singularity analysis of Closed-Loop Kinematic chains[END_REF]. A configuration where a finite tool velocity requires infinite joint rates is called a serial singularity or a type 1 singularity. A configuration where the tool cannot resist any effort and in turn, becomes uncontrollable, is called a parallel singularity or type 2 singularity. Parallel singularities are particularly undesirable because they cause the following problems:

-a high increase of forces in joints and links, that may damage the structure, -a decrease of the mechanism stiffness that can lead to uncontrolled motions of the tool though actuated joints are locked.

Figures 3a and3b show the singularities for a "biglide" mechanism, which is a 2-PRR mechanism with prismatic actuated joints. Its legs are made of fixed length struts with gliding node points. In Fig. 3a, we have a serial singularity. The velocity amplification factor along the vertical direction is null and the force amplification factor is infinite.

Figure 3b shows a parallel singularity. The velocity amplification factor is infinite along the vertical direction and the force amplification factor is close to zero. Note that a high velocity amplification factor is not necessarily desirable because the actuator encoder resolution is amplified and thus the accuracy is lower.

The Orthoglide project

The Orthoglide project aims at building a prototype of a parallel kinematic machine tool for HSM with a kinematic behaviour similar to the one of a classical serial 3-axis machine. First, we present some research works on the structural synthesis of 3-DOF translational parallel mechanisms that helped our choice of the Orthoglide architecture. Then we show how isotropic and manipulability constraints led to modifications of the basis architecture as well as of the legs architecture. The study of forces inside the legs concludes this study.

Choice of a suitable kinematic architecture 2.1 Design of a 3-DOF translational parallel mechanism

Many studies have been conducted on the design of parallel mechanisms. Hervé proposed in [START_REF] Hervé | Structural Synthesis of Parallel Robots Generating Spatial Translation[END_REF] a tool for the synthesis of parallel robots based on the mathematical group theory. In [START_REF] Hervé | Star, a New Concept in Robotics[END_REF], this tool was applied to the design of a 3-DOF translational parallel manipulator called Y-STAR (Fig. 4). The author in [START_REF] Leguay-Durand | Conception et Optimisation de Mécanismes parallèles à Mobilités Restreintes[END_REF] explored tools for the design and optimization of parallel mechanisms with constrained DOFs. Recently, Kong proposed in [START_REF] Kong | Generation of Parallel Manipulators with Three Translational Degrees of Freedom based on Screw Theory[END_REF] the generation of translational parallel manipulators based on screw theory [START_REF] Ball | A Treatise on the Screw Theory[END_REF]. The Y-Star robot, with helical actuated joints replaced by prismatic actuated joints followed by passive revolute joints is a convenient choice regarded to these constraints. We chose it as the basis mechanism of the Orthoglide project.

The structural synthesis is now achieved : an ordered set of joints is available for each leg. We now have to adjust it regarded to isotropic and manipulability constraints.

Optimization of the leg architecture

Conditioning index and manipulability

For a serial mechanism, the velocity and force transmission ratios are constant in the workspace. For a parallel mechanism, in contrast, these ratios may vary significantly in the workspace because the displacement of the tool is not linearly related to the displacement of the actuators. In some parts of the workspace, the maximal velocities and forces measured at the tool may differ significantly from the maximal velocities and forces that the actuators can produce. This is particularly true in the vicinity of singularities. At a singularity, the velocity, accuracy and force ratios reach extreme values.

Let ρ  be referred to as the vector of actuated joint rates and p  as the velocity vector of point P. ρ  and p  are related through the Jacobian matrix J as follows :

ρ p   J 
J also relates the static tool efforts to the actuator efforts. For parallel manipulators, it is more convenient to study the conditioning of the Jacobian matrix that is related to the inverse transformation, J -1 . To evaluate the ability of a parallel mechanism to transmit forces or velocities from the actuators to the tool, two complementary kinetostatic performance criteria can be used : the conditioning of the inverse Jacobian matrix J -1 , called conditioning index, and the manipulability ellipsoid associated with J -1 [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems[END_REF]. The conditioning index is defined as the ratio of the highest to the smallest eigenvalue of J -1 . The conditioning index varies from 1 to infinity. At a singularity, the index is infinity. It is 1 at another special configuration called isotropic configuration. At this configuration, the tool velocity and stiffness are equal in all directions. The conditioning index measures the uniformity of the distribution of the velocities and efforts around one given configuration but it does not inform about the magnitude of the velocity amplification or effort factors. The manipulability ellipsoid is defined by its principal axes as the eigenvectors of (J J T ) -1 and by the lengths of the principal axes as the square roots of the eigenvalues of (J J T ) -1 . The eigenvalues are associated with the velocity (or force) amplification factors along the principal axes of the manipulability ellipsoid [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems[END_REF].

Design constraints imposed to cope with this problem

To design a translational PKM with kinematic behaviour close to the one of a serial 3-axis machine tool, we impose the following conditions : (i) there is one point in the workspace for which the velocity transmission ratio is the same in every direction, (ii) and its value is one at this configuration. In [START_REF] Wenger | Kinematic Analysis of a New Parallel Machine Tool : the Orthoglide[END_REF] and [START_REF] Chablat | Architecture Optimization of the Orthoglide, a 3-DOF Parallel Mechanism for Machining Applications[END_REF] the geometric conditions implied by these constraints are described in a more rigorous way.

Condition (i) means that there is an isotropic configuration in the workspace. Condition (ii) means that for this configuration, the velocity amplification factors along the principal axes of the manipulability ellipsoid are equal to 1.

Geometrical arrangement of the legs

These two kinetostatic conditions lead to new geometric conditions on the Y-Star legs : -(i) implies that for each leg (Figure 5), orientation between the axis of the linear joint T i and the axis of parallelogram W i must be the same for each leg i, and that all vectors W i must be orthogonal to each other [START_REF] Wenger | Kinematic Analysis of a New Parallel Machine Tool : the Orthoglide[END_REF].

P

Ti Wi

Fig. 5 : Y-Star leg -(ii) implies that for each leg, T i and W i must be collinear [START_REF] Wenger | Kinematic Analysis of a New Parallel Machine Tool : the Orthoglide[END_REF]. Since, at the isotropic configuration, W i vectors are orthogonal, this implies that T i vectors are orthogonal, i.e. the linear joints are orthogonal (Figure 6).

x y z Fig. 6 : New arrangement of Y-Star robot

Rearrangement of legs architecture

The new geometrical arrangement of the machine legs leads to a singularity of the parallelogram (Figure 7) which becomes an antiparallelogram [START_REF] Vischer | Argos : A Novel 3-DOF Parallel Wrist mechanism[END_REF] : a passive rotation appears around an axis orthogonal to the parallelogram plane. A solution to this problem is to change the leg architecture by rearranging the joints, while keeping the same degree of freedom.

Ti Wi

P

Fig. 7 : Parallelogram singularity at isotropic configuration

A second version of the leg architecture is then proposed (Figure 8a).

Fig. 8a : Second version of the leg architecture

The parallelogram singularity at the isotropic configuration is avoided but another problem arises : a special singularity of the leg at the isotropic configuration. There is a passive rotation around the T i axis. It appears that this particular singularity is not detected by the method described in [START_REF] Gosselin | Singularity analysis of Closed-Loop Kinematic chains[END_REF]. [START_REF] Zlatanov | Singularity analysis of mechanism and Robots via a Velocity Equation Model of the Instantaneous Kinematics[END_REF] proposes a way to find it, and from [START_REF] Zlatanov | Generalized Singularity Analysis of Mechanisms[END_REF] this type of singularity is a (RPM, IO, II) singularity. RPM means that a Redundant Passive Motion is possible. This motion is the rotation of the parallelogram around the T i axis (Figure 8b). IO and II mean that in this configuration we have an Impossible Output ( p  = 0 ) and an Impossible Input ( ρ  = 0 ), respectively.

At the isotropic configuration, each leg can passively transmit a force whose axis is orthogonal to the parallelogram plane which means that no translation of point P along this axis is possible (IO and II, see figure 8b).

Furthermore, to have a pure translational 3-DOF mechanism (the tool cannot rotate), the parallelograms must be orthogonal to one another at the isotropic configuration. Thus, the mechanism gets locked at this configuration because no translation nor rotation of the tool is possible.

P

Ti Wi

Impossible translations

Fig. 8b : Leg singularity at isotropic configuration

The last version of Orthoglide legs (Figure 9) avoids all previous problems mentioned at the isotropic configuration : no parallelogram singularity, and no leg singularity.

P

Ti

Ui

W i

Fig. 9 : Current version of Orthoglide legs

Leg orientation for a pure translational mechanism

Now that a correct leg architecture has been defined, we have to choose the parallelograms orientation in the machine frame. The screw theory [START_REF] Ball | A Treatise on the Screw Theory[END_REF] allows a geometric explanation of the conditions on legs orientation that lead to a pure translational motion of the tool : the wrench system (forces or torques that can be passively transmitted by a kinematic chain) of each leg is composed of two torques (Figure 9) : the first torque axis is T i , and the second torque axis is perpendicular to the plane (T i , U i ). The tool can rotate when its wrench system (which is the union of all legs wrench systems) does not contain any torque. Consequently, the three legs wrench systems must include all torques, i.e., the three parallelograms planes must be orthogonal to one another. The result is shown on Figure 10. Note that the constant orientation of parallelograms in the whole workspace makes the Orthoglide free from constraint singularities 1 . When the parallelogram approaches its singularity, we have :   90° and Fb  . The tension / compression forces in bars 1 and 2 at joints limits have to be checked. In [START_REF] Chablat | Architecture Optimization of the Orthoglide, a 3-DOF Parallel Mechanism for Machining Applications[END_REF] the maximum value of  was calculated as : max = 14°.

Our design gave us : d = 100 mm. An approximated value of the torque C exerted on the tool regarded to machining conditions expected from our prototype is : C  10 Nm. Thus, the tension / compression force in bars 1 and 2 is : F b = 103 N. The section of bars 1 and 2 is : S = 144 mm². The maximum tensile stress in bars 1 and 2 is then :  max = 0,7 Mpa. This result is far less than the legs material maximum tensile strength (aluminium).

Conclusions

This paper describes the design of a new 3-axis machine tool based on a translational parallel mechanism : the Orthoglide. We chose a convenient architecture regarded to technological constraints. To have a kinematic behaviour close to the one of a serial 3-axis machine tool, we optimized the kinematic architecture to fit with isotropic and manipulability constraints. Several leg architectures were convenient. We chose the correct architecture by

Fig. 1 :

 1 Fig. 1 : A serial 3-axis Machine Tool

Fig. 2 :

 2 Fig. 2 : A 3-RPR parallel mechanism

Fig

  Fig. 3a : A serial singularity Fig. 3b : A parallel singularity

Fig. 4 :

 4 Fig. 4 : The Y-Star manipulator

Fig. 10 :Fig. 11 :i

 1011 Fig. 10 : Orthoglide legs orientation

Constraint singularities were discovered recently[15] [16]. They may arise for parallel mechanisms with less than 6 DOFs. At a constraint singularity, the moving platform gains a new DOF.

10 eliminating all legs singularities, particularly those that could not be detected by the classical velocities equations. Tensile strength in the parallelograms bars was evaluated for the most penalizing configurations.


Marianne Janowski

Nicolas Loget

Cécile Gautheron

Jocelyn Barbarand

Nicolas Bellahsen

Jean Van Den Driessche

Julien Babault

Bertrand Meyer

Neogene exhumation and relief evolution in the eastern Betics (SE Spain): insights from the Sierra de Gador

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Exhumation of rocks and topographic evolution are two parameters that are closely related in orogenic systems (e.g. [START_REF] England | Surface uplift, uplift of rocks, and exhumation of rocks[END_REF]. In active orogenic wedges, uplift and exhumation of accreted rocks are generally associated with a pattern of high-elevation and highrelief topography [START_REF] Willett | On steady states in mountain belts[END_REF]. Then, progressive removal of the lithospheric root together with surface erosion or tectonic unroofing leads to topographic decrease in postorogenic settings. The final stage tends towards low-elevation and low-relief topographies associated to low-temperature plateaus in cooling paths [START_REF] Calvet | Flat-topped mountain ranges: Their global distribution and value for understanding the evolution of mountain topography[END_REF]. The occurrence of planar landforms at the top of mountain ranges illustrates a topographic transient stage linked to a large-scale tectonic perturbation or local base-level changes during orogenic evolution (e.g. [START_REF] Babault | High elevation of low-relief surfaces in mountain belts: does it equate to post-orogenic surface uplift?[END_REF][START_REF] Calvet | Flat-topped mountain ranges: Their global distribution and value for understanding the evolution of mountain topography[END_REF]. The Betics in southern Spain (Fig. 1) are an example of such orogenic disruption (e.g. [START_REF] Platt | Extensional collapse of thickened continental lithosphere: A working hypothesis for the Alboran Sea and Gibraltar arc[END_REF][START_REF] Lonergan | Origin of the Betic-Rif mountain belt[END_REF][START_REF] Jolivet | Subduction, convergence and the mode of backarc extension in the Mediterranean region[END_REF], where low-relief surfaces are described at the top of ranges [START_REF] Farines | The summit erosion surfaces of the inner Betic Cordillera: Their value as tools for reconstructing the chronology of topographic growth in southern Spain[END_REF].

The Eocene-Oligocene mountain building is disrupted by an Oligocene-Miocene extensional event associated with the collapse of the internal zones. Major detachments enabled the exhumation of metamorphosed stacked nappes as the Alpujarride complex and the underlying Nevado-Filabride complex (Fig. 1). The system has been inverted and the Betic crust shortened since the Late Tortonian, as demonstrated by the folding of Miocene deposits (e.g. [START_REF] Weijermars | Uplift history of a Betic fold nappe inferred from Neogene-Quaternary sedimentation and tectonics (in the Sierra Alhamilla and Almeria, Sorbas and Tabernas basins of the Betic Cordilleras, SE Spain)[END_REF][START_REF] De Galdeano | Stratigraphic record and palaeogeographical context of the Neogene basins in the Betic Cordillera, Spain[END_REF]. Although the cooling event associated with the Miocene extensional event is well known, the late story, related to the final extensive stage and the following compressional event, remains poorly constrained, especially for the Alpujarride domain. Yet, this is the main basement area where surface uplift and topographic building can be constrained thanks to uplifted Miocene platform deposits [START_REF] Braga | Patterns and average rates of late Neogene-Recent uplift of the Betic Cordillera, SE Spain[END_REF]. This study is focused on the Sierra de Gador, which forms the largest antiform of the Alpujarride domain and is still covered by remnants of Miocene platform deposits overlying a smooth topography.

Thanks to low-temperature thermochronology (LTT) data (fission tracks and (U-Th)/He on apatite), we show that the current topography is inherited from a Middle-Upper Miocene planation event coeval with the late extensional stage of the eastern Betics. The post-Tortonian regional inversion, characterized by a strong surface uplift, is undetected by LTT data, suggesting that the Sierra de Gador topography is transient in nature.

Geological setting

The Betics are located at the western edge of the peri-Mediterranean alpine ranges and form the northern part of the Betic-Rif arcuate mountain belt surrounding the Alboran Sea (Fig. 1). This orogenic arc results from the Meso-Cenozoic convergence between Africa and Iberia. The Betic ranges can be subdivided into non-metamorphic external zones and metamorphic internal zones. The external zones are composed of folded and deformed sediments of the Iberian paleomargin thrust over the autochthonous Iberian platform, forming the Guadalquivir foreland basin, which is filled by Neogene sediments [START_REF] Michard | How does the Alpine belt end between Spain and Morocco? Bull[END_REF][START_REF] Frizon De Lamotte | TRANSMED Transect I: Iberian Meseta-Guadalquivir Basin-Betic Cordillera-Alboran Sea-Rif-Moroccan Meseta-High Atlas-Sahara Platform[END_REF]. The internal zones (or Alboran domain) are composed of a stack of three nappes of different metamorphic grade; from bottom to top they are the Nevado-Filabride, the Alpujarride and the Malaguide complexes. These nappes were accreted during the pre-Miocene collisional stage and then mainly exhumed during a Miocene extensional event (Martínez-Martínez and [START_REF] Martínez-Martínez | Mode of extensional tectonics in the southeastern Betics (SE Spain): Implications for the tectonic evolution of the peri-Alborán orogenic system[END_REF][START_REF] Azañón | Exhumation during a continental collision inferred from the tectonometamorphic evolution of the Alpujarride Complex in the central Betics (Alboran Domain, SE Spain)[END_REF]. During this period, the development of lowangle normal faults induced the separation of the three complexes, as observed currently. This extensional stage within the internal zones is coeval with thrusting and shortening in the external zones. This pattern can be explained by deep tectonic processes such as convective removal, delamination or slab rollback (e.g. [START_REF] Calvert | Geodynamic evolution of the lithosphere and upper mantle beneath the Alboran region of the western Mediterranean: Constraints from travel time tomography[END_REF]. Lithospheric imaging and volcanic data support a westward slab rollback model [START_REF] Lonergan | Origin of the Betic-Rif mountain belt[END_REF][START_REF] Duggen | Magmatic evolution of the Alboran region: The role of subduction in forming the western Mediterranean and causing the Messinian Salinity Crisis[END_REF][START_REF] Faccenna | Lateral slab deformation and the origin of the western Mediterranean arcs[END_REF][START_REF] Spakman | A tomographic view on western Mediterranean geodynamics[END_REF]. The extensional stage induced a major thinning of the Alboran domain and resulted in the formation of marine sedimentary basins including the Alboran basin (Sanz de [START_REF] De Galdeano | Stratigraphic record and palaeogeographical context of the Neogene basins in the Betic Cordillera, Spain[END_REF][START_REF] Comas | The origin and tectonic history of the Alboran Basin: insights from Leg 161 results[END_REF].

Since the Late Tortonian, the internal zones have undergone N-S to NW-SE compression, inducing large-scale folding and the emergence of a part of their overlying marine basins [START_REF] Weijermars | Uplift and subsidence history of the Alboran Basin and a profile of the Alboran Diapir (W-Mediterranean)[END_REF][START_REF] Comas | The origin and tectonic history of the Alboran Basin: insights from Leg 161 results[END_REF][START_REF] Marín-Lechado | Mountain Front Development by Folding and Crustal Thickening in the Internal Zone of the Betic Cordillera-Alboran Sea Boundary[END_REF][START_REF] Iribarren | Sediment supply from the Betic-Rif orogen to basins through Neogene[END_REF].

Exhumation and topographic features of the eastern Betics

The topography of the Betics can be described by a succession of ranges separated by depressions where Miocene marine and continental basins are preserved (Fig. 1). The elevations of these ranges reach more than 3000 m in the Sierra Nevada and more than 2000 m in the Sierra de Gador. These ranges correspond to large E-W antiforms of the exhumed Alboran domain. The relief of the ranges corresponds to incised topographies, such as in the Sierra Nevada [START_REF] Pérez-Peña | Active tectonics in the Sierra Nevada (Betic Cordillera, SE Spain): Insights from geomorphic indexes and drainage pattern analysis[END_REF], or, more remarkably, to low-relief surfaces at the top of ranges, as in Sierra de Gador (Figs. 2 et 3;[START_REF] Farines | The summit erosion surfaces of the inner Betic Cordillera: Their value as tools for reconstructing the chronology of topographic growth in southern Spain[END_REF]. In the Sierra Nevada, composed of Nevado-Filabride units, AFT and AHe ages range from 2.7 ± 0.6 to 10.5 ± 0.9 Ma (Fig. 2) and from 6.2 ± 0.4 to 8.8 ± 0.5 Ma respectively [START_REF] Johnson | The role of extension in the Miocene denudation of the Nevado-Filabride Complex, Betic Cordillera (SE Spain)[END_REF][START_REF] Reinhardt | Tectonic denudation and topographic development in the Spanish Sierra Nevada[END_REF][START_REF] Clark | The record of tectonic denudation and erosion in an emerging orogen: an apatite fission-track study of the Sierra Nevada, southern Spain[END_REF][START_REF] Vázquez | Deciphering tectonic-and erosiondriven exhumation of the Nevado-Filábride Complex (Betic Cordillera, Southern Spain) by low-temperature thermochronology: deciphering tectonic-and erosion-driven exhumation[END_REF]. AFT ages show a westward rejuvenation interpreted as the consequence of the top-to-the-WSW extension along the tectonic contact between the Nevado-Filabride and the Alpujarride complexes [START_REF] Johnson | Resolving denudational histories in orogenic belts with apatite fission-track thermochronology and structural data: An example from southern Spain[END_REF][START_REF] Johnson | The role of extension in the Miocene denudation of the Nevado-Filabride Complex, Betic Cordillera (SE Spain)[END_REF]. Thermal history indicates a rapid cooling (660 to 60°C) between ~16.5 and 6-8 Ma, followed by a slow cooling to surface conditions, reached at around 3-5 Ma [START_REF] Vázquez | Deciphering tectonic-and erosiondriven exhumation of the Nevado-Filábride Complex (Betic Cordillera, Southern Spain) by low-temperature thermochronology: deciphering tectonic-and erosion-driven exhumation[END_REF]. Very young AFT ages in western Sierra Nevada are interpreted as the result of a fold core's late extensional collapse [START_REF] Reinhardt | Tectonic denudation and topographic development in the Spanish Sierra Nevada[END_REF]). In the Sierra de Gador, composed of Alpujarride units, no thermochronological data are available except in the western valley where AFT ages of ~16 Ma have been reported (Fig. 2, samples PB365 and PB367 from [START_REF] Platt | Timing of tectonic events in the Alpujárride Complex, Betic Cordillera, southern Spain[END_REF]. A coupling with Ar/Ar data (~25 Ma) suggests a rapid cooling event during the early Miocene (Fig. 2, samples PB369 and PB533 from [START_REF] Platt | Timing of tectonic events in the Alpujárride Complex, Betic Cordillera, southern Spain[END_REF]. The timing of the final exhumation of the Sierra de Gador cannot be determined due to the lack of LTT data (such as AHe). Remnants of inner-platform calcarenites deposited around 7.2 Ma (uppermost Tortonianlowermost Messinian) now at high elevation (up to 1600 m) indicate that an important surface uplift has occurred since the end of the Tortonian [START_REF] Braga | Patterns and average rates of late Neogene-Recent uplift of the Betic Cordillera, SE Spain[END_REF]. This uplift results from the combination of a small-wavelength uplift induced by folding and a greater one induced by regional dynamic topography [START_REF] Marín-Lechado | Mountain Front Development by Folding and Crustal Thickening in the Internal Zone of the Betic Cordillera-Alboran Sea Boundary[END_REF][START_REF] Garcia-Castellanos | Messinian salinity crisis regulated by competing tectonics and erosion at the Gibraltar arc[END_REF]. These deposits lie on a low-relief topography (<250 m; Figs. 2 and3), and the whole area has been affected by a Late Tortonian-Early Messinian planation event [START_REF] Farines | The summit erosion surfaces of the inner Betic Cordillera: Their value as tools for reconstructing the chronology of topographic growth in southern Spain[END_REF]. The low-relief summits now observed in the Sierra de Gador can be explained by this polygenic erosional process. Finally, although a Late Miocene-Early Pliocene exhumation in the Sierra Nevada associated with the post-Tortonian folding event has been proposed [START_REF] Vázquez | Deciphering tectonic-and erosiondriven exhumation of the Nevado-Filábride Complex (Betic Cordillera, Southern Spain) by low-temperature thermochronology: deciphering tectonic-and erosion-driven exhumation[END_REF], its equivalent has not been described in the Sierra de Gador.

LT thermochronology and thermal modelling

Eleven samples were initially collected for LT thermochronology analysis in the Alpujarride complex (sampled in quartzite and phyllite areas) and four AHe datings were performed (Ga-04, Ga-07, Ga-09 and Ga-1; Fig. 4 and Supporting Information). In this area, the scarcity of apatites and their low uranium content [START_REF] Johnson | The role of extension in the Miocene denudation of the Nevado-Filabride Complex, Betic Cordillera (SE Spain)[END_REF][START_REF] Platt | Timing of tectonic events in the Alpujárride Complex, Betic Cordillera, southern Spain[END_REF] led us to select Ga-04 (nearest sample from Tortonian calcarenites) for AFT dating to determine a Sierra de Gador cooling path (Fig. 4 and Supporting Information). Alpha-ejection-corrected AHe ages show apparent cooling AHe ages ranging from 9.2 ± 0.7 to 200 ± 16.2 Ma with no age-elevation relationship but a negative correlation between eU content and AHe apparent ages (Fig. 4). [START_REF] Spiegel | Apatite (U-Th-Sm)/He thermochronology of rapidly cooled samples: The effect of He implantation[END_REF] and [START_REF] Gautheron | Accounting for long alphaparticle stopping distances in (U-Th-Sm)/He geochronology: 3D modeling of diffusion, zoning, implantation, and abrasion[END_REF] have shown that the age-eU negative correlation can be interpreted as the result of He implantation from the surrounding material richer in U and Th than most of apatites. [START_REF] Murray | Effects of U-Th-rich grain boundary phases on apatite helium ages[END_REF] show also that implantation can affect AHe age even for a low U-content of the surrounding medium. Following this implantation model, the implantation-free age (or 'true' age) corresponds to the apparent age of the single grain with the highest eU content, which, in our study, corresponds to a AHe 'true' age of 9.2 ± 0.7 Ma (Fig. 4; Supporting Information). Sample Ga-04 shows an AFT central age of 16 ± 2 Ma corresponding to the AFT ages proposed for the Alpujarride complex in the western valley (PB365 dated at 16.1 ± 1.5 Ma and PB367 dated at 16.5 ± 1.5 Ma; [START_REF] Platt | Timing of tectonic events in the Alpujárride Complex, Betic Cordillera, southern Spain[END_REF] Figs. 2 and4). We propose a common exhumation history for the Sierra de Gador and its western valley on the basis of the similar AFT ages without age-elevation correlation (Fig. 4), and the similar levels of erosion of the Alpujarride units partially covered by Tortonian sediments.

Thermal modelling was performed using QTQt software (Gallagher, 2012) to infer the Neogene cooling history of the Sierra de Gador. In addition to the new AHe data (age of 9.2 ± 0.7 Ma) and AFT age (16 ± 2 Ma), published AFT (PB365 dated at 16.1 ± 1.5 Ma; PB367 dated at 16.5 ± 1.5 Ma) and muscovite Ar/Ar (PB369 dated at 22.7 ± 0.8 Ma; PB533 dated at 26.0 ± 1.9 Ma) data for the Alpujarride complex were used as a vertical profile [START_REF] Platt | Timing of tectonic events in the Alpujárride Complex, Betic Cordillera, southern Spain[END_REF]; Fig. 2). The resulting model (Fig. 5A) shows a polyphase thermal story with three clear stages: (1) a first stage of very fast cooling between 23 and 16 Ma (~45°C/Ma); (2) a second stage of strong cooling slowing down between 16 and 7.2 Ma (~11.5°C /Ma); and (3) a third stage of nearsurface conditions since 7.2 Ma without late reheating.

Discussion and conclusions

Our results show an unsteady exhumation since the early Miocene. To infer exhumation rates, we used a mean geothermal gradient of 49 ±11°C/km following the range published for the Alpujarride complex exhumation [START_REF] Argles | Attenuation and excision of a crustal section during extensional exhumation: the Carratraca Massif, Betic Cordillera, southern Spain[END_REF][START_REF] Azañón | Exhumation during a continental collision inferred from the tectonometamorphic evolution of the Alpujarride Complex in the central Betics (Alboran Domain, SE Spain)[END_REF].

During the first cooling stage (23-16 Ma), rocks were exhumed at a rate of ~0.9 mm/yr. Such a fast cooling event has been widely described for the Alpujarride complex [START_REF] Sosson | Late exhumation stages of the Alpujarride Complex (western Betic Cordilleras, Spain): new thermochronological and structural data on Los Reales and Ojen nappes[END_REF][START_REF] Comas | The origin and tectonic history of the Alboran Basin: insights from Leg 161 results[END_REF][START_REF] Platt | Timing of tectonic events in the Alpujárride Complex, Betic Cordillera, southern Spain[END_REF] and interpreted as a late stage of post-orogenic thinning leading to the exhumation of a previously deeply buried metamorphic alpine-type nappe-stack (Fig. 5B). Then, during the second cooling stage (16-7.2 Ma), Alpujarride rocks from the Sierra de Gador kept exhuming but at a lower rate of ~0.2 mm/yr, whereas the Nevado-Filabride units located further north, exhumed at a higher rate of ~1 mm/y [START_REF] Johnson | The role of extension in the Miocene denudation of the Nevado-Filabride Complex, Betic Cordillera (SE Spain)[END_REF][START_REF] Vázquez | Deciphering tectonic-and erosiondriven exhumation of the Nevado-Filábride Complex (Betic Cordillera, Southern Spain) by low-temperature thermochronology: deciphering tectonic-and erosion-driven exhumation[END_REF] (Fig. 5A). During the 16-7.2 period, the Alpujarride topography decreased toward sea-level, whereas the Nevado-Filabride unit likely remained in a higher position, as suggested by the current difference in crustal thickness [START_REF] Diaz | Moho topography beneath the Iberian-Western Mediterranean region mapped from controlled-source and natural seismicity surveys[END_REF]; Fig. 5B). These two units are separated by the Alpujarras corridor, which is mainly filled by Serravallian and Tortonian sediments eroded from Alpujarride-Malaguide and Nevado-Filabride units respectively [START_REF] Rodríguez Fernández | Le couloir des Alpujarras[END_REF][START_REF] Farines | The summit erosion surfaces of the inner Betic Cordillera: Their value as tools for reconstructing the chronology of topographic growth in southern Spain[END_REF]. However, the Sierra de Gador summit surfaces seem to have been formed during pre-Tortonian times [START_REF] Farines | The summit erosion surfaces of the inner Betic Cordillera: Their value as tools for reconstructing the chronology of topographic growth in southern Spain[END_REF]. According to our thermochronological results coupled with these surface and stratigraphic constraints, we propose that the current low relief at the top of the Sierra de Gador started to be shaped 16 Ma ago (Fig. 5). Indeed, an important decrease in the exhumation rate coeval with topography decrease toward sea-level strongly suggests a levelling process. Some works have shown that an important decrease in the cooling rate tending towards a low-temperature plateau is often associated with planation processes, as in the Sierra de los Filabres [START_REF] Calvet | Flat-topped mountain ranges: Their global distribution and value for understanding the evolution of mountain topography[END_REF][START_REF] Farines | The summit erosion surfaces of the inner Betic Cordillera: Their value as tools for reconstructing the chronology of topographic growth in southern Spain[END_REF]. At 7.2 Ma, surface temperatures were reached (Fig. 5A) and the low topography was then flooded and reworked by the sea, and inner-platform calcarenites were deposited (Fig. 5B). Low-relief summits, described at a larger scale in the Betics, can be due to Miocene erosion events, as described in the Sierra de Gador [START_REF] Farines | The summit erosion surfaces of the inner Betic Cordillera: Their value as tools for reconstructing the chronology of topographic growth in southern Spain[END_REF]. After 7.2 Ma, the Sierra de Gador was uplifted and the Alpujarride rocks have remained at surface temperature conditions. This non-resetting of thermochronometers implies that the upper Tortonian cover was thin, in good agreement with the 100-200 m thickness depicted in the preserved Poniente basin further south [START_REF] Pedrera | Smooth folds favoring gypsum precipitation in the Messinian Poniente marginal basin (Western Mediterranean)[END_REF]. We also cannot distinguish exhumation of rocks after 7.2 Ma, contrary to what has been described for the Sierra Nevada [START_REF] Johnson | The role of extension in the Miocene denudation of the Nevado-Filabride Complex, Betic Cordillera (SE Spain)[END_REF][START_REF] Reinhardt | Tectonic denudation and topographic development in the Spanish Sierra Nevada[END_REF][START_REF] Vázquez | Deciphering tectonic-and erosiondriven exhumation of the Nevado-Filábride Complex (Betic Cordillera, Southern Spain) by low-temperature thermochronology: deciphering tectonic-and erosion-driven exhumation[END_REF]. In particular, some authors have proposed a stage of accelerated exhumation around 4-5 Ma linked to a denudation affecting the western Sierra Nevada [START_REF] Johnson | The role of extension in the Miocene denudation of the Nevado-Filabride Complex, Betic Cordillera (SE Spain)[END_REF][START_REF] Reinhardt | Tectonic denudation and topographic development in the Spanish Sierra Nevada[END_REF]. This denudation could be the result of focused crustal folding accompanied by gravity sliding along surface detachments [START_REF] Reinhardt | Tectonic denudation and topographic development in the Spanish Sierra Nevada[END_REF]. [START_REF] Marín-Lechado | Mountain Front Development by Folding and Crustal Thickening in the Internal Zone of the Betic Cordillera-Alboran Sea Boundary[END_REF] proposed that the elevation decrease of the ranges southward is related to a decrease of the folding amplitude. Moreover, the dynamic topographic effect described in the eastern Betics (Garcia-Castellanos and Villaseñor, 2011) induces a superimposed uplift signal, which may be higher in the Sierra Nevada as it is located directly above the Iberian slab tear [START_REF] Mancilla | Slab rupture and delamination under the Betics and Rif constrained from receiver functions[END_REF]. One can suppose that the resulting uplift in the Sierra de Gador was not high enough to induce important denudation after 7.2 Ma. The degree of fluvial erosion indicates roughly whether there has been equilibrium between uplift and erosion of landscapes over the last million years (e.g., [START_REF] Whipple | Fluvial landscape response time: how plausible is steady-state denudation?[END_REF]. The presence of minimally dissected summit surfaces in the Sierra de Gador argues for a transient state of the range's topography, whereas equilibrium seems to have been reached for the Sierra Nevada [START_REF] Azañón | Relief and drainage evolution during the exhumation of the Sierra Nevada (SE Spain): Is denudation keeping pace with uplift?[END_REF]. This difference could be explained either by a later uplift stage of the Sierra de Gador and/or by the resetting of the topography due to the Tortonian marine transgression. Indeed, the absence of initial topography increases the landscape response time following an uplift [START_REF] Lague | Laboratory experiments simulating the geomorphic response to tectonic uplift[END_REF]. The current landscape of the Sierra de Gador results from inherited low relief, which was mainly initiated during Late Miocene, then uplifted and slightly dissected since 7.2 Ma. Finally, the topography of this range appears to be transient, indicating that transient landforms can develop and persist for several Myr during orogenic evolution. [START_REF] Johnson | The role of extension in the Miocene denudation of the Nevado-Filabride Complex, Betic Cordillera (SE Spain)[END_REF][START_REF] Platt | Timing of tectonic events in the Alpujárride Complex, Betic Cordillera, southern Spain[END_REF][START_REF] Reinhardt | Tectonic denudation and topographic development in the Spanish Sierra Nevada[END_REF][START_REF] Clark | The record of tectonic denudation and erosion in an emerging orogen: an apatite fission-track study of the Sierra Nevada, southern Spain[END_REF][START_REF] Vázquez | Deciphering tectonic-and erosiondriven exhumation of the Nevado-Filábride Complex (Betic Cordillera, Southern Spain) by low-temperature thermochronology: deciphering tectonic-and erosion-driven exhumation[END_REF]

Figure 1 .

 1 Figure 1. Simplified geological map of the eastern Betics showing the main tectonic units and

Figure 2 .

 2 Figure 2. Shaded relief map of the eastern Betics showing the locations of the

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Photograph (looking north) showing low-relief surfaces overlain by thin calcarenite

Acknowledgements

The authors thank editor J. Braun, associate editor C. Spiegel, reviewers T. Dempster, U.

Glasmacher and an anonymous reviewer for their constructive comments. We also thank Rosella Pinna-Jaume for the U-Th chemistry preparation. Kerry Gallagher is thanked for assistance with the setup of QTQt runs. M.J., N.L., J.VDD. and J. Babault collected the samples. M.J., C.G and J. Barbarand performed LTT analyses. All authors contributed to writing the paper. This work benefitted from the support of the French INSU-CNRS project Action-Marges.


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

Stéphane Caro
email: stephane.caro@irccyn.ec-nantes.fr

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

Jorge Angeles
email: angeles@cim.mcgill.ca

U M R 6597

THE ISOCONDITIONING LOCI OF PLANAR THREE-DOF PARALLEL MANIPULATORS

Keywords: parallel manipulator, optimum design, isoconditioning loci, characteristic length

The subject of this paper is a special class of parallel manipulators. First, we analyze a family of threedegree-of-freedom manipulators. Two Jacobian matrices appear in the kinematic relations between the jointrate and the Cartesian-velocity vectors, which are called the "inverse kinematics" and the "direct kinematics" matrices. The singular configurations of these matrices are studied. The isotropic configurations are then studied based on the characteristic length of this manipulator. The isoconditioning loci of all Jacobian matrices are computed to define a global performance index to compare the different working modes.

Introduction

Various performance indices have been devised to assess the kinetostatic performances of serial and parallel manipulators. The literature on performance indices is extremely rich to fit in the limits of this paper, the interested reader are invited to look at it in the rather recent references cited here. A dimensionless quality index was recently introduced by Lee, Duffy, and Hunt [START_REF] Lee | A Pratical Quality Index Based on the Octahedral Manipulator[END_REF] based on the ratio of the Jacobian determinant to its maximum absolute value, as applicable to parallel manipulators. This index does not take into account the location of the operation point of the end-effector, because the Jacobian determinant is independent of this location. The proof of the foregoing result is available in [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems[END_REF], as pertaining to serial manipulators, its extension to their parallel counterparts being straightforward. The condition number of a given matrix, on the other hand, is well known to provide a measure of invertibility of the matrix [START_REF] Golub | Matrix Computations[END_REF]. It is thus natural that this concept found its way in this context. Indeed, the condition number of the Jacobian matrix was proposed by Salisbury and Craig [START_REF] Salisbury | Articulated Hands: Force Control and Kinematic Issues[END_REF] as a figure of merit to minimize when designing manipulators for maximum accuracy. In fact, the condition number gives, for a square matrix, a measure of the relative roundoff-error amplification of the computed results [START_REF] Golub | Matrix Computations[END_REF] with respect to the data roundoff error. As is well known, however, the dimensional inhomogeneity of the entries of the Jacobian matrix prevents the straightforward application of the condition number as a measure of Jacobian invertibility. The characteristic length was introduced in [START_REF] Angeles | Kinematic Isotropy and the Conditioning Index of Serial Manipulators[END_REF] to cope with the abovementioned inhomogeneity.

In this paper, we use the characteristic length to normalize the Jacobian matrix of a threedegree-of-freedom (dof) planar manipulator and to calculate the isoconditioning loci for all its working modes.

Preliminaries

A planar three-dof manipulator, with three parallel PRR chains, the object of this paper, is shown in Fig. 1. This manipulator has been frequently studied, in particular in [START_REF] Merlet | Les robots parallèles[END_REF][START_REF] Gosselin | Solutions polynomiales au problème de la cinématique des manipulateurs parallèles plans à trois degré de liberté[END_REF]. The actuated joint variables are the displacements of the three prismatic joints. The Cartesian variables are the position P and the orientation  of the platform.

The points A 1 , A 2 and A 3 are located in the vertices of an equilateral triangle whose geometric center is the point O, as well as the points C 1 , C 2 and C 3 , whose geometric center is the point P. Point P, moreover, is the operation point of the manipulator. To reduce the number of design variables, we set R=R 1 =R 2 =R 3 , with R i denoting the lengths of A i O, l=l 1 =l 2 =l 3 with l i denoting the lengths of B i C i and r=r 1 =r 2 =r 3 with r i denoting the lengths of C i P.

Kinematic Relations

The velocity . p of the point P, of position vector p, can be obtained in three different forms, depending on the direction in which the loop is traversed, namely, .

p = . b 1 + .  1 E (c 1 -b 1 ) + .  E (p -c 1 ) (1a)
.

p = . b 2 + .  2 E (c 2 -b 2 ) + .  E (p -c 2 ) (1b) . p = . b 2 + .  3 E (c 3 -b 3 ) + .  E (p -c 3 ) (1c) with matrix E defined as E=     0 -1 1 0 The velocity . b i of points B i is given by . b i = .  i       cos( i ) sin( i ) = .  i  i
where  i are unit vectors, collinear with the prismatic joints.

We would like to eliminate the three idle joint rates .  1 , .  2 and .  3 from eqs.(1a-c), which we do upon dot-multiplying the former by (c i -b i ) T , thus obtaining

(c 1 -b 1 ) T . p = (c 1 -b 1 ) T . b 1 + (c 1 -b 1 ) T .  E (p -c 1 ) (2a) (c 2 -b 2 ) T . p = (c 2 -b 2 ) T . b 2 + (c 2 -b 2 ) T .  E (p -c 2 ) (2b) (c 3 -b 3 ) T . p = (c 3 -b 3 ) T . b 2 + (c 3 -b 3 ) T .  E (p -c 3 ) (2c
) Equations (2a-c) can now be cast in vector form, namely,

A t = B .  with t =         . p .  , .  =       .  1 .  2 .  3 (3)
and .  defined as the vector of actuated joint rates. Moreover A and B are, respectively, the direct-kinematics and the inverse-kinematics matrices of the manipulator, defined as

A =       (c 1 -b 1 ) T -(c 1 -b 1 ) T E (p -c 1 ) (c 2 -b 2 ) T -(c 2 -b 2 ) T E (p -c 2 ) (c 3 -b 3 ) T -(c 3 -b 3 ) T E (p -c 3 ) (4a) and B =       (c 1 -b 1 ) T  1 0 0 0 (c 2 -b 2 ) T  2 0 0 0 (c 3 -b 3 ) T  3 (4b)
When A and B are nonsingular, we obtain the relations

t = J .  with J = A -1 B and .  = K t, with K = B -1 A

Parallel Singularities

Parallel singularities occur when the determinant of matrix A vanishes [START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF][START_REF] Gosselin | Singularity Analysis of Closed-Loop Kinematic Chains[END_REF]. In this configuration, it is possible to move locally the operation point P with the actuators locked. These singularities are to be avoided, because the structure cannot resist arbitrary forces, and control is lost. To avoid any performance deterioration, it is necessary to have a Cartesian workspace free of parallel singularities. For the planar manipulator studied, such configurations are reached whenever the axes B 1 C 1 , B 2 C 2 and B 3 C 3 intersect, possibly at infinity, as depicted in Fig. 2. In such configurations, the manipulator cannot resist a force applied at the intersection point. These singular configurations are located inside the Cartesian workspace and form the boundaries of the joint workspace [START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF].

Serial Singularities

Serial singularities occur when det(B) = 0. In the presence of theses singularities, there is a direction along which no Cartesian velocity can be produced. Serial singularities define the boundary of the Cartesian workspace. For the topology under study, the serial singularities occur whenever

A 1 B 1  B 1 C 1 or A 2 B 2  B 2 C 2 or A 3 B 3  B 3 C 3
as depicted in Fig. 3.

Isoconditioning Loci

The Matrix Condition Number

We derive below the loci of equal condition number of the direct-and inverse-kinematics matrices. To do this, we first recall the definition of condition number (M) of an mn matrix M, with m  n. This number can be defined in various ways; for our purposes, we define (M) as the ratio of the largest,  l , to the smallest,  s , singular values of M, namely,

(M) =  l  s (5)
The singular values { i } m 1 of matrix M are defined, in turn, as the square roots of the nonnegative eigenvalues of the positive definite mm matrix MM T .

Non-Homogeneous Direct-Kinematics Matrix

It will prove convenient to partition A into a 23 block A 1 and a 13 block A 2 , defined as

A 1 =       (c 1 -b 1 ) T (c 2 -b 2 ) T (c 3 -b 3 ) T and A 2 =       -(c 1 -b 1 ) T E (p -c 1 ) -(c 2 -b 2 ) T E (p -c 2 ) -(c 3 -b 3 ) T E (p -c 3 ) (6)
Therefore, while the entries of A 1 have units of length, those of A 2 have units of square length. It is thus apparent that the singular values of A have different dimensions and hence, it is impossible to compute (A) because the singular values of A cannot be ordered from smallest to largest. The normalization of the Jacobian for purposes of rendering it dimensionless has been judged to be dependent on the normalizing length [START_REF] Paden | Optimal Kinematic Design of 6R Manipulator[END_REF][START_REF] Li | Geometrical Consideration of Robot Kinematics[END_REF]. As a means to avoid the arbitrariness of the choice of that normalizing length, the characteristic length L was introduced in [START_REF] Ranjbaran | The Mechanical Design of a Seven-Axes Manipulator with Kinematic Isotropy[END_REF].

To yield the matrix A homogeneous, each term of the third column of A is divided by the characteristic length L, thereby deriving its normalized counterpart _ A:

_ A =       (c 1 -b 1 ) T -(c 1 -b 1 ) T E (p -c 1 )/L (c 2 -b 2 ) T -(c 2 -b 2 ) T E (p -c 2 )/L (c 3 -b 3 ) T -(c 3 -b 3 ) T E (p -c 3 )/L (7)
which is calculated so as to minimize ( _ A), along with the posture variables  1 ,  2 and  3 .

Isotropic Configuration

In this section, we write the isotropy condition in J to define the geometric parameters of the manipulator. We shall obtain also the value L of the characteristic length. To simplify A and B, we use the notation

l i = c i -b i (8a) k i =(c i -b i ) T E (p -c i ) (8b) m i = (c i -b i ) T  i (8c)  i = B i C i P (8d) We can write matrices _ A, B and B -1 as, _ A=       l T 1 k 1 /L l T 2 k 2 /L l T 3 k 3 /L , B=       m 1 0 0 0 m 2 0 0 0 m 3 and B -1 =       1/m 1 0 0 0 1/m 2 0 0 0 1/m 3
Whenever matrix B is nonsingular, that is when m i  0, for i=1,2,3. By using the previous simplifications, we have We can notice that the first and the last conditions have already been proposed in § 2.

_ K=         l T 1 /m 1 -k 1 /(L m 1 ) l T 2 /m 2 -k 2 /(L m 2 ) l T 3 /m 3 -k 3 /(L m 3 ) Matrix _ J, the normalized J, is isotropic if and only if _ K _ K T = 2 1 33 for  > 0, i.e., (l T 1 l 1 +k 2 1 /L 2 )/m 2 1 = 2 (9a) (l T 2 l 2 +k 2 2 /L 2 )/m 2 2 = 2 (9b) (l T 3 l 3 +k 2 3 /L 2 )/m 2 3 = 2 (9c) (l T 1 l 2 +k 1 k 2 /L 2 )/(m 1 m 2 )=0 (9d) (l T 1 l 3 +k 1 k 3 /L 2 )/(m 1 m 3 )=0 (9e) (l T 2 l 3 +k 2 k 3 /L 2 )/(m 2 m 3 )=0 (

Characteristic Length

The characteristic length is defined at the isotropic configuration. From eqs.(9d-f), we determine the value of the characteristic length as, This means, that the manipulator under study possesses several isotropic configurations (Fig. 4a-b), whereas the characteristic length L of the manipulator is unique [START_REF] Angeles | Fundamentals of Robotic Mechanical Systems[END_REF]. When  is equal to /2 (Fig. 4a), we obtain the condition:

L = -k 1 k 2 l T
 B i C i is perpendicular to C i G.
This condition yields a configuration furthest away from parallel singularities. To have a isotropic configuration furthest away from serial singularities, we have two conditions:

 A i B i is collinear with B i C i ;  r=R/2.

Working Modes

The manipulator under study has a diagonal inverse-kinematics matrix B, as shown in eq.(4b), the vanishing of one of its diagonal entries thus indicating the occurrence of a serial singularity. The set of manipulator postures free of this kind of singularity is termed a working mode. The different working modes are thus separated by a serial singularity, with a set of postures in different working modes corresponding to an inverse kinematics solution.

The formal definition of the working mode is detailed in [START_REF] Chablat | Working Modes and Aspects in Fully-Parallel Manipulator[END_REF]. For the manipulator at hand, there are eight working modes, as depicted in Fig. 5.

Figure 5: The eight working modes of the 3PRR manipulator

However, because of symmetry, we can restrict our study to only two working modes, if there are no joint limits. Indeed, the working mode 1 is similar to the working mode 5, because in a case the signs m i of the diagonal entries of B are either all positive, or all negative. We can associate the working modes 2-6, 3-7 and 4-8 likewise the working modes 3-4 and 7-8 can be derived from the working modes 2 and 6 by a rotation of 120° and 240°, respectively. Therefore, only the working modes 1 and 2 are studied.

Isoconditioning Loci

Moreover, for each Jacobian matrix and for all the poses of the end-effector, we calculate the optimum conditioning according to the orientation of the end-effector. At the other end of the spectrum, the minimum conditioning is always associated with a singular configuration.

To calculate the condition number of matrix _ A, we need the product _ A _ A T . Figure 6 depicts the isoconditioning loci of this matrix.

(B) =  max  min
We depict in Fig. 7, the isoconditioning loci of matrix B. We notice that the loci of both working modes are identical. This is due to the absence of joint limits on the actuated joints. For one configuration, only the signs of m i change from a working mode to an other, and the condition number  is computed a the absolute values of m i . The shapes of the isoconditioning loci of _ K (Fig. 8) are similar to those of the isconditioning loci of _ A; only the numerical values change. We can thus conclude that in our example, the singularities due to the matrix B are less important than those due to matrix The average of  can be accepted as a global performance index. So, this index can allow us to compare the working modes (Table 1 For the first working mode, the condition number decreases regularly around the isotropic configuration. The isoconditioning loci resemble concentric circles. However, for the second working mode, the isoconditioning loci resemble elliptical curves. The average of  is larger in the second working mode. Therefore, the prescribed trajectories should lie in an rectangular zone rather than in a square zone.

Conclusions

We produced the isoconditioning loci of the Jacobian matrices of a three-PRR parallel manipulator. This method being general, it can be applied to any three-dof planar parallel manipulator. To solve the problem of nonhomogeneity of the Jacobian matrix, we used the notion of characteristic length. This length was defined for the isotropic configuration of the manipulator. The isoconditioning curves thus obtained characterize, for every posture of the manipulator, the optimum conditioning for all possible orientations of the end-effector. This work can be used to choose the working mode which is best suited to prescribed trajectories or as a global performance index when we study the optimum design of this kind of manipulators.

Figure 1 :

 1 Figure 1: A three-dof parallel manipulator

Figure 2 :

 2 Figure 2: Parallel singularity Figure 3: Serial singularity

  9f) From eqs. (9a-f), we can derive the conditions below: ||l 1 || = ||l 2 || = ||l 3 || (10a) ||p-c 1 || = ||p-c 2 || = ||p-c 3 m 1 m 2 = m 1 m 3 = m 2 m 3 (10d) In summary, the constraints defined in eqs.(10a-d) are  Pivots C i should be placed in the vertices of an equilateral triangle, i.e. r=r 1 = r 2 = r 3 ;  Segments A i B i are the sides of an equilateral triangle;  Segments B i C i are the sides of an equilateral triangle;  Segments B i C i are equal, i.e. l=l 1 = l 2 = l 3 .

1 l 2

 12 By applying the constraints defined in eqs.(10a-d), we can write the characteristic length in terms of angle , i.e. L= 2 r sin() where = 1 = 2 = 3 are defined in eq. (8d) and   [0 , 2].

Figure 4

 4 depicts two isotropic configurations.

Figure 4 :

 4 Figure 4: Two isotropic configurations with two values of 

Figure 6 :

 6 Figure 6: Isoconditioning loci of the matrix _ A with r=100mm, R=200mm and l=200mm By virtue of the diagonal form of matrix B, its singular values  1 ,  2 and  3 are simply the absolute values of its diagonal entries, namely,  1 = |m 1 |,  2 = |m 2 | and  3 = |m 3 | The condition number  of matrix B is thus

Figure 7 :

 7 Figure 7: Isoconditioning loci of the matrix B with r=100mm, R=200mm and l=200mm

Figure 8 :

 8 Figure 8: Isoconditioning loci of the matrix _ K with r=100mm, R=200mm and l=200mm

Table 1 :

 1 ). The average value of the condition number in the Cartesian workspace

			_			_
	Working modes	(	A)	(B)	(	K)
	1	0.354	0.709	0.329
	2	0.511	0.709	0.460


Muhammad Ahsan Bashir

Muhammad Ahsan

come